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class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_no_clonaci%C3%B3" title="Teorema de no clonació – Catalan" lang="ca" hreflang="ca" data-title="Teorema de no clonació" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/No-Cloning-Theorem" title="No-Cloning-Theorem – German" lang="de" hreflang="de" data-title="No-Cloning-Theorem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_no_clonaci%C3%B3n" title="Teorema de no clonación – Spanish" lang="es" hreflang="es" data-title="Teorema de no clonación" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Impossibilit%C3%A9_du_clonage_quantique" title="Impossibilité du clonage quantique – French" lang="fr" hreflang="fr" data-title="Impossibilité du clonage quantique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B5%EC%A0%9C_%EB%B6%88%EA%B0%80%EB%8A%A5%EC%84%B1_%EC%A0%95%EB%A6%AC" title="복제 불가능성 정리 – Korean" lang="ko" hreflang="ko" data-title="복제 불가능성 정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_no-cloning_quantistico" title="Teorema di no-cloning quantistico – Italian" lang="it" hreflang="it" data-title="Teorema di no-cloning quantistico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%90%D7%99-%D7%A9%D7%9B%D7%A4%D7%95%D7%9C" title="משפט האי-שכפול – Hebrew" lang="he" hreflang="he" data-title="משפט האי-שכפול" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E8%A4%87%E8%A3%BD%E4%B8%8D%E5%8F%AF%E8%83%BD%E5%AE%9A%E7%90%86" title="量子複製不可能定理 – Japanese" lang="ja" hreflang="ja" data-title="量子複製不可能定理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zakaz_klonowania" title="Zakaz klonowania – Polish" lang="pl" hreflang="pl" data-title="Zakaz klonowania" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_da_n%C3%A3o-clonagem" title="Teorema da não-clonagem – Portuguese" lang="pt" hreflang="pt" data-title="Teorema da não-clonagem" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D0%B7%D0%B0%D0%BF%D1%80%D0%B5%D1%82%D0%B5_%D0%BA%D0%BB%D0%BE%D0%BD%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F" title="Теорема о запрете клонирования – Russian" lang="ru" hreflang="ru" data-title="Теорема о запрете клонирования" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BF%D1%80%D0%BE_%D0%B7%D0%B0%D0%B1%D0%BE%D1%80%D0%BE%D0%BD%D1%83_%D0%BA%D0%BB%D0%BE%D0%BD%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Теорема про заборону клонування – Ukrainian" lang="uk" hreflang="uk" data-title="Теорема про заборону клонування" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_kh%C3%B4ng_nh%C3%A2n_b%E1%BA%A3n" title="Định lý không nhân bản – Vietnamese" lang="vi" hreflang="vi" data-title="Định lý không nhân bản" 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href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table><p>In <a href="/wiki/Physics" title="Physics">physics</a>, the <b>no-cloning theorem</b> states that it is impossible to create an independent and identical copy of an arbitrary unknown <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a>, a statement which has profound implications in the field of <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computing</a> among others. The theorem is an evolution of the 1970 <a href="/wiki/No-go_theorem" title="No-go theorem">no-go theorem</a> authored by James Park,<sup id="cite_ref-park_1-0" class="reference"><a href="#cite_note-park-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by <a href="/wiki/William_Wootters" title="William Wootters">William Wootters</a> and <a href="/wiki/Wojciech_H._Zurek" title="Wojciech H. Zurek">Wojciech H. Zurek</a><sup id="cite_ref-wootterszurek_2-0" class="reference"><a href="#cite_note-wootterszurek-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> as well as <a href="/wiki/Dennis_Dieks" title="Dennis Dieks">Dennis Dieks</a><sup id="cite_ref-dieks_3-0" class="reference"><a href="#cite_note-dieks-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> the same year). The aforementioned theorems do not preclude the state of one system becoming <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entangled</a> with the state of another as cloning specifically refers to the creation of a <a href="/wiki/Separable_state" title="Separable state">separable state</a> with identical factors. For example, one might use the <a href="/wiki/Controlled_NOT_gate" title="Controlled NOT gate">controlled NOT gate</a> and the <a href="/wiki/Hadamard_transform#Quantum_computing_applications" title="Hadamard transform">Walsh–Hadamard gate</a> to entangle two <a href="/wiki/Qubit" title="Qubit">qubits</a> without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem (as generally understood) concerns only <a href="/wiki/Pure_state" class="mw-redirect" title="Pure state">pure states</a> whereas the generalized statement regarding <a href="/wiki/Mixed_state_(physics)" class="mw-redirect" title="Mixed state (physics)">mixed states</a> is known as the <a href="/wiki/No-broadcast_theorem" class="mw-redirect" title="No-broadcast theorem">no-broadcast theorem</a>. The no-cloning theorem has a time-reversed <a href="/wiki/Dual_(category_theory)" title="Dual (category theory)">dual</a>, the <a href="/wiki/No-deleting_theorem" title="No-deleting theorem">no-deleting theorem</a>. </p><meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=No-cloning_theorem&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to <a href="/wiki/Asher_Peres" title="Asher Peres">Asher Peres</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/David_Kaiser_(physicist)" title="David Kaiser (physicist)">David Kaiser</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> the publication of the 1982 proof of the no-cloning theorem by <a href="/wiki/William_Wootters" title="William Wootters">Wootters</a> and <a href="/wiki/Wojciech_H._Zurek" title="Wojciech H. Zurek">Zurek</a><sup id="cite_ref-wootterszurek_2-1" class="reference"><a href="#cite_note-wootterszurek-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and by <a href="/wiki/Dennis_Dieks" title="Dennis Dieks">Dieks</a><sup id="cite_ref-dieks_3-1" class="reference"><a href="#cite_note-dieks-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> was prompted by a proposal of <a href="/wiki/Nick_Herbert_(physicist)" title="Nick Herbert (physicist)">Nick Herbert</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> for a <a href="/wiki/Superluminal_communication" title="Superluminal communication">superluminal communication</a> device using quantum entanglement, and <a href="/wiki/Giancarlo_Ghirardi" title="Giancarlo Ghirardi">Giancarlo Ghirardi</a><sup id="cite_ref-Ghirardi2013_7-0" class="reference"><a href="#cite_note-Ghirardi2013-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> had proven the theorem 18 months prior to the published proof by Wootters and Zurek in his referee report to said proposal (as evidenced by a letter from the editor<sup id="cite_ref-Ghirardi2013_7-1" class="reference"><a href="#cite_note-Ghirardi2013-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>). However, Juan Ortigoso<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> pointed out in 2018 that a complete proof along with an interpretation in terms of the lack of simple nondisturbing measurements in quantum mechanics was already delivered by Park in 1970.<sup id="cite_ref-park_1-1" class="reference"><a href="#cite_note-park-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Theorem_and_proof">Theorem and proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=No-cloning_theorem&action=edit&section=2" title="Edit section: Theorem and proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose we have two quantum systems <i>A</i> and <i>B</i> with a common Hilbert space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=H_{A}=H_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=H_{A}=H_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34dc291e40f7943f1ed9a7f7e876dc3ad2b0c02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.067ex; height:2.509ex;" alt="{\displaystyle H=H_{A}=H_{B}}"></span>. Suppose we want to have a procedure to copy the state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42dbf68834d1562fc06f25f1ab73368e2a6cadcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.402ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{A}}"></span> of quantum system <i>A</i>, over the state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |e\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |e\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38947560f5a3cdcdb588d0dd86369cd29a23ccd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.115ex; height:2.843ex;" alt="{\displaystyle |e\rangle _{B}}"></span> of quantum system <i>B,</i> for any original state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42dbf68834d1562fc06f25f1ab73368e2a6cadcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.402ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{A}}"></span> (see <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a>). That is, beginning with the state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f44460c23bfca2192bc258c6377d57a5a3a7613b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.357ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}}"></span>, we want to end up with the state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{A}\otimes |\phi \rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{A}\otimes |\phi \rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17a52e2a3bf2f590fecafd2eea0506911f9d2228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.659ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{A}\otimes |\phi \rangle _{B}}"></span>. To make a "copy" of the state <i>A</i>, we combine it with system <i>B</i> in some unknown initial, or blank, state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |e\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |e\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38947560f5a3cdcdb588d0dd86369cd29a23ccd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.115ex; height:2.843ex;" alt="{\displaystyle |e\rangle _{B}}"></span> independent of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42dbf68834d1562fc06f25f1ab73368e2a6cadcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.402ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{A}}"></span>, of which we have no prior knowledge. </p><p>The state of the initial composite system is then described by the following <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/341dfb9e2e1a53aef8efab0acd92de23fa86acd5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.004ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}.}"></span> (in the following we will omit the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }"></span> symbol and keep it implicit). </p><p>There are only two permissible <a href="/wiki/Quantum_operation" title="Quantum operation">quantum operations</a> with which we may manipulate the composite system: </p> <ul><li>We can perform an <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">observation</a>, which irreversibly <a href="/wiki/Wave_function_collapse" title="Wave function collapse">collapses</a> the system into some <a href="/wiki/Eigenstate" class="mw-redirect" title="Eigenstate">eigenstate</a> of an <a href="/wiki/Observable" title="Observable">observable</a>, corrupting the information contained in the <a href="/wiki/Qubit" title="Qubit">qubit(s)</a>. This is obviously not what we want.</li> <li>Alternatively, we could control the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> of the <i>combined</i> system, and thus the <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">time-evolution operator</a> <i>U</i>(<i>t</i>), e.g. for a time-independent Hamiltonian, <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(t)=e^{-iHt/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>H</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(t)=e^{-iHt/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17a17e78953fa7d61788cc8bc41a25ef254fc8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.49ex; height:3.343ex;" alt="{\displaystyle U(t)=e^{-iHt/\hbar }}"></span>.</span> Evolving up to some fixed time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}"></span> yields a <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operator</a> <i>U</i> on <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\otimes H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊗</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\otimes H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276e49fb8f55731e87a532ad84769bf6166896b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.968ex; height:2.343ex;" alt="{\displaystyle H\otimes H}"></span>,</span> the Hilbert space of the combined system. However, no such unitary operator <i>U</i> can clone all states.</li></ul> <p>The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator <i>U</i>, acting on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{A}\otimes H_{B}=H\otimes H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mi>H</mi> <mo>⊗</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{A}\otimes H_{B}=H\otimes H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecdd029c29ce25c8158cb9485b31577d03e9d42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.713ex; height:2.509ex;" alt="{\displaystyle H_{A}\otimes H_{B}=H\otimes H}"></span>, under which the state the system B is in always evolves into the state the system A is in, <i>regardless</i> of the state system A is in? </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>There is no unitary operator <i>U</i> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\otimes H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊗</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\otimes H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276e49fb8f55731e87a532ad84769bf6166896b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.968ex; height:2.343ex;" alt="{\displaystyle H\otimes H}"></span> such that for all normalised states <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42dbf68834d1562fc06f25f1ab73368e2a6cadcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.402ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{A}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |e\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |e\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38947560f5a3cdcdb588d0dd86369cd29a23ccd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.115ex; height:2.843ex;" alt="{\displaystyle |e\rangle _{B}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(|\phi \rangle _{A}|e\rangle _{B})=e^{i\alpha (\phi ,e)}|\phi \rangle _{A}|\phi \rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(|\phi \rangle _{A}|e\rangle _{B})=e^{i\alpha (\phi ,e)}|\phi \rangle _{A}|\phi \rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5eb0bdbeed50366a083359fea12416fa1b5ec7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.443ex; height:3.343ex;" alt="{\displaystyle U(|\phi \rangle _{A}|e\rangle _{B})=e^{i\alpha (\phi ,e)}|\phi \rangle _{A}|\phi \rangle _{B}}"></span> for some real number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> depending on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>. </p> </div> <p>The extra phase factor expresses the fact that a quantum-mechanical state defines a normalised vector in Hilbert space only up to a phase factor i.e. as an element of <a href="/wiki/Projective_space" title="Projective space">projectivised Hilbert space</a>. </p><p>To prove the theorem, we select an arbitrary pair of states <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42dbf68834d1562fc06f25f1ab73368e2a6cadcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.402ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{A}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/435c58e1961f1e68f74f8734418e7c386efd5ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.529ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{A}}"></span> in the Hilbert space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. Because <i>U</i> is supposed to be unitary, we would have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi |\psi \rangle \langle e|e\rangle \equiv \langle \phi |_{A}\langle e|_{B}|\psi \rangle _{A}|e\rangle _{B}=\langle \phi |_{A}\langle e|_{B}U^{\dagger }U|\psi \rangle _{A}|e\rangle _{B}=e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |_{A}\langle \phi |_{B}|\psi \rangle _{A}|\psi \rangle _{B}\equiv e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |\psi \rangle ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mi>e</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mi>e</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>≡</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi |\psi \rangle \langle e|e\rangle \equiv \langle \phi |_{A}\langle e|_{B}|\psi \rangle _{A}|e\rangle _{B}=\langle \phi |_{A}\langle e|_{B}U^{\dagger }U|\psi \rangle _{A}|e\rangle _{B}=e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |_{A}\langle \phi |_{B}|\psi \rangle _{A}|\psi \rangle _{B}\equiv e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |\psi \rangle ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2501d7ee1e65b80f61adb054c3219e5de649dd11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:115.984ex; height:3.343ex;" alt="{\displaystyle \langle \phi |\psi \rangle \langle e|e\rangle \equiv \langle \phi |_{A}\langle e|_{B}|\psi \rangle _{A}|e\rangle _{B}=\langle \phi |_{A}\langle e|_{B}U^{\dagger }U|\psi \rangle _{A}|e\rangle _{B}=e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |_{A}\langle \phi |_{B}|\psi \rangle _{A}|\psi \rangle _{B}\equiv e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |\psi \rangle ^{2}.}"></span> Since the quantum state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |e\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |e\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/485e88c0a857a3244c5e513c2e6fc663cb2ad1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.635ex; height:2.843ex;" alt="{\displaystyle |e\rangle }"></span> is assumed to be normalized, we thus get <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle \phi |\psi \rangle |^{2}=|\langle \phi |\psi \rangle |.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle \phi |\psi \rangle |^{2}=|\langle \phi |\psi \rangle |.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ead32eba5a0776bdd8735946a86e2b5f81ba67b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.096ex; height:3.343ex;" alt="{\displaystyle |\langle \phi |\psi \rangle |^{2}=|\langle \phi |\psi \rangle |.}"></span> </p><p>This implies that either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle \phi |\psi \rangle |=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle \phi |\psi \rangle |=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58835b7013a17c1ae442d1561123ed349f972983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.909ex; height:2.843ex;" alt="{\displaystyle |\langle \phi |\psi \rangle |=1}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle \phi |\psi \rangle |=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle \phi |\psi \rangle |=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cddafa4da0a423898ce5a2aa80c3c36e55bfd97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.909ex; height:2.843ex;" alt="{\displaystyle |\langle \phi |\psi \rangle |=0}"></span>. Hence by the <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a> either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle =e^{i\beta }|\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>β</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle =e^{i\beta }|\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce1e9008a5fb57755102acd981de846067d26820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.925ex; height:3.176ex;" alt="{\displaystyle |\phi \rangle =e^{i\beta }|\psi \rangle }"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span> is <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span>. However, this cannot be the case for two <i>arbitrary</i> states. Therefore, a single universal <i>U</i> cannot clone a <i>general</i> quantum state. This proves the no-cloning theorem. </p><p>Take a qubit for example. It can be represented by two <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, called <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitudes</a> (<a href="/wiki/Qubit#Qubit_states" title="Qubit">normalised to 1</a>), that is three real numbers (two polar angles and one radius). Copying three numbers on a classical computer using any <a href="/wiki/Copy_and_paste" class="mw-redirect" title="Copy and paste">copy and paste</a> operation is trivial (up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the <a href="/wiki/Quantum_logic_gate#Hadamard_gate" title="Quantum logic gate">Hadamard quantum gate</a>) to be polarised (which <a href="/wiki/Unitary_operator" title="Unitary operator">unitary transformation</a> is a <a href="/wiki/Isometry#Introduction" title="Isometry">surjective isometry</a>). In such a case the qubit can be represented by just two real numbers (one polar angle and one radius equal to 1), while the value of the third can be arbitrary in such a representation. Yet a <a href="/wiki/Qubit#Physical_implementations" title="Qubit">realisation</a> of a qubit (polarisation-encoded photon, for example) is capable of storing the whole qubit information support within its "structure". Thus no single universal unitary evolution <i>U</i> can clone an arbitrary quantum state according to the no-cloning theorem. It would have to depend on the transformed qubit (initial) state and thus would not have been <i>universal</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalization">Generalization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=No-cloning_theorem&action=edit&section=3" title="Edit section: Generalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the statement of the theorem, two assumptions were made: the state to be copied is a <a href="/wiki/Pure_state" class="mw-redirect" title="Pure state">pure state</a> and the proposed copier acts via unitary time evolution. These assumptions cause no loss of generality. If the state to be copied is a <a href="/wiki/Mixed_state_(physics)" class="mw-redirect" title="Mixed state (physics)">mixed state</a>, it can be <a href="/wiki/Purification_of_quantum_state" class="mw-redirect" title="Purification of quantum state">"purified</a>," i.e. treated as a pure state of a larger system. Alternately, a different proof can be given that works directly with mixed states; in this case, the theorem is often known as the no-broadcast theorem.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Similarly, an arbitrary <a href="/wiki/Quantum_operation" title="Quantum operation">quantum operation</a> can be implemented via introducing an <a href="/wiki/Ancilla_bit" title="Ancilla bit">ancilla</a> and performing a suitable unitary evolution.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="What other quantum operations are there? The phrasing of the proof in the previous section contains the implicit claim that we had them all when we had unitary ones and Hermitian ones. Now there are more? These new operations should be spelled out. The linked article is a grab bag. (May 2015)">clarification needed</span></a></i>]</sup> Thus the no-cloning theorem holds in full generality. </p> <div class="mw-heading mw-heading2"><h2 id="Consequences">Consequences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=No-cloning_theorem&action=edit&section=4" title="Edit section: Consequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The no-cloning theorem prevents the use of certain classical <a href="/wiki/Error_correction" class="mw-redirect" title="Error correction">error correction</a> techniques on quantum states. For example, backup copies of a state in the middle of a <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computation</a> cannot be created and used for correcting subsequent errors. Error correction is vital for practical quantum computing, and for some time it was unclear whether or not it was possible. In 1995, <a href="/wiki/Peter_Shor" title="Peter Shor">Shor</a> and <a href="/wiki/Andrew_Steane" title="Andrew Steane">Steane</a> showed that it is, by independently devising the first <a href="/wiki/Quantum_error_correction" title="Quantum error correction">quantum error correcting</a> codes, which circumvent the no-cloning theorem.</li> <li>Similarly, cloning would violate the <a href="/wiki/No-teleportation_theorem" title="No-teleportation theorem">no-teleportation theorem</a>, which says that it is impossible to convert a quantum state into a sequence of classical bits (even an infinite sequence of bits), copy those bits to some new location, and recreate a copy of the original quantum state in the new location. This should not be confused with <a href="/wiki/Quantum_teleportation" title="Quantum teleportation">entanglement-assisted teleportation</a>, which does allow a quantum state to be destroyed in one location, and an exact copy to be recreated in another location.</li> <li>The no-cloning theorem is implied by the <a href="/wiki/No-communication_theorem" title="No-communication theorem">no-communication theorem</a>, which states that quantum entanglement cannot be used to transmit classical information (whether superluminally, or slower). That is, cloning, together with entanglement, would allow such communication to occur. To see this, consider the <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR thought experiment</a>, and suppose quantum states could be cloned. Assume parts of a <a href="/wiki/Maximally_entangled_state" class="mw-redirect" title="Maximally entangled state">maximally entangled</a> <a href="/wiki/Bell_state" title="Bell state">Bell state</a> are distributed to Alice and Bob. Alice could send bits to Bob in the following way: If Alice wishes to transmit a "0", she measures the spin of her electron in the <b>z</b> direction, collapsing Bob's state to either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z+\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>+</mo> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z+\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd76cfa243e574c8521dbdaab97f0ca6593ae59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.927ex; height:2.843ex;" alt="{\displaystyle |z+\rangle _{B}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z-\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z-\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19df96ae272c5a91e5a25e24ab208f436abe4340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.927ex; height:2.843ex;" alt="{\displaystyle |z-\rangle _{B}}"></span>. To transmit "1", Alice does nothing to her qubit. Bob creates many copies of his electron's state, and measures the spin of each copy in the <b>z</b> direction. Bob will know that Alice has transmitted a "0" if all his measurements produce the same result; otherwise, his measurements will have outcomes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z+\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>+</mo> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z+\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd76cfa243e574c8521dbdaab97f0ca6593ae59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.927ex; height:2.843ex;" alt="{\displaystyle |z+\rangle _{B}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z-\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z-\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19df96ae272c5a91e5a25e24ab208f436abe4340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.927ex; height:2.843ex;" alt="{\displaystyle |z-\rangle _{B}}"></span> with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across <a href="/wiki/Space-like" class="mw-redirect" title="Space-like">space-like</a> separations, violating <a href="/wiki/Causality" title="Causality">causality</a>).</li> <li>The no cloning theorem prevents an interpretation of the <a href="/wiki/Holographic_principle" title="Holographic principle">holographic principle</a> for <a href="/wiki/Black_hole" title="Black hole">black holes</a> as meaning that there are two copies of information, one lying at the <a href="/wiki/Event_horizon" title="Event horizon">event horizon</a> and the other in the black hole interior. This leads to more radical interpretations, such as <a href="/wiki/Black_hole_complementarity" title="Black hole complementarity">black hole complementarity</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Imperfect_cloning">Imperfect cloning</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=No-cloning_theorem&action=edit&section=5" title="Edit section: Imperfect cloning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Even though it is impossible to make perfect copies of an unknown quantum state, it is possible to produce imperfect copies. This can be done by coupling a larger auxiliary system to the system that is to be cloned, and applying a <a href="/wiki/Unitary_transformation" title="Unitary transformation">unitary transformation</a> to the combined system. If the unitary transformation is chosen correctly, several components of the combined system will evolve into approximate copies of the original system. In 1996, V. Buzek and M. Hillery showed that a universal cloning machine can make a clone of an unknown state with the surprisingly high fidelity of 5/6.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Imperfect <a href="/wiki/Quantum_cloning" title="Quantum cloning">quantum cloning</a> can be used as an <a href="/wiki/Eavesdropping_attack" class="mw-redirect" title="Eavesdropping attack">eavesdropping attack</a> on <a href="/wiki/Quantum_cryptography" title="Quantum cryptography">quantum cryptography</a> protocols, among other uses in quantum information science. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=No-cloning_theorem&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 26em;"> <ul><li><a href="/wiki/Fundamental_Fysiks_Group" title="Fundamental Fysiks Group">Fundamental Fysiks Group</a></li> <li><a href="/wiki/Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li> <li><a href="/wiki/No-broadcast_theorem" class="mw-redirect" title="No-broadcast theorem">No-broadcast theorem</a></li> <li><a href="/wiki/No-communication_theorem" title="No-communication theorem">No-communication theorem</a></li> <li><a href="/wiki/No-deleting_theorem" title="No-deleting theorem">No-deleting theorem</a></li> <li><a href="/wiki/No-hiding_theorem" title="No-hiding theorem">No-hiding theorem</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Quantum entanglement</a></li> <li><a href="/wiki/Quantum_cloning" title="Quantum cloning">Quantum cloning</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li> <li><a href="/wiki/Stronger_uncertainty_relations" title="Stronger uncertainty relations">Stronger uncertainty relations</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty principle</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=No-cloning_theorem&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-park-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-park_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-park_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPark1970" class="citation journal cs1">Park, James (1970). "The concept of transition in quantum mechanics". <i><a href="/wiki/Foundations_of_Physics" title="Foundations of Physics">Foundations of Physics</a></i>. <b>1</b> (1): 23–33. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1970FoPh....1...23P">1970FoPh....1...23P</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.623.5267">10.1.1.623.5267</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00708652">10.1007/BF00708652</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:55890485">55890485</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=The+concept+of+transition+in+quantum+mechanics&rft.volume=1&rft.issue=1&rft.pages=23-33&rft.date=1970&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.623.5267%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A55890485%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00708652&rft_id=info%3Abibcode%2F1970FoPh....1...23P&rft.aulast=Park&rft.aufirst=James&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-wootterszurek-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-wootterszurek_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-wootterszurek_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoottersZurek1982" class="citation journal cs1">Wootters, William; Zurek, Wojciech (1982). "A Single Quantum Cannot be Cloned". <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>299</b> (5886): 802–803. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982Natur.299..802W">1982Natur.299..802W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F299802a0">10.1038/299802a0</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4339227">4339227</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=A+Single+Quantum+Cannot+be+Cloned&rft.volume=299&rft.issue=5886&rft.pages=802-803&rft.date=1982&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4339227%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F299802a0&rft_id=info%3Abibcode%2F1982Natur.299..802W&rft.aulast=Wootters&rft.aufirst=William&rft.au=Zurek%2C+Wojciech&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-dieks-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-dieks_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dieks_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDieks1982" class="citation journal cs1">Dieks, Dennis (1982). "Communication by EPR devices". <i><a href="/wiki/Physics_Letters_A" class="mw-redirect" title="Physics Letters A">Physics Letters A</a></i>. <b>92</b> (6): 271–272. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982PhLA...92..271D">1982PhLA...92..271D</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.654.7183">10.1.1.654.7183</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0375-9601%2882%2990084-6">10.1016/0375-9601(82)90084-6</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1874%2F16932">1874/16932</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+A&rft.atitle=Communication+by+EPR+devices&rft.volume=92&rft.issue=6&rft.pages=271-272&rft.date=1982&rft_id=info%3Ahdl%2F1874%2F16932&rft_id=info%3Adoi%2F10.1016%2F0375-9601%2882%2990084-6&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.654.7183%23id-name%3DCiteSeerX&rft_id=info%3Abibcode%2F1982PhLA...92..271D&rft.aulast=Dieks&rft.aufirst=Dennis&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeres2003" class="citation journal cs1">Peres, Asher (2003). "How the No-Cloning Theorem Got its Name". <i><a href="/wiki/Fortschritte_der_Physik" class="mw-redirect" title="Fortschritte der Physik">Fortschritte der Physik</a></i>. <b>51</b> (45): 458–461. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0205076">quant-ph/0205076</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003ForPh..51..458P">2003ForPh..51..458P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fprop.200310062">10.1002/prop.200310062</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16588882">16588882</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fortschritte+der+Physik&rft.atitle=How+the+No-Cloning+Theorem+Got+its+Name&rft.volume=51&rft.issue=45&rft.pages=458-461&rft.date=2003&rft_id=info%3Aarxiv%2Fquant-ph%2F0205076&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16588882%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2Fprop.200310062&rft_id=info%3Abibcode%2F2003ForPh..51..458P&rft.aulast=Peres&rft.aufirst=Asher&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaiser2011" class="citation book cs1"><a href="/wiki/David_Kaiser_(physicist)" title="David Kaiser (physicist)">Kaiser, David</a> (2011). <i>How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival</i>. <a href="/wiki/W._W._Norton" class="mw-redirect" title="W. W. Norton">W. W. Norton</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-07636-3" title="Special:BookSources/978-0-393-07636-3"><bdi>978-0-393-07636-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+the+Hippies+Saved+Physics%3A+Science%2C+Counterculture%2C+and+the+Quantum+Revival&rft.pub=W.+W.+Norton&rft.date=2011&rft.isbn=978-0-393-07636-3&rft.aulast=Kaiser&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerbert1982" class="citation journal cs1">Herbert, Nick (1982). "FLASH—A superluminal communicator based upon a new kind of quantum measurement". <i><a href="/wiki/Foundations_of_Physics" title="Foundations of Physics">Foundations of Physics</a></i>. <b>12</b> (12): 1171–1179. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982FoPh...12.1171H">1982FoPh...12.1171H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00729622">10.1007/BF00729622</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123118337">123118337</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=FLASH%E2%80%94A+superluminal+communicator+based+upon+a+new+kind+of+quantum+measurement&rft.volume=12&rft.issue=12&rft.pages=1171-1179&rft.date=1982&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123118337%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00729622&rft_id=info%3Abibcode%2F1982FoPh...12.1171H&rft.aulast=Herbert&rft.aufirst=Nick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Ghirardi2013-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ghirardi2013_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ghirardi2013_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGhirardi2013" class="citation cs2">Ghirardi, GianCarlo (2013), "Entanglement, Nonlocality, Superluminal Signaling and Cloning", in Bracken, Paul (ed.), <i>Advances in Quantum Mechanics</i>, IntechOpen (published April 3, 2013), <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1305.2305">1305.2305</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5772%2F56429">10.5772/56429</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-953-51-1089-7" title="Special:BookSources/978-953-51-1089-7"><bdi>978-953-51-1089-7</bdi></a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118778014">118778014</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Entanglement%2C+Nonlocality%2C+Superluminal+Signaling+and+Cloning&rft.btitle=Advances+in+Quantum+Mechanics&rft.pub=IntechOpen&rft.date=2013&rft_id=info%3Aarxiv%2F1305.2305&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118778014%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.5772%2F56429&rft.isbn=978-953-51-1089-7&rft.aulast=Ghirardi&rft.aufirst=GianCarlo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOrtigoso2018" class="citation journal cs1">Ortigoso, Juan (2018). "Twelve years before the quantum no-cloning theorem". <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>. <b>86</b> (3): 201–205. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1707.06910">1707.06910</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018AmJPh..86..201O">2018AmJPh..86..201O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.5021356">10.1119/1.5021356</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119192142">119192142</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Twelve+years+before+the+quantum+no-cloning+theorem&rft.volume=86&rft.issue=3&rft.pages=201-205&rft.date=2018&rft_id=info%3Aarxiv%2F1707.06910&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119192142%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.5021356&rft_id=info%3Abibcode%2F2018AmJPh..86..201O&rft.aulast=Ortigoso&rft.aufirst=Juan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarnumCavesFuchsJozsa1996" class="citation journal cs1">Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin (1996-04-08). "Noncommuting Mixed States Cannot Be Broadcast". <i>Physical Review Letters</i>. <b>76</b> (15): 2818–2821. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9511010">quant-ph/9511010</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996PhRvL..76.2818B">1996PhRvL..76.2818B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.76.2818">10.1103/PhysRevLett.76.2818</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10060796">10060796</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11724387">11724387</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Noncommuting+Mixed+States+Cannot+Be+Broadcast&rft.volume=76&rft.issue=15&rft.pages=2818-2821&rft.date=1996-04-08&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11724387%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1996PhRvL..76.2818B&rft_id=info%3Aarxiv%2Fquant-ph%2F9511010&rft_id=info%3Apmid%2F10060796&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.76.2818&rft.aulast=Barnum&rft.aufirst=Howard&rft.au=Caves%2C+Carlton+M.&rft.au=Fuchs%2C+Christopher+A.&rft.au=Jozsa%2C+Richard&rft.au=Schumacher%2C+Benjamin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKalevHen2008" class="citation journal cs1">Kalev, Amir; Hen, Itay (2008-05-29). "No-Broadcasting Theorem and Its Classical Counterpart". <i>Physical Review Letters</i>. <b>100</b> (21): 210502. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0704.1754">0704.1754</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008PhRvL.100u0502K">2008PhRvL.100u0502K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.100.210502">10.1103/PhysRevLett.100.210502</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18518590">18518590</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40349990">40349990</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=No-Broadcasting+Theorem+and+Its+Classical+Counterpart&rft.volume=100&rft.issue=21&rft.pages=210502&rft.date=2008-05-29&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A40349990%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2008PhRvL.100u0502K&rft_id=info%3Aarxiv%2F0704.1754&rft_id=info%3Apmid%2F18518590&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.100.210502&rft.aulast=Kalev&rft.aufirst=Amir&rft.au=Hen%2C+Itay&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBužekHillery1996" class="citation journal cs1">Bužek, V.; Hillery, M. (1996). "Quantum Copying: Beyond the No-Cloning Theorem". <i>Phys. Rev. A</i>. <b>54</b> (3): 1844–1852. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9607018">quant-ph/9607018</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996PhRvA..54.1844B">1996PhRvA..54.1844B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.54.1844">10.1103/PhysRevA.54.1844</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9913670">9913670</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1446565">1446565</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+A&rft.atitle=Quantum+Copying%3A+Beyond+the+No-Cloning+Theorem&rft.volume=54&rft.issue=3&rft.pages=1844-1852&rft.date=1996&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1446565%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1996PhRvA..54.1844B&rft_id=info%3Aarxiv%2Fquant-ph%2F9607018&rft_id=info%3Apmid%2F9913670&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.54.1844&rft.aulast=Bu%C5%BEek&rft.aufirst=V.&rft.au=Hillery%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANo-cloning+theorem" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Other_sources">Other sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=No-cloning_theorem&action=edit&section=8" title="Edit section: Other sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>V. Buzek and M. Hillery, <i>Quantum cloning</i>, Physics World 14 (11) (2001), pp. 25–29.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output 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td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_information_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_information" title="Template:Quantum information"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_information" title="Template talk:Quantum information"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_information" title="Special:EditPage/Template:Quantum information"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_information_science" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/DiVincenzo%27s_criteria" title="DiVincenzo's criteria">DiVincenzo's criteria</a></li> <li><a href="/wiki/Noisy_intermediate-scale_quantum_era" title="Noisy intermediate-scale quantum era">NISQ era</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulation</a></li> <li><a href="/wiki/Qubit" title="Qubit">Qubit</a> <ul><li><a href="/wiki/Physical_and_logical_qubits" title="Physical and logical qubits">physical vs. logical</a></li></ul></li> <li><a href="/wiki/List_of_quantum_processors" title="List of quantum processors">Quantum processors</a> <ul><li><a href="/wiki/Cloud-based_quantum_computing" title="Cloud-based quantum computing">cloud-based</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's</a></li> <li><a href="/wiki/Eastin%E2%80%93Knill_theorem" title="Eastin–Knill theorem">Eastin–Knill</a></li> <li><a href="/wiki/Gleason%27s_theorem" title="Gleason's theorem">Gleason's</a></li> <li><a href="/wiki/Gottesman%E2%80%93Knill_theorem" title="Gottesman–Knill theorem">Gottesman–Knill</a></li> <li><a href="/wiki/Holevo%27s_theorem" title="Holevo's theorem">Holevo's</a></li> <li><a href="/wiki/No-broadcasting_theorem" title="No-broadcasting theorem">No-broadcasting</a></li> <li><a class="mw-selflink selflink">No-cloning</a></li> <li><a href="/wiki/No-communication_theorem" title="No-communication theorem">No-communication</a></li> <li><a href="/wiki/No-deleting_theorem" title="No-deleting theorem">No-deleting</a></li> <li><a href="/wiki/No-hiding_theorem" title="No-hiding theorem">No-hiding</a></li> <li><a href="/wiki/No-teleportation_theorem" title="No-teleportation theorem">No-teleportation</a></li> <li><a href="/wiki/PBR_theorem" class="mw-redirect" title="PBR theorem">PBR</a></li> <li><a href="/wiki/Quantum_speed_limit_theorems" class="mw-redirect" title="Quantum speed limit theorems">Quantum speed limit</a></li> <li><a href="/wiki/Threshold_theorem" title="Threshold theorem">Threshold</a></li> <li><a href="/wiki/Solovay%E2%80%93Kitaev_theorem" title="Solovay–Kitaev theorem">Solovay–Kitaev</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93HJW_theorem" title="Schrödinger–HJW theorem">Purification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum<br />communication</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_capacity" title="Classical capacity">Classical capacity</a> <ul><li><a href="/wiki/Entanglement-assisted_classical_capacity" title="Entanglement-assisted classical capacity">entanglement-assisted</a></li> <li><a href="/wiki/Quantum_capacity" title="Quantum capacity">quantum capacity</a></li></ul></li> <li><a href="/wiki/Entanglement_distillation" title="Entanglement distillation">Entanglement distillation</a></li> <li><a href="/wiki/Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li> <li><a href="/wiki/LOCC" title="LOCC">LOCC</a></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a> <ul><li><a href="/wiki/Quantum_network" title="Quantum network">quantum network</a></li></ul></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a> <ul><li><a href="/wiki/Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a></li></ul></li> <li><a href="/wiki/Superdense_coding" title="Superdense coding">Superdense coding</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Quantum_cryptography" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/Quantum_coin_flipping" title="Quantum coin flipping">Quantum coin flipping</a></li> <li><a href="/wiki/Quantum_money" title="Quantum money">Quantum money</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a> <ul><li><a href="/wiki/BB84" title="BB84">BB84</a></li> <li><a href="/wiki/SARG04" title="SARG04">SARG04</a></li> <li><a href="/wiki/List_of_quantum_key_distribution_protocols" title="List of quantum key distribution protocols">other protocols</a></li></ul></li> <li><a href="/wiki/Quantum_secret_sharing" title="Quantum secret sharing">Quantum secret sharing</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a></li> <li><a href="/wiki/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani</a></li> <li><a href="/wiki/BHT_algorithm" title="BHT algorithm">BHT</a></li> <li><a href="/wiki/Boson_sampling" title="Boson sampling">Boson sampling</a></li> <li><a href="/wiki/Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa</a></li> <li><a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's</a></li> <li><a href="/wiki/HHL_algorithm" title="HHL algorithm">HHL</a></li> <li><a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">Hidden subgroup</a></li> <li><a href="/wiki/Quantum_annealing" title="Quantum annealing">Quantum annealing</a></li> <li><a href="/wiki/Quantum_counting_algorithm" title="Quantum counting algorithm">Quantum counting</a></li> <li><a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li> <li><a href="/wiki/Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization</a></li> <li><a href="/wiki/Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li> <li><a href="/wiki/Simon%27s_problem" title="Simon's problem">Simon's</a></li> <li><a href="/wiki/Variational_quantum_eigensolver" title="Variational quantum eigensolver">VQE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum<br />complexity theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BQP" title="BQP">BQP</a></li> <li><a href="/wiki/Exact_quantum_polynomial_time" title="Exact quantum polynomial time">EQP</a></li> <li><a href="/wiki/QIP_(complexity)" title="QIP (complexity)">QIP</a></li> <li><a href="/wiki/QMA" title="QMA">QMA</a></li> <li><a href="/wiki/PostBQP" title="PostBQP">PostBQP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum <br /> processor benchmarks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a></li> <li><a href="/wiki/Quantum_volume" title="Quantum volume">Quantum volume</a></li> <li><a href="/wiki/Randomized_benchmarking" title="Randomized benchmarking">Randomized benchmarking</a> <ul><li><a href="/wiki/Cross-entropy_benchmarking" title="Cross-entropy benchmarking">XEB</a></li></ul></li> <li><a href="/wiki/Relaxation_(NMR)" title="Relaxation (NMR)">Relaxation times</a> <ul><li><a href="/wiki/Spin%E2%80%93lattice_relaxation" title="Spin–lattice relaxation"><i>T</i><sub>1</sub></a></li> <li><a href="/wiki/Spin%E2%80%93spin_relaxation" title="Spin–spin relaxation"><i>T</i><sub>2</sub></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum<br /><a href="/wiki/Model_of_computation" title="Model of computation">computing models</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adiabatic_quantum_computation" title="Adiabatic quantum computation">Adiabatic quantum computation</a></li> <li><a href="/wiki/Continuous-variable_quantum_information" title="Continuous-variable quantum information">Continuous-variable quantum information</a></li> <li><a href="/wiki/One-way_quantum_computer" title="One-way quantum computer">One-way quantum computer</a> <ul><li><a href="/wiki/Cluster_state" title="Cluster state">cluster state</a></li></ul></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a> <ul><li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gate</a></li></ul></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a> <ul><li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">quantum neural network</a></li></ul></li> <li><a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">Quantum Turing machine</a></li> <li><a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">Topological quantum computer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum<br />error correction</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Codes <ul><li><a href="/wiki/CSS_code" title="CSS code">CSS</a></li> <li><a href="/wiki/Quantum_convolutional_code" title="Quantum convolutional code">quantum convolutional</a></li> <li><a href="/wiki/Stabilizer_code" title="Stabilizer code">stabilizer</a></li> <li><a href="/wiki/Shor_code" class="mw-redirect" title="Shor code">Shor</a></li> <li><a href="/wiki/Bacon%E2%80%93Shor_code" title="Bacon–Shor code">Bacon–Shor</a></li> <li><a href="/wiki/Steane_code" title="Steane code">Steane</a></li> <li><a href="/wiki/Toric_code" title="Toric code">Toric</a></li> <li><a href="/wiki/Gnu_code" title="Gnu code"><i>gnu</i></a></li></ul></li> <li><a href="/wiki/Entanglement-assisted_stabilizer_formalism" title="Entanglement-assisted stabilizer formalism">Entanglement-assisted</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical<br />implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cavity_quantum_electrodynamics" title="Cavity quantum electrodynamics">Cavity QED</a></li> <li><a href="/wiki/Circuit_quantum_electrodynamics" title="Circuit quantum electrodynamics">Circuit QED</a></li> <li><a href="/wiki/Linear_optical_quantum_computing" title="Linear optical quantum computing">Linear optical QC</a></li> <li><a href="/wiki/KLM_protocol" title="KLM protocol">KLM protocol</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ultracold_atom" title="Ultracold atom">Ultracold atoms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Neutral_atom_quantum_computer" title="Neutral atom quantum computer">Neutral atom QC</a></li> <li><a href="/wiki/Trapped-ion_quantum_computer" title="Trapped-ion quantum computer">Trapped-ion QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spin_(physics)" title="Spin (physics)">Spin</a>-based</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kane_quantum_computer" title="Kane quantum computer">Kane QC</a></li> <li><a href="/wiki/Spin_qubit_quantum_computer" title="Spin qubit quantum computer">Spin qubit QC</a></li> <li><a href="/wiki/Nitrogen-vacancy_center" title="Nitrogen-vacancy center">NV center</a></li> <li><a href="/wiki/Nuclear_magnetic_resonance_quantum_computer" title="Nuclear magnetic resonance quantum computer">NMR QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">Superconducting</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Charge_qubit" title="Charge qubit">Charge qubit</a></li> <li><a href="/wiki/Flux_qubit" title="Flux qubit">Flux qubit</a></li> <li><a href="/wiki/Phase_qubit" title="Phase qubit">Phase qubit</a></li> <li><a href="/wiki/Transmon" title="Transmon">Transmon</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum<br />programming</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/OpenQASM" title="OpenQASM">OpenQASM</a>–<a href="/wiki/Qiskit" title="Qiskit">Qiskit</a>–<a href="/wiki/IBM_Quantum_Experience" class="mw-redirect" title="IBM Quantum Experience">IBM QX</a></li> <li><a href="/wiki/Quil_(instruction_set_architecture)" title="Quil (instruction set architecture)">Quil</a>–<a href="/wiki/Rigetti_Computing" title="Rigetti Computing">Forest/Rigetti QCS</a></li> <li><a href="/wiki/Cirq" title="Cirq">Cirq</a></li> <li><a href="/wiki/Q_Sharp" title="Q Sharp">Q#</a></li> <li><a href="/wiki/Libquantum" title="Libquantum">libquantum</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">many others...</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Quantum_information_science" title="Category:Quantum information science">Quantum information 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