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typeof="BreadcrumbList" class="sectionLinks" style="font-family:"Encode Sans Semi Condensed", sans-serif;padding-top:0;" data-v-6794a9ad><li property="itemListElement" typeof="ListItem" data-v-6794a9ad><a href="/science-research" property="item" typeof="WebPage" style="color:#333e48;" data-v-6794a9ad><span property="name" data-v-6794a9ad>Science & Research</span></a><meta property="position" content="1" data-v-6794a9ad></li></ol><div class="main-header"><h1 class="title-text">Quantum entanglement free of paradoxes</h1></div><div class="date-summary" style="flex-direction:column-reverse;"><div class="date-wrapper"><div class="date" style="color:#666666;font-family:"Encode Sans Semi Condensed", sans-serif;">Dec. 1, 2022</div><div style="display:none;" class="comment-count"><span></span></div></div><div><div class="teaser-text">While the well-known philosophical path toward quantum entanglement is conceptually attractive, it is fraught with hidden variable theories and paradoxes.</div><div class="author" style="font-family:"Encode Sans Semi Condensed", sans-serif;display:flex;"><div><a href="/home/contact/14286237/f-j-duarte" style="color:#2472ae;">F. J. Duarte</a></div></div></div></div></div></div></div></div></div></div><div style="margin-bottom:10px;padding-top:10px;display:block;flex:;align-items:;" class="block-wrapper 80907e01-e9da-459e-af73-7081c0b84be9" data-v-f6470316><div class="block-2 block" data-v-f6470316 data-v-18405aaa><div class="above-cta" style="background-color:;border:;padding:0;border-radius:0;box-shadow:;" data-v-18405aaa><div class="content" style="border-radius: 0 ;padding:10px;" data-v-18405aaa><div class="ebm-share-wrapper" block-id="80907e01-e9da-459e-af73-7081c0b84be9" data-v-18405aaa data-v-fe74701c><ul class="ebm-share-list collapsed" data-v-fe74701c><li class="ebm-share-item printer-icon" data-v-fe74701c><a href="/print/content/14286266" target="_blank" data-v-fe74701c><div style="padding:8px;" data-v-fe74701c><div class="ebm-icon mdi:printer" style="min-width:24px;min-height:24px;color:white;" data-v-fe74701c data-v-b1fb1343><span class="iconify i-mdi:printer" aria-hidden="true" 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data-v-18405aaa><div class="above-cta" style="background-color:;border:;padding:0;border-radius:0;box-shadow:;" data-v-18405aaa><div class="content" style="border-radius: 0 ;padding:10px;" data-v-18405aaa><div class="ebm-content-teaser-image" style="width:100%;" block-id="5e4619a7-5013-4e2f-923e-686ce2977f75" data-v-18405aaa><span class="ebm-content-teaser-image__small"><div class="web-image" credit-position="top"><figure class="ebm-image-wrapper"><div to target="_blank" aria-label class="ebm-image container" style="padding:0;"><div class="image-content"><div src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/12/FJ_Duarte_LFW_Figure.638dfee1dc61c.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="Simplified visible optical version of the quantum entanglement diagram disclosed by Pryce and Ward in 1947." height width="250" title="Simplified visible optical version of the quantum entanglement diagram disclosed by Pryce and Ward in 1947." style="border-radius:0;box-shadow:none;"><img src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/12/FJ_Duarte_LFW_Figure.638dfee1dc61c.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="Simplified visible optical version of the quantum entanglement diagram disclosed by Pryce and Ward in 1947." height="" width="250" title="Simplified visible optical version of the quantum entanglement diagram disclosed by Pryce and Ward in 1947." loading="lazy" width="250" height="" style="display: block; max-width: 100%; height: auto; margin-left: undefined; margin-right: undefined;" /></div></div><figcaption class="ebm-image-caption" style="font-family:"Encode Sans Semi Condensed", sans-serif;max-width:100%;text-align:left;">Simplified visible optical version of the quantum entanglement diagram disclosed by Pryce and Ward in 1947.</figcaption></div></figure></div></span><div class="ebm-button-wrapper solid view-gallery-btn" 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#onetrust-reject-all-handler, #onetrust-consent-sdk #onetrust-pc-btn-handler.cookie-setting-link { background-color: #26abe2 !important; border-color: #26abe2 !important; } #onetrust-consent-sdk .onetrust-pc-btn-handler { color: #26abe2 !important; border-color: #26abe2 !important; background-color: undefined !important; } </style></div><div id="fb-root"></div><div class="web-body-blocks page-contents__content-body"><div class="block text" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"></div><div style="width:100%;text-align:left;" class="text"><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><p>The popular press and the specialized literature often exalt the “strangeness” and “weirdness” of quantum mechanics. Certainly, quantum entanglement is the source of such pervasive and engrained opinions. That literature may be forgiven since it follows the philosophical path toward quantum entanglement as initiated by Einstein et al. (EPR),<sup>1</sup> and Schrödinger,<sup>2</sup> in 1935.</p><p>EPR’s opinion toward quantum mechanics as an “incomplete theory” was a powerful statement that promoted hidden variable theories (HVTs).<sup>3-5</sup> In the 1950s, Bohm linked the philosophical discussion with knowledge of <i>γ</i>-ray quantum entanglement experiments<sup>6</sup> and introduced the concept of an EPR paradox.<sup>3, 4</sup></p><div class="ebm-ad__embed" id="gam-5e91-3ed6-e20-1279"></div><p>The EPR paradox and HVTs became central to the philosophical path and resulted in the introduction of Bell’s theorem,<sup>7</sup> which reinforced von Neumann’s opinion<sup>8</sup> classifying HTVs as incompatible with quantum mechanics.<sup>7, 8</sup> In turn, Bell’s theorem stimulated the early experiments on quantum entanglement, in the visible optical regime<sup>9</sup> that provided the necessary evidence to convince a large portion of the physics community that HTVs were indeed incompatible with quantum mechanics.†</p><p>The philosophical path can be summarized as EPR <span>→ </span>Bohm and Aharonov <span>→ </span>Bell.</p><p>The physical path to quantum entanglement began with Dirac’s pair theory<sup>10</sup> in 1930 and was reinforced via a 1946 statement from Wheeler, relevant to a hypothetical e<sup>+</sup>e<sup>- </sup><span>→ </span><i>γ</i><sub>1</sub><i>γ</i><sub>2</sub> experiment, which provided the very first definition of what quantum entanglement is: <span>“</span>According to the pair theory, if one of these photons is linearly polarized in one plane, then the photon that goes off in the opposite direction with equal momentum is linearly polarized in the perpendicular plane.<span>”</span><sup>11</sup> This is the essence of quantum entanglement.</p><p>Pryce and Ward in 1947 provided the first diagram of a quantum entanglement experiment.<sup>12</sup> They also derived the quantum mechanical cross-sections to determine the scattering coincidence ratio caused by a quanta pair, with orthogonal polarizations (| <i>x</i> ⟩ and | <i>y</i> ⟩), propagating in opposite directions.<sup>12</sup> The cross-section equation was a function of the now iconic angular term cos 2 (<i>φ</i><sub>1</sub> <span>–</span> <i>φ</i><sub>2</sub>) so ubiquitous in quantum entanglement probability calculations. The prediction of Pryce and Ward was soon confirmed<sup>13</sup> and scattering experiments powered by<i> γ</i>-ray quanta pairs followed in 1948-1950.<sup>6, 14, 15</sup></p><p>Bohm and Aharonov<sup>5</sup> argued for the<i> γ</i>-ray experiments to be considered applicable to the EPR paradox situation. However, contemporaneous researchers opined that those experiments, even though in agreement with quantum mechanics, did not exclude HTVs.</p><div class="ebm-ad__embed" id="gam-9abe-2f92-bb1-0597"></div><p>In retrospect, that opinion was misguided†: “the two photons are entangled, and according to local realism, their polarizations planes should become independent… a typical EPR situation. Already in 1948, observations agreed with quantum mechanics… not with local realism.”<sup>16</sup></p><p>Here, it is important to emphasize that the physical path of Dirac <span>→ </span>Wheeler <span>→ </span>Pryce and Ward, which led to the <i>γ</i>-ray experiments, was developed in a complete vacuum from the philosophical path.</p><p>The essence of quantum entanglement is its probability amplitude expressed in Dirac’s notation:<sup>12,17</sup></p></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div class="block image" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"><div class="ebm-content-teaser-image" style="width:100%;"><span class="ebm-content-teaser-image__small"><div class="web-image" credit-position="top"><figure class="ebm-image-wrapper"><div to target="_blank" aria-label class="ebm-image container" style="padding:0;"><div class="image-content"><div src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq1.63852c9bec005.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq1" height width="250" title="2301 Lfw Dua Eq1" style="border-radius:0;box-shadow:none;"><img src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq1.63852c9bec005.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq1" height="" width="250" title="2301 Lfw Dua Eq1" loading="lazy" width="250" height="" style="display: block; max-width: 100%; height: auto; margin-left: undefined; margin-right: undefined;" /></div></div></div></figure></div></span></div></div></div><div class="block text" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"></div><div style="width:100%;text-align:left;" class="text"><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><p>that, following normalization, becomes:</p></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div class="block image" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"><div class="ebm-content-teaser-image" style="width:100%;"><span class="ebm-content-teaser-image__small"><div class="web-image" credit-position="top"><figure class="ebm-image-wrapper"><div to target="_blank" aria-label class="ebm-image container" style="padding:0;"><div class="image-content"><div src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq2.63852d0480e99.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq2" height width="250" title="2301 Lfw Dua Eq2" style="border-radius:0;box-shadow:none;"><img src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq2.63852d0480e99.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq2" height="" width="250" title="2301 Lfw Dua Eq2" loading="lazy" width="250" height="" style="display: block; max-width: 100%; height: auto; margin-left: undefined; margin-right: undefined;" /></div></div></div></figure></div></span></div></div></div><div class="block text" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"></div><div style="width:100%;text-align:left;" class="text"><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><p>Indeed, Pryce and Ward used this equation in their derivation of the correct quantum scattering cross-section disclosed in their 1947 paper. Ward went on to provide a semi-heuristic derivation, using conservation arguments, of this iconic probability amplitude in his doctoral thesis.<sup>17</sup></p><p>In 1965, Feynman introduced elements applicable to its derivation from a fairly abstract Hamiltonian perspective.<sup>18</sup> The Diracian probability amplitude of quantum entanglement was absent from the literature of the philosophical path but appeared out of the blue, without mention of its origin, in the early 1990s.</p><div class="ebm-ad__embed" id="gam-dd59-005a-321-aedc"></div><p>Depending on the sign and combination of polarization states, for a quanta pair and two channels of propagation, there are four probability amplitudes, or states:</p></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div class="block image" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"><div class="ebm-content-teaser-image" style="width:100%;"><span class="ebm-content-teaser-image__small"><div class="web-image" credit-position="top"><figure class="ebm-image-wrapper"><div to target="_blank" aria-label class="ebm-image container" style="padding:0;"><div class="image-content"><div src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq3.63852d6b0b350.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq3" height width="250" title="2301 Lfw Dua Eq3" style="border-radius:0;box-shadow:none;"><img src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq3.63852d6b0b350.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq3" height="" width="250" title="2301 Lfw Dua Eq3" loading="lazy" width="250" height="" style="display: block; max-width: 100%; height: auto; margin-left: undefined; margin-right: undefined;" /></div></div></div></figure></div></span></div></div></div><div class="block text" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"></div><div style="width:100%;text-align:left;" class="text"><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><p>The four states for quantum entanglement can be derived in a transparent mathematical approach using the interferometric Dirac-Feynman superposition probability amplitude,<sup>18, 19</sup> so that:<sup>20-22</sup></p></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div class="block image" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"><div class="ebm-content-teaser-image" style="width:100%;"><span class="ebm-content-teaser-image__small"><div class="web-image" credit-position="top"><figure class="ebm-image-wrapper"><div to target="_blank" aria-label class="ebm-image container" style="padding:0;"><div class="image-content"><div src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq4.63852dce28e92.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq4" height width="250" title="2301 Lfw Dua Eq4" style="border-radius:0;box-shadow:none;"><img src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq4.63852dce28e92.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq4" height="" width="250" title="2301 Lfw Dua Eq4" loading="lazy" width="250" height="" style="display: block; max-width: 100%; height: auto; margin-left: undefined; margin-right: undefined;" /></div></div></div></figure></div></span></div></div></div><div class="block text" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"></div><div style="width:100%;text-align:left;" class="text"><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><p>It can be shown that Pauli matrices can also be used to derive the four states of quantum entanglement:<sup>22, 23</sup></p></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div class="block image" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"><div class="ebm-content-teaser-image" style="width:100%;"><span class="ebm-content-teaser-image__small"><div class="web-image" credit-position="top"><figure class="ebm-image-wrapper"><div to target="_blank" aria-label class="ebm-image container" style="padding:0;"><div class="image-content"><div src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq5.63852e404c119.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq5" height width="250" title="2301 Lfw Dua Eq5" style="border-radius:0;box-shadow:none;"><img src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq5.63852e404c119.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq5" height="" width="250" title="2301 Lfw Dua Eq5" loading="lazy" width="250" height="" style="display: block; max-width: 100%; height: auto; margin-left: undefined; margin-right: undefined;" /></div></div></div></figure></div></span></div></div></div><div class="block text" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"></div><div style="width:100%;text-align:left;" class="text"><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><p>Since the equations of quantum mechanics are reversible, the probability amplitudes of quantum entanglement can be used to derive both the Dirac-Feynman interferometric principle and the Pauli matrices.<sup>22</sup></p><p>It is important to emphasize that the N-slit interferometric derivation<sup>20-22</sup> of:</p></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div class="block image" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"><div class="ebm-content-teaser-image" style="width:100%;"><span class="ebm-content-teaser-image__small"><div class="web-image" credit-position="top"><figure class="ebm-image-wrapper"><div to target="_blank" aria-label class="ebm-image container" style="padding:0;"><div class="image-content"><div src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq6.63852ebe5fd15.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq6" height width="250" title="2301 Lfw Dua Eq6" style="border-radius:0;box-shadow:none;"><img src="https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq6.63852ebe5fd15.png?auto=format,compress&fit=max&q=45&w=250&width=250" alt="2301 Lfw Dua Eq6" height="" width="250" title="2301 Lfw Dua Eq6" loading="lazy" width="250" height="" style="display: block; max-width: 100%; height: auto; margin-left: undefined; margin-right: undefined;" /></div></div></div></figure></div></span></div></div></div><div class="block text" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"></div><div style="width:100%;text-align:left;" class="text"><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><p>is completely free of paradoxes, independent of EPR arguments, and independent of Bell’s theorem.<sup>22</sup></p><p>Going back to the EPR paper: their crucial premise is an exact measurement of momentum <i>p </i>that leads to an “all values” spread in Δ<i>x</i>. This is what led EPR to conclude that quantum mechanics was an <span>“</span>incomplete theory.<span>”</span><sup>1</sup> However, “an absolutely exact measurement of momentum <i>p</i> with Δ<i>p </i>= 0 is physically impossible.”<sup>21</sup> There is always a non-zero Δ<i>p</i>. Once this is accepted, Heisenberg’s uncertainty principle Δ<i>x</i> Δ<i>p </i>≈ <i>h</i> applies and the “all values” is no longer a concern.<sup>21</sup></p><p>Central to the indeterminism of quantum mechanics is the experimentally well-proven nonlocality of the photon.<sup>22</sup> Indeterminism, as advocated by Born,<sup>24</sup> is crucial to quantum concepts. While discussing quantum mechanics, Feynman stated:<sup>18</sup> “Already in classical mechanics there was indeterminability.”</p><p>† HTVs, as proposed by Bohm,<sup>4</sup> are not quantum mechanical since they cease to obey Born’s rule and sought to overcome Δ<i>x</i> Δ<i>p</i> ≈ <i>h</i>.<sup>22</sup></p><p><b>REFERENCES </b></p><p>1. A. Einstein et al., <i>Phys. Rev.</i>, 47, 777–780 (1935). </p><p>2. E. Schrödinger, <i>Math. Proc. Camb. Phil. Soc.</i>, 31, 555<span>–</span>563 (1935). </p><p>3. D. Bohm, <i>Quantum Theory</i> (Prentice-Hall, Englewood Cliffs, 1951). </p><p>4. D. Bohm, <i>Phys. Rev.</i>, 85, 166–179 (1952). </p><p>5. D. Bohm and Y. Aharonov, <i>Phys. Rev.</i>, 108, 1070–1076 (1957). </p><p>6. C. S. Wu and I. Shaknov, <i>Phys. Rev.</i>, 77, 136 (1950). </p><p>7. J. S. Bell, <i>Physics</i>, 1, 195–200 (1964). </p><p>8. J. von Neumann, <i>Mathematische Grundlagen der Quanten-Mechanik</i> (Springer, Berlin, 1932). </p><p>9. A. Aspect et al., <i>Phys. Rev. Lett.</i>, 47, 460–463 (1981). </p><p>10. P. A. M. Dirac, <i>Math. Proc. Camb. Phil. Soc.</i>, 2, 361–375 (1930). </p><p>11. J. A. Wheeler, <i>Ann. New York Acad. Sci.</i>, 48, 219–238 (1946). </p><p>12. M. L. H. Pryce and J. C. Ward, <i>Nature</i>, 160, 435 (1947). </p><p>13. H. S. Snyder et al., <i>Phys. Rev.</i>, 73, 440–448 (1948). </p><p>14. R. C. Hanna, <i>Nature</i>, 162, 332 (1948). </p><p>15. E. Bleuler and H. L. Bradt, <i>Phys. Rev.</i>, 73, 1398 (1948). </p><p>16. R. H. Dalitz and F. J. Duarte, <i>Physics Today</i>, 53, 99 (2000). </p><p>17. J. C. Ward, <i>Some Properties of the Elementary Particles</i> (Oxford University, Oxford, 1949). </p><p>18. R. P. Feynman, <i>The Feynman Lectures on Physics</i> (Addison-Wesley, Reading, 1965). </p><p>19. P. A. M. Dirac, <i>The Principles of Quantum Mechanics</i>, 4th edition (Oxford University, Oxford, 1958). </p><p>20. F. J. Duarte, <i>J. Mod. Opt.</i>, 60, 1585–1587 (2013). </p><p>21. F. J. Duarte, <i>Quantum Optics for Engineers</i> (CRC, Boca Raton, 2014). </p><p>22. F. J. Duarte, <i>Fundamentals of Quantum Entanglement</i>, 2nd edition (Institute of Physics, Bristol, 2022). </p><p>23. F. J. Duarte et al., <i>Opt. Quantum Electron.</i>, 52, 106 (2020). </p><p>24. M. Born, <i>Natural Philosophy of Cause and Chance</i> (Clarendon, Oxford, 1949).</p></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div class="block text" style="--primary-font:"Encode Sans Semi Condensed", sans-serif;--secondary-font:Georgia, serif;--link-color:#2472ae;--color-primary-main:#26abe2;background-color:white;border:;"><div class="embed natural" style="max-width:autopx;"></div><div style="width:100%;text-align:left;" class="text"><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><div style="line-height:normal;" class="native-ad-embed"><div class="content-list left" style="--four-columns: 1fr;--three-columns: 1fr;--two-columns: 1fr;" data-v-836d1e34><div class="items-wrapper" data-v-836d1e34></div></div></div></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div style="display:block;"><div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div><div></div></div></div></div></div><div><div><div class="ebm-web-content-body-sidebars right"></div></div></div></div></div></div></div></div></div><div style="margin-bottom:10px;padding-top:10px;display:block;flex:;align-items:;" class="block-wrapper b849-1652-cb6-b266 hide-bottom-border" data-v-f6470316><div class="block-5 block" data-v-f6470316 data-v-18405aaa><div class="above-cta" style="background-color:;border:;padding:0;border-radius:0;box-shadow:;" data-v-18405aaa><div class="content" style="border-radius: 0 ;padding:10px;" data-v-18405aaa><div class="ebm-about-author wide" display-all-authors="true" block-id="b849-1652-cb6-b266" data-v-18405aaa data-v-d0888b06><div class="author-item" style="border-top-style:solid;border-top-color:#333e48;border-top-width:2px;padding-top:10px;border-bottom-style:solid;border-bottom-color:#333e48;border-bottom-width:2px;padding-bottom:10px;" data-v-d0888b06><div class="author-content" data-v-d0888b06><div class="heading" data-v-d0888b06> About the Author </div><h3 class="author-name-title" style="font-family:"Encode Sans Semi Condensed", sans-serif;" data-v-d0888b06>F. J. Duarte | Research Physicist, Interferometric Optics</h3><div class="author-bio" data-v-d0888b06><div style="width:100%;text-align:left;" data-v-d0888b06><section style="font-size:1.1rem;line-height:1.8rem;" class="web-html"><div class="html"><p><b>F. J. Duarte</b> is a research physicist at Interferometric Optics (Jonesborough, TN), an Optica Fellow (1993), and a Fellow of the Australian Institute of Physics (1987).</p></div><div style="display:none;" class="preview-overlay"></div></section></div></div><div class="ebm-button-wrapper hollow author-email" style="--button-accent-color:#26abe2;--hollow-hover-background:#26abe2;" data-v-d0888b06 data-v-f66234c9><a href="/cdn-cgi/l/email-protection#0d627d79646e7e6762787f636c614d6a606c6461236e6260" class="ebm-button ebm-button--hollow ebm-button--medium" style="background-color:white;color:#26abe2;border-width:1px;border-color:#26abe2;" target="_self" data-v-f66234c9><div class="content icon-left" data-v-f66234c9> Email </div></a></div></div></div></div></div></div></div></div><div style="margin-bottom:10px;padding-top:10px;display:block;flex:;align-items:;" class="block-wrapper 584f8a9f-7f80-47f4-a077-dfcf2de8db08 hide-bottom-border" data-v-f6470316><div class="block-6 block" data-v-f6470316 data-v-18405aaa><div class="above-cta" style="background-color:#ffffff;border:;padding:0;border-radius:0;box-shadow:;" data-v-18405aaa><h2 class="title" style="background-color:white;color:#26abe2;border-radius:0;padding:10px 10px;font-weight:600;" data-v-18405aaa><div style="display:inline-block;border-bottom:4px solid #26abe2;padding-bottom:8px;" data-v-18405aaa><div data-v-18405aaa>Continue Reading</div></div></h2><div class="content" style="border-radius: 0 ;padding:10px;" data-v-18405aaa><div class="blueconic-recommendations" data-v-18405aaa><div class="content-list above grid" style="--four-columns: 1fr 1fr;--three-columns: 1fr 1fr;--two-columns:1fr 1fr;" data-v-836d1e34><div class="items-wrapper" data-v-836d1e34><div class="item small" data-v-836d1e34><div class="ebm-content-item image-above section-top content-item" style="display:block;flex-direction:row;--image-width:100%;--image-height:auto;" title-font-size="16" some-items-have-images="true" data-v-836d1e34><div class="image-wrapper"><div class="web-image" credit-position="top"><figure class="ebm-image-wrapper"><a href="https://informa.blueconic.net/rest/v2/recommendations/redirect?storeId=0a8c11ea-8a56-4832-89b6-50fe636162b7&profileId=&itemId=www.laserfocusworld.com%2Fquantum%2Farticle%2F14310480%2Fvescent-photonics-inc-is-your-quantum-20-solution-stable-enough" rel="noopener noreferrer" target="_blank" aria-label="redirect?storeId=0a8c11ea 8a56 4832 89b6 50fe636162b7&profileId=&itemId=www.laserfocusworld.com%2Fquantum%2Farticle%2F14310480%2Fvescent photonics inc is your quantum 20 solution stable enough" class="ebm-image container full-teaser" style="padding:0;"><div class="image-content"><div src="https://informa.blueconic.net/rest/v2/contentStores/0a8c11ea-8a56-4832-89b6-50fe636162b7/items/www.laserfocusworld.com%252Fquantum%252Farticle%252F14310480%252Fvescent-photonics-inc-is-your-quantum-20-solution-stable-enough/image?auto=format,compress&fit=crop&q=45&h=139&height=139&w=250&width=250" alt height="139" width="250" title style="border-radius:0;box-shadow:none;"><img src="https://informa.blueconic.net/rest/v2/contentStores/0a8c11ea-8a56-4832-89b6-50fe636162b7/items/www.laserfocusworld.com%252Fquantum%252Farticle%252F14310480%252Fvescent-photonics-inc-is-your-quantum-20-solution-stable-enough/image?auto=format,compress&fit=crop&q=45&h=139&height=139&w=250&width=250" alt="" height="139" width="250" title="" loading="lazy" width="250" height="139" style="display: block; 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Certainly, quantum entanglement is the source of such pervasive and engrained opinions. That literature may be forgiven since it follows the philosophical path toward quantum entanglement as initiated by Einstein et al. (EPR),\u003Csup>1\u003C/sup> and Schrödinger,\u003Csup>2\u003C/sup> in 1935.\u003C/p>\u003Cp>EPR’s opinion toward quantum mechanics as an “incomplete theory” was a powerful statement that promoted hidden variable theories (HVTs).\u003Csup>3-5\u003C/sup> In the 1950s, Bohm linked the philosophical discussion with knowledge of \u003Ci>γ\u003C/i>-ray quantum entanglement experiments\u003Csup>6\u003C/sup> and introduced the concept of an EPR paradox.\u003Csup>3, 4\u003C/sup>\u003C/p>\u003Cp>The EPR paradox and HVTs became central to the philosophical path and resulted in the introduction of Bell’s theorem,\u003Csup>7\u003C/sup> which reinforced von Neumann’s opinion\u003Csup>8\u003C/sup> classifying HTVs as incompatible with quantum mechanics.\u003Csup>7, 8\u003C/sup> In turn, Bell’s theorem stimulated the early experiments on quantum entanglement, in the visible optical regime\u003Csup>9\u003C/sup> that provided the necessary evidence to convince a large portion of the physics community that HTVs were indeed incompatible with quantum mechanics.†\u003C/p>\u003Cp>The philosophical path can be summarized as EPR \u003Cspan>→ \u003C/span>Bohm and Aharonov \u003Cspan>→ \u003C/span>Bell.\u003C/p>\u003Cp>The physical path to quantum entanglement began with Dirac’s pair theory\u003Csup>10\u003C/sup> in 1930 and was reinforced via a 1946 statement from Wheeler, relevant to a hypothetical e\u003Csup>+\u003C/sup>e\u003Csup>- \u003C/sup>\u003Cspan>→ \u003C/span>\u003Ci>γ\u003C/i>\u003Csub>1\u003C/sub>\u003Ci>γ\u003C/i>\u003Csub>2\u003C/sub> experiment, which provided the very first definition of what quantum entanglement is: \u003Cspan>“\u003C/span>According to the pair theory, if one of these photons is linearly polarized in one plane, then the photon that goes off in the opposite direction with equal momentum is linearly polarized in the perpendicular plane.\u003Cspan>”\u003C/span>\u003Csup>11\u003C/sup> This is the essence of quantum entanglement.\u003C/p>\u003Cp>Pryce and Ward in 1947 provided the first diagram of a quantum entanglement experiment.\u003Csup>12\u003C/sup> They also derived the quantum mechanical cross-sections to determine the scattering coincidence ratio caused by a quanta pair, with orthogonal polarizations (| \u003Ci>x\u003C/i> ⟩ and | \u003Ci>y\u003C/i> ⟩), propagating in opposite directions.\u003Csup>12\u003C/sup> The cross-section equation was a function of the now iconic angular term cos 2 (\u003Ci>φ\u003C/i>\u003Csub>1\u003C/sub> \u003Cspan>–\u003C/span> \u003Ci>φ\u003C/i>\u003Csub>2\u003C/sub>) so ubiquitous in quantum entanglement probability calculations. The prediction of Pryce and Ward was soon confirmed\u003Csup>13\u003C/sup> and scattering experiments powered by\u003Ci> γ\u003C/i>-ray quanta pairs followed in 1948-1950.\u003Csup>6, 14, 15\u003C/sup>\u003C/p>\u003Cp>Bohm and Aharonov\u003Csup>5\u003C/sup> argued for the\u003Ci> γ\u003C/i>-ray experiments to be considered applicable to the EPR paradox situation. However, contemporaneous researchers opined that those experiments, even though in agreement with quantum mechanics, did not exclude HTVs.\u003C/p>\u003Cp>In retrospect, that opinion was misguided†: “the two photons are entangled, and according to local realism, their polarizations planes should become independent… a typical EPR situation. Already in 1948, observations agreed with quantum mechanics… not with local realism.”\u003Csup>16\u003C/sup>\u003C/p>\u003Cp>Here, it is important to emphasize that the physical path of Dirac \u003Cspan>→ \u003C/span>Wheeler \u003Cspan>→ \u003C/span>Pryce and Ward, which led to the \u003Ci>γ\u003C/i>-ray experiments, was developed in a complete vacuum from the philosophical path.\u003C/p>\u003Cp>The essence of quantum entanglement is its probability amplitude expressed in Dirac’s notation:\u003Csup>12,17\u003C/sup>\u003C/p>",{"type":442,"settings":443,"data":446},"image",{"text":45,"source":444,"imageId":445},"internal","63852c9cfb34a7aa1a8b45ee",{"image":447},{"src":448,"alt":449,"credit":394,"caption":394,"processingOptions":394,"metaData":450},"https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq1.63852c9bec005.png?auto=format%2Ccompress&w=320","2301 Lfw Dua Eq1",{"pixelWidth":451,"pixelHeight":452},592,78,{"type":438,"settings":454},{"text":455},"\u003Cp>that, following normalization, becomes:\u003C/p>",{"type":442,"settings":457,"data":459},{"text":45,"source":444,"imageId":458},"63852d0484f25b661b8b45ca",{"image":460},{"src":461,"alt":462,"credit":394,"caption":394,"processingOptions":394,"metaData":463},"https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq2.63852d0480e99.png?auto=format%2Ccompress&w=320","2301 Lfw Dua Eq2",{"pixelWidth":464,"pixelHeight":465},788,144,{"type":438,"settings":467},{"text":468,"addHtml":9},"\u003Cp>Indeed, Pryce and Ward used this equation in their derivation of the correct quantum scattering cross-section disclosed in their 1947 paper. Ward went on to provide a semi-heuristic derivation, using conservation arguments, of this iconic probability amplitude in his doctoral thesis.\u003Csup>17\u003C/sup>\u003C/p>\u003Cp>In 1965, Feynman introduced elements applicable to its derivation from a fairly abstract Hamiltonian perspective.\u003Csup>18\u003C/sup> The Diracian probability amplitude of quantum entanglement was absent from the literature of the philosophical path but appeared out of the blue, without mention of its origin, in the early 1990s.\u003C/p>\u003Cp>Depending on the sign and combination of polarization states, for a quanta pair and two channels of propagation, there are four probability amplitudes, or states:\u003C/p>",{"type":442,"settings":470,"data":472},{"text":45,"source":444,"imageId":471},"63852d6bdcebc001198b45ee",{"image":473},{"src":474,"alt":475,"credit":394,"caption":394,"processingOptions":394,"metaData":476},"https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq3.63852d6b0b350.png?auto=format%2Ccompress&w=320","2301 Lfw Dua Eq3",{"pixelWidth":477,"pixelHeight":478},518,106,{"type":438,"settings":480},{"text":481},"\u003Cp>The four states for quantum entanglement can be derived in a transparent mathematical approach using the interferometric Dirac-Feynman superposition probability amplitude,\u003Csup>18, 19\u003C/sup> so that:\u003Csup>20-22\u003C/sup>\u003C/p>",{"type":442,"settings":483,"data":485},{"text":45,"source":444,"imageId":484},"63852dce84f25b90198b45e9",{"image":486},{"src":487,"alt":488,"credit":394,"caption":394,"processingOptions":394,"metaData":489},"https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq4.63852dce28e92.png?auto=format%2Ccompress&w=320","2301 Lfw Dua Eq4",{"pixelWidth":490,"pixelHeight":491},1009,191,{"type":438,"settings":493},{"text":494},"\u003Cp>It can be shown that Pauli matrices can also be used to derive the four states of quantum entanglement:\u003Csup>22, 23\u003C/sup>\u003C/p>",{"type":442,"settings":496,"data":498},{"text":45,"source":444,"imageId":497},"63852e4084f25b671b8b45dd",{"image":499},{"src":500,"alt":501,"credit":394,"caption":394,"processingOptions":394,"metaData":502},"https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq5.63852e404c119.png?auto=format%2Ccompress&w=320","2301 Lfw Dua Eq5",{"pixelWidth":503,"pixelHeight":465},884,{"type":438,"settings":505},{"text":506,"addHtml":9},"\u003Cp>Since the equations of quantum mechanics are reversible, the probability amplitudes of quantum entanglement can be used to derive both the Dirac-Feynman interferometric principle and the Pauli matrices.\u003Csup>22\u003C/sup>\u003C/p>\u003Cp>It is important to emphasize that the N-slit interferometric derivation\u003Csup>20-22\u003C/sup> of:\u003C/p>",{"type":442,"settings":508,"data":510},{"text":45,"source":444,"imageId":509},"63852ebe84f25b5b198b45e7",{"image":511},{"src":512,"alt":513,"credit":394,"caption":394,"processingOptions":394,"metaData":514},"https://img.laserfocusworld.com/files/base/ebm/lfw/image/2022/11/2301LFW_dua_eq6.63852ebe5fd15.png?auto=format%2Ccompress&w=320","2301 Lfw Dua Eq6",{"pixelWidth":515,"pixelHeight":516},560,132,{"type":438,"settings":518},{"text":519,"addHtml":9},"\u003Cp>is completely free of paradoxes, independent of EPR arguments, and independent of Bell’s theorem.\u003Csup>22\u003C/sup>\u003C/p>\u003Cp>Going back to the EPR paper: their crucial premise is an exact measurement of momentum \u003Ci>p \u003C/i>that leads to an “all values” spread in Δ\u003Ci>x\u003C/i>. This is what led EPR to conclude that quantum mechanics was an \u003Cspan>“\u003C/span>incomplete theory.\u003Cspan>”\u003C/span>\u003Csup>1\u003C/sup> However, “an absolutely exact measurement of momentum \u003Ci>p\u003C/i> with Δ\u003Ci>p \u003C/i>= 0 is physically impossible.”\u003Csup>21\u003C/sup> There is always a non-zero Δ\u003Ci>p\u003C/i>. Once this is accepted, Heisenberg’s uncertainty principle Δ\u003Ci>x\u003C/i> Δ\u003Ci>p \u003C/i>≈ \u003Ci>h\u003C/i> applies and the “all values” is no longer a concern.\u003Csup>21\u003C/sup>\u003C/p>\u003Cp>Central to the indeterminism of quantum mechanics is the experimentally well-proven nonlocality of the photon.\u003Csup>22\u003C/sup> Indeterminism, as advocated by Born,\u003Csup>24\u003C/sup> is crucial to quantum concepts. While discussing quantum mechanics, Feynman stated:\u003Csup>18\u003C/sup> “Already in classical mechanics there was indeterminability.”\u003C/p>\u003Cp>† HTVs, as proposed by Bohm,\u003Csup>4\u003C/sup> are not quantum mechanical since they cease to obey Born’s rule and sought to overcome Δ\u003Ci>x\u003C/i> Δ\u003Ci>p\u003C/i> ≈ \u003Ci>h\u003C/i>.\u003Csup>22\u003C/sup>\u003C/p>\u003Cp>\u003Cb>REFERENCES\n\u003C/b>\u003C/p>\u003Cp>1. A. Einstein et al., \u003Ci>Phys. Rev.\u003C/i>, 47, 777–780 (1935).\n\u003C/p>\u003Cp>2. E. Schrödinger, \u003Ci>Math. Proc. Camb. Phil. Soc.\u003C/i>, 31, 555\u003Cspan>–\u003C/span>563 (1935). \n\u003C/p>\u003Cp>3. D. Bohm, \u003Ci>Quantum Theory\u003C/i> (Prentice-Hall, Englewood Cliffs, 1951).\n\u003C/p>\u003Cp>4. D. Bohm, \u003Ci>Phys. Rev.\u003C/i>, 85, 166–179 (1952).\n\u003C/p>\u003Cp>5. D. Bohm and Y. Aharonov, \u003Ci>Phys. Rev.\u003C/i>, 108, 1070–1076 (1957).\n\u003C/p>\u003Cp>6. C. S. Wu and I. Shaknov, \u003Ci>Phys. Rev.\u003C/i>, 77, 136 (1950).\n\u003C/p>\u003Cp>7. J. S. Bell, \u003Ci>Physics\u003C/i>, 1, 195–200 (1964).\n\u003C/p>\u003Cp>8. J. von Neumann, \u003Ci>Mathematische Grundlagen der Quanten-Mechanik\u003C/i> (Springer, Berlin, 1932).\n\u003C/p>\u003Cp>9. A. Aspect et al., \u003Ci>Phys. Rev. Lett.\u003C/i>, 47, 460–463 (1981).\n\u003C/p>\u003Cp>10. P. A. M. Dirac, \u003Ci>Math. Proc. Camb. Phil. Soc.\u003C/i>, 2, 361–375 (1930).\n\u003C/p>\u003Cp>11. J. A. Wheeler, \u003Ci>Ann. New York Acad. Sci.\u003C/i>, 48, 219–238 (1946).\n\u003C/p>\u003Cp>12. M. L. H. Pryce and J. C. Ward, \u003Ci>Nature\u003C/i>, 160, 435 (1947).\n\u003C/p>\u003Cp>13. H. S. Snyder et al., \u003Ci>Phys. Rev.\u003C/i>, 73, 440–448 (1948).\n\u003C/p>\u003Cp>14. R. C. Hanna, \u003Ci>Nature\u003C/i>, 162, 332 (1948).\n\u003C/p>\u003Cp>15. E. Bleuler and H. L. Bradt, \u003Ci>Phys. Rev.\u003C/i>, 73, 1398 (1948).\n\u003C/p>\u003Cp>16. R. H. Dalitz and F. J. Duarte, \u003Ci>Physics Today\u003C/i>, 53, 99 (2000).\n\u003C/p>\u003Cp>17. J. C. Ward, \u003Ci>Some Properties of the Elementary Particles\u003C/i> (Oxford University, Oxford, 1949).\n\u003C/p>\u003Cp>18. R. P. Feynman, \u003Ci>The Feynman Lectures on Physics\u003C/i> (Addison-Wesley, Reading, 1965).\n\u003C/p>\u003Cp>19. P. A. M. Dirac, \u003Ci>The Principles of Quantum Mechanics\u003C/i>, 4th edition (Oxford University, Oxford, 1958).\n\u003C/p>\u003Cp>20. F. J. Duarte, \u003Ci>J. Mod. Opt.\u003C/i>, 60, 1585–1587 (2013). \n\u003C/p>\u003Cp>21. F. J. Duarte, \u003Ci>Quantum Optics for Engineers\u003C/i> (CRC, Boca Raton, 2014).\n\u003C/p>\u003Cp>22. F. J. Duarte, \u003Ci>Fundamentals of Quantum Entanglement\u003C/i>, 2nd edition (Institute of Physics, Bristol, 2022).\n\u003C/p>\u003Cp>23. F. J. Duarte et al., \u003Ci>Opt. Quantum Electron.\u003C/i>, 52, 106 (2020).\n\u003C/p>\u003Cp>24. M. Born, \u003Ci>Natural Philosophy of Cause and Chance\u003C/i> (Clarendon, Oxford, 1949).\u003C/p>","Dec 1st 2022",1669870800000,[],[],{"title":434,"aiKeywords":525,"aiIndustries":551},[526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550],"Quantum entanglement","Hidden variable theories","Paradoxes","Philosophical path","Weirdness","Specialized literature","Einstein","Schrödinger","Bohm","Aharonov","Bell's theorem","von Neumann","Dirac","Wheeler","Pryce","Ward","Probability amplitude","Feynman","Pauli matrices","N-slit interferometric derivation","Indeterminism","Born's rule","Heisenberg's uncertainty principle","Nonlocality","Quantum mechanics",[552,553,531,554,555,550,527,556,557,558,559,560,561,562,563,544,545,564,536,565,546,566,548,547,567],"Quantum physics research","Philosophy","Physics community","Optical technology","Gamma-ray experiments","Quantum entanglement experiments","Probability calculations","Scattering experiments","Quantum scattering","Interferometric principles","Quantum optics","Hamiltonian perspective","EPR arguments","Nonlocality of photons","Classical mechanics","Mathematical foundations",{"surveyType":394,"surveyId":394},{"isRequired":44,"accessLevels":570},[],{"edges":572},[],{"edges":574},[],{"edges":576},[577],{"node":578},{"id":579,"name":580,"title":581,"type":18,"alias":582},14286237,"F. 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