Abstract Physical Traces

ABSTRACT PHYSICAL TRA CES SAMSON ABRAMSKY AND BOB COECKE A BSTRACT . W e revise our ‘Physical T races’ pap er [Abramsky and Coeck e CTCS‘02] in the light of the r esults in [Abra msky and Co eck e LiCS‘04]. The key fact is that the no tion of a str ongly c o mp a ct close d c ate gory allo ws abstract notions of adjoint, bipartite pro jector and inner pro duct to b e defined, a nd their key prop erties to b e prov ed. In this pap er w e improv e on the definition of strong compact closur e as compared to the one presented in [Abramsky and Co ec ke LiCS‘04]. This mo dification enables an e legan t c haracter iz ation of str ong compact closure in terms of a dj oints and a Y a nking axiom, a nd a b etter treatment of bipa rtite pro jectors. 1. In t ro duction In [Abramsky and Co ec k e CTCS‘02] we sho wed tha t ve ctor space pr oje c t ors P : V ⊗ W → V ⊗ W whic h ha ve a one-dimensional subspace of V ⊗ W as fixed-p oin t s , suffice to implemen t a n y linear map, and also the cat ego rical trace [Jo y al, Street and V erit y 1996] of the category ( FdV ec K , ⊗ ) o f finite-dimensional v ector spaces and linear ma ps o v er a base field K . The in terest o f this is that pro jectors of this kind arise naturally in quantum mec hanics (for K = C ), and play a ke y role in infor mation proto cols suc h as [quan tum telep ortation 199 3 ] and [en tanglemen t swapping 1993], and a lso in measuremen t- ba s ed sc hemes for quan tum computation. W e sho w ed how b oth the category ( FdHilb , ⊗ ) o f finite-dimensional c om- plex Hilbert s paces and linear maps, and the category ( Rel , × ) of r elatio ns with the cartesian pro duct a s tensor, can b e ph ysically realized in this sense. In [Abramsky and Co ec k e LiCS‘04] w e sho w ed that suc h pro jectors can b e defined and their crucial prop erties prov ed at the abstract lev el of s t r ongly c omp ac t close d c ate- gories . This categorical structure is a ma jor ingredien t of the categorical axiomatization in [Abramsky and Co ec ke LiCS‘04] of quan tum theory [v on Neumann 1932]. It captures quan tum en tanglement and its b eha vioral prop erties [Co ec k e 2003]. In this pap er w e will impro ve on the definition of strong compact closure, enabling a characterization in terms of adjo ints - in the linear alg ebra sense, suitably abstracted - and y anking, without ex- plicit reference to compact closure, and enabling a nicer treatment of bipartite pro jectors, coheren t with the treatmen t of arbitra ry pro jectors in [Abramsky and Co ec k e LiCS‘04]. Rick Blute, Sam Bra unstein, Vincent Danos, Mar tin Hyland a nd P rak ash P anangaden provided useful feed-bac k. 2000 Mathematics Sub ject Cla ssification: 15 A0 4,15A90,18B1 0 ,18C50,18D10,81P10,81P68. c  Samson Abramsk y and Bob Co ec ke, 20 04. Permission to copy for priv a te use gra n ted. 1 2 W e ar e then able to show that the constructions in [Abramsky and Co ec ke CTCS‘02] for realizing arbitra ry morphisms and the trace by pro jectors a ls o carry ov er to the ab- stract lev el, and tha t these constructions admit a n infor mation-flo w interpretation in the spirit of the one f or additiv e traces [Abramsky 1 996, Abramsky , Haghv erdi and Scott 20 02 ]. It is the information flo w due to (strong) compact closure whic h is crucial for the abstract form ulation, and for the pro ofs of correctness of prot ocols suc h as quan t um telep ortation [Abramsky and Co ec k e LiCS‘04]. A concise presen tation of (v ery) basic quan tum mec hanics whic h supp orts the dev el- opmen ts in t his pap er can b e found in [Abramsky and Co ec ke CTCS‘02, Co ec k e 2003]. Ho wev er, the reader with a sufficien t categor ical background might find the a bs tract pre- sen tation in [Abramsky and Co ec ke LiCS‘04] more enligh tening. 2. Strong ly co mpact closed ca tegories As sho wn in [Kelly and Laplaza 1980], in an y monoidal category C , the endomorphism monoid C ( I , I) is commutativ e. F urthermore an y s : I → I induces a natural transforma- tion s A : A ≃ ✲ I ⊗ A s ⊗ 1 A ✲ I ⊗ A ≃ ✲ A . Hence, setting s • f for f ◦ s A = s B ◦ f for f : A → B , w e hav e ( s • g ) ◦ ( r • f ) = ( s ◦ r ) • ( g ◦ f ) for r : I → I and g : B → C . W e call the morphisms s ∈ C (I , I) sc al a rs a nd s • − sc alar multiplic ation . In ( FdV ec K , ⊗ ), linear maps s : K → K are uniquely determined b y the image of 1, and hence correspo nd biuniquely to elemen ts of K . In ( Rel , × ), there are just t wo scalars, corresp onding to the Bo oleans B . Recall from [Kelly and Laplaza 1980] that a c omp a ct c l o se d c ate gory is a symmetric monoidal category C , in which, when C is view ed as a one-ob ject bicategory , ev ery one- cell A has a left adjoin t A ∗ . Explicitly this means that for eac h ob ject A of C there exists a dual obje c t A ∗ , a unit η A : I → A ∗ ⊗ A and a c ounit ǫ A : A ⊗ A ∗ → I, and that t he diagrams A ≃ ✲ A ⊗ I 1 A ⊗ η A ✲ A ⊗ ( A ∗ ⊗ A ) A 1 A ❄ ✛ ≃ I ⊗ A ✛ ǫ A ⊗ 1 A ( A ⊗ A ∗ ) ⊗ A ≃ ❄ (1) 3 and A ∗ ≃ ✲ I ⊗ A ∗ η A ⊗ 1 A ∗ ✲ ( A ∗ ⊗ A ) ⊗ A ∗ A ∗ 1 A ∗ ❄ ✛ ≃ A ∗ ⊗ I ✛ 1 A ∗ ⊗ ǫ A A ∗ ⊗ ( A ⊗ A ∗ ) ≃ ❄ (2) b oth comm ute. Alternatively , a compact closed category ma y b e defined as a ∗ -autono mo us category [Barr 19 79 ] with a self-dual tensor, hence a mo del of ‘degenerate’ linear logic [Seely 19 9 8 ]. F or each mo r phism f : A → B in a compact closed category we can construct a dual f ∗ , a name p f q and a c on a me x f y , resp ectiv ely as B ∗ ≃ ✲ I ⊗ B ∗ η A ⊗ 1 B ∗ ✲ A ∗ ⊗ A ⊗ B ∗ A ∗ f ∗ ❄ ✛ ≃ A ∗ ⊗ I ✛ 1 A ∗ ⊗ ǫ B A ∗ ⊗ B ⊗ B ∗ 1 A ∗ ⊗ f ⊗ 1 B ∗ ❄ A ∗ ⊗ A 1 A ∗ ⊗ f ✲ A ∗ ⊗ B I I η A ✻ p f q ✲ A ⊗ B ∗ f ⊗ 1 B ∗ ✲ x f y ✲ B ⊗ B ∗ ǫ B ✻ In particular, the assignmen t f 7→ f ∗ extends A 7→ A ∗ in to a contra v ariant endofunctor with A ≃ A ∗∗ . In any compact closed category , we ha v e C ( A ⊗ B ∗ , I) ≃ C ( A, B ) ≃ C ( I , A ∗ ⊗ B ) , so ‘elemen ts’ o f A ⊗ B a re in biunique correspondence with names/conames of morphisms f : A → B T ypical examples are ( Rel , × ) where X ∗ = X and where for R ⊆ X × Y , p R q = { ( ∗ , ( x, y )) | xR y , x ∈ X, y ∈ Y } x R y = { (( x, y ) , ∗ ) | xR y , x ∈ X, y ∈ Y } and, ( FdV ec K , ⊗ ) where V ∗ is the dual ve ctor space of linear functiona ls v : V → K and where for f : V → W with matrix ( m ij ) in ba ses { e V i } i = n i =1 and { e W j } j = m j =1 of V and W 4 resp e ctiv ely w e hav e p f q : K → V ∗ ⊗ W :: 1 7→ i,j = n,m X i,j =1 m ij · ¯ e V i ⊗ e W j x f y : V ⊗ W ∗ → K :: e V i ⊗ ¯ e W j 7→ m ij . where { ¯ e V i } i = n i =1 is the base of V ∗ satisfying ¯ e V i ( e V j ) = δ ij , and similarly for W . Another example is the category n Cob of n -dimensional c ob or disms whic h is regularly considered in mathematical phy sics, e.g. [Ba ez 20 04]. Eac h compact closed category admits a categorical trace, that is, for ev ery morphism f : A ⊗ C → B ⊗ C a t r ace T r C A,B ( f ) : A → B is specified and satisfies certain a xioms [Jo yal, Street and V erit y 1996]. Indeed, w e can set T r C A,B ( f ) := ρ − 1 B ◦ (1 B ⊗ ǫ C ) ◦ ( f ◦ 1 C ∗ ) ◦ (1 A ⊗ ( σ C ∗ ,C ◦ η C )) ◦ ρ A (3) where ρ X : X ≃ X ⊗ I and σ X,Y : X ⊗ Y ≃ Y ⊗ X . In ( R el , × ) this yields x T r Z X,Y ( R ) y ⇔ ∃ z ∈ Z . ( x, z ) R ( y , z ) for R ⊆ ( X × Z ) × ( Y × Z ) while in ( FdV ec K , ⊗ ) w e obtain T r U V ,W ( f ) : e V i 7→ X α m iαj α e W j where ( m ik j l ) is the matrix of f in bases { e V i ⊗ e U k } ik and { e W j ⊗ e U l } j l . 2.1. Definition. [Strong Compact Closure I] A str ongly c omp act close d c ate gory is a compact closed category C in whic h A = A ∗∗ and ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ , and whic h comes together with an in v olutive cov ariant compact closed functor ( ) ∗ : C → C whic h assigns eac h ob ject A to its dua l A ∗ . So in a strongly compact closed category w e hav e t w o inv olutiv e functors, namely a con trav ariant one ( ) ∗ : C → C and a co v aria nt one ( ) ∗ : C → C whic h coincide in their action o n ob jects. Recall that ( ) ∗ b eing c omp a c t clos e d functor means that it preserv es the monoidal structure strictly , and unit and counit i.e. p 1 A ∗ q = ( p 1 A q ) ∗ ◦ u − 1 I and x 1 A ∗ y = u I ◦ ( x 1 A y ) ∗ (4) where u I : I ∗ ≃ I. This in pa r t ic ular implies tha t ( ) ∗ comm utes with ( ) ∗ since ( ) ∗ is de- finable in terms o f the monoidal structure, η and ǫ — in [Abramsky and Co ec ke LiCS‘04] w e only assumed comm uta tion of ( ) ∗ and ( ) ∗ instead o f the stronger requiremen t of equations ( 4). F or eac h morphism f : A → B in a strongly compact closed category w e can define an adjoint — as in linear algebra — as f † := ( f ∗ ) ∗ = ( f ∗ ) ∗ : B → A . It turns out that w e can also define strong compact closure b y taking the adjo int to b e a primitiv e. 5 2.2. Theorem. [Strong Compact Closure I I] A str on g ly c om p a ct clo se d c ate gory c an b e e quiva l e ntly define d as a symmetric mo n oidal c ate gory C which c omes with 1. a monoidal invo l utive assignment A 7→ A ∗ on o bje cts, 2. an identity-on-obje cts, c ontr avaria nt, strict monoidal, inv olut ive functor f 7→ f † , and, 3. for e ac h obje ct A a unit η A : I → A ∗ ⊗ A with η A ∗ = σ A ∗ ,A ◦ η A and such that either the dia g r am A ≃ ✲ A ⊗ I 1 A ⊗ η A ✲ A ⊗ ( A ∗ ⊗ A ) A 1 A ❄ ✛ ≃ I ⊗ A ✛ ( η † A ◦ σ A,A ∗ ) ⊗ 1 A ( A ⊗ A ∗ ) ⊗ A ≃ ❄ (5) or the diagr am A ≃ ✲ I ⊗ A η A ⊗ 1 A ✲ ( A ∗ ⊗ A ) ⊗ A ≃ ✲ A ∗ ⊗ ( A ⊗ A ) A 1 A ❄ ✛ ≃ I ⊗ A ✛ η † A ⊗ 1 A ( A ∗ ⊗ A ) ⊗ A ✛ ≃ A ∗ ⊗ ( A ⊗ A ) 1 A ∗ ⊗ σ A,A ❄ (6) c ommutes, wher e σ A,A : A ⊗ A ≃ A ⊗ A is the twist map. While diag ram (5) is the analo gue to diagram (1) with η † A ◦ σ A,A ∗ pla ying the ro le of the coname, diagram (6) expresses yanking with resp ect to the canonical trace of the compact closed structure. W e only need one comm uting diagram a s compared to diagrams (1) and (2) in the definition of compact closure and hence in Definition 2.1 since due to the strictness assumption (i.e. A 7→ A ∗ b eing inv olutiv e) w e w ere able to replace the second diagram by η A ∗ = σ A ∗ ,A ◦ η A . Returning to the ma in issue of this pap er, w e are now able to construct a bip artite pr oje ctor ( i.e. a pro jector on a n ob ject of type A ⊗ B ) a s P f := p f q ◦ ( p f q ) † = p f q ◦ x f ∗ y : A ∗ ⊗ B → A ∗ ⊗ B , that is, we hav e an assignmen t P : C (I , A ∗ ⊗ B ) − → C ( A ∗ ⊗ B , A ∗ ⊗ B ) :: Ψ 7→ Ψ ◦ Ψ † 6 from bipartite elemen ts to bipartite pro jectors. No t e that the use of ( ) ∗ is ess en tial in order f o r P f to b e endomorphic. W e can normalize these pro jectors P f b y considering s f • P f for s f := ( x f ∗ y ◦ p f q ) − 1 (pro vided this in v erse exists in C (I , I)), yielding ( s f • P f ) ◦ ( s f • P f ) = s f • ( p f q ◦ ( s f • ( x f ∗ y ◦ p f q )) ◦ x f ∗ y ) = s f • P f , and a lso ( s f • P f ) ◦ p f q = p f q and x f ∗ y ◦ ( s f • P f ) = x f ∗ y . An y compact closed category in whic h ( ) ∗ is the iden tity on ob jects is trivially strong ly compact close d. Examples inclu de relations and finite-dimensional real inne r-pro duct spaces, and also the in tera ctio n category SPro c fro m []. So, imp ortan tly , are finite-dimensional c om plex Hilb ert spaces and linear maps ( F dHilb , ⊗ ). W e tak e H ∗ to b e the c onjugate sp ac e , that is, the Hilb ert space with the same elemen ts as H but with t he scalar multiplic ation and the inner-pro duct in H ∗ defined by α • H ∗ φ := ¯ α • H φ h φ | ψ i H ∗ := h ψ | φ i H , where ¯ α is the complex conjugate of α . Hence we can still tak e ǫ H to b e the se s q uiline ar inner-pro duct. Con ve rsely , an abstr act notion o f inn e r pr o duct can b e defined in any strongly compact closed catego r y . Giv en ‘elemen ts’ ψ , φ : I → A , w e define h ψ | φ i := ψ † ◦ φ ∈ C (I , I) . As an example, the inner- pro duct in ( Rel , × ) is, for x, y ⊆ {∗} × X , h x | y i = 1 I for x ∩ y 6 = ∅ and h x | y i = 0 I for x ∩ y = ∅ with 1 I := { ∗ } × {∗} ⊆ {∗} × {∗} and 0 I := ∅ ⊆ { ∗} × {∗} . Whe n defining unitarity of an isomorphism U : A → B b y U − 1 = U † w e can pro v e the defining prop erties b oth of inner-pro duct space adjoints a nd inner-pro duct space unitarity: h f † ◦ ψ | φ i B = ( f † ◦ ψ ) † ◦ φ = ψ † ◦ f ◦ φ = h ψ | f ◦ φ i A , h U ◦ ψ | U ◦ ϕ i B = h U † ◦ U ◦ ψ | ϕ i A = h ψ | ϕ i A , for ψ , ϕ : I → A , φ : I → B , f : B → A and U : A → B . As sho wn in [Abra ms ky and Co ec ke LiCS‘04], an alternative w ay to define the abstract inner-pro duct is I ρ I ✲ I ⊗ I 1 I ⊗ u I ✲ I ⊗ I ∗ φ ⊗ ψ ∗ ✲ A ⊗ A ∗ ǫ A ✲ I where u I : I ≃ I ∗ and ρ I : I ≃ I ⊗ I. Here the k ey data we use is the coname ǫ A : A ⊗ A ∗ → I , and a lso ( ) ∗ : cf. also the ab o v e examples of b oth real and complex inner- pro duct spaces where ǫ A := h− | −i . Hence it is fa ir to say that strong compact closure compact closure ≃ inner-pro duct space v ector space . 7 Finally , note that abstract bipartite pro jectors P f ha ve t wo comp onen ts: a ‘name’- comp onen t and a ‘coname’-comp onen t. While in most algebraic treatments inv olving pro jectors these are tak en to b e primitiv e, in our setting pro jectors are comp osite en tities, and this decomp osition will carry o v er to their crucial prop erties (see b elo w). W e depict names, conames, and pro jectors a s follows : p f q x f y P f := p f q x f ∗ y In this represen tation, diagrams (1 ) and (6) can b e expressed as the resp ec tiv e pictures ǫ A η A η † A η A σ A,A b eing equal to the iden tity . Belo w we will express equalities in this manner. 3. Informa tion-flo w through pro jecto rs 3.1. Lemma. [Comp ositionalit y - Abramsky and Co ec k e LiCS‘04] In a c omp a ct close d c ate gory λ − 1 C ◦ ( x f y ⊗ 1 C ) ◦ (1 A ⊗ p g q ) ◦ ρ A = g ◦ f for A f ✲ B g ✲ C , ρ A : A ≃ A ⊗ I and λ C : C ≃ I ⊗ C , i.e., x f y p g q = g f in o ur gr aphic al r epr esentation. F ollo wing [Abramsky and Co ec ke LiCS‘04, Co ec k e 2003] w e can think of the informa- tion flowing along the grey line in the diagra m b elo w, b eing acted on by t he morphisms whic h lab el the coname and the name resp ectiv ely . 8 x f y p g q W e refer to this as the information-flow interpr etation of c omp act closur e . Many v ariants can a ls o b e derived [Abra msky and Co ec k e LiCS‘04, Co ec k e 2003]. The pictures express- ing the non-tr ivial branc hes of diag rams (1) and (6) b ecome ǫ η η † η η σ Lemma 2 of [Abramsky and Co ec k e CTCS‘02 ], whic h states that w e can realize any linear map g : V → W using only ( FdHilb , ⊗ )-pro jectors, follow s trivially by setting f := 1 V while viewing b oth x 1 V y and p g q a s b eing parts of pro jectors — all this is up to a scalar m ultiple whic h dep e nds on the input of P g . Note that b y functoriality 1 V ∗ = (1 V ) ∗ and hence P (1 V ) ∗ = P 1 V ∗ . As discusse d in [Co ec k e 200 3 ] this f eat ure constitutes the core o f lo gic-gate telep ortation , whic h is a fault-to leran t univ ersal quan tum computational primitiv e [G ottesm an a nd Ch uang 1999]. Explicitly , 3.2. Lemma. In a str ongly c o mp act clos e d c ate gory C for f : A → B , f ⊗ ( p 1 A ∗ q ◦ x ξ y ) = s ( f , ξ ) • ( σ A,B ◦ (P 1 A ∗ ⊗ 1 B ) ◦ (1 A ⊗ P f )) wher e s ( f , ξ ) ∈ C (I , I) is a sc alar, σ A,B : A ⊗ A ∗ ⊗ B → B ⊗ A ∗ ⊗ A is symmetry, ξ : A ∗ → B ∗ is arb itr ary, an d s ( f , f ∗ ) = 1 I . Lemma 1 of [Abramsky and Co ec k e CTCS‘02], that is, w e can realize the ( FdHilb , ⊗ )- trace by means o f pro jectors trivially f ollo ws from eq.(3), noting that η = p 1 q and ǫ = x 1 y and a g ain viewing these as parts of pro jectors. Explicitly: 3.3. Lemma. In a str ongly c o mp act clos e d c ate gory C for f : A ⊗ C → B ⊗ C , T r C A,B ( f ) ⊗ ( p 1 C ∗ q ◦ x ξ y ) = s ( ξ ) • ((1 A ⊗ P 1 C ∗ ) ◦ ( f ⊗ 1 C ∗ ) ◦ (1 B ⊗ P 1 C ∗ )) wher e s ( ξ ) ∈ C (I , I) is a sc a l a r, ξ : C → C is a rbitr ary, an d s (1 C ) = 1 I . Indeed, since σ A ∗ ,A ◦ p 1 A q = p (1 A ) ∗ q = p 1 A ∗ q by functorialit y , eq.(3) is 9 f p 1 q x 1 y = T r( f ) In terestingly , using the information-flow in terpretation of compact closure, pro vided f itself admits an information-flow inte rpretation, this construction admits one to o, and can b e regarded as a feed-bac k construction. As an example, for f := ( g 1 ⊗ g 2 ) ◦ σ ◦ ( f 1 ⊗ f 2 ), w e hav e (use natura lit y of σ , t he definition of (co)name and comp ositionalit y) f 1 f 2 g 1 g 2 σ p 1 q x 1 y = f 1 g 2 f 2 g 1 When taking f itself to b e a pro jector P g = p g q ◦ x g ∗ y w e ha v e p f q x f ∗ y p 1 q x 1 y = f ∗ f ∗ using σ ◦ p f q = p f ∗ q , naturalit y of σ a nd comp ositionality . Note that the information- flo w in the lo op is in this case ‘forward’ as compared to ‘bac kw ard’ in the previous example. F or f of ty p e A ⊗ ( C 1 ⊗ . . . ⊗ C n ) → B ⊗ ( C 1 ⊗ . . . ⊗ C n ) we can hav e m ultiple lo oping: 10 P σ σ p 1 q x 1 y Note the resem blance b et w een this b eha vior and that of additive tr ac es [Abra msky 1996, Abramsky , Haghv erdi and Scott 2002] suc h as the one on ( R el , +) namely x T r Z X,Y ( R ) y ⇔ ∃ z 1 , . . . , z n ∈ Z .xRz 1 R . . . Rz n Ry for R ⊆ X + Z × Y + Z . In this case we can think of a particle tra veling tro ugh a net work where the elemen ts x ∈ X are the p ossible states of t he particle. The morphisms R ⊆ X × Y are pro cess es tha t imp ose a (non-deterministic) c hange of state x ∈ X to y ∈ R ( x ), empt yness of R ( x ) corresp onding to undefinedness. The sum X + Y is the disjoin t union of state sets and R + S represen ts parallel comp osition of pro cesses. The trace T r Z X,Y ( R ) is fe e db ack , that is, entering in a state x ∈ X the particle will either halt, exit at y ∈ Y or, exit at z 1 ∈ Z in whic h case it is fed ba c k into R at the Z en trance, and so on, un t il it halt s or exits a t y ∈ Y . F or a more conceptual view of the ma t t er, note that the examples illustrat ed ab o v e all live in the fr e e c o m p act clo s e d c ate gory g e nerated b y a suitable category in the sense of [Kelly and L a plaza 198 0 ]. Indeed our diagra ms , whic h are essen tially ‘pro of nets for compact closed logic’ [Abramsky a nd Duncan 2004], giv e a presen tation of this free cat- egory . Of course, these diag r ams will then ha v e represen tations in any compact closed category . F or a detailed discussion o f free con tsructions for traced and strongly compact closed catego r ies, see the fo rthcoming pap er [ ? ]. 4. ( FRel , × , T r) fr om ( FdHilb , ⊗ , T r) In [Abramsky and Co ec ke CTCS‘02] § 3.3 w e provide d a lax functorial passage from the category ( FdHilb , ⊗ , T r) to the category of finite sets and relations ( FRel , × , T r). This passage in v olve d c ho osing a base f o r each Hilb ert space. When restricting the morphisms of FdHilb to those for whic h the matrices in the chose n bases are R + -v alued w e obtain a true functor. The results in [Abramsky and Co ec k e L iCS‘04 ], to gethe r with the ideas dev elop ed in this pap er, pro vide a b etter understanding o f this passage. In any mono idal category , 11 C (I , I) is an a belian monoid [Kelly and Lapla za 19 8 0 ] (Prop. 6.1 ) . If C has a zero ob ject 0 and bipro ducts I ⊕ ... ⊕ I w e obtain an ab elian semiring with zero 0 I : I → I and sum − + − : ∇ I ◦ ( − ⊕ − ) ◦ ∆ I : I → I. When in suc h a catego ry ev ery ob ject is isomorphic to one of the form I ⊕ · · · ⊕ I (finitar y ), a s is the case for b oth ( FdHilb , ⊗ ) and ( FRel , × ), then this category is equiv a len t (as a monoidal catego r y ) to the category of C (I , I)-v alued matrices with the usual matrix op erations. Note that this equiv a lenc e inv olv es choosing a basis isomorphism for eac h ob ject. F or ( FdHilb , ⊗ ) w e ha v e C (I , I) ≃ C and for ( FRel , × ) w e hav e C (I , I) ≃ B , the semiring of b o oleans. Suc h a category of matrices is trivially strongly compact closed for ( L i = n i =1 I) ∗ := L i = n i =1 I, η := ( δ i,j ) i,j : I → i = n M i =1 I ! ⊗ j = n M j =1 I ! (using distributivity and I ⊗ I ≃ I), and ǫ : i = n M i =1 I ! ⊗ i = n M i =1 I ! → I :: ( ψ , φ ) 7→ φ T ◦ ψ where φ T denotes the tra ns p ose of ψ . In the case of ( FRel , × ), this yields the strong compact closed structure desc rib ed ab o ve. If the ab elian semiring C ( I , I) also admits a non-trivial in v olutio n ( ) ∗ , an alternativ e compact closed structure arises b y defining ǫ :: ( ψ , φ ) 7→ ( φ T ) ∗ ◦ ψ , where ( ) ∗ is applied p oin t wise. The corresponding strong compact closed structure inv o lves defining the a djoin t of a matrix M to b e M T ∗ , i . e. the inv o lut io n is applied comp onen t wise to the transp ose of M . In this w a y w e obtain (up to categorical equiv alence) the strong compact closed structure on ( FdHilb , ⊗ ) describ ed ab o v e, t aking ( ) ∗ to b e complex conjugatio n. No w w e can relate t r ace preserving a nd (strongly) compact closed functors to (in volu- tion preserving) semiring homomorphisms. An y suc h homomorphism h : R → S lifts to a functor on the categories of matrices. Moreov er, suc h a functor prese rv es compact closure (and strong compact closure if h preserv es t he giv en in v olution), a nd hence also the trace. 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