Universal properties in Quantum Theory SYCO 2, University of Strathclyde, Glasgow Mathieu Huot Sam Staton Oxford University December 17, 2018
Table of contents 1. Introduction 2. Symmetric monoidal categories with discarding 3. Universality of CPTP 4. Affine completions, PROP and quantum circuits 5. Enriched setting 1
Introduction
Standard point of view • pure QM is not random, and is reversible • pure QM does not allow discarding composition tensor product | 0 y X X U X 2
Standard point of view • pure QM is not random, and is reversible • pure QM does not allow discarding • full QM: mixed states, quantum channels composition tensor product | 0 y X X U X 2
Von Neumann’s model: density matrices Completely Positive Trace Preserving (CPTP) maps Pure QM (+ ancillas) • M n p C q • state space C n • combination of systems: b • combination of systems: b • ancilla • ancilla (auxiliary system) • ad U : M ÞÑ UMU ˚ super-operator • unitary transformation U | j 0 � × H S T | j 1 � • H S • • | j 2 � × H Figure 1: Quantum Fourier transform on three qubits 3
Von Neumann’s model: density matrices Completely Positive Trace Preserving (CPTP) maps Pure QM (+ ancillas) • M n p C q • state space C n • combination of systems: b • combination of systems: b • ancilla • ancilla (auxiliary system) • ad U : M ÞÑ UMU ˚ super-operator • unitary transformation U • no global phase • allows discarding (trace) | j 0 � × H S T | j 1 � • H S • • | j 2 � × H Figure 1: Quantum Fourier transform on three qubits 3
❴ ❴ ✤✤✤✤✤✤✤ ✤✤✤✤✤✤✤ ❴ ❴ � ✙✙✙✙✙✙ � ✙✙✙✙✙✙ ❴ ❴ ❴ ❴ � ✙✙✙✙✙✙ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✤✤✤✤✤✤✤ ✤✤✤✤✤✤✤ ❴ ❴ ❴ ❴ ✤✤✤✤✤✤✤ ✤✤✤✤✤✤✤ ❴ ❴ ❴ ❴ Von Neumann’s model: density matrices Completely Positive Trace Preserving (CPTP) maps Pure QM (+ ancillas) • M n p C q • state space C n • combination of systems: b • combination of systems: b • ancilla • ancilla (auxiliary system) • ad U : M ÞÑ UMU ˚ super-operator • unitary transformation U • no global phase • allows discarding (trace) | j 0 � × H S T ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ | j 0 � = | 0 � • • • | j 1 � • H H H S ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ | j 1 � = | 0 � • • H S H • • | j 2 � × H ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ | j 2 � = | 0 � • H T S H | s 0 � Figure 1: Quantum Fourier U 4 U 2 U | s 1 � transform on three qubits Figure 2: Three-qubit phase estimation circuit with QFT and controlled-U 3
Today’s presentation Informally: The category of Completely Positive Trace Preserving (CPTP) is the simplest category that interprets PureQM with ancillas, quotients global phase and allows discarding. 4
Today’s presentation Informally: The category of Completely Positive Trace Preserving (CPTP) is the simplest category that interprets PureQM with ancillas, quotients global phase and allows discarding. CPTP is the universal monoidal category on PureQM whose unit is a terminal object: D ! Full QT V.N. E PureQM CPTP @ D ! ˆ F @ F @ D Pure QT PureQM 4
Outline Introduction Symmetric monoidal categories with discarding Universality of CPTP Affine completions, PROP and quantum circuits Enriched setting 5
Symmetric strict monoidal category A symmetric monoidal category p C , b , I , α, λ, σ q is symmetric strict monoidal when • α A , B , C : p A b B q b C “ A b p B b C q • λ A : I b A “ A “ A b I • σ A , B : A b B Ñ B b A ‰ id in general 6
The category Isometry (PureQM) We define the category Isometry as follows: • Objects: natural numbers n ( C n ) • Morphisms f : n Ñ m are linear maps f : C n Ñ C m that are isometries: @ v , || f p v q|| “ || v || • Composition: composition of linear maps • m b n : “ mn and f b g is the usual tensor product 7
The category Isometry (PureQM) We define the category Isometry as follows: • Objects: natural numbers n ( C n ) • Morphisms f : n Ñ m are linear maps f : C n Ñ C m that are isometries: @ v , || f p v q|| “ || v || • Composition: composition of linear maps • m b n : “ mn and f b g is the usual tensor product Equivalently morphisms are matrices V such that V ˚ V “ I and b is then the Kronecker product. 7
The category Isometry (PureQM) We define the category Isometry as follows: • Objects: natural numbers n ( C n ) • Morphisms f : n Ñ m are linear maps f : C n Ñ C m that are isometries: @ v , || f p v q|| “ || v || • Composition: composition of linear maps • m b n : “ mn and f b g is the usual tensor product Equivalently morphisms are matrices V such that V ˚ V “ I and b is then the Kronecker product. Examples: • the isometries V : n Ñ n are the unitaries • an isometry V : 1 Ñ n is a pure state 7
Discarding Monoidal category with discarding: 1 A (strict) symmetric monoidal category p C , b , I q has discarding when the unit of the tensor product I is a terminal object. ! A I f ! B 1 B. Jacobs (1994): Semantics of weakening and contraction , D. Walker (2002): Substructural Type Systems , P. Selinger & B. Valiron (2006): A lambda calculus for quantum computation with classical 8
The category CPTP (FullQM) We define the category CPTP of completely positive trace preserving maps as follows: • Objects are natural numbers n ( M n p C q ) • Morphisms f : n Ñ m are linear maps f : M n p C q Ñ M m p C q that are completely positive and trace preserving (quantum channels) • Composition: composition of linear maps • m b n : “ mn and f b g is again the tensor product 9
The category CPTP (FullQM) We define the category CPTP of completely positive trace preserving maps as follows: • Objects are natural numbers n ( M n p C q ) • Morphisms f : n Ñ m are linear maps f : M n p C q Ñ M m p C q that are completely positive and trace preserving (quantum channels) • Composition: composition of linear maps • m b n : “ mn and f b g is again the tensor product CPTP has discarding • ! n : n Ñ 1 is the trace operator • id m b ! n : m b n Ñ m b 1 “ m is the partial trace operator 9
Symmetric strict monoidal functor A symmetric monoidal functor F is symmetric strict monoidal when the isomorphisms • F p A q b F p B q – F p A b B q • I – F p I q are identities. 10
Symmetric strict monoidal functor A symmetric monoidal functor F is symmetric strict monoidal when the isomorphisms • F p A q b F p B q – F p A b B q • I – F p I q are identities. E : Isometry Ñ CPTP • E p n q : “ n • E p V q : “ ad V : M ÞÑ VMV ˚ 10
Outline Introduction Symmetric monoidal categories with discarding Universality of CPTP Affine completions, PROP and quantum circuits Enriched setting 11
Main theorem The category of Completely Positive Trace Preserving (CPTP) is the simplest category that interprets PureQM with ancillas, quotients global phase and allows discarding. 12
Main theorem The category of Completely Positive Trace Preserving (CPTP) is the simplest category that interprets PureQM with ancillas, quotients global phase and allows discarding. Theorem: universality of CPTP • @ D strict symmetric monoidal category with discarding • @ F : Isometry Ñ D symmetric strict monoidal functor There is a unique symmetric strict monoidal functor ˆ F : CPTP Ñ D such that: E Isometry CPTP ˆ F F D 12
Proof: key lemma Stinespring’s theorem 2 For every CPTP f there is a pair p V , a q such that: id b ! ad W ad W 1 id b ! f m n ad V 1 ad V n ¨ a id b ! 2 B. Coecke & A. Kissinger (2017): Picturing Quantum Processes. CUP. . 13
Proof: key lemma Stinespring’s theorem 2 For every CPTP f there is a pair p V , a q such that: n ¨ b id b ! ad W ad W 1 id b ! f m n ad V 1 ad V n ¨ a id b ! 2 B. Coecke & A. Kissinger (2017): Picturing Quantum Processes. CUP. . 13
Proof: key lemma Stinespring’s theorem 2 For every CPTP f there is a pair p V , a q such that: n ¨ b id b ! ad W 1 ad W id b ! f m n ¨ c n ad V ad V 1 n ¨ a id b ! p V , a ) is called a dilation for f . 2 B. Coecke & A. Kissinger (2017): Picturing Quantum Processes. CUP. . 13
Proof: uniqueness If any symmetric monoidal functor ˆ F is going to make diagram commute then it must be defined as • ˆ F p n q def “ F p n q as E is identity on objects ˆ F p f q f n m n m • If then ad V id b ! ˆ F p V q p id b ! q ˆ F p ad V q F p id b ! q n ¨ a n ¨ a 14
Proof: uniqueness If any symmetric monoidal functor ˆ F is going to make diagram commute then it must be defined as • ˆ F p n q def “ F p n q as E is identity on objects ˆ F p f q f n m n m • If then ad V id b ! ˆ F p V q p id b ! q ˆ F p ad V q F p id b ! q n ¨ a n ¨ a 14
Proof: well-definedness Only choice : F p n q def • ˆ “ F p n q • ˆ F pp id b ! q ˝ ad V q def “ p id b ! q ˝ F p V q 15
Proof: well-definedness Only choice : F p n q def • ˆ “ F p n q • ˆ F pp id b ! q ˝ ad V q def “ p id b ! q ˝ F p V q Independence of the choice of dilation p V , a q Given p W , b q another dilation, F p n q b F p a q F p n qb ! F p V q F p n qb F p V 1 q F p n qb ! F p m q F p n q F p n qb F p W 1 q F p W q F p n q b F p b q F p n qb ! 15
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