Complete positivity for Mixed Unitary Categories Robin Cockett, and - - PowerPoint PPT Presentation

Complete positivity for Mixed Unitary Categories

Robin Cockett, and Priyaa V. Srinivasan FMCS 2019

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Figure: Tiger’s Nest, Bhutan

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Motivation

Categorical quantum mechanics (CQM) studies quantum foundations using monoidal category theory. In CQM, finite dimensional quantum processes are described as completely positive maps with the Dagger compact closed categories (†-KCC) of FHilb, finite dimensional Hilbert Spaces and linear maps The CPM construction on a †-KCC chooses exactly the completely positive maps from the category. CPM(category of pure states) = category of mixed states What about infinite dimensions? Well! There have been efforts to generalize the existing structures and constructions to infinite dimensions.

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Exisiting approaches to infinite dimensions

H*-algebras Samson Abramsky, and Chris Heunen.“H∗-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics.” Clifford Lectures 71 (2012): 1-24. Using monoidal categories (CP∞ construction) Bob Coecke, and Chris Heunen.“Pictures of complete positivity in arbitrary dimension.” Information and Computation 250 (2016): 50-58. Using non-standard analysis Stefano Gogoiso, and Fabrizio Genovese.“Infinite-dimensional categorical quantum mechanics.” arXiv:1605.04305 (2016).

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Our approach

Find generalized †-compact closed categories and generalize the exisiting constructions to the new setting. with Cole and Priyaa ...

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Our progress

generalizes applies to characterize Mixed unitary categories CP∞ construction Environment structures †-monoidal categories

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Linearly distributive categories

Monoidal categories: (X, ⊗, I, a⊗, uL

⊗, uR ⊗)

Linearly distributive categories (LDC) 1: (X, ⊗, ⊤, a⊗, uL

⊗, uR ⊗)

(X, ⊕, ⊥, a⊕, uL

⊕, uR ⊕)

linked by linear distributors ∂L : A ⊗ (B ⊕ C) → (A ⊗ B) ⊕ C ∂R : (A ⊕ B) ⊗ C → A ⊕ (B ⊗ C) ∂L and ∂R are natural but not isomorphisms.

1Robin Cockett and Robert Seely (1997). Weakly distributive categories.

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Mix categories

Mix category 2: LDC with m : ⊥ − → ⊤ called the mix map with mxA,B : A ⊗ B − → A ⊕ B :=

⊥

m

⊤

=

⊥

m

⊤

(1 ⊕ (uL

⊕)−1)(1 ⊗ (m ⊕ 1))δL(uR ⊗ ⊕ 1)

mx is called a mixor. The mixor is a natural transformation. Isomix category: m : ⊥ − → ⊤ is an isomorphism

2Richard Blute, Robin Cockett, and Robert Seely (2000). ”Feedback for

linearly distributive categories: traces and fixpoints.”

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Variations of LDCs

Compact LDC: isomix category in which every mx is an isomorphism mxA,B : A ⊗ B ≃ A ⊕ B Monoidal category: isomix category with m = 1 and mx = 1

LDC Mix category m : ⊥ − → ⊤ Isomix category Compact LDC A ⊗ B

mx

− − − →

≃

A ⊕ B Monoidal category ⊥

m

− − →

≃

⊤ m = 1, mx = 1 (X, ⊗, ⊤) (X, ⊕, ⊥)

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Mixed unitary categories

†-monoidal categories: Monoidal categories with † : Xop − → X such that A† = A f †† = f (f ⊗ g)† = f † ⊗ g† All basic natural isomorphisms are unitary (i.e., a†

⊗ = a−1 ⊗ )

Mixed unitary categories . . .

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Forging the †

Dagger for LDCs minimally has to flip the tensor products: (A ⊗ B)† = A† ⊕ B† If not the linear distributors denegenerate to an associator: (δR)† : (A ⊕ (B ⊗ C))† − → ((A ⊕ B) ⊗ C)† (δR)† : A† ⊕ (B† ⊗ C †) − → (A† ⊕ B†) ⊗ C †

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†-LDC

A †-LDC is a LDC X with a dagger functor † : Xop − → X and the natural isomorphisms: tensor laxors: λ⊕ : A†⊕B† − → (A⊗B)† λ⊗ : A†⊗B† − → (A⊕B)† unit laxors: λ⊤ : ⊤ − → ⊥† λ⊥ : ⊥ − → ⊤† involutor: ι : A − → A†† such that certain coherence conditions hold.

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Unitary isomorphisms in †-LDCs

Unitary isomorphism in a †-monoidal category: (f † : B† → A† = f −1 : B − → A) What is a unitary isomorphism in a †-LDC?

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Unitary category

Unitary object: An object in a †-isomix category with an isomorphism satisfying certain coherences: A

ϕA

− − − →

≃

A† = Unitary isomorphism: A, B are unitary objects, A

ϕA f

• A†
• f †

B

ϕB B†

Unitary category: compact †-LDC in which every object is unitary and certain coherence conditions satisfied. Every unitary category is †-linearly equivalent to a †-monoidal category.

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Mixed unitary category

A mixed unitary category, M : U − → C, is †-isomix functor: unitary category − → †-isomix category % draw pictures

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An example of a MUC: R ⊂ C

Consider the discrete monoidal category C: Objects: c = a + ib ∈ C Maps: Identity maps only c = c Tensor: multiplication (a + ib) ⊗ (x + iy) := (ax − by) + i(ay + bx) Unit: 1 Dagger: (a + ib)† := a − ib C is a compact LDC (⊗ ≃ ⊕) with a non-stationary dagger functor. The subcategory R is a unitary category with the unitary structure map being the identity map. R ⊂ C is a mixed unitary category.

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An example of a MUC: Mat(C) ⊂ FMat(C)

A finiteness space, (X, A, B), consists of a set X and a subset A, B ⊆ P(X) such that B = A⊥, that is B = {b|b ∈ P(X) with for all a ∈ A, |a ∩ b| < ∞}, and A = B⊥. A finiteness relation, (X, A, B)

R

− − → (Y , A′, B′) is relation X

R

− − → Y such that ∀A ∈ A.AR ∈ A′ and ∀B′ ∈ B′.RB′ ∈ B Finiteness spaces with finiteness relation form a ∗-autonomous category.

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An example of a MUC: Mat(C) ⊂ FMat(C)

FMat(C) is defined as follows: Objects: Finiteness spaces (X, A, B) Maps: (X, A, B)

M

− − → (Y , A′, B′) is a matrix XxY

M

− − → C such that supp(M) := {(x, y)|x ∈ X, y ∈ Y and M(x, y) = 0} is a finiteness relation from (X, A, B) to (Y , A′, B′). Dagger: (X, A, B)† := (X, B, A) M† is the complex conjugate of M. Mat(C) is a full subcategory of FMat(C) which is determined by the objects, (X, P(X), P(X)), where X is a finite set. Mat(C) is a unitary category, indeed a well-known †-compact closed category. The inclusion Mat(C) ⊂ FMat(C) is a mixed unitary category.

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Next step

CPM construction

f∗ f U A∗ A B∗ B

in †- compact closed categories Selinger, 2007 CP∞ construction

f f † A U A B B

in †-SMCs Coecke and Heunen, 2011 in MUCs

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Kraus maps

A Kraus map (f , U) : A − → B in a mixed unitary category, M : U − → C, is a map f : A − → M(U) ⊕ B ∈ C for some U ∈ U and M(U) is called the ancillary system of f .

f A M(U) B

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Combinator

A Kraus map can be glued along the unitary structure map with the dagger of itself to get a combinator which takes positive maps to positive maps:

f f ρ A B B† M(U) M(U†) M(U)† A†

f f † A U A B B

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Test maps

A combinator built from a Kraus map (f , U) : A − → B acts on test maps, h : B ⊗ C − → M(V ) as follows:

h ρ h f f ρ M g A U g† A h h† C V C B

Test maps are glued along unitary structure map with its dagger to give a positive map.

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Equivalence relation

(f , U) ∼ (g, V ) : A − → B, if for all test maps maps h : B ⊗ C → V , the following holds:

h ρ h f f ρ M M

=

h ρ h g g ρ M M f A U f † A h h† C V C B

=

g A U g† A h h† C V C B

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CP∞ construction

CP∞(M : U − → C) is given as follows: Objects: Same as C Maps: A map [(f , U)] : A − → B ∈ CP∞(M : U − → C) is an equivalence class of Kraus maps (f , U) : A − → B ∈ C / ∼

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CP∞ construction cont...

Composition: [(f , U)][(g, V )] :=    

f g M

    [A f − → M(U) ⊕ B

1⊕g

− − → M(U) ⊕ (M(V ) ⊕ C)

a⊕

− → (M(U) ⊕ M(V ) ⊕ C

n−1

⊕ ⊕ 1

− − − − − − → M(U ⊕ V ) ⊕ C] Identity: 1A := [A

(uL

⊕)−1

− − − − → ⊥ ⊕ A

n−1

⊥ ⊕ 1

− − − − − − → M(⊥) ⊕ A]

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CP∞ construction cont...

CP∞(M : U − → C) has two tensor products: [(f , U)]⊗[(g, V )] :=     

f g

     [(f , U)]⊕[(g, V )] :=     

f g

     Unit of ⊗ is ⊤ and the unit of ⊕ is ⊥. Lemma: CP∞(M : U − → C) is an isomix category.

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CP∞(R ⊂ C)

Kraus maps in C are (=, r) : c − → c′ such that c = rc′. c′ = 0 then there is at most one Kraus map (=, r) : c − → c′, for all c ∈ C when c = rc′. c′ = 0 then c = 0 and for all r′ ∈ R, (=, r) ∼ (=, r′) : c − → c′.

R ι

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CP∞(Mat(C) ⊂ FMat(C))

A Kraus map (M, Cn) : (X, F, F⊥) − → (X ′, G′, G′⊥) gives a pure completely positive map:

n

• i

M (X, Fi) M† (X, F⊥

i )

N N† (Y , G) Cm (Y , G⊥)

Lemma: Every map in CP∞(Mat(C) ⊂ FMat(C)) can be written as a sum of pure completely positive maps.

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The canonical embedding

For any mixed unitary category, M : U − → C, there exists a canonical functor, Q : C ֒ → CP∞(M : U − → C); A

f

• B

→ A

[(f (uL

⊕)−1((nM ⊥)−1⊕1),⊥)]

• M(⊥) ⊕ B

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Characterizing the CP∞ construction for MUCs

Suppose X is an isomix cateogry, then what is the sufficient condition for: X ≃ CP∞(M : U − → C)?

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Environment structure

An environment structure for M : U − → C, consists of:

1 a strict isomix functor,

F : C − → D where D is any isomix category, and

2 a family of maps

U : F(M(U)) −

→ ⊥ indexed by objects U ∈ U such that:

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Environment structure cont...

For all unitary objects U, V ∈ U, the following holds: [Env.1a] F(M(U)) ⊗ F(M(V )) − → ⊥ :=

MF = ⊥

[Env.1b] F(M(U ⊕ V )) − → ⊥ := MF

⊥

=

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Environment structure cont...

[Env.2] (f , U) ∼ (g, V ) ∈ C if and only if the following equation holds:

f ⊥ F = g ⊥ F ⇔

f

∼

g

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Examples cont...

For the MUC, R ⊂ C,

r := (=, 1/r) : r −

→ 1 C ֒ → CP∞(R∗ ⊂ C) For the MUC, MatC − → FMat(C),

Cn : Cn −

→ C; ρ → Tr(ρ) FMat(C) ֒ → CP∞(Mat(C) ⊂ FMat(C))

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The purification axiom

An environment structure for M : U − → C satisfies purification axiom if F : C − → D is bijective on objects, and for all f : F(A) − → F(B) ∈ D, there exists a Kraus map (g, U) : A − → B ∈ C such that [Env.3] f =

g ⊥ F

Equationally, F(A)

f

− − → F(B) = F(A)

F(g)

− − − − → F(M(U)⊕B)

n⊕

− − − → M(F(U))⊕F(B)

⊕ 1

− − − − → ⊥ ⊕ F(B)

u⊕

− − − → F(B)

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Characterization of the CP∞ construction for MUCs

Proposition: Let D be any isomix category and M : U − → C. If F : C − → D with is an environment structure with purification, then D ≃ CP∞(M : U − → C)

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Thank you Cole and JS for many useful discussions on the CP∞ construction and the examples! Robin Cockett, Cole Comfort, and Priyaa Srinivasan. Dagger linear logic for categorical quantum mechanics. ArXiv e-prints, September 2018. Robin Cockett, and Priyaa Srinivasan. Complete positivity for mixed unitary categories. ArXiv e-prints, May 2019.

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