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 0/33

Figure: Tiger’s Nest, Bhutan 0/33

Mixed unitary catgories CP ∞ -construction Environment srtuctures 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 1/33 dimensions. 1 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures 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). 2/33 2 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Our approach Find generalized † -compact closed categories and generalize the exisiting constructions to the new setting. with Cole and Priyaa ... 3/33 3 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Our progress generalizes Mixed unitary categories † -monoidal categories applies to CP ∞ construction characterize Environment structures 4/33 4 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Linearly distributive categories Monoidal categories: ( X , ⊗ , I , a ⊗ , u L ⊗ , u R ⊗ ) Linearly distributive categories (LDC) 1 : ( X , ⊗ , ⊤ , a ⊗ , u L ⊗ , u R ( X , ⊕ , ⊥ , a ⊕ , u L ⊕ , u R ⊗ ) ⊕ ) 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. 5/33 1 Robin Cockett and Robert Seely (1997). Weakly distributive categories. 5 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Mix categories Mix category 2 : LDC with m : ⊥ − → ⊤ called the mix map with ⊥ ⊥ mx A , B : A ⊗ B − → A ⊕ B := = m m ⊤ ⊤ ⊕ ) − 1 )(1 ⊗ (m ⊕ 1)) δ L ( u R (1 ⊕ ( u L ⊗ ⊕ 1) mx is called a mixor . The mixor is a natural transformation. Isomix category : m : ⊥ − → ⊤ is an isomorphism 2 Richard Blute, Robin Cockett, and Robert Seely (2000). ”Feedback for 6/33 linearly distributive categories: traces and fixpoints.” 6 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Variations of LDCs Compact LDC: isomix category in which every mx is an isomorphism mx A , B : A ⊗ B ≃ A ⊕ B Monoidal category: isomix category with m = 1 and mx = 1 Compact LDC Mix category mx A ⊗ B − − − → A ⊕ B m : ⊥ − → ⊤ ≃ Isomix category Monoidal category LDC m ( X , ⊗ , ⊤ ) m = 1, mx = 1 ⊥ − − → ⊤ ≃ ( X , ⊕ , ⊥ ) 7/33 7 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Mixed unitary categories † -monoidal categories: Monoidal categories with † : X op − → 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 . . . 8/33 8 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures 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: ( δ R ) † : ( A ⊕ ( B ⊗ C )) † − → (( A ⊕ B ) ⊗ C ) † ( δ R ) † : A † ⊕ ( B † ⊗ C † ) − → ( A † ⊕ B † ) ⊗ C † 9/33 9 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures † -LDC A † -LDC is a LDC X with a dagger functor † : X op − → X and the natural isomorphisms: λ ⊕ : A † ⊕ B † − λ ⊗ : A † ⊗ B † − → ( A ⊗ B ) † → ( A ⊕ B ) † tensor laxors: → ⊥ † → ⊤ † unit laxors: λ ⊤ : ⊤ − λ ⊥ : ⊥ − → A †† involutor: ι : A − such that certain coherence conditions hold. 10/33 10 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures 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? 11/33 11 / 33

� � Mixed unitary catgories CP ∞ -construction Environment srtuctures Unitary category Unitary object: An object in a † -isomix category with an isomorphism satisfying certain coherences: A † = ϕ A A − − − → ≃ Unitary isomorphism: A , B are unitary objects, ϕ A � A † A f f † ϕ B � B † 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 12/33 category. 12 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Mixed unitary category A mixed unitary category , M : U − → C , is † -isomix functor: unitary category − → † -isomix category % draw pictures 13/33 13 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures 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. 14/33 14 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures 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 ⊥ . R → ( Y , A ′ , B ′ ) is relation X A finiteness relation , ( X , A , B ) − − R − − → Y such that ∀ A ∈ A . AR ∈ A ′ and ∀ B ′ ∈ B ′ . RB ′ ∈ B Finiteness spaces with finiteness relation form a ∗ -autonomous category. 15/33 15 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures An example of a MUC: Mat( C ) ⊂ FMat( C ) FMat( C ) is defined as follows: Objects: Finiteness spaces ( X , A , B ) M M → ( Y , A ′ , B ′ ) is a matrix XxY Maps: ( X , A , B ) − − − − → 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. 16/33 16 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Next step CP ∞ construction A CPM construction f A ∗ B U A f ∗ U f B B ∗ B f † in † - compact A in MUCs closed categories in † -SMCs Selinger, 2007 Coecke and Heunen, 2011 17/33 17 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures 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 . A f M ( U ) B 18/33 18 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures 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: A A f f M ( U ) B B M ( U † ) U ρ B † B M ( U ) † f † f 19/33 A † A 19 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Test maps A combinator built from a Kraus map ( f , U ) : A − → B acts on test maps , h : B ⊗ C − → M ( V ) as follows: A C f g h B h M ρ ρ U V h † h g † f A C Test maps are glued along unitary structure map with its dagger to give a positive map . 20/33 20 / 33

Mixed unitary catgories CP ∞ -construction Environment srtuctures Equivalence relation ( f , U ) ∼ ( g , V ) : A − → B , if for all test maps maps h : B ⊗ C → V , the following holds: A C A C g f g f h h B B h h M M M M ρ ρ ρ ρ = = U V U V h † h † h h g † f † g f A C A C 21/33 21 / 33