Superpositions and Categorical Quantum Reconstructions Sean Tull - PowerPoint PPT Presentation

Superpositions and Categorical Quantum Reconstructions Sean Tull sean.tull@cs.ox.ac.uk University of Oxford SYCO 2 University of Strathclyde December 17 2018

“Local and Global Phases in Categorical Quantum Theory”

The Plan 1. Motivation 2. Phased Biproducts 3. Relating Local and Global Phases 4. Quantum Reconstructions

1. Motivation

Two Categories for Quantum Theory Pure quantum theory is normally described via the category Hilb of Hilbert spaces and continous linear maps f : H → K .

Two Categories for Quantum Theory Pure quantum theory is normally described via the category Hilb of Hilbert spaces and continous linear maps f : H → K . But physically we only consider maps up to global phase: f ∼ g ⇐ ⇒ f = z · g for z ∈ C , | z | = 1 .

Two Categories for Quantum Theory Pure quantum theory is normally described via the category Hilb of Hilbert spaces and continous linear maps f : H → K . But physically we only consider maps up to global phase: f ∼ g ⇐ ⇒ f = z · g for z ∈ C , | z | = 1 . Hence it is really given by the category Hilb P := Hilb / ∼ where morphisms are ∼ -equivalence classes [ f ]: H → K .

Two Categories for Quantum Theory Pure quantum theory is normally described via the category Hilb of Hilbert spaces and continous linear maps f : H → K . But physically we only consider maps up to global phase: f ∼ g ⇐ ⇒ f = z · g for z ∈ C , | z | = 1 . Hence it is really given by the category Hilb P := Hilb / ∼ where morphisms are ∼ -equivalence classes [ f ]: H → K . Question 1: How is Hilb built from Hilb P ?

Idea: Superpositions A defining quantum feature are superpositions, corresponding to an addition operation f + g in Hilb .

Idea: Superpositions A defining quantum feature are superpositions, corresponding to an addition operation f + g in Hilb . This exists because Hilb has biproducts: κ 1 κ 2 H H ⊕ K K π 1 π 2

Idea: Superpositions A defining quantum feature are superpositions, corresponding to an addition operation f + g in Hilb . This exists because Hilb has biproducts: κ 1 κ 2 H H ⊕ K K π 1 π 2 This means that the κ i form a coproduct of H , K : κ 1 κ 2 H H ⊕ K K ∃ ! h g f L and the π i dually form a product, in a compatible way.

Idea: Superpositions A defining quantum feature are superpositions, corresponding to an addition operation f + g in Hilb . This exists because Hilb has biproducts: κ 1 κ 2 H H ⊕ K K π 1 π 2 This means that the κ i form a coproduct of H , K : κ 1 κ 2 H H ⊕ K K ∃ ! h g f L and the π i dually form a product, in a compatible way. However: H ⊕ K is not a biproduct in Hilb P .

Question 2: How is H ⊕ K described in Hilb P ?

Question 2: How is H ⊕ K described in Hilb P ? [ κ 1 ] [ κ 2 ] H H ⊕ K K [ h ] [ f ] [ g ] L Commutes when h ◦ κ 1 = z · f and h ◦ κ 2 = w · g for global phases z , w .

Question 2: How is H ⊕ K described in Hilb P ? [ κ 1 ] [ κ 2 ] H H ⊕ K K [ h ] [ f ] [ g ] L Commutes when h ◦ κ 1 = z · f and h ◦ κ 2 = w · g for global phases z , w . So [ h ] exists but is now only unique up to a phase: � id H � 0 [ U ] H ⊕ K H ⊕ K with U = 0 z · id K

2. Phased Biproducts

Phased Coproducts

Phased Coproducts Definition In any category, a phased coproduct of A , B is an object A ˙ + B along with morphisms κ A , κ B as below, called coprojections, such that:

Phased Coproducts Definition In any category, a phased coproduct of A , B is an object A ˙ + B along with morphisms κ A , κ B as below, called coprojections, such that: κ A κ B A ˙ A + B B h g f C 1. For all f , g as above there exists h making the diagram commute;

Phased Coproducts Definition In any category, a phased coproduct of A , B is an object A ˙ + B along with morphisms κ A , κ B as below, called coprojections, such that: U κ A κ B A ˙ A + B B h h ′ g f C 1. For all f , g as above there exists h making the diagram commute; 2. For any such h , h ′ we have h ′ = h ◦ U for some endomorphism U of A ˙ + B which is a phase, meaning that U ◦ κ A = κ A U ◦ κ B = κ B

Phased coproducts are surprisingly well-behaved.

Phased coproducts are surprisingly well-behaved. Lemma 1. They are unique up to (non-unique) isomorphism. 2. Any phase is an isomorphism. 3. Associativity holds: ( A ˙ + B ) ˙ + C ≃ A ˙ + B ˙ + C ≃ A ˙ + ( B ˙ + C ) 4. Having finite phased coproducts A 1 ˙ + · · · ˙ + A n ⇒ having binary ones A ˙ ⇐ + B and an initial object 0 .

Phased coproducts are surprisingly well-behaved. Lemma 1. They are unique up to (non-unique) isomorphism. 2. Any phase is an isomorphism. 3. Associativity holds: ( A ˙ + B ) ˙ + C ≃ A ˙ + B ˙ + C ≃ A ˙ + ( B ˙ + C ) 4. Having finite phased coproducts A 1 ˙ + · · · ˙ + A n ⇒ having binary ones A ˙ ⇐ + B and an initial object 0 . In a monoidal category, call them distributive when they interact well with ⊗ , via isomorphisms A ⊗ ( B ˙ + C ) ≃ ( A ⊗ B ) ˙ + ( A ⊗ C )

Phased coproducts are surprisingly well-behaved. Lemma 1. They are unique up to (non-unique) isomorphism. 2. Any phase is an isomorphism. 3. Associativity holds: ( A ˙ + B ) ˙ + C ≃ A ˙ + B ˙ + C ≃ A ˙ + ( B ˙ + C ) 4. Having finite phased coproducts A 1 ˙ + · · · ˙ + A n ⇒ having binary ones A ˙ ⇐ + B and an initial object 0 . In a monoidal category, call them distributive when they interact well with ⊗ , via isomorphisms A ⊗ ( B ˙ + C ) ≃ ( A ⊗ B ) ˙ + ( A ⊗ C ) Can define phased products ( A ← A ˙ × B → B ) dually, and even phased (co)limits more generally.

Phased Biproducts Definition In a category with zero morphisms, a phased biproduct of A , B is an object A ˙ ⊕ B which is both a phased coproduct and phased product: κ 1 κ 2 A ˙ A ⊕ B B π 1 π 2

Phased Biproducts Definition In a category with zero morphisms, a phased biproduct of A , B is an object A ˙ ⊕ B which is both a phased coproduct and phased product: κ 1 κ 2 A ˙ A ⊕ B B π 1 π 2 with the same phases for each, and satisfying � i = j id π i ◦ κ j = 0 i � = j

Phased Biproducts Definition In a category with zero morphisms, a phased biproduct of A , B is an object A ˙ ⊕ B which is both a phased coproduct and phased product: κ 1 κ 2 A ˙ A ⊕ B B π 1 π 2 with the same phases for each, and satisfying � i = j id π i ◦ κ j = 0 i � = j In a dagger category, a phased dagger biproduct also has π i = κ † i .

Phased Biproducts Definition In a category with zero morphisms, a phased biproduct of A , B is an object A ˙ ⊕ B which is both a phased coproduct and phased product: κ 1 κ 2 A ˙ A ⊕ B B π 1 π 2 with the same phases for each, and satisfying � i = j id π i ◦ κ j = 0 i � = j In a dagger category, a phased dagger biproduct also has π i = κ † i . Example Hilb P has phased dagger biproducts given by the direct sum H ⊕ K of Hilbert spaces.

3. Relating Local and Global Phases

From Global to Local Phases In any monoidal category ( C , ⊗ ), by a choice of global phases we mean a designated subgroup P of its invertible central scalars.

From Global to Local Phases In any monoidal category ( C , ⊗ ), by a choice of global phases we mean a designated subgroup P of its invertible central scalars. We then define f ∼ g ⇐ ⇒ f = p · g for some p ∈ P and write C P := C / ∼ .

From Global to Local Phases In any monoidal category ( C , ⊗ ), by a choice of global phases we mean a designated subgroup P of its invertible central scalars. We then define f ∼ g ⇐ ⇒ f = p · g for some p ∈ P and write C P := C / ∼ . Lemma If C has distributive (co,bi)products then C P has distributive phased (co,bi)products.

From Global to Local Phases In any monoidal category ( C , ⊗ ), by a choice of global phases we mean a designated subgroup P of its invertible central scalars. We then define f ∼ g ⇐ ⇒ f = p · g for some p ∈ P and write C P := C / ∼ . Lemma If C has distributive (co,bi)products then C P has distributive phased (co,bi)products. Examples Hilb P has phased biproducts as we’ve seen, arising from Hilb via the global phases P := { z ∈ C | | z | = 1 } . So does the quotient Vec P of Vec := k-vectors spaces and linear maps, via P := { λ ∈ k | λ � = 0 } .

From Local to Global Phases Observation: Linear maps f : H → K �� f �� 0 ⇐ ⇒ Equivalence classes : H ⊕ C → K ⊕ C 0 1

From Local to Global Phases Observation: Linear maps f : H → K �� f �� 0 ⇐ ⇒ Equivalence classes : H ⊕ C → K ⊕ C 0 1 Definition Let ( D , ⊗ ) have phased coproducts. We define a category GP( D ) by: ◮ objects are phased coproducts of the form A = A ˙ + I in D ; ◮ morphisms are those f : A → B in D with: f f A B A B κ A κ B κ I κ I A B I ∃ g

From Local to Global Phases Observation: Linear maps f : H → K �� f �� 0 ⇐ ⇒ Equivalence classes : H ⊕ C → K ⊕ C 0 1 Definition Let ( D , ⊗ ) have phased coproducts. We define a category GP( D ) by: ◮ objects are phased coproducts of the form A = A ˙ + I in D ; ◮ morphisms are those f : A → B in D with: f f A B A B κ A κ B κ I κ I A B I ∃ g

From Local to Global Phases We have reached our first main result.