BBENNETT AND STINESPRING, TOGETHER AT LAST
CHRIS HEUNEN ∗ AND ROBIN KAARSGAARD † Abstract.
We present a universal construction that relates reversible dynamics on open sys-tems to arbitrary dynamics on closed systems: the well-pointed restriction affine completion ofa monoidal restriction category. This categorical completion encompasses both quantum chan-nels, via Stinespring dilation, and classical computing, via Bennett’s method. Moreover, in thesetwo cases, we show how our construction can be ‘undone’ by a further universal construction.This shows how both mixed quantum theory and classical computation rest on entirely reversiblefoundations. Introduction
Two constructions relate reversible dynamics on open systems to arbitrary dynamics on closedsystems: • Stinespring dilation realises a quantum channel as a reversible process on a larger space [17]. • Bennett’s method makes a classical computer program reversible by allowing extra output [2,16].This paper presents a universal categorical construction encompassing both, making precise how therelationship between pure and mixed quantum theory resembles the relationship between reversibleand conventional classical computation.The construction has three phases: allowing additional constant input, leakage of output, andmaking it extensional. The first two phases adjoin auxiliary systems to the processes in question. Theancilla input can be seen as a form of temporary storage, while the output ancilla is not consideredpart of the desired output, and therefore is sometimes called garbage . However, the garbage cannotbe discarded without altering the function. The third phase of the construction ensures that at leastthe garbage is extensional , that is, that equality of morphisms is judged solely on their observableinput-output behaviour.We can also go in the converse direction by taking the cofree inverse category. All four phaseshave universal properties. On the whole, this shows how both mixed quantum theory and classicalcomputation rest on entirely reversible foundations.Reversible dynamics on open systems Arbitrary dynamics on closed systemspartial injectionsbetween sets partial functionsbetween setscompletely positive trace-preserving maps betweenfin-dim C*-algebrasunitaries betweenfin-dim Hilbert spaces Ext ◦ Aux ◦ InpInvExt ◦ Aux ◦ InpInv ∗ Supported by EPSRC Fellowship EP/R044759/1. † Supported by
DFF | Natural Sciences
International Postdoctoral Fellowship 0131-00025B.. a r X i v : . [ qu a n t - ph ] M a r C. HEUNEN AND R. KAARSGAARD
There are some idiosyncracies among the four phases. The Inp-construction leaves the category ofpartial injections invariant, whereas it turns unitaries into isometries. The Ext-construction leavesthe category of completely positive trace-preserving maps invariant, because minimal Stinespringdilations exist. That is, Stinespring dilation allows an extensional choice of auxiliary system, whereasreversibilising embeddings are intensional. There are several (canonical) methods to make irreversibleprograms reversible. For example, Bennett’s method stores the input and returns it in full alongwith the output, while the
Landauer embedding [1, 14] additionally returns a trace of all instructionsand attendant intermediate states.
Related work
Both Stinespring dilation and Bennett’s method have seen categorical presenta-tions. Despite the similarity of their statements, these categorical completions are surprisingly dis-similar. The universal construction of completely positive trace-preserving map from isometries andunitaries is due to Huot and Staton [10, 11]. A different categorical approach to Stinespring’s dilationtheorem as a universal construction is given by Westerbaan and Westerbaan [18]. The equivalenceof discrete cartesian restriction categories and discrete inverse categories is due to Giles [5], thoughlater recast by Comfort [4] as a counital completion of inverse categories with chosen semi-Frobeniusalgebras. Our Aux-construction generalises a result by Hermida and Tennent [8]. Combining itwith our Ext-construction gives the well-pointed completion of a monoidal restriction category thatgeneralises both Huot-Staton and Giles.
Future work
Following Giles, we conjecture that there is an equivalence between a category ofcertain monoidal inverse categories and certain well-pointed monoidal restriction categories. Anotherinteresting question is whether there is a minimal set that can be adjoined to any partial functionto make it injective. Such a minimal Bennett embedding, as the miniminal Stinespring dilation,could be used to measure the degree to which a map is reversible. It may relate to the informationtheoretic characterisation of reversible maps as those that preserve entropy [14].
Overview
We assume familiarity with basic category theory. Section 2 briefly recalls restrictioncategories and inverse categories. In Section 3, we present the Aux-construction and show that itis the affine completion of a restriction monoidal category. Next, Section 4 introduces the Ext-construction, and shows that it is governed by a universal property. The constructions are put towork in Section 5 by showing that Ext ◦ Aux completes isometries to quantum channels and partialinjective functions to partial functions. In Section 6, we use the dual
Inp of the Aux-constructionto show how quantum channels and partial functions can be universally constructed from unitariesand partial injections, respectively, and further that the latter can be recovered from the former bythe Inv-construction. Appendix A holds proofs that would distract in the main body of the article.2.
Restriction categories and inverse categories
While we assume basic familiarity with category theory, and in particular monoidal categories [9],we briefly summarise restriction categories and inverse categories, which relatively less well-known.Restriction categories [3] axiomatise partially defined morphisms. The idea is to record for eachmorphism f its restriction idempotent f , a partial identity defined precisely where f is defined. Definition 1. A restriction category is a category equipped with a choice of endomorphism f : A → A for each morphism f : A → B satisfying:(i) f ◦ f = f ;(ii) f ◦ g = g ◦ f ;(iii) g ◦ f = g ◦ f ;(iv) g ◦ f = f ◦ g ◦ f .The restriction idempotent f measures ‘how partial’ f is. If f = id, we call f total . Any categorybecomes a restriction category when endowed with the trivial choice f = id, but many other choicesmay be possible. When working with a restriction category, we often leave implicit which choice ismade, just like the choice of tensor product making a category monoidal. When we speak of the ENNETT AND STINESPRING, TOGETHER AT LAST 3 following categories, we will use the trivial restriction structure:
Unitary has finite-dimensionalHilbert spaces as objects and unitary linear maps as morphisms;
Isometry has finite-dimensionalHilbert spaces as objects and isometric linear maps as morphisms;
CPTP has finite-dimensionalC*-algebras as objects and completely positive trace-preserving maps as morphisms.But there are also nontrivial choices of restriction structure. On the category
Pfn of sets andpartial functions, we will choose the restriction idempotent of a partial function f : A → B as follows: f ( x ) = (cid:40) x if f is defined at x undefined otherwiseThus a partial function f is total in the usual sense precisely when it is total in the abstract sense.A functor F : C → D between restriction categories is a restriction functor when F ( f ) = F ( f ).A (symmetric) monoidal restriction category is a restriction category which that is also (symmetric)monoidal, such that the monoidal product is a restriction bifunctor: f ⊗ g = f ⊗ g .Similarly, restriction limits and colimits are ones that respect the restriction structure, thoughespecially limits tend to be quite different. A restriction terminal object is an object 1 such that eachobject A allows a unique total morphism A →
1. Restriction terminal objects need not be terminalin the usual sense; for example, any singleton set is restriction terminal but not terminal in
Pfn ,because there is (at least) also the nowhere defined function A → Lemma 2. [3]
For all appropriate f and g in a restriction category:(i) g ◦ f = g ◦ f ;(ii) g ◦ f = f if g is total;(iii) f = id if f is invertible. A morphism f : A → B in a restriction category is a partial isomorphism if there is a morphism f ◦ : B → A such that f ◦ ◦ f = f and f ◦ f ◦ = f ◦ . Such partial inverses are unique whenever theyexist. In Pfn , the partial isomorphisms are precisely the partial injective functions.Recall that in a dagger category , every morphisms f : A → B has a partner f † : B → A such that f †† = f , id † = id, and ( g ◦ f ) † = f † ◦ g † [9]. Proposition 3. [3]
The following are equivalent:(i) C is a restriction category in which each morphism is a partial isomorphism;(ii) C is an inverse category : a dagger category with f ◦ f † ◦ f = f and f † ◦ f ◦ g † ◦ g = g † ◦ g ◦ f † ◦ f . Inverse categories were originally conceived as a categorical extension of inverse semigroups [13],but have recently seen applications as categorical models of classical reversible computation [5, 12, 6,7]. Examples of inverse categories include the category
PInj of sets and partial injective functions, aswell as any groupoid (such as
Unitary ). The connection between restriction and inverse categoriesgeneralises that between mere categories and groupoids.
Proposition 4. [12]
The wide subcategory
Inv( C ) of all partial isomorphisms of a (monoidal) re-striction category C is its cofree (monoidal) inverse category: any inverse category D with a (strictmonoidal) functor D → C allows a unique (strict monoidal) functor D → Inv( C ) making the fol-lowing diagram commute: D Inv( C ) C If C in the above is a trivial restriction category, then Inv( C ) is its core , that is, its cofree groupoid. C. HEUNEN AND R. KAARSGAARD The
Aux -construction
This section is dedicated to the Aux-construction, a generalisation of Hermida and Tennent’sconstruction [8] to (symmetric monoidal) restriction categories. After introducing Aux( C ), we showstep by step that it is an affine monoidal restriction category. Here, a monoidal restriction categoryis affine when its tensor unit I is restriction terminal. The crowning theorem shows that Aux( C ) isin fact the restriction affine completion of C . Definition 5.
Define a relation (cid:46) on the morphisms of a symmetric monoidal restriction categoryas follows. For f : A → B ⊗ E and f (cid:48) : A → B ⊗ E (cid:48) , set f (cid:46) f (cid:48) if and only if f = f (cid:48) and there is a mediator h : E → E (cid:48) making the triangle commute: AB ⊗ E B ⊗ E (cid:48) f (cid:48) f id ⊗ h This is a preorder: reflexivity follows by mediating with identities; transitivity follows by com-posing mediators. However, the relation need not be symmetric, for example if dim( E ) < dim( E (cid:48) )in Isometry . Definition 6.
Write ∼ for the equivalence relation generated by (cid:46) . Explicitly, for f : A → B ⊗ E and f (cid:48) : A → B ⊗ E (cid:48) , we have f ∼ f (cid:48) if and only if there are intermediate morphisms f , . . . , f n − with f = f = · · · = f n − = f (cid:48) and mediators E h −→ E h ←− E h −→ · · · h n ←−− E (cid:48) making the followingdiagram commute: AB ⊗ E B ⊗ E B ⊗ E · · · B ⊗ E (cid:48) id ⊗ h id ⊗ h id ⊗ h id ⊗ h n f (cid:48) f f ...f Definition 7.
For a symmetric monoidal restriction category C , define a category Aux( C ): • objects are those of C ; • morphisms [ f, E ] : A → B are ∼ -equivalence classes represented by morphisms f : A → B ⊗ E in C ; • composition of [ f, E ] : A → B and [ g, E (cid:48) ] : B → C is [ α ◦ ( g ⊗ id) ◦ f, E (cid:48) ⊗ E ] : A → C ; • identities are [ ρ − , I ] : A → A .The previous definition differs from [8] only by the additional requirement that f = f (cid:48) if f ∼ f (cid:48) .It follows that the two are the same when C is a trivial restriction category, making Aux a genuinegeneralisation. Remark 8.
Morphisms in Aux( C ) are often given by composing chains of morphisms in C , furtherquotiented by a nontrivial equivalence relation. This can quickly become unintelligible. Therefore wewill always make the zig-zag path of mediators from Definition 6 explicit in equivalence arguments.To indicate which part of a diagram in C corresponds to which morphism in Aux( C ), we will usesquiggly grey ‘ghost’ arrows: A B ⊗ E ( C ⊗ E (cid:48) ) ⊗ EC ⊗ ( E (cid:48) ⊗ E ) f g ⊗ id α [ g,E (cid:48) ] ◦ [ f,E ] This ghost arrow is not a part of the commutative diagram. It merely indicates that α ◦ ( g ⊗ id) ◦ f corresponds precisely to [ g, E (cid:48) ] ◦ [ f, E ] in Aux( C ). ENNETT AND STINESPRING, TOGETHER AT LAST 5
Notation settled, we now set out to show that this actually defines a restriction symmetricmonoidal category. We proceed in three steps: first we show that it is a category; then that itinherits a restriction structure; and finally that it inherits a symmetric monoidal structure in a waythat respects restriction. The proofs of the following three propositions are deferred to Appendix Aas they would distract from the main development.
Proposition 9.
Aux( C ) is a category. Proposition 10.
Aux( C ) inherits a restriction structure from C with [ f, E ] = [ ρ − ◦ f , I ] . Proposition 11. If C is a restriction symmetric monoidal category, then so is Aux( C ) : • the tensor unit and tensor product of objects are as in C ; • the tensor product of [ f, E ] : A → B and [ f (cid:48) , E (cid:48) ] : A (cid:48) → B (cid:48) is [ ϑ ◦ ( f ⊗ f (cid:48) ) , E ⊗ E (cid:48) ] : A ⊗ A (cid:48) → B ⊗ B (cid:48) ;where ϑ is the canonical isomorphism ( B ⊗ E ) ⊗ ( B (cid:48) ⊗ E (cid:48) ) (cid:39) ( B ⊗ B (cid:48) ) ⊗ ( E ⊗ E (cid:48) ) in C . Having established that Aux( C ) is a restriction symmetric monoidal category, our next goal is toshow that it is the restriction affine completion of C . Again we proceed in steps. First we show thatthere is a strict monoidal functor C → Aux( C ). Then we show that the unit in Aux( C ) is restrictionterminal, so that the tensor product has total projections. From this we derive a factorisationtheorem for morphisms in Aux( C ), which finally lets us institute Aux( C ) as the restriction affinecompletion of C . Proposition 12. If C is a restriction symmetric monoidal category, there is a strict monoidalrestriction functor E : C → Aux( C ) given by E ( A ) = A on objects and by E ( f ) = [ ρ − ◦ f, I ] onmorphisms.Proof. To see E is functorial, compute E (id) = [ ρ − ◦ id , I ] = [ ρ − , I ] = id. Composition is preservedbecause E ( g ) ◦ E ( f ) = α ◦ ( ρ − ⊗ id) ◦ ( g ⊗ id) ◦ ρ − ◦ f = α ◦ ( ρ − ⊗ id) ◦ ρ − ◦ g ◦ f = g ◦ f = ρ − ◦ g ◦ f = E ( g ◦ f )and the diagram below commutes: B B ⊗ I C ⊗ I ( C ⊗ I ) ⊗ I C ⊗ ( I ⊗ I ) A C ⊗ IB C C ⊗ I f αf g ρ − id ⊗ idid ⊗ ρF ( g ) ◦ F ( f ) F ( g ◦ f ) ρ − g ⊗ id ρ − ⊗ id The functor E preserves restriction idempotents: E ( f ) = [ ρ − ◦ ρ − ◦ f , I ] = [ ρ − ◦ f , I ] = E ( f ).That it is a strict monoidal functor follows from E ( A ⊗ B ) = A ⊗ B = E ( A ) ⊗ E ( B ), E ( I ) = I , E ( f ⊗ g ) = E ( f ) ⊗E ( g ) (shown entirely analogously to showing [ ρ − ◦ β, I ] ⊗ [ ρ − ◦ φ, I ] ∼ [ ρ − ◦ ( β ⊗ φ ) , I ]for coherences β and φ in Proposition 11, see Appendix A), and the fact that coherence isomorphismsin C are precisely of the form [ ρ − ◦ β, I ] = E ( β ) for each coherence isomorphism β of C . (cid:3) Proposition 13.
The tensor unit in
Aux( C ) is restriction terminal.Proof. First note I is weakly terminal: there is a morphism from each object A into I , namely[ λ − , A ]. Furthermore, this morphism is total since [ λ − , A ] = [ ρ − ◦ λ − , I ] = [ ρ − ◦ id , I ] = C. HEUNEN AND R. KAARSGAARD [ ρ − , I ] = id. Because I ⊗ EA I ⊗ E I ⊗ ( I ⊗ E ) I ⊗ A fλ − id ⊗ λ − id ⊗ fλ − f any total morphism [ f, E ] : A → I satisfies [ f, E ] ∼ [ λ − , A ]. (cid:3) We will simply write ! for the unique morphism [ λ − , A ] : A → I from now on. Remark 14.
An important property of restriction affine monoidal categories is that they have totalmaps π : A ⊗ B → A and π : A ⊗ B → B . These can be defined as A ⊗ B id ⊗ ! −−−→ A ⊗ I ρ −→ A andsymmetrically, and are total since ρ ◦ (id ⊗ !) = (id ⊗ !) = id ⊗ ! = id ⊗ id = id, and similarly for thesecond projection.These total projections are crucial in showing the following factorisation of morphisms in Aux( C ),based on Hermida and Tennent’s expansion-raw morphism factorisation [8, Lemma 2.8]. Lemma 15.
Every morphism [ f, E ] : A → B of Aux( C ) factors as π ◦ E ( f ) . This factorisation isunique in the sense that if [ f, E ] ∼ π ◦ E ( f (cid:48) ) for any f (cid:48) , then [ f, E ] ∼ [ f (cid:48) , E (cid:48) ] .Proof. Let [ f, E ] : A → B be a morphism of Aux( C ). First, π ◦ E ( f ) = E ( f ) = E ( f ) = [ ρ − ◦ f , I ] =[ f, E ]. That [ f, E ] ∼ π ◦ E ( f ) then follows by commutativity of the diagram below. B ⊗ E ( B ⊗ E ) ⊗ I B ⊗ ( E ⊗ I ) A B ⊗ EB ⊗ E f ρ − αf id ⊗ ρ id ⊗ id[ f,E ] π ◦E ( f ) Now suppose [ f, E ] ∼ π ◦ E ( f (cid:48) ) for some f (cid:48) : A → B ⊗ E (cid:48) in C . Similarly as before, [ f (cid:48) , E (cid:48) ] ∼ π ◦ E ( f (cid:48) ), so it simply follows by transitivity that [ f, E ] ∼ π ◦ E ( f (cid:48) ) ∼ [ f (cid:48) , E (cid:48) ]. (cid:3) We have finally arrived at the main theorem of this section.
Theorem 16.
Aux( C ) is the restriction affine completion of a restriction symmetric monoidalcategory C : given any other restriction affine symmetric monoidal category D and strict monoidalrestriction functor F : C → D , there is a unique functor ˆ F : Aux( C ) → D with F = ˆ F ◦ E . C Aux( C ) D E F ˆ F Proof.
Define ˆ F : Aux( C ) → D by ˆ F ( A ) = F ( A ) on objects, on a morphism [ f, E ] : A → B by:ˆ F ( A ) = F ( A ) F ( f ) −−−→ F ( B ⊗ E ) = F ( B ) ⊗ F ( E ) π −→ F ( B ) = ˆ F ( B )This makes the diagram commute since ˆ F ( E ( A )) = ˆ F ( A ) = F ( A ) on objects, and on morphismsˆ F ( E ( f )) = ˆ F ([ ρ − ◦ f, I ]) = π ◦ F ( ρ − ◦ f ) = π ◦ F ( ρ − ) ◦ F ( f ) = π ◦ ρ − ◦ F ( f ) = F ( f )because π ◦ ρ − is just the identity by definition of π . The functor ˆ F is strict monoidal sinceˆ F ( A ⊗ B ) = F ( A ⊗ B ) = F ( A ) ⊗ F ( B ) and since all coherence isomorphisms Aux( C ) are of the ENNETT AND STINESPRING, TOGETHER AT LAST 7 form E ( β ) for a coherence isomorphism β of C , so that ˆ F ( β ) = ˆ F ( E ( β )) = F ( β ) = β . Also, ˆ F is arestriction functor since F is: ˆ F ([ f, E ]) = π ◦ F ( f ) = F ( f ) = F ( f ) = ˆ F ( E ( f )) = ˆ F ([ ρ − ◦ f , I ]) =ˆ F ([ f, E ]).To see that ˆ F is unique, suppose G : Aux( C ) → D is a strict monoidal restriction functor makingthe triangle commute. First, ˆ F and G agree on objects as G ( A ) = ˆ F ( E ( A )) = ˆ F ( A ). If [ f, E ] : A → B is a morphism of Aux( C ), then Lemma 15 guarantees [ f, E ] ∼ π ◦ E ( f ), so: G ([ f, E ]) = G ( π ◦ E ( f )) = G ( π ) ◦ G ( E ( f )) = π ◦ F ( f ) = ˆ F ([ f, E ]) (cid:3) Extensionality
This section concerns the second phase of our completion: the Ext-construction. It quotients agiven category by an equivalence relation related to well-pointedness to make it extensional, which wewill show has a universal property. Combining this with the Aux-construction of Section 3, the mainresults of this section will show that Ext(Aux(
Isometry )) (cid:39) CPTP and Ext(Aux(
PInj )) (cid:39) Pfn .Say that a (restriction) category is pointed if it has a (restriction) terminal object, and that it is (restriction) well-pointed if additionally f = g as soon as f ◦ a = g ◦ a for all a : 1 → A . Both Pfn and
CPTP are restriction well-pointed.
Definition 17.
In a pointed restriction category, define a relation ≈ on parallel morphisms f, g : A → B by setting f ≈ g if and only if f ◦ a = g ◦ a for all a : 1 → A . Lemma 18.
The relation · ≈ · is a congruence, and so
Ext( C ) = C / ≈ is a well-defined category.Proof. Suppose that f, f (cid:48) : A → B and g, g (cid:48) : B → C satisfy f ≈ f (cid:48) and g ≈ g (cid:48) . Let a : 1 → A . Then f ◦ a = f (cid:48) ◦ a , and hence g ◦ f ◦ a = g (cid:48) ◦ f (cid:48) ◦ a . So g ◦ f ≈ g (cid:48) ◦ f (cid:48) . (cid:3) The congruence ≈ also respects restriction structure: if f, f (cid:48) : A → B satisfy f ≈ f (cid:48) , then also f ≈ f (cid:48) , by Definition 1(iv), for if a : 1 → A , then f ◦ a = a ◦ f ◦ a = a ◦ f (cid:48) ◦ a = f (cid:48) ◦ a . Therefore Ext( C )is a well-defined restriction category, and the quotient functor C → Ext( C ) sending a morphism toits equivalence class is a restriction functor.However, it is not clear whether ≈ is a monoidal congruence when the category is affine monoidal.If f ≈ f (cid:48) : A → C and g ≈ g (cid:48) : B → D , then ( f ⊗ g ) ◦ x = ( f (cid:48) ⊗ g (cid:48) ) ◦ x for all x : 1 → A ⊗ B of theform x = ( a ⊗ b ) ◦ λ − I for a : 1 → A and b : 1 → B . But what about entangled states x : 1 → A ⊗ B ?Luckily, in the examples below this holds, so Ext( C ) is again a well-defined monoidal category, and C → Ext( C ) a strict monoidal functor.By construction Ext( C ) is well-pointed, and the Ext-construction is universal in accomplishingthis. Definition 19.
Call a functor F : C → D between pointed restriction categories full on points ifeach p : 1 → F ( A ) in D is of the form F ( a ) for some a : 1 → A in C . Theorem 20.
Ext( C ) is the well-pointed completion of the pointed restriction category C : given awell-pointed restriction category D and restriction functor F : C → D that is full on points, there isa unique restriction functor ˆ F : Ext( C ) → D that is full on points and makes the triangle commute: C Ext( C ) D F ˆ F Proof.
Set ˆ F ( A ) = F ( A ) on objects and ˆ F ([ f ]) = F ( f ) on morphisms. To see that this is well-defined, suppose f ≈ g , that is f ◦ a = g ◦ a for all a : 1 → A in C . Then also F ( f ) ◦ p = F ( g ) ◦ p for all p : 1 → F ( A ) in D because F is full on points, and so F ( f ) = F ( g ) since D is well-pointed. C. HEUNEN AND R. KAARSGAARD
Moreover, ˆ F is a restriction functor since ˆ F ([ f ]) = F ( f ) = F ( f ) = ˆ F ([ f ]), and it is full on pointssince F is. Now ˆ F ◦ Q = F directly. It remains to show that ˆ F is the unique such functor. Suppose G ◦ Q = F for a functor G : Ext( C ) → D that is full on points. But then G ( A ) = F ( A ), and since Q ( f ) = [ f ], we must also have G ([ f ]) = F ( f ) = ˆ F ([ f ]). (cid:3) When the functor C → Ext( C ) is strict monoidal, as is the case for both Isometry and
PInj , itcompletes restriction affine monoidal categories to restriction well-pointed monoidal categories.5.
Quantum channels and classical functions as completions
This section instantiates the theory of the previous ones for our main examples. The quantumcase is quickly established thanks to Huot and Staton.
Proposition 21.
Ext(Aux(
Isometry )) (cid:39) CPTP .Proof.
Since
Isometry is a trivial restriction category,
CPTP (cid:39) L ( Isometry ) (cid:39) Aux(
Isometry )by [10, Corollary 7]. Also,
CPTP is already well-pointed, so Ext(
CPTP ) (cid:39) CPTP . (cid:3) Above, the Ext-phase was trivial, but this is not always the case. Consider the (intensional)category Aux(
PInj ): objects are sets, and morphisms A → B are partial injective functions A → B × E that are identified f ∼ f (cid:48) when there is a partial injective function h : E → E (cid:48) such that f ( x ) = ( y, e ) implies f (cid:48) ( x ) = ( y, h ( e )) for all x ∈ A . The environment E is often thought of as the garbage produced by the function because, being injective, it cannot actually discard any information.However, the Aux-construction allows it to instead place the garbage off to the side, demarcatingit from the desired output. In reversible computation, such garbage is unavoidable (since not allcomputable functions, and even not all interesting such, happen to be injective), so it is importantthat it is managed properly.Garbage is ideally extensional: we should be able to compare functions by looking only at theirinput-output behavior, even when some of it is designated as garbage. But unless you are careful,this might not be the case. Consider the successor function n (cid:55)→ n + 1 on natural numbers. Wecan consider many different ways to vary the environment: for example f : N → N × {∗} given by n (cid:55)→ ( n + 1 , ∗ ); but also f : N → N × N given by n (cid:55)→ ( n + 1 , n ). These two functions effect theexact same behaviour when disregarding garbage. But they are in different equivalence classes asmorphisms N → N in Aux( PInj ) because their garbage is so different.How to mend this? First notice that points 1 → A in Aux( PInj ) correspond to those in
Pfn (see Lemma 22 below). Even though f and f are different in Aux( PInj ), they do agree on each n : 1 → N : build the partial function h n : N → defined only on n by h n ( n ) = ∗ ; this mediatesbecause f ( n ) = ( n + 1 , ∗ ) = ( n + 1 , h n ( n )) and f ( n ) = ( n + 1 , n ). So, garbage is intensional inAux( PInj ) because the category is not well-pointed. Because
Pfn is well-pointed, it is necessary toidentify morphisms when they agree on all points, which is exactly what the Ext-construction does.Why was this not an issue in the quantum case? There, extensionality arises from minimal
Stinespring dilations. Minimality gives is a unique minimal (up to unitary) auxiliary system wecan adjoin to realise any CPTP-map as conjugation by an isometry, thus taking away the choice ofenvironment E that sparked the trouble in Aux( PInj ). Lemma 22.
The global points → A in Aux(
PInj ) coincide with those in Pfn .Proof.
Points in Aux(
PInj ) are partial injective functions x : 1 → A × E modulo identification.However, any such point can always be identified with one of the form y : 1 → A × x ( ∗ ) = ( a, e ) then the point ∗ (cid:55)→ e mediates 1 → E to witness ( x, E ) ∼ ( y, E is the empty set,the nowhere defined function trivially mediates. (cid:3) It follows from the previous Lemma that the functor Aux(
PInj ) → Ext(Aux(
PInj )) is full onpoints. So is Aux(
Isometry ) → Ext(Aux(
Isometry )), but in a trivial way: because
ENNETT AND STINESPRING, TOGETHER AT LAST 9
Aux(
Isometry ) (cid:39) CPTP by [10], and
CPTP is already well-pointed, this functor is an isomor-phism of categories.
Proposition 23.
Ext(Aux(
PInj )) (cid:39) Pfn .Proof.
Define F : Pfn → Ext(Aux(
PInj )) by F ( A ) = A on objects, and on morphisms f : A → B by F ( f ) = [ b f , A ], where b f is the Bennett embedding of f given by b f ( x ) = ( f ( x ) , x ).We argue first that this is functorial: F (id) is b id ( x ) = ( x, x ), but the chosen identity is (theequivalence class of) ρ − ( x ) = ( x, (cid:63) ). However, on each point p , simply choose p itself to mediateto see b id ≈ ρ − . Likewise, whereas F ( g ◦ f ) is b g ◦ f ( x ) = ( g ( f ( x )) , x ) and F ( g ) ◦ F ( f ) is b (cid:48) ( x ) = (cid:0) g ( f ( x )) , ( f ( x ) , x ) (cid:1) , for each point x , mediate that point by h x : A → B × A given by: h x ( a ) = (cid:40) ( f ( x ) , x ) if a = x undefined otherwiseThus F ( g ◦ f ) ≈ F ( g ) ◦ F ( f ). Since Pfn and Ext(Aux(
PInj )) have the same objects, it remainsonly to be seen that F is full and faithful.For fullness, let a partial injective f : A → B × E represent a morphism in Ext(Aux( PInj )). Since[ f, E ] and [ f (cid:48) , E (cid:48) ] are identified if and only if for all x ∈ A there exists a partial injective function h x : E → E (cid:48) such that f ( x ) = ( y, e ) implies f (cid:48) ( x ) = ( y, h x ( e )), either way π ◦ f = π ◦ f (cid:48) as partialfunctions. Consider now the Bennett embedding of π ◦ f , that is, the partial injective function b π ◦ f : A → B × A given by x (cid:55)→ ( π ( f ( x )) , x ), and compare it to f : A → B × E . For any x ∈ A , itfollows that if f ( x ) = ( y, e ) then b π ◦ f ( x ) = ( π ( f ( x )) , x ) = ( π ( y, e ) , x ) = ( y, x ), so the two agreein the first component. Define a one-point mediator h x : A → E for x given by: h x ( a ) = (cid:40) e if a = x undefined otherwiseThus b π ◦ f ≈ f and F is full.Towards faithfulness, suppose F ( f ) ≈ F ( g ), so b f ≈ b g for some f, g : A → B . Thus b f ( x ) =( f ( x ) , x ) for some partial function f , and similarly b g ( x ) = ( g ( x ) , x ). That b f ≈ b g means thatfor each a ∈ A there exists h a : A → A (necessarily the identity) such that b f ( a ) = ( y, a ) implies b g ( a ) = ( y, h a ( a )) = ( y, a ). But since y = f ( a ) by definition of b f , and since the above holds for all a ∈ A , it thus follows that f ( x ) = g ( x ) for all x ∈ A , which in turn implies f = g in Pfn byextensionality. So F is faithful. (cid:3) Corollary 24. CPTP is the restriction monoidal well-pointed completion of
Isometry , and
Pfn is the restriction monoidal well-pointed completion of
PInj .Proof.
Combine Theorems 16 and 20 with Propositions 21 and 23. (cid:3) Cofree reversible foundations
While
CPTP and
Pfn both arise as completions of ‘reversible’ categories
Isometry and
PInj ,it is difficult to pinpoint the features which make them reversible. For example,
PInj is an inversecategory, but
Isometry is not even a dagger category. Following [10], we peel off another layer toreveal the inverse category underneath using the Inp-construction, the dual to Aux. Thus we canshow that both
CPTP and
Pfn arise via the same universal constructions on the inverse categories
Unitary and
PInj . We go on to show that this amalgamation of constructions is itself invertibleby universal means, allowing us to reconstruct
PInj and
Unitary from
Pfn and
CPTP as theircofree (monoidal) inverse categories.
Definition 25.
For a symmetric monoidal inverse category C , define Inp( C ) = Aux( C op ) op . Proposition 26.
When C is a symmetric monoidal inverse category, Inp( C ) is a coaffine symmetricmonoidal restriction category. Proof.
Inverse categories are self-dual, C (cid:39) C op , so Inp( C ) = Aux( C op ) op (cid:39) Aux( C ) op . HenceAux( C ) is an affine symmetric monoidal restriction category, and Inp( C ) is a coaffine symmetricmonoidal corestriction category. It is also a symmetric monoidal restriction category under [ f, E ] op =[ ρ − ◦ f † , I ], because in an inverse category C morphisms f have (monoidal) corestriction f † . (cid:3) The Aux-construction (and, by duality, the Inp-construction) is conservative : if a monoidal cate-gory is already affine, the construction does nothing (up to isomorphism).
Proposition 27. If C is a restriction affine symmetric monoidal category, then Aux( C ) (cid:39) C .Proof. It suffices to show that each morphism is equivalent to one of the form E ( f (cid:48) ). Let [ f, E ] : A → B be a morphism of Aux( C ). Then E ( π ◦ f ) = E ( π ◦ f ) = E ( f ) = [ ρ − ◦ f , I ] = [ f, E ] and: B ⊗ IB ⊗ E B B ⊗ IA B ⊗ IB ⊗ E f π ρ − f id ⊗ !id ⊗ id E ( π ◦ f )[ f,E ]id ⊗ ! ρ id So E ( π ◦ f ) ∼ [ f, E ]. (cid:3) We can now show that
Pfn and
CPTP arise as completions of the inverse categories
PInj and
Unitary . The quantum case relies on Huot and Staton’s characterisation of
Isometry as acompletion of
Unitary [11] making initial the unit of the direct sum . We consider
PInj and
Unitary as inverse rig categories, using the Inp-construction to make the unit of the direct sum initial, andthen the Aux-construction to make the tensor unit terminal. In this bimonoidal setting, we will usesubscripts to clarify which monoidal structure a construction acts on.
Theorem 28.
Ext(Aux ⊗ (Inp ⊕ ( PInj ))) (cid:39)
Pfn and
Ext(Aux ⊗ (Inp ⊕ ( Unitary ))) (cid:39)
CPTP .Proof.
First, that Ext(Aux ⊗ (Inp ⊕ ( Unitary )) (cid:39) Ext( L ⊗ ( R ⊕ ( Unitary ))) (cid:39)
CPTP follows fromthe fact that R ⊕ ( Unitary ) (cid:39) Isometry by [11, III.3] and Proposition 21. NowExt(Aux ⊗ (Inp ⊕ ( PInj ))) (cid:39)
Pfn follows from the unit 0 of the disjoint sum ⊕ in PInj already being(restriction) initial, so Inp ⊕ ( PInj ) (cid:39) PInj by dualising Proposition 27 and finallyExt(Aux ⊗ (Inp ⊕ ( PInj ))) (cid:39)
Ext(Aux ⊗ ( PInj )) (cid:39) Pfn . (cid:3) Finally, we show that, at least in these two cases, this construction can be undone by consideringtheir cofree inverse categories (see Proposition 4).
Theorem 29.
Inv(
Pfn ) (cid:39) PInj and
Inv(
CPTP ) (cid:39) Unitary .Proof.
That Inv(
Pfn ) (cid:39) PInj is well known; see for example [3]. With
CPTP a trivial restrictioncategory, we show that
Unitary is its cofree groupoid. It suffices to show that isomorphisms in
CPTP just conjugate with a unitary.Let Λ : B ( H ) → B ( K ) be an isomorphism in CPTP , that is, a bijective CPTP map with a CPTPinverse. Notice first that since Λ is bijective and H and K finite-dimensional, they must in fact haveequal dimension. Second, notice that Λ must then preserve pure states, since if Λ( | φ (cid:105)(cid:104) φ | ) is somemixed state (cid:80) i α i ρ i then | φ (cid:105)(cid:104) φ | = Λ − (Λ( | φ (cid:105)(cid:104) φ | )) = Λ − ( (cid:80) i α i ρ i ) = (cid:80) i α i Λ − ( ρ i ), contradictingpurity of | φ (cid:105)(cid:104) φ | . But since id ⊗ Λ is then also an isomorphism, it too preserves pure states, and sothe Choi-state (id ⊗ Λ)( | Φ (cid:105)(cid:104) Φ | ) for Λ is pure, too. Recall that a Stinespring dilation of a CPTP mapcan be obtained by purifying its Choi-state, sending the result back through the Choi-Jamiolkowskiisomorphism, and tracing out the auxiliary system [15]. Since the Choi-state (id ⊗ Λ)( | Φ (cid:105)(cid:104) Φ | ) is ENNETT AND STINESPRING, TOGETHER AT LAST 11 already pure, Λ must then already be conjugation by some isometry V , which must in fact beunitary by surjectivity of Λ. (cid:3) Acknowledgements
We thank Frederik vom Ende for his clarifying comments on Theorem 29,and Cole Comfort for pointing out related work.
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Appendix A. Deferred proofs
Proposition 9.
Aux( C ) is a category. Proof.
We need to show that composition is associative, unital, and well-defined. Let [ f, E ] : A → B ,[ g, E (cid:48) ] : B → C , and [ h, E (cid:48)(cid:48) ] : C → D be morphisms of Aux( C ). That [ h, E (cid:48)(cid:48) ] ◦ ([ g, E (cid:48) ] ◦ [ f, E ]) isequivalent to ([ h, E (cid:48)(cid:48) ] ◦ [ g, E (cid:48) ]) ◦ [ f, E ] follows from α ◦ ( h ⊗ id) ◦ α ◦ g ⊗ id ◦ f = α ◦ α ◦ (( h ⊗ id) ⊗ id) ◦ g ⊗ id ◦ f = (( h ⊗ id) ⊗ id) ◦ g ⊗ id ◦ f = α ◦ ( α ⊗ id) ◦ (( h ⊗ id) ⊗ id) ◦ g ⊗ id ◦ f in C and commutativity of the following diagram in C : B ⊗ E ( C ⊗ E (cid:48) ) ⊗ E C ⊗ ( E (cid:48) ⊗ E ) ( D ⊗ E (cid:48)(cid:48) ) ⊗ ( E (cid:48) ⊗ E ) D ⊗ ( E (cid:48)(cid:48) ⊗ ( E (cid:48) ⊗ E )) A D ⊗ ( E (cid:48)(cid:48) ⊗ ( E (cid:48) ⊗ E )) D ⊗ (( E (cid:48)(cid:48) ⊗ E (cid:48) ) ⊗ E ) B ⊗ E ( C ⊗ E (cid:48) ) ⊗ E (( D ⊗ E (cid:48)(cid:48) ) ⊗ E (cid:48) ) ⊗ E ( D ⊗ ( E (cid:48)(cid:48) ⊗ E (cid:48) )) ⊗ E g ⊗ id ( h ⊗ id) ⊗ id α ⊗ id αff g ⊗ id α h ⊗ id α [ h,E (cid:48)(cid:48) ] ◦ ([ g,E (cid:48) ] ◦ [ f,E ])([ h,E (cid:48)(cid:48) ] ◦ [ g,E (cid:48) ]) ◦ [ f,E ] id ⊗ idid ⊗ α That id ◦ [ f, E ] ∼ [ f, E ] follows from α ◦ ( ρ − ⊗ id) ◦ f = f and commutativity in C of the diagram: B ⊗ E ( B ⊗ I ) ⊗ E B ⊗ ( I ⊗ E ) A B ⊗ EB ⊗ E ff ρ − ⊗ id α id ◦ [ f,E ] id ⊗ λ id ⊗ id[ f,E ] Similarly [ f, E ] ◦ id ∼ [ f, E ]. Finally, we show that composition is well-defined. Suppose [ f, E ] : A → B is equivalent to [ f (cid:48) , G ] : A → B by a zigzag of mediators E h −→ E h ←− E h −→ · · · h n ←−− G . Given[ g, E (cid:48) ] : B → C and intermediates f , f , . . . , f n − , to show [ g, E (cid:48) ] ◦ [ f, E ] ∼ [ g, E (cid:48) ] ◦ [ f (cid:48) , G ] we seefirst that the diagram below commutes in C : AB ⊗ E B ⊗ E B ⊗ E . . . B ⊗ G ( C ⊗ E (cid:48) ) ⊗ E ( C ⊗ E (cid:48) ) ⊗ E ( C ⊗ E (cid:48) ) ⊗ E . . . ( C ⊗ E (cid:48) ) ⊗ GC ⊗ ( E (cid:48) ⊗ E ) C ⊗ ( E (cid:48) ⊗ E ) C ⊗ ( E (cid:48) ⊗ E ) . . . C ⊗ ( E (cid:48) ⊗ G ) f f (cid:48) αg ⊗ id g ⊗ id α id ⊗ h id ⊗ h id ⊗ h n id ⊗ h id ⊗ (id ⊗ h ) id ⊗ (id ⊗ h ) id ⊗ (id ⊗ h ) id ⊗ (id ⊗ h n ) g ⊗ id g ⊗ id α α id ⊗ h id ⊗ h n id ⊗ h id ⊗ h f ··· f ENNETT AND STINESPRING, TOGETHER AT LAST 13
There is no room in the diagram above for ghost arrows, but each downward path α ◦ ( g ⊗ id) ◦ f i corresponds to [ g, E (cid:48) ] ◦ [ f i , E i ], and likewise for [ f, E ] and [ f (cid:48) , G ] instead of f i . We have left to showthat α ◦ ( g ⊗ id) ◦ f = α ◦ ( g ⊗ id) ◦ f (cid:48) . Now id ⊗ h ◦ f = f ◦ (id ⊗ h ) ◦ f = f ◦ f = f ◦ f = f , so: α ◦ ( g ⊗ id) ◦ f = ( g ⊗ id) ◦ f = ( g ⊗ id) ◦ id ⊗ h ◦ f = ( g ⊗ id) ◦ (id ⊗ h ) ◦ f = ( g ⊗ h ) ◦ f = g ⊗ h ◦ f = ( g ⊗ h ) ◦ f = ( g ⊗ id) ◦ (id ⊗ h ) ◦ f = ( g ⊗ id) ◦ f = α ◦ ( g ⊗ id) ◦ f By induction eventually α ◦ ( g ⊗ id) ◦ f = α ◦ ( g ⊗ id) ◦ f (cid:48) .Pre-composition is similarly well-defined, though the condition on restriction idempotents followsmore readily by ( g ⊗ id) ◦ f = ( g ⊗ id) ◦ f = ( g (cid:48) ⊗ id) ◦ f = ( g (cid:48) ⊗ id) ◦ f . (cid:3) Proposition 10.
Aux( C ) inherits a restriction structure from C with [ f, E ] = [ ρ − ◦ f , I ]. Proof.
We establish the axioms of Definition 1 in order. That [ f, E ] ◦ [ f, E ] = [ f, E ] for each[ f, E ] : A → B in Aux( C ) follows by commutativity of the following diagram in C : B ⊗ EA A ⊗ I ( B ⊗ E ) ⊗ I B ⊗ ( E ⊗ I ) A B ⊗ EB ⊗ E ff ρ − f ⊗ id α id ⊗ ρ id ⊗ id[ f,E ] ◦ [ f,E ][ f,E ] f ρ − f ρ To see that [ f, E ] ◦ [ g, E (cid:48) ] = [ g, E (cid:48) ] ◦ [ f, E ] for [ f, E ] : A → B and [ g, E (cid:48) ] : A → C in Aux( C ): A A ⊗ I A ⊗ I ( A ⊗ I ) ⊗ IA A A ⊗ ( I ⊗ I ) A A ⊗ ( I ⊗ I ) A A A ⊗ ( I ⊗ I ) A A ⊗ I A ⊗ I ( A ⊗ I ) ⊗ I fg ρ − fg ρ − f ⊗ id ρ − g ⊗ id ρ − ρ − ⊗ id ρ − ⊗ id αα [ g,E (cid:48) ] ◦ [ f,E ][ f,E ] ◦ [ g,E (cid:48) ] id ⊗ idid ⊗ id f gg f To show [ g, E (cid:48) ] ◦ [ f, E ] = [ g, E (cid:48) ] ◦ [ f, E ] for all [ f, E ] : A → B and [ g, E (cid:48) ] : A → C of Aux( C ), firstcompute:[ g, E (cid:48) ] ◦ [ f, E ] = [ g, E (cid:48) ] ◦ ( ρ − ◦ f , I ) = ( α ◦ ( g ⊗ id) ◦ ρ − ◦ f , I ) = ( α ◦ ρ − ◦ g ◦ f , I )= ( ρ − ◦ α ◦ ρ − ◦ g ◦ f , I ) = ( ρ − ◦ g ◦ f , I ) Now the diagram below commutes in C because g ◦ f = g ◦ f : AA A ⊗ I A ⊗ I ( A ⊗ I ) ⊗ I A ⊗ ( I ⊗ I ) A A ⊗ IA A ⊗ I fg ◦ f ρ − ρ − g ⊗ id ρ − ⊗ id α id ⊗ ρ id ⊗ id g ρ − [ g,E (cid:48) ] ◦ [ f,E ][ g,E (cid:48) ] ◦ [ f,E ] Finally, for [ f, E ] : A → B and [ g, E (cid:48) ] : B → C we have [ g, E (cid:48) ] ◦ [ f, E ] = [ f, E ] ◦ [ g, E (cid:48) ] ◦ [ f, E ]because [ g, E (cid:48) ] ◦ [ f, E ] = [ α ◦ ( g ⊗ id) ◦ f , I ] = [( g ⊗ id) ◦ f , I ] = [( g ⊗ id) ◦ f , I ] and the diagrambelow commutes: B ⊗ E B ⊗ E ( B ⊗ I ) ⊗ E B ⊗ ( I ⊗ E ) A B ⊗ EA B ⊗ E ( B ⊗ E ) ⊗ I B ⊗ ( E ⊗ I ) A ⊗ I f α ( g ⊗ id) ◦ f ρ − f ⊗ id α id ⊗ λ id ⊗ ρf ρ − g ⊗ id ρ − ⊗ id[ f,E ] ◦ [ g,E (cid:48) ] ◦ [ f,E ][ g,E (cid:48) ] ◦ [ f,E ] Here ( g ⊗ id) ◦ f = g ⊗ id ◦ f = f ◦ ( g ⊗ id) ◦ f by the corresponding axiom in C . (cid:3) Proposition 11. If C is a restriction symmetric monoidal category, then so is Aux( C ): • the tensor unit and tensor product of objects are as in C ; • the tensor product of [ f, E ] : A → B and [ f (cid:48) , e (cid:48) ] : A (cid:48) → B (cid:48) is [ ϑ ◦ ( f ⊗ f (cid:48) ) , E ⊗ E (cid:48) ] : A ⊗ A (cid:48) → B ⊗ B (cid:48) ;where ϑ is the canonical isomorphism ( B ⊗ E ) ⊗ ( B (cid:48) ⊗ E (cid:48) ) (cid:39) ( B ⊗ B (cid:48) ) ⊗ ( E ⊗ E (cid:48) ) in C . Proof.
Coherence isomorphisms ρ : A → B of C lift to Aux( C ) as [ ρ − ◦ β, I ] : A → B . For example,the symmetry γ : A ⊗ B → B ⊗ A in C becomes [ ρ − ◦ γ, I ] : A ⊗ B → B ⊗ A in Aux( C ). Composingcoherence isomorphisms [ ρ − ◦ β, I ] : A → B and [ ρ − ◦ φ, I ] : B → C in Aux( C ) is equivalent to firstcomposing them in C and then lifting to Aux( C ): B B ⊗ I C ⊗ I ( C ⊗ I ) ⊗ I C ⊗ ( I ⊗ I ) A C ⊗ IB C C ⊗ I β ρ − φ ⊗ id ρ − ⊗ id αβ φ ρ − id ⊗ ρ id ⊗ id[ ρ − ◦ φ ◦ β,I ][ ρ − ◦ φ,I ] ◦ [ ρ − ◦ β,I ]ENNETT AND STINESPRING, TOGETHER AT LAST 15 Similarly, tensoring coherences β and φ in C and then lifting is equivalent to first lifting themindividually and then tensoring them in Aux( C ) by B ⊗ B (cid:48) ( B ⊗ I ) ⊗ ( B (cid:48) ⊗ I ) ( B ⊗ B (cid:48) ) ⊗ ( I ⊗ I ) A ⊗ A (cid:48) ( B ⊗ B (cid:48) ) ⊗ IB ⊗ B (cid:48) ( B ⊗ B (cid:48) ) ⊗ I β ⊗ φ ρ − id ⊗ idid ⊗ ρβ ⊗ φ ρ − ⊗ ρ − ϑ [ ρ − ◦ β,I ] ⊗ [ ρ − ◦ φ,I ][ ρ − ◦ ( β ⊗ φ ) ,I ] In this way, coherence of the monoidal structure in Aux( C ) follows from that of C . It remains toshow is that the tensor product of morphisms is well-defined, and that it respects restrictions.Suppose that [ f, E ] ∼ [ g, G ] via mediators E h −→ E h ←− . . . h n ←−− G and intermediates f , . . . , f n − with f = f = · · · = f n − = g . Then ϑ ◦ ( f ⊗ f (cid:48) ) = f ⊗ f (cid:48) = f ⊗ f (cid:48) = g ⊗ f (cid:48) = g ⊗ f (cid:48) = ϑ ◦ ( g ⊗ f (cid:48) )since ϑ is an isomorphism (and so total). Also [ f ⊗ f (cid:48) , E ⊗ E (cid:48) ] ∼ [ g ⊗ f (cid:48) , G ⊗ E (cid:48) ]: A ⊗ A (cid:48) ( B ⊗ E ) ⊗ ( B (cid:48) ⊗ E (cid:48) ) ( B ⊗ E ) ⊗ ( B (cid:48) ⊗ E (cid:48) ) · · · ( B ⊗ B (cid:48) ) ⊗ ( G ⊗ E (cid:48) )( B ⊗ B (cid:48) ) ⊗ ( E ⊗ E (cid:48) ) ( B ⊗ B (cid:48) ) ⊗ ( E ⊗ E (cid:48) ) · · · ( B ⊗ B (cid:48) ) ⊗ ( G ⊗ E (cid:48) ) ϑ f ⊗ f (cid:48) ϑ ϑg ⊗ f (cid:48) id ⊗ ( h ⊗ id) id ⊗ ( h n ⊗ id)(id ⊗ h n ) ⊗ id(id ⊗ h ) ⊗ id (id ⊗ h ) ⊗ idid ⊗ ( h ⊗ id) f ⊗ f (cid:48) ··· Similarly [ f (cid:48) ⊗ f, E (cid:48) ⊗ E ] ∼ [ f (cid:48) ⊗ g, E (cid:48) ⊗ G ]. Finally,[ f, E ] ⊗ [ f (cid:48) , E (cid:48) ] = [ ρ − ◦ ϑ ◦ ( f ⊗ f (cid:48) ) , I ] = [ ρ − ◦ f ⊗ f (cid:48) , I ] = [ ρ − ◦ ( f ⊗ f (cid:48) ) , I ]and the diagram below commutes: A ⊗ A (cid:48) ( A ⊗ I ) ⊗ ( A (cid:48) ⊗ I ) ( A ⊗ A (cid:48) ) ⊗ ( I ⊗ I ) A ⊗ A (cid:48) ( A ⊗ A (cid:48) ) ⊗ IA ⊗ A (cid:48) ( A ⊗ A (cid:48) ) ⊗ I f ⊗ f (cid:48) ρ − f ⊗ f (cid:48) ρ − ⊗ ρ − ϑ id ⊗ ρ id ⊗ id[ f,E ] ⊗ [ f (cid:48) ,E (cid:48) ][ f,E ] ⊗ [ f (cid:48) ,E (cid:48) ] This shows that [ f, E ] ⊗ [ f (cid:48) , E (cid:48) ] = [ f, E ] ⊗ [ f (cid:48) , E (cid:48) ]. (cid:3) University of Edinburgh
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