Stinespring's construction as an adjunction

Given a representation of a unital $C^*$-algebra $\mathcal{A}$ on a Hilbert space $\mathcal{H}$, together with a bounded linear map $V:\mathcal{K}\to\mathcal{H}$ from some other Hilbert space, one obtains a completely positive map on $\mathcal{A}$ via restriction using the adjoint action associated to $V$. We show this restriction forms a natural transformation from a functor of $C^*$-algebra representations to a functor of completely positive maps. We exhibit Stinespring's construction as a left adjoint of this restriction. Our Stinespring adjunction provides a universal property associated to minimal Stinespring dilations and morphisms of Stinespring dilations. We use these results to prove the purification postulate for all finite-dimensional $C^*$-algebras.


Introduction and outline
Given a unital C * -algebra A and a state (a positive and unital linear functional) on A, the Gelfand-Naimark-Segal (GNS) construction produces a cyclic representation of A in a natural way compatible with the operation that produces a state from a representation together with a unit vector. This naturality has been expressed as an adjunction in a certain 2-category of functors from the (opposite of the) category of C * -algebras to the 2category of locally small categories 1 [21]. Stinespring's construction can be viewed as a generalization of the GNS construction by replacing states with operator-valued completely positive (OCP) maps. In the present work, we extend our GNS adjunction in Theorem 5.8 to include such OCP maps, showing that Stinespring's construction can also be viewed as an adjunction (1.2) in the same 2-category of functors. There are several subtle differences between the two adjunctions (1.1) and (1.2). The most notable one is that the category of OCP maps for a given C * -algebra is no longer discrete as it is for the GNS construction. This is a consequence of relaxing the unitality assumption on the commutes. When T is an isometry, commutativity of (2.10) implies commutativity of (2.8) and commutativity of (2.8) implies commutativity of (2.11). When T * is an isometry, these implications are reversed. In particular, when T is unitary, all three are equivalent. To see the first implication when T is an isometry, diagram (2. We will see that commutativity of (2.10) is too strong a requirement and commutativity of (2.11) is too weak a requirement for the purposes sought out in this work (these points will be explained in footnotes). No such simple comparison can be made for OCP maps. 2 These subtle points and the universal properties that will be discussed in this work would have been missed if we demanded T to be unitary.
morphism T : (C, ϕ) G G (C, ψ) of OCP maps consists of a linear map T : C G G C. Such a linear map must be of the form T (λ) = zλ for all λ ∈ C for some unique z ∈ C. If z = 0, since ϕ and ψ are linear, the diagram commutes for all a ∈ A if and only if ϕ = ψ. Notice that when T is an isometry, it is of the form T (λ) = e iθ λ for some θ ∈ [0, 2π). In this case, (2.11) and (2.10) (and hence (2.8) as well) are equivalent because T is also unitary.
Example 2.14. Fix m, p ∈ N and define the tracial map where tr denotes the standard (un-normalized) trace. A quick computation shows τ is unital. To see that it is CP, first note that since the trace is positive, the trace is CP because every positive map into C is automatically CP (cf. Theorem 3 in Stinespring [29]). Second, note that τ equals the composite of CPU maps (2.16) where the second map is the unique unital linear map sending 1 to 1 p . Therefore, τ is CPU by Example 2.4 and Lemma 2.6. Given q ∈ N, let σ : M m (C) G G M q (C) denote the tracial map for these dimensions and let T : C p G G C q be any linear transformation. Then (C p , τ ) T − → (C q , σ) is a morphism of OCP maps. (2.18) Then ϕ and ψ are operator states and T : (C, ϕ) G G (C 2 , ψ) defines a morphism of operator states by Example 2.14. Note that T * : C 2 G G C is given by  Then ϕ and ψ are operator states (cf. Examples 2.14 and 2.4). Note that T * : C 2 G G C is given by and ψ a b c d = 1 2 a + d b + c b + c a + d . (2.25) Therefore, although (2.11) holds, show that condition (2.8) fails.
The following proposition provides the general structure of morphisms of operator states. Proof. you found me! (⇒) For the forward direction, set L 1 := T (K). Since T is an isometry, L 1 is a closed subspace of L. Set L 2 := L ⊥ 1 , the orthogonal complement of L 1 inside L, U := π 1 T, and ψ j (a) := π j ψ(a)i j for all a ∈ A, where π j : L G G L j denotes the projection onto the j-th factor and i j : L j G G L denotes the inclusion of the j-th factor. Since T is an isometry, U is unitary. Since T is a morphism of operator states, ψ(a)L 1 ⊆ L 1 for all a ∈ A. Furthermore, since ψ is positive, ψ(a) * = ψ(a * ) so that T * ψ(a * ) = ϕ(a * )T * for all a ∈ A upon taking the adjoint of (2.8). Since * is an involution on A, this is equivalent to T * ψ(a) = ϕ(a)T * for all a ∈ A. Thus, ψ(a)L 2 ⊆ L 2 . These two facts imply ψ(a) = ψ 1 (a) ⊕ ψ 2 (a) for all a ∈ A. Finally, U ϕ(a)U * = π 1 T ϕ(a)T * i 1 by definition of U = π 1 T T * ψ(a)T T * i 1 by (2.8) and Remark 2.9 = π 1 ψ(a)i 1 since T is an isometry onto L 1 = ψ 1 (a) by definition of ψ j (2.28) for all a ∈ A.
(⇐) To see the reverse direction, set T := i 1 U . Then T is a morphism of operator states since the diagram commutes for all a ∈ A.
OCP maps and operator states together with their morphisms form categories.
commutes for all a ∈ A. Second, the diagrams both commute. When V, W, T, and L are isometries, then (T, L) is said to be a morphism of preserving anchored representations.

Remark 3.4.
For preserving anchored representations, commutativity of the right diagram in (3.3) holds whenever T is unitary and the left diagram in (3.3) commutes. To see this, the left diagram says W T = LV. Taking the adjoint of this condition gives T * W * = V * L * . Applying T on the left and L on the right gives W * L = T V * , which is the diagram on the right in (3.3).
When the bounded linear maps in these definitions are isometries, the following lemma shows the isometry T : K G G L is redundant and can be constructed from L : H G G I.
if and only if there exists a (necessarily unique) isometry T : K G G L such that the diagrams in (3.3) commute.
In (3.6), W (L) ⊥ stands for the orthogonal complement of W (L) ⊆ I and likewise for V (K) ⊥ ⊆ H.
Proof of Lemma 3.5. First note that since V and W are isometries, their images are closed. Hence, (3.7) (⇒) Assume L satisfies (3.6). Then the diagram commutes by construction. In this diagram, π W (L) and i V (K) denote projection and inclusion maps, respectively. Note that π W (L) Li V (K) is an isometry because L V (K) ⊆ W (L) by (3.7). Setting T := W * LV , it follows from (3.8) that T is the composite of the two unitary maps π V (K) V and (π W (L) W ) * and the isometry π W (L) Li V (K) . Therefore, T is an isometry. Finally, the diagrams in (3.3) commute because they are given by respectively, where P W (L) is the projection onto W (L) inside I, and similarly for P V (K) .
(⇐) Assume an isometry T satisfying (3.3) exists. Commutativity of the diagram on the left requires L V (K) ⊆ W (L), while commutativity of the diagram on the right requires Notation 3.10. A morphism of preserving anchored representations will be denoted by the pair (T, L) : (K, H, π, V ) G G (L, I, ρ, W ), even though the isometry T is uniquely determined by L as illustrated in Lemma 3.5. This is because T need not be uniquely determined by L in the general case of anchored representations and because we will use T to relate anchored representations to OCP maps in this work.
Accepted in Compositionality on 2019-07-24. Click on the title to verify. Example 3.11. In the special case K = C, an anchored representation of A consists of a *representation π : A G G B(H) and a linear map V : C G G H. Such a map is uniquely characterized by the vector Ω := V (1) ∈ H, which is a unit vector if and only if V is an isometry. Hence, a preserving anchored representation of the form (C, H, π, V ) is equivalent to a pointed representation as introduced in Definition 5.1 in [21]. If (C, I, ρ, W ) is another preserving anchored representation, then a morphism (C, H, π, V ) (T,L) −−−→ (C, I, ρ, W ) of preserving anchored representations consists of isometries T : C G G C and L : H G G I satisfying (3.2) and (3.3). T must be of the form T (λ) = λe iθ for all λ ∈ C for some θ ∈ [0, 2π). L is an intertwiner of representations by (3.2). Let Ξ := W (1). The left diagram in (3.3) entails L(Ω) = e iθ Ξ. Since V * = Ω, · and W * = Ξ, · , the right diagram in (3.3) entails e iθ Ω, · = Ξ, L( · ) as linear functionals on H. However, this second condition is implied by the first one in (3.3) by Remark 3.4. Hence, this reproduces the morphisms of pointed representations in [21] up to a phase.
defines a functor. Furthermore, it restricts to a functor PAnRep(A)

Lemma 3.15. The assignment
defines a functor. The same is true for

The restriction natural transformation
The follow proposition illustrates how to construct an OCP map from an anchored representation. Here, (Ad V * • π)(a) := V * π(a)V for all a ∈ A (cf. Example 2.4).
Proof of Proposition 4.1. Let (K, H, π, V ) be an anchored representation of A. Then Ad V * • π is an OCP map because it is the composite of the CP map Ad V * and the * -homomorphism π. Since Ad V * is unital if and only if V is an isometry, Ad V * • π is an operator state when (K, H, π, V ) is a preserving anchored representation. Let (K, H, π, V ) (T,L) −−−→ (L, I, ρ, W ) be a morphism of anchored representations. In order for (K, Ad V * • π) T − → (L, Ad W * • ρ) to be a morphism of OCP maps, the diagram must commute for all a ∈ A. Expanding out the definition of the adjoint action map provides the diagram This diagram commutes because the diagram

OCP(A) OCP(A )
of functors commutes (on the nose). A similar statement holds for the subcategories obtained from PAnRep and OpSt. In the construction of a left adjoint to rest, some preliminary facts will be needed.
is a positive linear functional.
Proof. Suppose A ∈ M n (A) is positive. Then, because ϕ is completely positive, ϕ n (A) ≥ 0. Hence, Linearity of s ϕ, v follows from linearity of ϕ n and linearity of the inner product in the right variable.
commutes. Here, X/N is the quotient space of X modulo N and X X/N is the quotient map.

commutes.
Proof. For the first fact, see Theorem 1.41 and Exercise 9 in Chapter 1 of Rudin [25]. The second fact is a consequence of the first.  iii. For a fixed * -homomorphism f : iv. Prove that Stine is an oplax-natural transformation (cf. Definition B.1).
v. For a fixed C * -algebra A, construct the appropriate natural transformation m A : In all of the above steps, justifications for reducing to operator states and preserving anchored representations will be provided. In what follows, if a proof for any claim is not supplied, it is because the justification is analogous to the standard GNS construction arguments or it follows easily from the definitions. The reader is referred to [21] for more details.
i. The construction of an anchored representation from an OCP map will be Stinespring's construction (cf. the proof of sufficiency of Theorem 1 in Stinespring [29]). Let A be a C * -algebra and let (K, ϕ) be an OCP map on A. Recall, this means ϕ : A ⊗ K denote the vector space tensor product of A with K. In particular, elements of A ⊗ K are finite sums of tensor products of vectors in A and vectors in K (in fact, all sums that follow are finite). The function is conjugate bilinear in the first A × K factor and bilinear in the second A × K factor. Hence, by the universal property of the algebraic tensor product (cf. Chapter IV Section 5 in Hungerford [17]), the assignment is well-defined, conjugate linear in the first variable, and linear in the second variable. Furthermore, · , · ϕ satisfies 5 Since the matrix in M n (A) is positive for all a 1 , . . . , a n ∈ A, This is the Cauchy-Schwarz inequality for such sesquilinear forms (cf. Construction 3.1 in [21]). Thus, · , · ϕ is a sesquilinear form whose associated seminorm endows A ⊗ K with the structure of a topological vector space. In general, · , · ϕ is not positive semi-definite. Hence, set N ϕ := ζ ∈ A ⊗ K : ζ, ζ ϕ = 0 (5. 16) to be its null-space. From the Cauchy-Schwarz inequality (5.15), it follows that Using this, one can show that N ϕ is a closed vector subspace of A ⊗ K (it is closed since it is defined as the inverse image of {0} under a continuous map).
By the universal property of the tensor product, for each a ∈ A, the map is a well-defined linear transformation. If we write End(V ) for the algebra of linear transformations from a vector space V to itself, then π ϕ : A G G End(A⊗K) defines a representation of the algebra A on the vector space A⊗K. Furthermore, for each a ∈ A and ξ = n i=1 a i ⊗v i ∈ A⊗K, In this calculation, v := (v 1 , . . . , v n ) ∈ K ⊕ · · · ⊕ K and the norm a of a is the one from the C * -algebra A. The third line in (5.19) follows from Lemma 5.1 and the inequality |ω(y * xy)| ≤ x ω(y * y) for all x, y in a C * -algebra and ω a positive linear functional on that C * -algebra (see Proposition 2.1.5. part (ii) in Dixmier [8] for a proof of this inequality). In this case, this inequality is applied to the positive linear functional ω : This is because every injective *homomorphism of C * -algebras is an isometry (cf. Propositions 1.3.7 and 1.8.1 in Dixmier [8]). In this case, the * -homomorphism is given by the function A G G M n (A) sending a ∈ A to diag(a, . . . , a). The last line of (5.19) follows from the C * -identity for C * -algebras and the definitions of ξ, ξ ϕ and s ϕ, v . Thus, (5.19) shows that π ϕ (a) is bounded/continuous. If we write B(V ) for the algebra of bounded operators on a seminormed vector space V , then π ϕ (a) ∈ B(A ⊗ K) for all a ∈ A.
Furthermore, (5.19) shows that N ϕ is an invariant subspace under the π ϕ action, meaning π ϕ (a)ζ ∈ N ϕ for all ζ ∈ N ϕ and a ∈ A. Therefore, the quotient space By (5.17), the sesquilinear form · , · ϕ descends to a well-defined inner product The fact that · , · ϕ is positive definite follows from (5.14) and the definition of N ϕ in (5.16). Let denote the completion of this topological vector space with respect to the inner product · , · ϕ . Since π ϕ (a) is a bounded linear operator on (A ⊗ K)/N ϕ , it extends uniquely to a bounded linear operator, also denoted by π ϕ (a), on H ϕ . Furthermore, since π ϕ (a) ∈ B(H ϕ ), it has an adjoint π ϕ (a) * . This adjoint satisfies Then, for all w ∈ K. The last inequality in (5.25) follows from Cauchy-Schwarz for · , · K and positivity of ϕ. This proves that V ϕ is bounded. Note that if ϕ is unital, the inequality in (5.25) becomes an equality, which proves V ϕ is an isometry.
This concludes Stinespring's construction of an anchored representation from an OCP map, i.e. the functor be the corresponding Stinespring anchored representations from the first step. Set

which itself extends uniquely to a bounded linear map
Note that when T is an isometry, L T is also an isometry. Commutativity of follow directly from the definitions. However, commutativity of the last of the conditions in Definition 2.7, requires an argument. First, to find the formula Commutativity of the diagram (5.30) then follows from 6 This concludes the definition of the assignment on morphisms of OCP maps on A. Note that Stine A (id K ) equals id K , id Hϕ for any OCP map (K, ϕ). Furthermore, for a composable pair (K, ϕ) iii. Let f : A G G A be a * -homomorphism of C * -algebras. Two diagrams associated with the constructions preceding this are given by and similarly for OpSt and AnRep. The diagram of functors on the left commutes (on the nose) by Lemma 4.6. However, the diagram on the right does not (this is analogous to what happens in the GNS construction-cf. Construction 3.3 and diagram (3.21) in [21]). 7 Nevertheless, there is a natural transformation defined as follows. Given an OCP map (K, ϕ) on A, applying OCP f followed by Stine A provides the Stinespring anchored representation (K, The morphism Stine f (K, ϕ) from the first to the latter is given by In fact, this calculation shows L f : iv. Oplax-naturality of Stine holds because Stine id A is the identity natural transformation for every C * -algebra A and the two natural transformations (after composition in the diagram on Accepted in Compositionality on 2019-07-24. Click on the title to verify. the right) v. Fix a C * -algebra A and let (K, H, π, V ) be an anchored representation of A. Applying the functor rest A followed by Stine A to this representation gives where the second map is the canonical action of B(H) on H.
Thus, m π,V defines an isometry m π,V : H Ad V * •π G G H by Lemma 5.4. Note that m π,V is an isometry even though (K, H, π, V ) need not be a preserving anchored representation. Commutativity of the diagrams of anchored representations follow directly from the definitions (for the last diagram, apply (5.32) to ϕ := Ad V * • π). Set m A to be the assignment be a morphism of anchored representations. Then, the diagram commutes by conditions (3.2) and (3.3) in the definition of a morphism of anchored representations.
vi. To see that the assignment sending a C * -algebra A to m A defines a modification 8

AnRep(A) AnRep(A)
of morphisms of anchored representations of A must commute. This follows directly from the definitions. Hence, m is a modification, which also restricts to a modification when working with OpSt and PAnRep by (5.44).
vii. To see that commutes, first fix a C * -algebra A. Commutativity in (5.53) requires that

OCP(A)
must commute. On objects, this translates to ϕ = Ad V * ϕ • π ϕ for every OCP map (K, ϕ) on A, which follows from the definitions of V ϕ and π ϕ . Since a morphism (K, ϕ) Commutativity of (5.53) also requires that for every * -homomorphism f : A G G A,

OCP(A) OCP(A)
i.e. to every OCP map (K, ϕ), the diagram of morphisms of OCP maps on A must commute, which it clearly does. Commutativity of (5.53) with OpSt and PAnRep follows from this as well.
viii. By the final remark in the appendix of [21], it suffices to prove

AnRep(A)
for each object A of C * -Alg op . The equality in (5.57) follows from the equality of morphisms of OCP maps from (K, Ad V * • π) to itself for every anchored representation (K, H, π, V ) of A. To see the equality in (5.58), consider an OCP map (K, ϕ).
In order for (5.58) to hold, it should be the case that is dense in H.

Remark 6.3.
In terms of the functors and natural transformations introduced, (K, H, π, V ) is a Stinespring representation of (K, ϕ) on A if and only if rest A (K, H, π, V ) = (K, ϕ). Minimality will be addressed in Corollary 6.9.
The following corollaries explain the meaning of Theorem 5.8 more concretely. Proof. Since (Stine A , rest A , id OpSt(A) , m A ) is an adjunction, existence and uniqueness follows from the universal property of adjunctions in part i of Lemma B.28. In the notation of that lemma, c = (K, ϕ), d = (L, I, ρ, W ), and g = T. This unique morphism is given by Recall, m ρ,W L T : H ϕ G G I is uniquely determined by the assignment When T is a morphism of operator states, both T and m ρ,W L T are isometries. Hence, T defines a morphism of preserving anchored representations.
The following is a special case of Corollary 6.4. It will be used to provide the relationship to minimal Stinespring representations in Corollary 6.9. Proof. This follows from Corollary 6.4 for T = id K . The fact that T consists of two isometries follows from the fact that m π,V is an isometry for any anchored representation (K, H, π, V ) by (5.44). We now state the relationship between our adjunction and minimal Stinespring representations. Proof. you found me! (⇒) By Remark 6.3, (K, H ϕ , π ϕ , V ϕ ) is a Stinespring representation due to part vii in the proof of Theorem 5.8. By the construction in part i, Theorem 5.8 entails the following with regards to morphisms and functoriality.
Corollary 6.11. Let (K, ϕ) be an OCP map on a C * -algebra A and let f : In fact, Furthermore, the assignment f → Stine f is functorial in the sense that

− → A of morphisms of C * -algebras and for all OCP maps
Proof. Since (K, H ϕ , π ϕ • f, V ϕ ) is a Stinespring representation of (K, ϕ • f ), Corollary 6.7 says there is a unique morphism from (K, and the source of this morphism is as we have, our results do not require our maps to be normal nor do we require our domain A to be a von Neumann algebra. As a result, we provide a universal property for minimal Stinespring dilations for all C * -algebras. Furthermore, our universal property highlights the relationship between morphisms of OCP maps and intertwiners of representations. This will have important implications, which will be discussed in Section 7. There is also a close connection between our work and that of categorical quantum mechanics [1], [27]. Proposition 6.20 below shows that OCP(A) and AnRep(A) are dagger categories. This fact, together with our Stinespring adjunction, has some interesting consequences such as Theorem 6.29, which states that there exists a unique minimal morphism of anchored representations between any two Stinespring representations of the same OCP map. *  -category (also called a dagger category) is a category C together with a functor * :

Definition 6.19. A
ii. (f * ) * = f for all morphisms f in C.
Explicitly, functoriality of * says id * x = id x for all objects x in C and (g • f ) * = f * • g * for all composable pairs of morphisms f and g in C. Note that OpSt(A) and PAnRep(A) are not * -categories with the same * operation because the adjoint of an isometry need not be an isometry. The adjoint of an isometry is, however, a co-isometry, and hence a partial isometry (cf. Chapter 15 in Halmos [14]). iff L * is an isometry. It is called a partial isometry iff L is an isometry when restricted to ker(L) ⊥ ⊆ H.
The following lemma includes several properties of partial isometries that will be used in this work. (a) If L is a partial isometry, it is an isometry when restricted to its initial space.
(c) The composite of a co-isometry followed by its adjoint is a partial isometry.
(d) Let H be another Hilbert space (whose dimension is at least 1). Then id H⊗ L : H⊗I G G H⊗J is a partial isometry if and only if L is a partial isometry. Here,⊗ denotes the completed tensor product (cf. Section I.2.3 in Dixmier [9]).
(e) If L : I G G J is a partial isometry, then there exists either an isometry or a co-isometry U : I G G J that agrees with L on its initial space. Proof. Most of these are adequately covered in Halmos [14], Halmos-McLaughlin [15], and Hines-Braunstein [16], with the exception of the forward implication in (d). Suppose id H⊗ L : H⊗I G G H⊗J is a partial isometry. Then id H⊗ L = (id H⊗ L)(id H⊗ L) * (id H⊗ L) = id H⊗ (LL * L) (6.25) by Corollary 3 in Section 127 of [14] and the property of adjoints with respect to the completed tensor product. Hence, id H⊗ (L − LL * L) = 0, which holds if and only if L = LL * L. Thus, L is a partial isometry by this same corollary.
Remark 6.26. The composite of partial isometries need not be a partial isometry (cf. Section 9.3 in Hines-Braunstein [16]).
Partial isometries have a natural partial ordering as described by Halmos and McLaughlin [15]. This partial ordering is slightly generalized below to include intertwiners of representations. In this case, the notation L M will be used when the representations are understood from the context. Whenever the notation or is used, it will be understood that the operators being compared are partial isometries.
It is straightforward to check the following. The relationship between and morphisms of anchored representations will be described in Lemma 6.34. But first, we provide a useful result that compares any two Stinespring representations of the same OCP map. The third condition is a uniqueness condition guaranteeing there exists a unique minimal partial isometry (minimal in the sense of the partial order on partial isometries) L for which (id K , L) is a morphism of anchored representations. This is justified in Lemma 6.34.
Proof of Theorem 6.29. By Corollary 6.7, we have a unique pair of morphisms of anchored representations consisting of isometries of the form  Remark 6.38. Although maximal elements for the partial order on partial isometries are either isometries or co-isometries (cf. [15]), this is not true in general of maximal elements for the partial order on intertwining partial isometries. This point will be addressed in Remark 7.41.

Examples and applications
The purification postulate has been used by Chiribella, D'Ariano, and Perinotti to classify finitedimensional quantum theories among all operational probabilistic theories (OPTs) [7], [6]. We do not need to review OPTs here, but will instead provide a definition of a purification of a process, our formulation of the purification postulate, and the standard finite-dimensional purification postulate of [7]. Our version of the purification postulate isolates some key assumptions made by [7] that are implicit from the tensor network (diagrammatic) perspective. We prove our purification postulate using our Stinespring adjunction and show how it reduces to the standard finite-dimensional one. We will first use our Stinespring adjunction to reproduce a Gelfand-Naimark-Segal (GNS) adjunction for states [21]. Applying Stine A to ω provides an anchored representation (C, H ω , π ω , V ω ). In this case, A⊗C ∼ = A so that N ω ∼ = {a ∈ A : ω(a * a) = 0} under this isomorphism. This agrees with the null-space from the usual GNS construction. Hence, the completion H ω := A/N ω and the representation π ω : A G G B(H ω ) agree with the usual GNS Hilbert space and representation. By Example 3.11, V ω : C G G H ω produces a unit vector Ω ω := V ω (1) ∈ H ω and V * ω = Ω ω , · : H ω G G C. Hence, can be identified with its evaluation at 1 and is equivalently described by the state Ω ω , · Ω ω : B(H ω ) G G C. Applying rest A entails ω = Ω ω , π( · )Ω ω , so that ω has been represented by a pure state. If (C, H, π, V ) is another anchored representation of ω, set Ω := V (1) ∈ H. By Corollary 6.7, there is a unique morphism of anchored representations of the H, π, V ), where m π,V is given by (7.2) which agrees with the modification from Construction 5.21 in [21]. The universal property of Stinespring's adjunction thus reproduces the minimality of the GNS construction in the sense that it reproduces the smallest cyclic representation of A on which ω can be realized as a pure state.  Here, V Ω : C G G H is the map that sends λ ∈ C to λΩ. It is not difficult to show that Σ A is a functor. Also, set

States(A)
whereω : A G G B(C) is defined by A a →ω(a) = ω(a) · , i.e. multiplication by ω(a) on the Hilbert space C. Since States(A) has only identity morphisms, this specifies the functor Υ A . Examples 2.12 and 3.11 show that the functors Σ A and Υ A are faithful but not full. The only reason these functors are not full is that the categories OpSt(A) and PAnRep(A) contain more data in their morphisms. However, the only added information for a morphism of states and pointed representations is a phase factor, which is a symmetry that can safely be ignored in the discussion of the GNS construction.
Given a * -homomorphism f : A G G A, the equalities also hold. Furthermore, in the diagram 10 although the equality rest Υ = Σ rest (7.8) holds, there is only an invertible modification since the two composites are not exactly equal but are canonically isomorphic. Indeed, for a fixed C * -algebra A, the resulting natural isomorphism

PAnRep(A) OpSt(A)
is defined by its evaluation on a state ω : A G G C by the morphism −−−−−→ C, Hω, πω, Vω , (7.11) where L ω : H ω G G Hω is defined as the unique extension of Similar calculations to the above show that this map is bounded and extends to a unitary intertwiner. Hence, (id C , L ω ) defines an isomorphism in the category PAnRep(A). The appropriate diagram also commutes when one considers a * -homomorphism f : This tells us our Stinespring adjunction reduces to the GNS adjunction by restricting to the images of Σ and Υ. For example, if ω : A G G C is a state, one can construct a pointed representation of it via GNS and view that pointed representation as a preserving anchored representation. Similarly, one can view ω as an operator state and apply Stinespring's construction to obtain another preserving anchored representation. The invertible modification in (7.9) says that these two preserving anchored representations are canonically isomorphic. Working out the details of Stinespring's adjunction in this present work has highlighted the importance of not restricting morphisms to be isometries so that our universal property is more robust and improves on our previous GNS adjunction in several respects. Hence, a purification of ω will be written as a triple (H, π, Ω). In other words, a purification of a state is Stinespring representation of that state. In other words, a purification of an OCP map is a Stinespring representation of that OCP map.
The following version of the purification postulate might seem unfamiliar, but we show that it is equivalent to the usual purification postulate (from Section VII.A. and VII.B. in [7]) when the algebras are finite-dimensional matrix algebras. This equivalence is worked out in detail in Lemma 7.18 and Theorem 7.30. Stinespring's theorem guarantees the existence of purifications. The existence of the unitary intertwiner in these postulates is referred to as the essential uniqueness of purifications. The purification postulate for processes implies the one for states by setting K = C. We will prove the purification postulate for processes on finite-dimensional matrix algebras in Theorem 7.30 and finite-dimensional C * -algebras in Corollary 7.38 after a few lemmas. Our version of the purification postulate is formulated without using traces or tensor products since these may be absent or ambiguous for general C * -algebras. This is partially achieved by using completely positive unital maps instead of the more common completely positive trace-preserving maps. In the finite-dimensional setting, these are equivalent and correspond to the Heisenberg and Schrödinger pictures, respectively. Completely positive unital maps are used to map observables to observables while their duals (adjoints with respect to the Hilbert-Schmidt inner product), completely positive trace-preserving maps, are used to map density matrices to density matrices. However, the category of C * -algebras does not have duals and therefore does not have symmetric purifications as defined in [26]. Since states supersede density matrices in the infinite-dimensional setting and the notion of a trace is not always available, it is sometimes more convenient to work within the Heisenberg picture. In other words, there exists an isomorphism (K, H, π, V ) Proof. This follows from a general fact about unital * -homomorphisms between finite-dimensional matrix algebras (cf. Section 1.1.2 in Fillmore [11]) and our definitions. The map C ! G G A is the unique unital linear map from C to A. Note that the algebras here are finite dimensional so the tensor product is the standard one. By choosing this convention, the direction of time is up. The direction of time is consistent with [26], while the direction of composition is opposite, because the dual maps on density matrices are used in [26]. due to Lemma 7.18. To prove the purification postulate for processes, we need to recall a few standard facts about the commutant (cf. Chapters 1 and 2 in Dixmier [9] and Chapter 4 in Fillmore [11]). Definition 7.24. Let S ⊆ A be a subset of a C * -algebra A. The commutant of S inside A is the unital algebra S := {a ∈ A : as = sa ∀ s ∈ S}. (7.25) Since the commutant depends on the embedding algebra, S will often be written as S ⊆ A. 11 Remark 7.26. If a subset S ⊆ A is * -closed (meaning a ∈ S implies a * ∈ S), then S is a unital * -algebra. In fact, S ⊆ A is a C * -subalgebra of A since S = s∈S {s} is the intersection of the kernels of the commutators [s, · ] : A G G A, which are all closed (cf. Chapter 2 in Topping [30]).
(if any of the integers n 1 , . . . , n t , c 1 , . . . , c t , are zero, terms corresponding to them are excluded from the above matrix). Then, the commutant of is used for 'internal' direct sum to distinguish it from the 'external' direct sum .
for all A ∈ M n (C) must be of the form 1 n ⊗ P for some partial isometry P : C q G G C p . Hence, one obtains a unique partial isometry P : C q G G C p such that RL * S * = 1 n ⊗ P . All intertwining extensions of this partial isometry must therefore also be of this form. Therefore, there exists either an isometry or a co-isometry M : C q G G C p satisfying 1 n ⊗P 1 n ⊗M by Lemma 6.24. Then U := S * (1 n ⊗M ) * R is the required partial isometry. Note that if dim H = dim I is assumed, then the representations are automatically unitarily equivalent since maximal partial isometries between finite-dimensional Hilbert spaces of equal dimension are unitary.
Our formulation of the purification postulate is also valid for arbitrary finite-dimensional C *algebras. Proof. Existence follows from Stinespring's theorem as before. What follows is a proof of the essential uniqueness of purifications. By the discussions preceding this, it suffices to consider the case K := C k and A := t j=1 M nj (C), (7.39) where t ∈ N and n j ∈ N for all j ∈ {1, . . . , t}. Given an OCP map ϕ : A G G B(K), it also suffices to consider two Stinespring representations of the form (K, C m , π, V ) and (K, C m , π, W ), where m ∈ N and the representation π : where the c 1 , . . . , c t are non-negative integers such that m = t j=1 n j c j . By similar arguments to those implemented in the proof of Theorem 7.30, there exist partial isometries (for the non-zero c j ) is a morphism of anchored representations from (K, C m , π, V ) to (K, C m , π, W ). These can be extended to unitaries U j : C cj G G C cj by finite dimensionality. Hence, id K , U := t j=1 (1 nj ⊗ U j ) is an isomorphism of anchored representations.
Remark 7.41. If one drops the assumptions that the finite-dimensional representations (H, π) and (I, ρ) are unitarily equivalent in Postulate 7.17 and Corollary 7.38, then U = t j=1 (1 nj ⊗ U j ) from the proof of Corollary 7.38 is replaced by an internal block sum of partial isometries that have been extended to isometries or co-isometries. In particular, even though U is maximal with respect to , it need not be an isometry nor a co-isometry. In more detail, let (K, C m , π, V ) and (K, C n , ρ, W ) be the two Stinespring representations, where n j c j , and n = t j=1 n j d j . (7.42) One can show L (from Theorem 6.29) must be of the form t j=1 (1 nj ⊗P j ). In fact, all intertwining extensions of L must also be of this form. By extending such intertwining partial isometries, one obtains an isometry or a co-isometry U j :

B 2-categorical preliminaries
We briefly recall the definitions of oplax-natural transformations and modifications. In addition, we include the universal property associated with adjunctions because it is used in explaining the Stinespring adjunction more concretely. For details on 2-categories and their pasting diagrams, we refer the reader to Bénabou's original work [4] as well as Kelly and Street's review [18]. For a more introductory take emphasizing string diagrams, see [22]. For other details on oplax-natural transformations and modifications, we refer the reader to Section 7.5 in Borceux [5].
Definition B.1. Let C and D be two (strict) 2-categories and let F, G : C G G D be two (strict) functors. An oplax-natural transformation ρ from F to G, written as ρ : F ⇒ G, consists of i. a function ρ : C 0 G G D 1 assigning a 1-morphism in D to each object x in C in the following ii. and a function ρ : C 1 G G D 2 assigning a 2-morphism in D to each 1-morphism y α ← − x in C in the following manner These data must satisfy the following conditions: . holds.
Remark B.8. We use the prefix "oplax" because for a lax-natural transformation (cf. Definition 7.5.2 in Borceux [5]) consists of a function m : C 0 G G D 2 assigning a 2-morphism in D to each object x in C in the following manner This assignment must satisfy the condition that for every 1-morphism y Accepted in Compositionality on 2019-07-24. Click on the title to verify.
Remark B.13. If one has a modification between lax-natural transformations, the diagram in (B.12) is modified appropriately. This will be used in Proposition B.31.
The composition of oplax-natural transformations and modifications are not changed as a result of these alterations to the usual definitions.
Definition B.14. The vertical composite of oplax-natural transformations is denoted using vertical concatenation as in and is defined by the assignments for each object x in C and (B.26) Conditions (B.25) and (B.26) are known as the zig-zag identities. An adjunction as above is typically written as a quadruple (f, g, η, ) and we say f is left adjoint to g and write f g. However, the notation The usual notion of an adjunction is one where the 2-category is that of categories, functors, and natural transformations. One may express adjunctions in terms of a universal property in this case.  Proof. This is an equivalent definition of an adjunction (cf. Chapter IV Section 1 in Mac Lane [19]).   Remark B.34. That ρ be a pseudo-natural transformation and not just a lax-or oplax-natural transformation is explicitly used in the proof.
Proof of Proposition B.31. you found me! (⇒) The forward direction was proved in the final Remark in [21]. (⇐) For the reverse direction, σ γ : σ y • F (γ) ⇒ G(γ) • σ x must be constructed for each morphism y γ ← − x in C. It is cumbersome to do this using globular diagrams, so we implement string diagrams to simplify the proof (see [22] for an introduction to string diagrams). By convention, string diagrams will be read from top to bottom and from right to left. Define σ γ by where ρ γ is the vertical inverse of ρ γ . To verify the oplax-naturality of σ, consider a composable   proves condition (B.7) for σ and concludes the proof that σ is an oplax-natural transformation. It remains to show that : id F σ ρ and η : ρ σ id G are modifications. For this, fix a 1-morphism where the last equality follows from the fact that ρ is the vertical inverse of ρ. This proves (B.12) for η, and therefore shows η is a modification. A similar proof shows is a modification. Finally, σ is unique up to canonical isomorphism by the uniqueness of adjoints in 2-categories (cf. Lemma A.4 in [21]).
Remark B.40. Proposition B.31 seems to be a useful fact for adjunctions in 2-categories of functors. It offers a slightly shorter proof of Theorem 5.8. One merely has to define the functors Stine A and rest A and define the natural transformation m A . Then one has to show Stine rest = id OCP and prove the zig-zag identities for (Stine A , rest A , id, m A ), the last of which were essentially tautologies. Proving oplax-naturality of Stine and that m is a modification is not necessary thanks to Proposition B.31.