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xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>String diagrams</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a> </li> </ul> With braiding <ul> <li> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> </li> </ul> With duals for objects <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them) </li> <li> <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have) </li> <li> <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a> </li> </ul> With duals for morphisms <ul> <li> monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a> </li> </ul> With traces <ul> <li> <a class="existingWikiWord" href="/nlab/show/trace">trace</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a> </li> </ul> Closed structure <ul> <li> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a> </li> </ul> Special sorts of products <ul> <li> <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a> </li> </ul> Semisimplicity <ul> <li> <a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a> </li> </ul> Morphisms <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a> (<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a> </li> </ul> Internal monoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a> </li> </ul> Examples <ul> <li> <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> </li> </ul> In higher category theory <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a> </li> </ul> </div></div> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/category">category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> </div> </div> <h1 id="string_diagrams">String diagrams</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#variants'>Variants</a></li> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#InLinearAlgebra'>In linear algebra</a></li> <li><a href='#in_quantum_computation'>In quantum computation</a></li> <li><a href='#InRepresentationTheory'>In representation theory</a></li> <li><a href='#ExamplesInLieTheory'>In Lie theory</a></li> <li><a href='#ExamplesInpQFT'>In perturbative quantum field theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#introduction_and_survey'>Introduction and survey</a></li> <li><a href='#OriginalArticles'>Original articles</a></li> <li><a href='#details'>Details</a></li> <li><a href='#software'>Software</a></li> <li><a href='#ReferencesQuantumInformationTheoryViaStringDiagrams'>Quantum information theory via String diagrams</a></li> <ul> <li><a href='#GeneralReferencesQuantumInformationTheoryViaStringDiagrams'>General</a></li> <li><a href='#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams'>Measurement & Classical structures</a></li> <li><a href='#ReferencesZXCalculus'>ZX-Calculus</a></li> </ul> </ul> </ul> </div> <h2 id="Idea">Idea</h2> String diagrams constitute a graphical calculus for expressing operations in <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>. <div class="float_right_image" style="margin: -30px 0px 20px 20px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/HotzFigure2.jpg" width="440px" /> <figcaption style="text-align: center">From <a href="#Hotz65">Hotz 65</a></figcaption> </figure> </div><div class="float_right_image" style="margin: -30px 0px 20px 20px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/PenroseDepictAContraction.jpg" width="440px" /> <figcaption style="text-align: center">From <a href="#Penrose71a">Penrose 71a</a></figcaption> </figure> </div><div class="float_right_image" style="margin: -30px 0px 20px 20px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/PenroseRindlerFigureA1.jpg" width="440px" /> <figcaption style="text-align: center">From <a href="#PenroseRindler84">Penrose-Rindler 84, Append.</a></figcaption> </figure> </div> In the archetypical case of the <a class="existingWikiWord" href="/nlab/show/Cartesian+monoidal+category">Cartesian monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a> this is Hotz’s notation (<a href="#Hotz65">Hotz 65</a>) for <a class="existingWikiWord" href="/nlab/show/automata">automata</a>, while for <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+spaces">finite-dimensional vector spaces</a> with their <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">usual tensor product</a> this is Penrose’s notation (<a href="#Penrose71a">Penrose 71a</a>, <a href="#PenroseRindler84">Penrose-Rindler 84</a>) for <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a>; but the same idea immediately applies more generally to any other <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> and yet more generally to <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a>, etc. The idea is roughly to think of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> in a monoidal category as “strings” and of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> from one <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> to another as a node which the source strings enter and the target strings exit. Further structure on the monoidal category is encoded in geometrical properties on these strings. For instance: <ul> <li> putting strings next to each other denotes the <a href="monoidal+category#eq:TensorProductFunctor">monoidal product</a>, and having no string at all denotes the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a>; </li> <li> braiding strings over each other corresponds to – yes, the <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">monoidal braiding</a> (if any); </li> <li> bending strings around corresponds to dualities on <a class="existingWikiWord" href="/nlab/show/dualizable+objects">dualizable objects</a> (if any). </li> </ul> Many operations in monoidal categories that look unenlightening in symbols become obvious in string diagram calculus, such as the <a class="existingWikiWord" href="/nlab/show/trace">trace</a>: an output wire gets bent around and connects to an input. String diagrams may be seen as dual (in the sense of <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a>) to <a class="existingWikiWord" href="/nlab/show/commutative+diagrams">commutative diagrams</a>. For instance, in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, an example of a string diagram for a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> (shown on the left) is shown on the right here: <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="281.219" height="120.449" viewBox="0 0 281.219 120.449"> <defs> <g> <g id="jJez_h2NiyvIQtT95mbV6JXRVFk=-glyph-0-0"> </g> <g id="jJez_h2NiyvIQtT95mbV6JXRVFk=-glyph-0-1"> <path d="M 2.03125 -1.328125 C 1.609375 -0.625 1.203125 -0.375 0.640625 -0.34375 C 0.5 -0.328125 0.40625 -0.328125 0.40625 -0.125 C 0.40625 -0.046875 0.46875 0 0.546875 0 C 0.765625 0 1.296875 -0.03125 1.515625 -0.03125 C 1.859375 -0.03125 2.25 0 2.578125 0 C 2.65625 0 2.796875 0 2.796875 -0.234375 C 2.796875 -0.328125 2.703125 -0.34375 2.625 -0.34375 C 2.359375 -0.375 2.125 -0.46875 2.125 -0.75 C 2.125 -0.921875 2.203125 -1.046875 2.359375 -1.3125 L 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transform="matrix(1, 0, 0, -1, 8.289, 60.806)"></path> <path fill="none" stroke-width="0.79701" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 215.894594 -42.389312 C 212.253969 -25.815094 216.480531 -12.510406 229.015688 -1.072906 " transform="matrix(1, 0, 0, -1, 8.289, 60.806)"></path> <path fill="none" stroke-width="0.79701" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 244.531313 -42.389312 C 248.171938 -25.815094 243.949281 -12.510406 231.414125 -1.072906 " transform="matrix(1, 0, 0, -1, 8.289, 60.806)"></path> <path fill="none" stroke-width="0.79701" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 230.214906 42.388031 L 230.214906 1.606781 " transform="matrix(1, 0, 0, -1, 8.289, 60.806)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use 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be represented by encoding further geometric properties of their string diagram calculus For instance: <ul> <li> in monoidal categories which are <a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon categories</a> the strings from above behave as if they have a small transversal extension which makes them behave as ribbons. Accordingly, there is a twist operation in the axioms of a ribbon category and graphically it corresponds to twisting the ribbons by 360 degrees. </li> <li> in a <a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a>, the trace can be represented by bending an output string around to connect to an input, even though if the objects are not dualizable the individual “bends” do not represent anything. </li> <li> in monoidal categories which are <a class="existingWikiWord" href="/nlab/show/spherical+category">spherical</a> all strings behave as if drawn on a sphere. </li> <li> in a <a class="existingWikiWord" href="/nlab/show/hypergraph+category">hypergraph category</a>, the string diagrams are labeled <a class="existingWikiWord" href="/nlab/show/hypergraphs">hypergraphs</a>. </li> <li> string diagrams can be extended to represent <a class="existingWikiWord" href="/nlab/show/monoidal+functors">monoidal functors</a> in several ways. One nice way is described in <a href="http://web.science.mq.edu.au/~mmccurdy/cms2010talk.pdf">these slides</a>, and can also be done with “3D regions” as drawn <a href="http://golem.ph.utexas.edu/category/2010/08/the_geometry_of_monoidal_fibra.html#c034428">here</a>. </li> <li> there is also a string diagram calculus for <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a>, which extends that for monoidal categories regarded as one-object bicategories. Thus, the strings now represent 1-cells and the nodes 2-cells, leaving the two-dimensional planar regions cut out by the strings to represent the 0-cells. This makes it manifest that in general, string diagram notation is Poincaré dual to the <a class="existingWikiWord" href="/nlab/show/globe">globular</a> notation: where one uses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional symbols, the other uses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>−</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2-d)</annotation></semantics></math>-dimensional symbols. </li> <li> string diagrams for bicategories can be generalized to string diagrams for <a class="existingWikiWord" href="/nlab/show/double+categories">double categories</a> and <a class="existingWikiWord" href="/nlab/show/proarrow+equipments">proarrow equipments</a> by distinguishing between “vertical” and “horizontal” strings. </li> <li> One can also <a class="existingWikiWord" href="/nlab/show/categorification">categorify</a> string diagrams to “<a class="existingWikiWord" href="/nlab/show/surface+diagram">surface diagrams</a>”, which are dual to commutative diagrams in <a class="existingWikiWord" href="/nlab/show/3-category">3-categories</a> (including <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategories</a>, and <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal categories</a>. </li> <li> Yet more generally, one can <a class="existingWikiWord" href="/nlab/show/categorification">categorify</a> string diagrams to diagrams that are dual to commutative diagrams in <a class="existingWikiWord" href="/nlab/show/n-category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-categories</a>, which yields the notion of <a class="existingWikiWord" href="/nlab/show/manifold+diagram">manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-diagrams</a>. </li> <li> As explained <a href="http://sbseminar.wordpress.com/2007/07/12/the-operadic-periodic-table/">here</a>, in the presence of certain levels of duality it may be better to work with diagrams on cylinders or spheres rather than in boxes. This relates to <a class="existingWikiWord" href="/nlab/show/planar+algebras">planar algebras</a> and canopolises<a href="/nlab/new/canopolis">?</a>. </li> <li> A string diagram calculus for <a class="existingWikiWord" href="/nlab/show/monoidal+fibrations">monoidal fibrations</a> can be obtained as a generalization of C.S. Peirce’s “existential graphs.” The ideas are essentially contained in (<a href="#BradyTrimble98">Brady-Trimble 98</a>) and developed in (<a href="#PontoShulman12">Ponto-Shulman 12</a>), and was discussed <a href="http://golem.ph.utexas.edu/category/2010/08/the_geometry_of_monoidal_fibra.html">here</a>. </li> <li> String diagrams for <a class="existingWikiWord" href="/nlab/show/closed+monoidal+categories">closed monoidal categories</a> (see also at <a class="existingWikiWord" href="/nlab/show/Kelly-Mac+Lane+graph">Kelly-Mac Lane graph</a>) are similar to those for <a class="existingWikiWord" href="/nlab/show/autonomous+categories">autonomous categories</a>, but a bit subtler, involving “boxes” to separate parts of the diagram. They were used informally by Baez and Stay <a href="#BaezQG06">here</a> and <a href="#Rosetta">here</a>, but can also be done in essentially the same way as the <a class="existingWikiWord" href="/nlab/show/proof+nets">proof nets</a> used in <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic</a> <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>; see <a href="#Lamarche08">Lamarche</a>. </li> <li> Proof nets for multiplicative <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> with negation similarly give string diagrams for <a class="existingWikiWord" href="/nlab/show/%2A-autonomous+categories">*-autonomous categories</a>, or more generally <a class="existingWikiWord" href="/nlab/show/linearly+distributive+categories">linearly distributive categories</a>; see <a href="#BCST">Blute-Cockett-Seely-Trimble</a>. </li> <li> Proof surfaces for noncommutative multiplicative <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> with negation, see <a href="#DunnVicary17">Dunn-Vicary</a> </li> <li> Sheet diagrams, string diagrams drawn on a branching surface, may be used for <a class="existingWikiWord" href="/nlab/show/rig+categories">rig categories</a>, see <a href="#CDH">Comfort-Delpeuch-Hedges</a>. </li> </ul> See also <a href="#Selinger09">Selinger 09</a> for a review of different string diagram formalisms. <h2 id="Examples">Examples</h2> <h3 id="InLinearAlgebra">In linear algebra</h3> String diagram calculus in <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>: <ul> <li id="PenroseNotation"> String diagrams in <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+spaces">finite-dimensional vector spaces</a> with the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">usual tensor product</a> are Penrose’s notation (<a href="#Penrose71a">Penrose 71a</a>, <a href="#Penrose71b">71b</a>, <a href="#Penrose72">72</a>, <a href="#PenroseRindler84">Penrose-Rindler 84, Append.</a>) for working with <a class="existingWikiWord" href="/nlab/show/tensors">tensors</a>. </li> <li> More recently, string diagrams in this category have come to be known as <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a>, especially so in application to <a class="existingWikiWord" href="/nlab/show/condensed+matter+physics">condensed matter physics</a> and also in <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a> and in particular in <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a>. </li> </ul> <h3 id="in_quantum_computation">In quantum computation</h3> <ul> <li>In <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit+diagrams">quantum circuit diagrams</a> are a form of string diagrams in <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+spaces">finite-dimensional</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>. See also at <a class="existingWikiWord" href="/nlab/show/finite+quantum+mechanics+in+terms+of+dagger-compact+categories">finite quantum mechanics in terms of dagger-compact categories</a>.</li> </ul> <h3 id="InRepresentationTheory">In representation theory</h3> String diagram calculus in <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>: <a href="#Mandula81">Mandula 81</a>, <a href="#Cvitanovic08">Cvitanović 08</a> <ul> <li>String diagrams for the monoidal category of finite-dimensional <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> of the group <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a> on complex vector spaces are called <a class="existingWikiWord" href="/nlab/show/spin+networks">spin networks</a>.</li> </ul> <h3 id="ExamplesInLieTheory">In Lie theory</h3> For string diagrams calculus in <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> see at: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie algebra object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/metric+Lie+algebra">metric Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/metric+Lie+representation">metric Lie representation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+weight+system">Lie algebra weight system</a>. </li> <li> <a class="existingWikiWord" href="/nlab/show/M2-brane+3-algebra">M2-brane 3-algebra</a> </li> </ul> <h3 id="ExamplesInpQFT">In perturbative quantum field theory</h3> For applications of string diagram calculus in <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, see at <ul> <li><a class="existingWikiWord" href="/nlab/show/%27t+Hooft+double+line+notation">'t Hooft double line notation</a></li> </ul> (…) <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/sharing+graph">sharing graph</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/surface+diagram">surface diagram</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kelly-Mac+Lane+diagram">Kelly-Mac Lane diagram</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/proof+net">proof net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/neural+network">neural network</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+weight+system">Lie algebra weight system</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/linguistics">natural language syntax</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Petri+net">Petri net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string-net+model">string-net model</a> </li> </ul> <h2 id="references">References</h2> <h3 id="introduction_and_survey">Introduction and survey</h3> Discussion in <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>: <ul> <li id="Mandula81"> <a class="existingWikiWord" href="/nlab/show/Jeffrey+Ellis+Mandula">Jeffrey Ellis Mandula</a>, Diagrammatic techniques in group theory, Southampton Univ. Phys. Dept. (1981) (<a href="https://cds.cern.ch/record/129911">cds:129911</a>, <a href="https://cds.cern.ch/record/129911/files/SHEP%2080-81-7.pdf">pdf</a>) </li> <li id="Cvitanovic08"> <a class="existingWikiWord" href="/nlab/show/Predrag+Cvitanovi%C4%87">Predrag Cvitanović</a>, Group Theory: Birdtracks, Lie’s, and Exceptional Groups, Princeton University Press July 2008 (<a href="https://press.princeton.edu/books/paperback/9780691202983/group-theory">PUP</a>, <a href="http://birdtracks.eu/">birdtracks.eu</a>, <a href="http://www.birdtracks.eu/version9.0/GroupTheory.pdf">pdf</a>) <blockquote> (aimed at <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> and <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>) </blockquote> </li> <li id="Peigné22"> <a class="existingWikiWord" href="/nlab/show/St%C3%A9phane+Peign%C3%A9">Stéphane Peigné</a>, Introduction to color in QCD: Initiation to the birdtrack pictorial technique [<a href="https://arxiv.org/abs/2302.07574">arXiv:2302.07574</a>] <blockquote> (further focus on <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>) </blockquote> </li> </ul> Introductions to and surveys of string diagram calculus: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, Categorical structures, in: M. Hazewinkel (ed.), Handbook of algebra – Volume 1, Elsevier (1996) [<a href="http://maths.mq.edu.au/~street/45.pdf">pdf</a>, <a href="https://www.elsevier.com/books/handbook-of-algebra/hazewinkel/978-0-444-82212-3">978-0-444-82212-3</a>] <blockquote> (in the generality of <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a>) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, Low dimensional topology and higher-order categories (<a href="http://www.mta.ca/~cat-dist/CT95Docs/LowDim.ps">ps</a>) <blockquote> (on <a class="existingWikiWord" href="/nlab/show/surface+diagrams">surface diagrams</a>) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, QG Seminar Fall 2000 [<a href="http://math.ucr.edu/home/baez/qg-fall2000/">web</a>), Winter 2001 (<a href="http://math.ucr.edu/home/baez/qg-winter2001/">web</a>), Fall 2006 (<a href="http://math.ucr.edu/home/baez/qg-fall2006/index.html#computation">web</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/The+Catsters">The Catsters</a>: String Diagrams, video lectures (2008) [playlist: <a href="https://www.youtube.com/playlist?list=PL50ABC4792BD0A086">YT</a>] </li> <li id="BaezStay11"> <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Stay">Mike Stay</a>: Physics, Topology, Logic and Computation: A Rosetta Stone, in: New Structures for Physics, Lecture Notes in Physics 813 Springer (2011) 95-174 [<a href="http://arxiv.org/abs/0903.0340">arXiv:0903.0340</a>, <a href="https://doi.org/10.1007/978-3-642-12821-9_2">doi:10.1007/978-3-642-12821-9_2</a>] </li> <li id="Selinger09"> <a class="existingWikiWord" href="/nlab/show/Peter+Selinger">Peter Selinger</a>, A survey of graphical languages for monoidal categories, in: <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a> (ed.) New Structures for Physics, Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg (2010) (<a href="http://arxiv.org/abs/0908.3347">arXiv:0908.334</a>, <a href="https://doi.org/10.1007/978-3-642-12821-9_4">doi:10.1007/978-3-642-12821-9_4</a>) </li> <li> Joseph M. Landsberg, §2.11 in: Tensors: Geometry and Applications, Graduate Studies in Mathematics 128, American Mathematical Society (2011) [ISBN:978-0-8218-6907-9, <a href="https://bookstore.ams.org/gsm-128">ams:gsm-128</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing (<a href="https://arxiv.org/abs/1203.0202">arXiv:1203.0202</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Robin+Piedeleu">Robin Piedeleu</a>, <a class="existingWikiWord" href="/nlab/show/Fabio+Zanasi">Fabio Zanasi</a>: An Introduction to String Diagrams for Computer Scientists [<a href="https://arxiv.org/abs/2305.08768">arXiv:2305.08768</a>] <blockquote> (in <a class="existingWikiWord" href="/nlab/show/computer+science">computer science</a>) </blockquote> </li> </ul> From the point of view of <a class="existingWikiWord" href="/nlab/show/finite+quantum+mechanics+in+terms+of+dagger-compact+categories">finite quantum mechanics in terms of dagger-compact categories</a>: <ul> <li id="CoeckePicturalism"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Quantum Picturalism (<a href="http://arxiv.org/abs/0908.1787">arXiv:0908.1787</a>) </li> <li id="CoeckeIntroducing"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Introducing categories to the practicing physicist (<a href="http://arxiv.org/abs/0808.1032">arXiv:0808.1032</a>) </li> <li id="CoeckePractising"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Categories for the practising physicist (<a href="http://arxiv.org/abs/0905.3010">arXiv:0905.3010</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Duncan">Ross Duncan</a>, Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13 (2011) 043016 (<a href="http://arxiv.org/abs/0906.4725">arXiv:0906.4725</a>) </li> <li id="HeunenVicary12"> <a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Jamie+Vicary">Jamie Vicary</a>, Lectures on categorical quantum mechanics, 2012 (<a href="https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf">pdf</a>) </li> </ul> From the point of view of <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a> in <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Jacob+Biamonte">Jacob Biamonte</a>, Ville Bergholm, Tensor Networks in a Nutshell, Contemporary Physics (<a href="https://arxiv.org/abs/1708.00006">arxiv:1708.00006</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Jacob+Biamonte">Jacob Biamonte</a>, Lectures on Quantum Tensor Networks (<a href="https://arxiv.org/abs/1912.10049">arXiv:1912.10049</a>) </li> </ul> Some philosophical discussion is given in <ul> <li><a class="existingWikiWord" href="/nlab/show/David+Corfield">David Corfield</a>, Section 10.4 of: Towards a Philosophy of Real Mathematics, CUP, 2003.</li> </ul> <h3 id="OriginalArticles">Original articles</h3> The development and use of string diagram calculus pre-dates its graphical appearance in print, due to the difficulty of printing non-text elements at the time. <blockquote> Many calculations in earlier works were quite clearly worked out with string diagrams, then painstakingly copied into equations. Sometimes, clearly graphical structures were described in some detail without actually being drawn: e.g. the construction of free compact closed categories in Kelly and Laplaza’s 1980 “Coherence for compact closed categories”. (<a href="http://angg.twu.net/categories-2017may02.html#.1">Pawel Sobocinski, 2 May 2017</a>) This idea that string diagrams are, due to technical issues, only useful for private calculation, is said explicitly by Penrose. Penrose and Rindler’s book “Spinors and Spacetime” (CUP 1984) has an 11-page appendix full of all sorts of beautiful, carefully hand-drawn graphical notation for tensors and various operations on them (e.g. anti-symmetrization and covariant derivative). On the second page, he says the following: “The notation has been found very useful in practice as it grealy simplifies the appearance of complicated tensor or spinor equations, the various interrelations expressed being discernable at a glance. Unfortunately the notation seems to be of value mainly for private calculations because it cannot be printed in the normal way.” (<a href="http://angg.twu.net/categories-2017may02.html#.2">Alex Kissinger, 2 May 2017</a>) </blockquote> The first formal definition of string diagrams in the literature appears to be in <ul> <li id="Hotz65"><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Hotz">Günter Hotz</a>, Eine Algebraisierung des Syntheseproblems von Schaltkreisen, EIK, Bd. 1, (185-205), Bd, 2, (209-231) 1965 (<a href="https://www.magentacloud.de/lnk/LiPMlYfh">part I</a>, <a href="https://www.magentacloud.de/lnk/YivslUWJ">part II</a>, <a class="existingWikiWord" href="/nlab/files/HotzSchaltkreise.pdf" title="pdf">pdf</a>)</li> </ul> Application of string diagrams to <a class="existingWikiWord" href="/nlab/show/tensor">tensor</a>-calculus in <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a> (hence for the case that the ambient <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> is that of <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> equipped with the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a>) was propagated by <a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a>, whence <a class="existingWikiWord" href="/nlab/show/physics">physicists</a> know string diagrams as Penrose notation for tensor calculus: <ul> <li id="Penrose71a"> <a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a>, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications, Academic Press (1971) (<a href="http://www2.math.uic.edu/~kauffman/Penrose.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/PenroseNegativeDimensionalTensors.pdf" title="pdf">pdf</a>) </li> <li id="Penrose71b"> <a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a>, Angular momentum: An approach to combinatorial spacetime, in Ted Bastin (ed.) Quantum Theory and Beyond, Cambridge University Press (1971), pp.151-180 (<a class="existingWikiWord" href="/nlab/files/PenroseAngularMomentum71.pdf" title="pdf">pdf</a>) </li> <li id="Penrose72"> <a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a>, On the nature of quantum geometry, in: J. Klauder (ed.) Magic Without Magic, Freeman, San Francisco, 1972, pp. 333–354 (<a href="http://inspirehep.net/record/74082">spire:74082</a>, <a class="existingWikiWord" href="/nlab/files/PenroseQuantumGeometry.pdf" title="pdf">pdf</a>) </li> <li id="PenroseRindler84"> <a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a>, <a class="existingWikiWord" href="/nlab/show/Wolfgang+Rindler">Wolfgang Rindler</a>, appendix (p. 424-434) of: Spinors and space-time – Volume 1: Two-spinor calculus and relativistic fields, Cambridge University Press 1984 (<a href="https://doi.org/10.1017/CBO9780511564048">doi:10.1017/CBO9780511564048</a>) </li> </ul> See also <ul> <li>Wikipedia, <a href="https://en.wikipedia.org/wiki/Penrose_graphical_notation">Penrose graphical notation</a></li> </ul> From the point of view of <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, an early description of string diagram calculus (without actually depicting any string diagrams, see the above comments) is in: <ul> <li id="KellyLaplaza80"> <a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <a class="existingWikiWord" href="/nlab/show/M.+L.+Laplaza">M. L. Laplaza</a>, Coherence for compact closed categories, Journal of Pure and Applied Algebra, 19 193-213 (1980) [<a hef="https://doi.org/10.1016/0022-4049(80)90101-2">doi:10.1016/0022-4049(80)90101-2</a>, <a href="https://core.ac.uk/download/pdf/82696829.pdf">pdf</a>] (proving the <a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence for compact closed categories</a>) </li> </ul> following <ul> <li id="Kelly72"> <a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, Many-variable functorial calculus I, in: <a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, M. Laplaza , <a class="existingWikiWord" href="/nlab/show/L.+Gaunce+Lewis%2C+Jr.">L. Gaunce Lewis, Jr.</a>, <a class="existingWikiWord" href="/nlab/show/Saunders+Mac+Lane">Saunders Mac Lane</a> (eds.) Coherence in Categories, Lecture Notes in Mathematics, vol 281. Springer, Berlin, Heidelberg 1972 (<a href="https://doi.org/10.1007/BFb0059556">doi:10.1007/BFb0059556</a>) (which does include the hand-drawn diagrams that are missing in <a href="#KellyLaplaza80">Kelly-Laplaza 80</a>!) </li> </ul> and in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, The geometry of tensor calculus I, Advances in Math. 88 (1991) 55-112; MR92d:18011 (<a href="https://core.ac.uk/download/pdf/82659437.pdf">pdf</a>, <a href="https://doi.org/10.1016/0001-8708(91)90003-P">doi:10.1016/0001-8708(91)90003-P</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a> and <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, The geometry of tensor calculus II (<a href="http://www.math.mq.edu.au/~street/GTCII.pdf">pdf</a>) </li> </ul> String diagram calculus was apparently popularized by its use in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Louis+Kauffman">Louis Kauffman</a>, Knots and physics, Series on Knots and Everything, Volume 1, World Scientific, 1991 (<a href="https://doi.org/10.1142/1116">doi:10.1142/1116</a>) (in the context of <a class="existingWikiWord" href="/nlab/show/knot+theory">knot theory</a>) </li> </ul> Probably <a class="existingWikiWord" href="/nlab/show/David+Yetter">David Yetter</a> was the first (at least in public) to write string diagrams with “coupons” (a term used by <a class="existingWikiWord" href="/nlab/show/Nicolai+Reshetikhin">Nicolai Reshetikhin</a> and <a class="existingWikiWord" href="/nlab/show/Turaev">Turaev</a> a few months later) to represent maps which are not inherent in the (braided or symmetric compact closed) monoidal structure. See also these: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a>, <a class="existingWikiWord" href="/nlab/show/David+Yetter">David Yetter</a>, Braided compact closed categories with applications to low dimensional topology Advances in Mathematics, 77:156–182, 1989. </li> <li> <a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a> and <a class="existingWikiWord" href="/nlab/show/David+Yetter">David Yetter</a>, Coherence theorems via knot theory. Journal of Pure and Applied Algebra, 78:49–76, 1992. </li> <li> <a class="existingWikiWord" href="/nlab/show/David+Yetter">David Yetter</a>, Framed tangles and a theorem of Deligne on braided deformations of tannakian categories In M. Gerstenhaber and <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a> (eds.) Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemporary Mathematics 134, pages 325–349. Americal Mathematical Society, 1992. </li> <li id="Mellies06"> <a class="existingWikiWord" href="/nlab/show/Paul-Andr%C3%A9+Melli%C3%A8s">Paul-André Melliès</a>, Functorial boxes in string diagrams, Procceding of Computer Science Logic 2006 in Szeged, Hungary. 2006 (<a href="https://dumas.ccsd.cnrs.fr/PPS/hal-00154243">hal:00154243</a>, <a href="https://hal.archives-ouvertes.fr/hal-00154243/document">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MelliesFunctorialBoxesInStringDiagrams.pdf" title="pdf">pdf</a>) <blockquote> (see also <a class="existingWikiWord" href="/nlab/show/computational+trilogy">computational trilogy</a>) </blockquote> </li> </ul> For more on the history of the notion see the bibliography in (<a href="#Selinger09">Selinger 09</a>). <h3 id="details">Details</h3> String diagrams for <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> are discussed in: <ul> <li> <a href="#Hotz65">Hotz 65</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> and <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, The geometry of tensor calculus I, Advances in Math. 88 (1991) 55-112; MR92d:18011. (<a href="http://tqft.net/other-papers/Geometry%20of%20Tensor%20Calculus%20-%20Joyal%20&%20Street.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> and <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, The geometry of tensor calculus II. (<a href="http://www.math.mq.edu.au/~street/GTCII.pdf">pdf</a>. </li> <li> <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> and <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, Planar diagrams and tensor algebra, available <a href="http://www.math.mq.edu.au/~street/PlanarDiags.pdf">here</a>. </li> </ul> and for discussion of <a class="existingWikiWord" href="/nlab/show/coherence+and+strictification+for+symmetric+monoidal+categories">coherence and strictification for symmetric monoidal categories</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Wilson">Paul Wilson</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Ghica">Dan Ghica</a>, <a class="existingWikiWord" href="/nlab/show/Fabio+Zanasi">Fabio Zanasi</a>: String Diagrams for Strictification and Coherence, Logical Methods in Computer Science (2024) [<a href="https://arxiv.org/abs/2201.11738">arXiv:2201.11738</a>]</li> </ul> For <a class="existingWikiWord" href="/nlab/show/1-categories">1-categories</a> in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Dan+Marsden">Dan Marsden</a>, Category Theory Using String Diagrams, (<a href="https://arxiv.org/abs/1401.7220">arXiv:1401.7220</a>). (therein: many explicit calculations, colored illustrations, avoiding the common practice of indicating 0-cells by non-filled circles) </li> </ul> For <a class="existingWikiWord" href="/nlab/show/traced+monoidal+categories">traced monoidal categories</a> in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a> and <a class="existingWikiWord" href="/nlab/show/Dominic+Verity">Verity</a>, Traced monoidal categories. </li> <li> David I. Spivak, Patrick Schultz, Dylan Rupel, String diagrams for traced and compact categories are oriented 1-cobordisms, <a href="https://arxiv.org/abs/1508.01069">arxiv</a> </li> </ul> For <a class="existingWikiWord" href="/nlab/show/closed+monoidal+categories">closed monoidal categories</a> in <ul id="BaezQG06"> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, Quantum Gravity Seminar - Fall 2006. <http://math.ucr.edu/home/baez/qg-fall2006/index.html#computation></li> </ul> <ul id="Lamarche08"> <li id="Rosetta"> <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a> and <a class="existingWikiWord" href="/nlab/show/Mike+Stay">Mike Stay</a>, Physics, Topology, Logic and Computation: A Rosetta Stone, <a href="https://arxiv.org/abs/0903.0340">arxiv</a> </li> <li> Francois Lamarche, Proof Nets for Intuitionistic Linear Logic: Essential nets, 2008 <a href="http://hal.inria.fr/docs/00/34/73/36/PDF/prfnet1.pdf">pdf</a> </li> </ul> <ul> <li> Ralf Hinze, Kan Extensions for Program Optimisation, Or: Art and Dan Explain an Old Trick, <a href="http://www.cs.ox.ac.uk/ralf.hinze/Kan.pdf">pdf</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dan+Ghica">Dan Ghica</a>, <a class="existingWikiWord" href="/nlab/show/Fabio+Zanasi">Fabio Zanasi</a>, Hierarchical string diagrams and applications (<a href="https://arxiv.org/abs/2305.18945">arXiv:2305.18945</a>) </li> </ul> For <a class="existingWikiWord" href="/nlab/show/biclosed+monoidal+categories">biclosed monoidal categories</a> in <ul> <li><a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Edward Grefenstette, and <a class="existingWikiWord" href="/nlab/show/Mehrnoosh+Sadrzadeh">Mehrnoosh Sadrzadeh</a>, Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus, 2013 <a href="https://www.sciencedirect.com/science/article/pii/S0168007213000626">link</a></li> </ul> For <a class="existingWikiWord" href="/nlab/show/linearly+distributive+categories">linearly distributive categories</a> in <ul id="BCST"> <li><a class="existingWikiWord" href="/nlab/show/Richard+Blute">Richard Blute</a> and <a class="existingWikiWord" href="/nlab/show/Robin+Cockett">Robin Cockett</a> and <a class="existingWikiWord" href="/nlab/show/R.A.G.+Seely">R.A.G. Seely</a> and <a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a>, Natural deduction and coherence for weakly distributive categories.</li> </ul> <ul> <li id="DunnVicary17">Lawrence Dunn, <a class="existingWikiWord" href="/nlab/show/Jamie+Vicary">Jamie Vicary</a>, Surface Proofs for Nonsymmetric Linear Logic (<a href="http://arxiv.org/abs/1701.04917">arXiv:1701.04917</a>)</li> </ul> For <a class="existingWikiWord" href="/nlab/show/indexed+monoidal+categories">indexed monoidal categories</a> in <ul> <li id="BradyTrimble98"> Geraldine Brady, <a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a>, <a class="existingWikiWord" href="/nlab/show/A+string+diagram+calculus+for+predicate+logic">A string diagram calculus for predicate logic</a> (1998) </li> <li id="PontoShulman12"> <a class="existingWikiWord" href="/nlab/show/Kate+Ponto">Kate Ponto</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, Duality and traces for indexed monoidal categories, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 (<a href="http://arxiv.org/abs/1211.1555">arXiv:1211.1555</a>) </li> </ul> For <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/traced+monoidal+categories">traced monoidal categories</a> in <ul> <li>George Kaye, The Graphical Language of Symmetric Traced Monoidal Categories, (<a href="https://arxiv.org/abs/2010.06319">arXiv:2010.06319</a>)</li> </ul> The generalization of string diagrams to one dimension higher is discussed in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a>, <a href="/toddtrimble/published/Surface+diagrams">Surface diagrams</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/John+Barrett">John Barrett</a>, <a class="existingWikiWord" href="/nlab/show/Catherine+Meusburger">Catherine Meusburger</a>, <a class="existingWikiWord" href="/nlab/show/Gregor+Schaumann">Gregor Schaumann</a>, Gray categories with duals and their diagrams, available <a href="http://arxiv.org/abs/1211.0529">here</a>. </li> </ul> The generalization to arbitrary dimension in terms of <a class="existingWikiWord" href="/nlab/show/opetope">opetopic</a> “zoom complexes” is due to <ul> <li id="KockJoyalBataninMascari07"><a class="existingWikiWord" href="/nlab/show/Joachim+Kock">Joachim Kock</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Batanin">Michael Batanin</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Fran%C3%A7ois+Mascari">Jean-François Mascari</a>, Polynomial functors and opetopes (<a href="http://arxiv.org/abs/0706.1033">arXiv:0706.1033</a>)</li> </ul> The generalization to arbitrary dimension in terms of <a class="existingWikiWord" href="/nlab/show/manifold+diagrams">manifold diagrams</a> (generalizing, in particular, <a class="existingWikiWord" href="/nlab/show/opetope">opetopic shapes</a>) is due to <ul> <li id="DornDouglas22"><a class="existingWikiWord" href="/nlab/show/Christoph+Dorn">Christoph Dorn</a> and <a class="existingWikiWord" href="/nlab/show/Christopher+Douglas">Christopher Douglas</a>, Manifold diagrams and tame tangles, 2022 (<a href="https://arxiv.org/abs/2208.13758">arXiv</a>, <a href="https://cxdorn.github.io/manifold-diagram-paper/">latest</a>)</li> </ul> Discussion of string diagram calculus for (<a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual</a>) <a class="existingWikiWord" href="/nlab/show/double+categories">double categories</a> and (<a class="existingWikiWord" href="/nlab/show/virtual+equipment">virtual</a>) <a class="existingWikiWord" href="/nlab/show/2-category+equipped+with+proarrows">pro-arrow equipments</a>: <ul> <li id="Myers16"> <a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, String Diagrams For Double Categories and (Virtual) Equipments [<a href="https://arxiv.org/abs/1612.02762">arXiv:1612.02762</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, String Diagrams for (Virtual) Proarrow Equipments (2017) [slides: <a href="http://www.mat.uc.pt/~ct2017/slides/myers_d.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Myers-StringDiagrams2017.pdf" title="pdf">pdf</a>] </li> </ul> See also at <a class="existingWikiWord" href="/nlab/show/opetopic+type+theory">opetopic type theory</a>. Discussion of sheet diagrams for <a class="existingWikiWord" href="/nlab/show/rig+categories">rig categories</a> is in <ul> <li id="CDH"><a class="existingWikiWord" href="/nlab/show/Cole+Comfort">Cole Comfort</a>, <a class="existingWikiWord" href="/nlab/show/Antonin+Delpeuch">Antonin Delpeuch</a>, <a class="existingWikiWord" href="/nlab/show/Jules+Hedges">Jules Hedges</a>, Sheet diagrams for bimonoidal categories, (<a href="https://arxiv.org/abs/2010.13361">arXiv:2010.13361</a>)</li> </ul> Discussion of the use of string diagrams to treat <a class="existingWikiWord" href="/nlab/show/universal+constructions">universal constructions</a> such as <a class="existingWikiWord" href="/nlab/show/limits">limits</a>, <a class="existingWikiWord" href="/nlab/show/Kan+extensions">Kan extensions</a>, and <a class="existingWikiWord" href="/nlab/show/ends">ends</a>: <ul> <li>Kenji Nakahira, Diagrammatic category theory [<a href="https://arxiv.org/abs/2307.08891">arXiv:2307.08891</a>]</li> </ul> A book on higher-categorical diagrams: <ul> <li><a class="existingWikiWord" href="/nlab/show/Amar+Hadzihasanovic">Amar Hadzihasanovic</a>, Combinatorics of higher-categorical diagrams [<a href="https://arxiv.org/abs/2404.07273">arXiv:2404.07273</a>]</li> </ul> <h3 id="software">Software</h3> The higher dimensional string diagrams (“zoom complexes” (<a href="#KockJoyalBataninMascari07">Kock-Joyal-Batanin-Mascari 07</a>)) used for presenting <a class="existingWikiWord" href="/nlab/show/opetopes">opetopes</a> in the context of <a class="existingWikiWord" href="/nlab/show/opetopic+type+theory">opetopic type theory</a> are introduced in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, Opetopic Diagrams 1 - Basics (<a href="http://www.youtube.com/watch?v=OANwLohwJqk">video</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, Opetopic Diagrams 2 - Geometry (<a href="http://www.youtube.com/watch?v=E7OvuA1jRKM">video</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Globular">Globular</a> is a web-based proof assistant for finitely-presented semistrict globular higher categories. It allows one to formalize higher-categorical proofs in finitely-presented n-categories and visualize them as string diagrams. </li> </ul> <div> <h3 id="ReferencesQuantumInformationTheoryViaStringDiagrams">Quantum information theory via String diagrams</h3> <h4 id="GeneralReferencesQuantumInformationTheoryViaStringDiagrams">General</h4> The observation that a natural language for <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a> and <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>, specifically for <a class="existingWikiWord" href="/nlab/show/quantum+circuit+diagrams">quantum circuit diagrams</a>, is that of <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/%E2%80%A0-compact+categories">†-compact categories</a> (see <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">quantum information theory via dagger-compact categories</a>): <ul> <li id="AbramskyCoecke04"> <a class="existingWikiWord" href="/nlab/show/Samson+Abramsky">Samson Abramsky</a>, <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/quant-ph/0402130">arXiv:quant-ph/0402130</a>, <a href="https://doi.org/10.1109/LICS.2004.1319636">doi:10.1109/LICS.2004.1319636</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="AbramskyCoecke05"> <a class="existingWikiWord" href="/nlab/show/Samson+Abramsky">Samson Abramsky</a>, <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Abstract Physical Traces, Theory and Applications of Categories, 14 6 (2005) 111-124. [<a href="http://www.tac.mta.ca/tac/volumes/14/6/14-06abs.html">tac:14-06</a>, <a href="https://arxiv.org/abs/0910.3144">arXiv:0910.3144</a>] </li> <li id="AbramskyCoecke08"> <a class="existingWikiWord" href="/nlab/show/Samson+Abramsky">Samson Abramsky</a>, <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Categorical quantum mechanics, in <a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Logic+and+Quantum+Structures">Handbook of Quantum Logic and Quantum Structures</a>, Elsevier (2008) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/0808.1023">arXiv:0808.1023</a>, <a href="https://www.sciencedirect.com/book/9780444528698/">ISBN:9780080931661</a>, <a href="https://doi.org/10.1109/LICS.2004.1319636">doi:10.1109/LICS.2004.1319636</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="Coecke07"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure, Proceedings of the <a href="https://www.mathstat.dal.ca/~selinger/qpl2005/">3rd International Workshop on Quantum Programming Languages (2005)</a>, Electronic Notes in Theoretical Computer Science 170 (2007) 49-72 [<a href="https://doi.org/10.1016/j.entcs.2006.12.011">doi:10.1016/j.entcs.2006.12.011</a>, <a href="https://arxiv.org/abs/quant-ph/0506134">arXiv:quant-ph/0506134</a>] </li> </ul> On the relation to <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a>/<a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>: <ul> <li id="AbramskyDuncan05"> <a class="existingWikiWord" href="/nlab/show/Samson+Abramsky">Samson Abramsky</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Duncan">Ross Duncan</a>, A Categorical Quantum Logic, Mathematical Structures in Computer Science 16 3 (2006) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/quant-ph/0512114">arXiv:quant-ph/0512114</a>, <a href="https://doi.org/10.1017/S0960129506005275">doi:10.1017/S0960129506005275</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="Duncan06"> <a class="existingWikiWord" href="/nlab/show/Ross+Duncan">Ross Duncan</a>, Types for quantum mechanics, 2006 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://personal.strath.ac.uk/ross.duncan/papers/rduncan-thesis.pdf">pdf</a>, <a href="http://www.cs.ox.ac.uk/people/ross.duncan/talks/2005/pps-22-05-2005.pdf">slides</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> Early exposition with introduction to <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>: <ul> <li id="CoeckeKindergarten"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Kindergarten quantum mechanics <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/quant-ph/0510032">arXiv:quant-ph/0510032</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="CoeckeIntroducing"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Introducing categories to the practicing physicist <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/0808.1032">arXiv:0808.1032</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="BaezStay09"> <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Stay">Mike Stay</a>, Physics, topology, logic and computation: a rosetta stone in: New Structures for Physics, <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a> (ed.), Lecture Notes in Physics 813, Springer (2011) 95-174 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/0903.0340">arxiv/0903.0340</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="CoeckePractising"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Oliver+Paquette">Eric Oliver Paquette</a>, Categories for the practising physicist, in: New Structures for Physics, Lecture Notes in Physics 813, Springer (2010) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/0905.3010">arXiv:0905.3010</a>, <a href="https://doi.org/10.1007/978-3-642-12821-9_3">doi:10.1007/978-3-642-12821-9_3</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="CoeckePicturalism"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Quantum Picturalism, Contemporary Physics 51 1 (2010) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/0908.1787">arXiv:0908.1787</a>, <a href="https://doi.org/10.1080/00107510903257624">doi:10.1080/00107510903257624</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> Review in contrast to <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a>: <ul> <li id="AbramskyCoecke2007"><a class="existingWikiWord" href="/nlab/show/Samson+Abramsky">Samson Abramsky</a>, <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, Physics from Computer Science: a Position Statement, <a href="https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/">International Journal of Unconventional Computing 3 3 (2007)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.cs.ox.ac.uk/files/349/YORKIJUC.pdf">pdf</a>, <a href="https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/ijuc-3-3-p-179-197/">ijuc-3-3-p-179-197</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> and with emphasis on <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Jamie+Vicary">Jamie Vicary</a>, The Topology of Quantum Algorithms, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science (2013) 93-102 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1209.3917">arXiv:1209.3917</a>, <a href="https://doi.org/10.1109/LICS.2013.14">doi:10.1109/LICS.2013.14</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> Generalization to <a class="existingWikiWord" href="/nlab/show/quantum+operations">quantum operations</a> on <a class="existingWikiWord" href="/nlab/show/mixed+states">mixed states</a> (<a class="existingWikiWord" href="/nlab/show/completely+positive+maps">completely positive maps</a> of <a class="existingWikiWord" href="/nlab/show/density+matrices">density matrices</a>): <ul> <li id="Selinger"> <a class="existingWikiWord" href="/nlab/show/Peter+Selinger">Peter Selinger</a>, Dagger compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer Science 170 (2007) 139-163 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/j.entcs.2006.12.018">doi:10.1016/j.entcs.2006.12.018</a>, <a href="http://www.mscs.dal.ca/~selinger/papers.html#dagger">web</a>, <a href="http://www.mscs.dal.ca/~selinger/papers/dagger.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, Pictures of complete positivity in arbitrary dimension, Information and Computation 250 50-58 (2016) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1110.3055">arXiv:1110.3055</a>, <a href="https://doi.org/10.1016/j.ic.2016.02.007">doi:10.1016/j.ic.2016.02.007</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, Categories of Quantum and Classical Channels, EPTCS 158 (2014) 1-14 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1408.0049">arXiv:1408.0049</a>, <a href="https://doi.org/10.4204/EPTCS.158.1">doi:10.4204/EPTCS.158.1</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> Textbook accounts (with background on relevant <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>): <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, Picturing Quantum Processes – A First Course in Quantum Theory and Diagrammatic Reasoning, Cambridge University Press (2017) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.cambridge.org/ae/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/picturing-quantum-processes-first-course-quantum-theory-and-diagrammatic-reasoning?format=HB&isbn=9781107104228">ISBN:9781107104228</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="HeunenVicary19"> <a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Jamie+Vicary">Jamie Vicary</a>: Categories for Quantum Theory, Oxford University Press (2019) [<a href="https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616">ISBN:9780198739616</a>] based on: <a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Jamie+Vicary">Jamie Vicary</a>, Lectures on categorical quantum mechanics (2012) [<a href="https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/HeunenVicary-QuantumLectures.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Stefano+Gogioso">Stefano Gogioso</a>, Quantum in Pictures, <a href="https://www.quantinuum.com/publications">Quantinuum Publications</a> (2023) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.amazon.co.uk/dp/1739214714">ISBN 978-1739214715</a>, <a href="https://www.quantinuum.com/news/quantum-in-pictures">Quantinuum blog</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> (focus on <a class="existingWikiWord" href="/nlab/show/ZX-calculus">ZX-calculus</a>) </blockquote> </li> </ul> <h4 id="MeasurementReferencesQuantumInformationTheoryViaStringDiagrams">Measurement & Classical structures</h4> Formalization of <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> via <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a>-<a class="existingWikiWord" href="/nlab/show/structures">structures</a> (“<a class="existingWikiWord" href="/nlab/show/classical+structures">classical structures</a>”): <ul> <li id="CoeckePavlović08"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Du%C5%A1ko+Pavlovi%C4%87">Duško Pavlović</a>, Quantum measurements without sums, in <a class="existingWikiWord" href="/nlab/show/Louis+Kauffman">Louis Kauffman</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+Lomonaco">Samuel Lomonaco</a> (eds.), Mathematics of Quantum Computation and Quantum Technology, Taylor & Francis (2008) 559-596 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/quant-ph/0608035">arXiv:quant-ph/0608035</a>, <a href="https://doi.org/10.1201/9781584889007">doi:10.1201/9781584889007</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="CoeckePaquette08"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Oliver+Paquette">Eric Oliver Paquette</a>, POVMs and Naimark’s theorem without sums, Electronic Notes in Theoretical Computer Science 210 (2008) 15-31 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/quant-ph/0608072">arXiv:quant-ph/0608072</a>, <a href="https://doi.org/10.1016/j.entcs.2008.04.015">doi:10.1016/j.entcs.2008.04.015</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="CoeckePaquettePavlović09"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Oliver+Paquette">Eric Oliver Paquette</a>, <a class="existingWikiWord" href="/nlab/show/Du%C5%A1ko+Pavlovi%C4%87">Duško Pavlović</a>, Classical and quantum structuralism, in: Semantic Techniques in Quantum Computation, Cambridge University Press (2009) 29-69 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/0904.1997">arXiv:0904.1997</a>, <a href="https://doi.org/10.1017/CBO9781139193313.003">doi:10.1017/CBO9781139193313.003</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Du%C5%A1ko+Pavlovi%C4%87">Duško Pavlović</a>, <a class="existingWikiWord" href="/nlab/show/Jamie+Vicary">Jamie Vicary</a>, A new description of orthogonal bases, Mathematical Structures in Computer Science 23 3 (2012) 555- 567 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/0810.0812">arXiv:0810.0812</a>, <a href="https://doi.org/10.1017/S0960129512000047">doi:10.1017/S0960129512000047</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> and the evolution of the “<a class="existingWikiWord" href="/nlab/show/classical+structures">classical structures</a>”-monad into the “spider”-diagrams (terminology for <a href="Frobenius+algebra#NormalFormAndSpiderTheorem">special Frobenius normal form</a>, originating in <a href="#CoeckePaquette08">Coecke & Paquette 2008, p. 6</a>, <a href="#CoeckeDuncan08">Coecke & Duncan 2008, Thm. 1</a>) of the <a class="existingWikiWord" href="/nlab/show/ZX-calculus">ZX-calculus</a>: <ul> <li id="CoeckeDuncan08"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Duncan">Ross Duncan</a>, §3 in: Interacting Quantum Observables, in Automata, Languages and Programming. ICALP 2008, Lecture Notes in Computer Science 5126, Springer (2008) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-3-540-70583-3_25">doi:10.1007/978-3-540-70583-3_25</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, §§2 in: Graph Rewrite Systems for Classical Structures in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>†</mo></mrow><annotation encoding="application/x-tex">\dagger</annotation></semantics></math>-Symmetric Monoidal Categories, MSc thesis, Oxford (2008) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.cs.ox.ac.uk/people/bob.coecke/Aleks.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Kissinger-CLassicalStructures.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, §4 in: Exploring a Quantum Theory with Graph Rewriting and Computer Algebra, in: Intelligent Computer Mathematics. CICM 2009, Lecture Notes in Computer Science 5625 (2009) 90-105 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-3-642-02614-0_12">doi:10.1007/978-3-642-02614-0_12</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="CoeckeDuncan11"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Duncan">Ross Duncan</a>, Def. 6.4 in: Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13 (2011) 043016 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/0906.4725">arXiv:0906.4725</a>, <a href="https://doi.org/10.1088/1367-2630/13/4/043016">doi:10.1088/1367-2630/13/4/043016</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> <h4 id="ReferencesZXCalculus">ZX-Calculus</h4> Evolution of the “classical structures”-Frobenius algebra (<a href="#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams">above</a>) into the “spider”-ingredient of the <a class="existingWikiWord" href="/nlab/show/ZX-calculus">ZX-calculus</a> for specific control of <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a>-diagrams: <ul> <li id="CoeckeDuncan08"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Duncan">Ross Duncan</a>, §3 in: Interacting Quantum Observables, in Automata, Languages and Programming. ICALP 2008, Lecture Notes in Computer Science 5126, Springer (2008) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-3-540-70583-3_25">doi:10.1007/978-3-540-70583-3_25</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, Graph Rewrite Systems for Classical Structures in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>†</mo></mrow><annotation encoding="application/x-tex">\dagger</annotation></semantics></math>-Symmetric Monoidal Categories, MSc thesis, Oxford (2008) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.cs.ox.ac.uk/people/bob.coecke/Aleks.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Kissinger-CLassicalStructures.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, Exploring a Quantum Theory with Graph Rewriting and Computer Algebra, in: Intelligent Computer Mathematics. CICM 2009, Lecture Notes in Computer Science 5625 (2009) 90-105 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-3-642-02614-0_12">doi:10.1007/978-3-642-02614-0_12</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li id="CoeckeDuncan11"> <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Duncan">Ross Duncan</a>, Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13 (2011) 043016 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/0906.4725">arXiv:0906.4725</a>, <a href="https://doi.org/10.1088/1367-2630/13/4/043016">doi:10.1088/1367-2630/13/4/043016</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> Relating the <a class="existingWikiWord" href="/nlab/show/ZX-calculus">ZX-calculus</a> to <a class="existingWikiWord" href="/nlab/show/braided+fusion+categories">braided fusion categories</a> for <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a> <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Fatimah+Rita+Ahmadi">Fatimah Rita Ahmadi</a>, <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2211.03855">arXiv:2211.03855</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> </div></body></html> </div> <div class="revisedby"> Last revised on October 8, 2024 at 12:22:41. 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