Quantum graphs as quantum relations

The"noncommutative graphs"which arise in quantum error correction are a special case of the quantum relations introduced in [N. Weaver, Quantum relations, Mem. Amer. Math. Soc. 215 (2012), v-vi, 81-140]. We use this perspective to interpret the Knill-Laflamme error-correction conditions [E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55 (1997), 900-911] in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke's noncommutative graph homomorphisms [D. Stahlke, Quantum source-channel coding and non-commutative graph theory, arXiv:1405.5254] and Duan, Severini, and Winter's noncommutative bipartite graphs [R. Duan, S. Severini, and A. Winter, Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovasz number, IEEE Trans. Inform. Theory 59 (2013), 1164-1174], and to realize the noncommutative confusability graph associated to a quantum channel as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the"mixed"setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.


Quantum graphs
"Quantum" or "noncommutative" graphs appear in the setting of quantum error correction [2,7,8]. The usual construction starts with a quantum channel, which in the Schrodinger picture is modelled by a completely positive trace preserving (CPTP) map Φ : M m → M n . Here M m is the set of m × m complex matrices and a CPTP map is concretely realized as a linear map of the form where ρ ∈ M m and the Kraus matrices K i are a finite family of n × m matrices satisfying K * i K i = I m (the m × m identity matrix). Positive matrices with unit trace represent mixed states, and pure states appear as the special case of matrices of the form |α α| for |α a unit vector in C m . Thus quantum channels transform mixed states to mixed states, and in error correction problems one is interested in determining which input states can be distinguished with certainty after passing through the channel. The condition that the images of two pure states |α and |β after transmission must be orthogonal can be expressed as α|B|β = 0 for all B ∈ V Φ = span{K * i K j } (see, e.g., [2]).
The space V Φ ⊆ M m is an operator system -a linear subspace of M m which is stable under the adjoint operation and contains the identity matrix (since we have assumed that K * i K i = I m ). In the analogous classical setting one would be dealing with a finite set of possible input states and one could create a graph by placing an edge between any pair of input states which might, after transmission through a noisy channel, be received as the same output state. This classical confusability graph is relevant to classical zero-error communication in something like the way Date: Dec. 27, 2015. Partially supported by NSF grant DMS-1067726. that the operator system V Φ is relevant to zero-error communication in the quantum setting. This led Duan, Severini, and Winter to term V Φ a noncommutative confusability graph [2].
Going further, since any operator system can arise in the above manner from a quantum channel, they suggested that operator systems generally could be considered "noncommutative graphs". This daring proposal was supported by the fact that they were able to define a "quantum Lovász ϑ function" for any operator system, in analogy to the classical Lovász ϑ function of a graph.

Quantum relations on M m
At around the same time, the notion of a "quantum relation" was introduced in [9]. This notion gives rise to natural definitions of such things as "quantum equivalence relations" and "quantum partial orders", and it is also the basis of the "quantum metrics" and "quantum uniform structures" which were studied in [4], an earlier project from which the notion of a quantum relation was extracted.
The identification of operator systems as "quantum graphs" was also made in [9], but not pursued further there. 1 However, it is worth investigating this connection, as under the quantum relations point of view basic aspects of the theory of zeroerror quantum communication become conceptually transparent.
The core idea is that an operator space -a linear subspace of M m -can be thought of as a quantum analog of a relation on a finite set. (In infinite dimensions, this becomes a weak* closed operator space, but I will stick to the finite-dimensional setting in this paper.) Classically, a relation on a set X is a subset R of X × X, and we write xRy to indicate that the pair (x, y) belongs to the relation. The relation R is said to be • reflexive if xRx, for all x ∈ X • symmetric if xRy implies yRx, for all x, y ∈ X • antisymmetric if xRy and yRx imply x = y, for all x, y ∈ X • transitive if xRy and yRz imply xRz, for all x, y, z ∈ X.
This can be expressed more algebraically by letting ∆ be the diagonal relation ∆ = {(x, x) : x ∈ X}, letting R t be the transpose relation R t = {(y, x) : (x, y) ∈ R}, and letting RR ′ be the product relation RR ′ = {(x, z) : (x, y) ∈ R and (y, z) ∈ R ′ for some y ∈ X}. We can then say that R is The analogous definitions for an operator space V ⊆ M m characterize V as The expression "quantum graph" unhappily conflicts with an earlier, unrelated use of this term, and also with the "noncommutative graph" terminology used in [2]. But in a setting that also includes quantum relations, quantum metrics, and so on, it is still the simple and obvious choice.
We define C · I m to be the diagonal quantum relation on M m , so that V is reflexive if and only if C · I m ⊆ V, in closer analogy with the classical case. In the above, V * = {A * : A ∈ V} is the set of Hermitian adjoints of matrices in V and V 2 = span{AB : A, B ∈ V} is a special case of the product of two operator spaces. Graphs appear in this framework by regarding a classical graph as a set of vertices equipped with a reflexive, symmetric relation. The elements of the relation represent edges, and symmetry expresses the fact that edges are undirected. Reflexivity corresponds to the convention that there is a loop at each vertex. This makes sense in the error correction setting: if we place an edge between any two states which might be confused, then it is natural to include an edge between any state and itself. (More pointedly, it is unnatural, and creates unnecessary complication, not to do this.) Of course, in other settings we may not wish to impose this requirement, in which case we could drop reflexivity and define a quantum graph to merely be a symmetric quantum relation. This was the approach taken in [8]. For the sake of definiteness, I will use the term quantum graph to mean a reflexive, symmetric quantum relation, i.e., an operator system, as in [2] and [9]; however, the main results of this paper apply to quantum relations generally, and hence also to noncommutative graphs in the broader sense of [8].

Restrictions, pushforwards, and pullbacks
In the quantum relations setting there are natural notions of restriction, pushforward, and pullback. Suppose we are given quantum relations on M m and M n , i.e., linear subspaces V ⊆ M m and W ⊆ M n . If E is any projection in M m , meaning that E = E 2 = E * , and Φ : M m → M n is a CPTP map expressed as If rank(E) = r then EM m E can be identified with M r , and EVE with a linear subspace of M r , so that the restriction of V can be regarded as a quantum relation on a smaller space.
These definitions are simple and concrete. It is easy to check that if V and W are quantum graphs (i.e., operator systems) then so are EVE ⊆ M r , − → V ⊆ M n , and ← − W ⊆ M m . However, the Kraus matrices K i are not uniquely determined by the map Φ and it is not immediately apparent that the definitions of − → V and ← − W are independent of this choice. The definitions are also rather unmotivated. For instance, when V is a quantum graph its restriction is to be thought of as analogous to the induced subgraph construction in classical graph theory. But an induced subgraph is obtained by choosing a subset of the vertex set and throwing out all edges which extend out of this subset, whereas our definition of restriction involves compressing everything in V to the range of E. So the validity of the analogy is unclear.
These concerns will be addressed in the next section, when we discuss how the quantum relations point of view leads to the definitions given above. But first, let us explain how these operations relate to error correction.
Consider first the classical setting in which a channel is modelled by a probabilistic transition from an initial set of states S to a final set of states T . That is, each initial state has some (possibly zero) probability of going to each of the final states. Such a transition is represented by a stochastic matrix. The confusability graph is specified by placing an edge between two initial states if there exists a final state to which they each have a nonzero probability of transitioning.
As I mentioned earlier, since each initial state can certainly end up at the same state as itself, it is natural to include a loop at each vertex in this graph. A code in this classical setting is then an independent subset of S, i.e., a set of vertices with the property that the induced subgraph contains only loops, with no edges between distinct vertices. In the terminology of Section 2, the induced subgraph is diagonal. The quantum analog of this would be a projection E with the property that the induced quantum subgraph EVE is diagonal, i.e., EVE = C · E. If V Φ is the quantum confusability graph mentioned in Section 1, this statement exactly expresses the Knill-Laflamme error correction conditions [3]. So the statement that the range of E is a quantum code is equivalent to the statement that E induces a diagonal quantum subgraph, just as in the classical case a code is a subset of the confusability graph for which the induced subgraph is diagonal.
A more sophisticated way to specify the classical confusability graph for a probabilistic transition from S to T is to say that it is the pullback of the diagonal relation on T . Here we define the pullback to S of a graph on T by putting an edge between two elements of S if they have a nonzero probability of mapping to adjacent elements of T . The quantum analog of this construction would be the pullback along a CPTP map Φ : M m → M n of the diagonal quantum relation on M n . According to the definition of quantum pullback given above, this would be span{K * i K j : 1 ≤ i, j ≤ d} ⊆ M m , which is exactly the quantum confusability graph V Φ . That is, the quantum confusability graph V Φ associated to a CPTP map Φ : M m → M n is the pullback along Φ of the diagonal quantum relation on M n , just as the clasical confusability graph associated to a classical channel is the pullback of the diagonal relation.
The passage of a message through sequential channels provides a simple illustration of the value of the pullback construction. Suppose we are given classical channels from S to T and from T to U . Then their composition defines a channel from S to U , and the confusability graph of this composition is the pullback to S of the confusability graph for the T -to-U channel. In other words, it is the pullback of the pullback of the diagonal relation on U . The same statement can be made in the quantum setting, as one can see by a short computation.
A similar construction is the pushforward of the diagonal relation on S. This "dual" confusability graph classically includes an edge between two states in T if they might have originated in the same state of S. It might be used by the recipient of a signal which was sent through a noisy channel without the aid of a code, as a way to keep track of possible ambiguity. (This could also be a model of a noisy measurement process in which nature is the "sender".) The quantum analog would simply be the quantum pushforward of the diagonal quantum relation.
Pushforwards and pullbacks give rise to notions of "morphism". Namely, we may regard Φ as a morphism from V to W if − → V ⊆ W, or, alternatively, if V ⊆ ← − W. These two conditions are not equivalent, even in the classical case: the classical analog of the first says that any possible targets of two adjacent vertices in S must be adjacent in T , while the second says that any two adjacent vertices in S must have some possible targets which are adjacent in T . The quantum version of the first, stronger, condition is identical to Stahlke's notion of "noncommutative graph homomorphism" described in [8]:

Connecting states
Now let us see why the definitions of restrictions, pushforwards, and pullbacks given above are natural. The idea is to think of elements of an operator space as "connecting" states. We could say that two pure states |α , |β ∈ C m are connected by a quantum relation V ⊆ M m if α|B|β = 0 for some B ∈ V. However, quantum relations are not determined by this kind of information. For example, take V 1 to be the set of 2 × 2 matrices of the form a b c a with a, b, c ∈ C and take V 2 to be the full 2 × 2 matrix algebra M 2 . These are both quantum graphs on M 2 , i.e., operator systems. It is routine to check that |α , |β ∈ C 2 are connected by V 1 if and only if neither of them is the zero vector if and only if they are connected by V 2 . Thus, V 1 and V 2 are distinct quantum relations which connect the same pairs of states in C 2 . We must instead consider states not in C m but in C m ⊗ C k ∼ = C mk . That is, we consider states of a composite system formed from the original system and some other system. Then we can define |α , |β ∈ C mk to be connected by V if there exists B ∈ V such that α|(B ⊗ I k )|β = 0. It is not hard to show that V is indeed determined by which pairs of states it connects in C mk for arbitrary k; indeed, k = m suffices. See Lemma 6.2 below.
It is convenient to also consider mixed states. First of all, observe that is nonzero if and only if the scalar factor α|(B ⊗ I k )|β is nonzero. So we can also say that V connects |α and |β if and only if the preceding expresion is nonzero for some B ∈ V. More generally, say that V connects positive matrices Since V is already determined by the pairs of (composite) pure states that it connects, it is certainly determined by the pairs of mixed states that it connects. Indeed, any positive matrix can be expressed as a sum of positive rank one matrices, so once we know which pure states are connected by V, we also know which mixed state are connected.
This point of view makes the constructions described in the last section transparent. Let k be a natural number, let V ⊆ M m and W ⊆ M n be quantum relations, let E ∈ M m be a rank r projection, and let Φ : M m → M n be a CPTP map. Then and Φ(C). See Proposition 7.2, Theorem 8.4, and Theorem 9.2 below.
Informally, the mixed states that EVE connects are just the mixed states that live on E and are connected by V. This jibes better with the "induced subgraph" intuition: in order to restrict V to E, look at the pairs of states that are connected by V, and throw out any of them which do not lie under E.
Pushforward and pullback are also easily understood in terms of connection. The states connected by − → V are just the states whose images under Φ * are connected by V, and the states connected by ← − W are just the states whose images under Φ are connected by W. This characterization shows that the definitions of pushforward and pullback only depend on the map Φ, not the choice of Kraus matrices.
The whole point of the noncommutative confusability graph is that it connects mixed states A and C if and only if Φ(A)Φ(C) = 0, i.e., their image states could be confused. That is the same as saying that their image states are connected by the diagonal quantum relation.
We can also use the idea of connecting mixed states to give an intrinsic characterization of the "noncommutative (directed) bipartite graphs" of Duan, Severini, and Winter [2]. Given a CPTP map Φ : M m → M n with Kraus matrices K i , they defined this to be the span of the matrices K i . This span is no longer an operator system in general, but it is still an operator space and hence still counts, in our terminology, as a quantum relation. Its intrinsic characterization is simple: if V = span{K i } ⊆ M n,m , then for any mixed states A ∈ M m ⊗M k and C ∈ M n ⊗M k , there is a possibility of confusing the image of A with C.
One direction is trivial: if C(B ⊗ I k )A = 0 for all B ∈ V, then in particular C(K i ⊗ I k )A = 0 for all i; multiplying on the right by (K * i ⊗ I k ) and summing over i then yields CΦ(A) = 0. This is equivalent to Φ(A)C = 0 since Φ(A) and C are positive. The reverse direction follows from Lemma 8.3 (cf. the proof of Theorem 8.4).

General quantum relations
The definition of "quantum relations" given in [9] was more general than the one described above and actually encompasses both the classical and quantum cases.
By placing the notions of channel, confusability graph, code, etc., in this context we obtain a common generalization in which the classical and quantum cases are not merely analogous, but literally special cases of a single theory. This material will be presented more formally, with proofs of most results.
Let M be a unital * -subalgebra of M m . (In [9] it could be an arbitrary von Neumann algebra in infinite dimensions.) The two most important cases to keep in mind are M = M m , the full matrix algebra, and M = D m , the subalgebra of diagonal matrices. Also let be the commutant of M. The commutant of M m is the scalar algebra C · I m , and the commutant of D m is itself. Von Neumann's double commutant theorem states that M ′′ = M always holds.
Here we use the operator space product VW = span{AB : A ∈ V, B ∈ W}.  The proof is easy. Earlier we interpreted classical graphs as sets equipped with reflexive, symmetric relations, and defined quantum graphs to be operator systems. Both notions are subsumed in the following definition.  Thus, the notion of a quantum relation on M is effectively independent of the representation of M. We therefore expect that there should be an "intrinsic" characterization of them which does not reference the ambient matrix algebra. This can be achieved using the idea of connecting states introduced in Section 4.
At that point it was convenient to consider mixed states, since we wanted to push forward and pull back along a CPTP map, which can convert pure states to mixed states. For the purpose of abstract characterization, it is better to generalize pure states, which can be identified with rank one projections, to projections of arbitrary rank. A direct connection between the two approaches can be made by observing that for any positive matrices  Now Q|β = |β , so the range of (B ⊗ I m )Q contains the vector (B ⊗ I m )|β , which is not orthogonal to |α . Since |α belongs to the range of P , it follows that P (B ⊗ I m )Q = 0. However, if A ∈ V and A 1 , A 2 ∈ M ′ then A * 1 AA 2 ∈ V, so that (A 1 ⊗ I m )α|(A ⊗ I m )|(A 2 ⊗ I m )β = α|(A * 1 AA 2 ⊗ I m )|β = 0. Since this is true for any A 1 , A 2 ∈ M ′ , it follows that α ′ |(A ⊗ I m )|β ′ = 0 for all |α ′ and |β ′ in the ranges of P and Q, respectively. Thus P (A ⊗ I m )Q = 0.
Say that V connects projections P, Q ∈ M k (M) if there exists A ∈ V such that P (A ⊗ I k )Q = 0. The preceding result shows that V is determined by the pairs of projections it connects in this manner: we can tell whether a given B ∈ M m belongs to V by testing whether it connects any pair of projections that is not connected by V. Since this is a crucial point, let us emphasize it: if M is a unital * -subalgebra of M m then an M ′ -M ′ bimodule is determined by the pairs of projections in M m (M) that it connects.
In fact, quantum relations can be characterized abstractly in these terms. We give the relevant definition first, and then state the equivalence with Definition 5.1 as a theorem. To avoid confusion, we will now refer to quantum relations in the sense of Definition 5.1 as concrete quantum relations. In condition (ii) the join P i of a finite family of projections (P i ) is defined to be the orthogonal projection onto the span of their ranges. In (iii), recall that the notation [B] indicates the range projection of B.
R k is to be thought of as the pairs of projections in M k (M) which are connected by some concrete quantum relation V ⊆ M m . To say that each R k is open is to say that if two projections are connected then so are any two projections sufficiently close to them. Condition (i) is trivial, condition (ii) is the basic axiom characterizing connection, and condition (iii) is a statement about scalar compatibility that is typical of what one sees when working at matrix levels. The point is that if B is a scalar matrix then . Proposition 6.2 (b) shows us how to go from concrete quantum relations, as characterized by Definition 5.1, to intrinsic quantum relations, axiomatized as in Definition 6.3. Namely, given V, for each k ∈ N let R k be the set of pairs (P, Q) of projections in M k (M) such that P (A⊗I k )Q = 0 for some A ∈ V. Conversely, given an intrinsic quantum relation (R k ) one recovers the concrete quantum relation that corresponds to it as the set of A ∈ M m satisfying P (A ⊗ I k )Q = 0 for all k ∈ N and all (P, Q) ∈ R k , i.e., the set of matrices which do not connect any pair of projections they are not supposed to connect. The proof of Theorem 6.4 is somewhat complicated. Observe that the characterization of quantum relations provided by Definition 6.3 is "intrinsic" to M in the sense that it makes no reference to the ambient matrix algebra in which M is located. It is manifestly compatible with * -isomorphisms.

Restrictions
Theorem 6.4 allows us to pass back and forth between concrete and intrinsic quantum relations, and we will do this repeatedly in the sequel.
An M ′ -M ′ bimodule is a straightforward object, especially when M = M m and M ′ = C · I m . The value in having a more complicated intrinsic characterization in terms of connecting projections is that some constructions are more naturally understood in these terms. For instance, the natural notion of "subobject" is the following.
Definition 7.1. Let M be a unital * -subalgebra of M m , let (R k ) be an intrinsic quantum relation on M, and let E ∈ M be a projection of rank r. The restriction of (R k ) to EME ⊆ EM m E ∼ = M r is the intrinsic quantum relation (R k ) on EME defined by settingR It is straightforward to verify that (R k ) as defined above has the properties of an intrinsic quantum relation described in Definition 6.3. So according to Theorem 6.4, if (R k ) is associated to the concrete quantum relation V ⊆ M m , its restriction (R k ) must be associated to a concrete quantum relationṼ ⊆ M r on EME. This concrete restriction has a simple direct characterization: Proposition 7.2. Let V be a concrete quantum relation on M ⊆ M m and let E ∈ M be a projection. Then the restrictionṼ of V to EME is concretely given asṼ = EVE. for A ∈ V and B ∈ M ′ now show that EVE is a bimodule over (EME) ′ = EM ′ E, i.e., it is a quantum relation on EME. Now let (R k ) be the intrinsic quantum relation on M corresponding to V, (R k ) the restriction of (R k ) to EME according to Definition 7.1, and (R ′ k ) the intrinsic quantum relation on EME corresponding to EVE. We must show that (R k ) = (R ′ k ). Fix k ∈ N. In one direction, if (P, Q) ∈R ′ k then there exists EAE ∈ EVE such that P (EAE ⊗ I k )Q = 0. But since P, Q ≤ E ⊗ I k and

Proof. First observe that the commutant of EME in EM
this implies that P (A ⊗ I k )Q = 0, so that (P, Q) belongs to R k and therefore tõ R k . Conversely, if (P, Q) ∈R k then (P, Q) ∈ R k and so P (A ⊗ I k )Q = 0 for some A ∈ V. But since P, Q ≤ E ⊗ I k , this implies that P (EAE ⊗ I k )Q = 0, and we therefore have (P, Q) ∈R ′ k . This completes the proof of the desired equality. Although this concrete description of the restriction of V to EME is very simple, it is the intrinsic formulation given in Definition 7.1 which brings out its role as a "restriction".
The following definition now becomes natural.
Definition 7.3. Let V be a quantum graph (a reflexive, symmetric quantum relation) on M ⊆ M m and let E ∈ M be a projection. Then E is independent if the restriction of V to EME is the diagonal quantum relation on EME.
In the case M = D m , the projection E corresponds to a subset of {1, . . . , m}, and E is independent in the above sense if and only if the classical graph corresponding to V has no nontrivial edges in this subset. That is, Definition 7.3 generalizes the classical notion of an independent subset of a graph. In the case M = M m , the independence condition simply states that which, as we noted earlier, expresses the Knill-Laflamme error correction conditions. So the notion of independence yields a common generalization of classical and quantum codes.

Pushforwards
Classically, if f : X → Y is a function between sets then we can push any relation R on X forward to a relation on Y , namely, the relation {(f (x), f (y)) : (x, y) ∈ R}. Similarly, any relation R ′ on Y can be pulled back to the relation {(x, y) : (f (x), f (y)) ∈ R ′ } on X. We now seek quantum versions of these constructions.
The first point to make is that the classical analog of a quantum channel is not an actual function between sets, but a classical channel which maps points in the domain to probability distributions in the range (representing the likelihood of the given input state being received as various output states). In this context the pushforward of a relation R on X would consist of the pairs (x ′ , y ′ ) ∈ Y 2 such that there exists a pair (x, y) ∈ R for which the transition probabilities x → x ′ and y → y ′ are both nonzero. The pullback of a relation S on Y would consist of the pairs (x, y) ∈ X 2 such that there exists a pair (x ′ , y ′ ) ∈ S for which the transition probabilities x → x ′ and y → y ′ are both nonzero.
Since we are working with unital * -algebras, it is natural to adopt the Heisenberg picture in which algebras of observables transform. Mathematically, this means that instead of the CPTP map Φ : ρ → K i ρK * i from M m to M n mentioned in Section 1, which acts on states, we consider the adjoint map Ψ : A → K * i AK i from M n to M m , which acts on observables. The adjoint of a CPTP map is a unital CP (unital completely positive) map. Taking adjoints reverses arrows, so that pushforwards become pullbacks and vice versa; consequently, to maintain consistency with Section 3 we will continue to take the CPTP map Φ : M → N as primary, even though at this point it becomes less natural. The map Φ really should be understood as a map from the predual of M to the predual of N whose adjoint unital CP map Ψ = Φ * takes the * -algebra N to the * -algebra M, but any finite-dimensional * -algebra can be identified with its predual via the pairing (A, B) → Tr(AB), so we need not make this distinction.
The unital CP maps which correspond to actual functions between sets are the unital * -homomorphisms, linear maps Ψ : N → M which preserve the identity and respect the product and adjoint operations. If Ψ = Φ * is a * -homomorphism and (R k ) is an intrinsic quantum relation on M, there is an obvious way to push forward a quantum relation (R k ) on M along Φ to a quantum relation (R k ) on N . Namely, for each k ∈ N letR k consist of those pairs of projections P, Q ∈ M k (N ) with the property that (Ψ(P ), Ψ(Q)) ∈ R k . (Here we abuse notation and also denote by Ψ the map from M k (N ) to M k (M) which applies Ψ entrywise.) This definition makes sense because the * -homomorphism property ensures that Ψ(P ) and Ψ(Q) are projections. It is easy to check that the preceding construction does yield an intrinsic quantum relation on N ([9], Proposition 2.25 (b)).
But we are interested in general CPTP maps, not just those whose adjoint maps are * -homomorphisms. The construction just described no longer works because the image of a projection under such a map need not be a projection. However, there is a simple solution to this difficulty. Let us consider two positive matrices A, B ∈ M n to be equivalent if [A] = [B]. Since the range of a Hermitian matrix is the orthocomplement of its kernel, this condition could also be stated as ker(A) = ker(B). This notion of equivalence is suitable here because whether positive matrices are connected by a quantum relation depends only on their range projections. Since A and B are positive, this implies that α|A|α = α|B|α = 0 and therefore that A|α = B|α = 0. So |α ⊥ ran(A) and |α ⊥ ran(B), and therefore |α ⊥ ran(A) + ran(B). This shows that ran(A) + ran(B) ⊆ ran(A + B), and so the first claim is proven.
We can now prove the lemma. We have Ψ(C) = K * i CK i for some finite family of n × m matrices K i . So, using the two claims, we have We note that a version of Lemma 8.1 for normal CP maps between von Neumann algebras can be proven using the normal Stinespring theorem ( [1], Theorem III.2.2.4).
We can now describe the appropriate version of the pushforward construction for CP maps. Here we return to the "connecting mixed states" point of view; note that if A, C ∈ M ⊗ M k are positive then since Ψ is completely positive, Ψ(A) and Ψ(C) will also be positive. Lemma 8.1 shows that CP maps preserve the relevant notion of equivalence between positive matrices. In order to justify this definition, we must check that ( − → R k ) satisfies the axioms given in Definition 6.3. This can be done directly using Lemma 8.1, but according to Theorem 6.4, it can also be done by finding a concrete quantum relation W on N with the property that (P, Q) ∈ ← − R k if and only if P (A ⊗ I k )Q = 0 for some A ∈ W. This will be achieved in Theorem 8.4 below. Thus, that theorem will simultaneously establish that the pushforward construction is well-defined and identify its concrete formulation.
We require two simple facts about positive matrices. Proof. (a) The reverse implication is trivial. For the forward implication let D = BK * 2 CK 2 and suppose K * 1 AK 1 D = 0. Then 0 = D * K * 1 AK 1 D = (A 1/2 K 1 D) * (A 1/2 K 1 D), so A 1/2 K 1 D = 0 and therefore AK 1 D = 0, i.e., AK 1 BK * 2 CK 2 = 0. Applying the same argument to the adjoint of the expression AK 1 BK * 2 CK 2 then yields the conclusion AK 1 BK * 2 C = 0. (b) Again, the reverse implication is trivial. For the forward implication, we claim that if (X 1 + X 2 )B = 0 with X 1 , X 2 ≥ 0 then X 1 B = X 2 B = 0. This inductively implies the same statement with any finite number of X i 's. Taking B = A( Y j ) in the statement of the lemma then yields X i A( Y j ) = 0 for all i, and applying the same argument to the adjoint of each of these expressions produces the desired conclusion.
To verify the claim, suppose (X 1 + X 2 )B = 0. Then and since both B * X 1 B and B * X 2 B are positive, this implies that both are zero. It follows that X 1 B = X 2 B = 0, as claimed.
is a concrete quantum relation on M. Then its pushforward is concretely given as the N ′ -N ′ bimodule generated by Proof. Let W be the N ′ -N ′ bimodule generated by the matrices K i AK * j for A ∈ V and 1 ≤ i, j ≤ d. We must show that for any k ∈ N and any projections P, Q ∈ M k (N ), we have P (B ⊗ I k )Q = 0 for some B ∈ W if and only if [Φ * (P )](A ⊗ I k )[Φ * (Q)] = 0 for some A ∈ V.
Since P and Q commute with anything in N ′ × I k , the condition obtains if and only if P (K i AK * j ⊗ I k )Q = 0 for some A ∈ V and some i, j. Equivalently, P (K i ⊗ I k )(A ⊗ I k )(K * j ⊗ I k )Q = 0 for some A ∈ V and some i, j, which by Lemma 8.3 (a) is equivalent to which is trivially equivalent to as desired.
If N = M n then its commutant is the set of scalar matrices, so that the pushforward described in Theorem 8.4 is just the linear span of the matrices K i AK * j . Once we know how to push forward quantum relations, it is easy to say what the appropriate notion of "morphism" should be: if M and N are both equipped with quantum relations V and W, then a CPTP map from M to N should be considered a morphism if the pushforward − → V of V is contained in W. The classical version (which is recovered as the case where M = D m and N = D n ) would be a classical channel from a set S of size m to a set T of size n for which the pushforward of a given relation on S is contained in a given relation on T .
Various definitions of quantum graph homomorphisms were proposed in [5,6,8]. Here the term "homomorphism" conflicts somewhat with classical usage, where a homomorphism between graphs is usually taken to be an actual function between the vertex sets, not a channel which could map vertices to probability distributions. An actual map between classical sets generalizes in the quantum setting to a *homomorphism from N to M. In the more general setting of CP maps we prefer the term "CP morphism": Definition 8.5. Let M ⊆ M m and N ⊆ M n be unital * -subalgebras equipped with intrinsic quantum relations (R k ) and (S k ), respectively. A CP morphism from M to N is then a CP map Φ : M → N with the property that − → R k ⊆ S k for all k.
In terms of concrete quantum relations V and W on M and N , respectively, the condition would be that − → V ⊆ W, where − → V is the pushforward of V. In particular, if M and N are matrix algebras and V and W are quantum graphs (i.e., operator systems), the concrete formulation given in Theorem 8.4 states that the condition for Φ to be a CP morphism is K i VK * j ⊆ W for all i and j, which is Stahlke's condition [8]. However, the − → R k ⊆ S k forulation is manifestly intrinsic.

Pullbacks
There is also a way to pull quantum relations back via a CP map. Since we already know how to push quantum relations forward, one obvious solution is just to push forward using the adjoint map. This makes perfect sense in the finitedimensional setting, but it fails in infinite dimensions when von Neumann algebras can no longer be identified with their preduals. However, there is an alternative approach which does generalize to infinite dimensions. We describe this construction now.
The key question is how to use a CP map Ψ = Φ * : N → M to turn a projection in M into a projection in N . We can do this using hereditary cones. A hereditary cone in M is a nonempty set C of positive matrices in M with the properties (i) if A ∈ C then aA ∈ C for all a ≥ 0 (ii) if A, B ∈ C then A + B ∈ C (iii) if A ∈ C and 0 ≤ B ≤ A then B ∈ C. If P is a projection in M then C P = {A ∈ M : A ≥ 0 and P A = 0} is a hereditary cone, and it is not hard to check that every hereditary cone in M has this form. As it is easy to see that the inverse image under any CP map Ψ : N → M of a hereditary cone in M is a hereditary cone in N , this shows us how to use Ψ to turn a projection P ∈ M into a projection ← − Ψ (P ) in N : take ← − Ψ (P ) to satisfy C← − Ψ (P ) = Ψ −1 (C P ).
We can now define the pushforward of a quantum relation via a CP map. We continue to abuse notation by letting ← − Ψ also denote the matrix level map which takes projections in M k (M) to projections in M k (N ). As with pushforwards, we must justify this definition by showing that ( ← − S k ) satisfies the axioms for an intrinsic quantum relation, and as in that case we will accomplish this by identifying the concrete quantum relation that corresponds to ( ← − S k ). for some B ∈ W and some i, j. This equivalent to saying that P (K * i ⊗ I k )(B ⊗ I k )(K j ⊗ I k )Q = 0, i.e., P (K * i BK j ⊗ I k )Q = 0, for some B ∈ W. Since P and Q commute with A ⊗ I k for every A ∈ M ′ , this last condition is equivalent to the statement that (P, Q) belongs to the intrinsic quantum relation associated to the M ′ -M ′ -bimodule generated by the matrices K * i BK j . We conclude that the latter bimodule is the concrete form of the pullback of W.
Again, in the case where M = M m , this pullback would simply be the linear span of the matrices K * i BK j . The pullback construction gives rise to an alternative version of CP morphism which is weaker than the one proposed in Definition 8.5. Namely, instead of requiring − → R k ⊆ S k for all k we could require R k ⊆ ← − S k for all k. In concrete terms, the condition that K i VK * j ⊆ W for all i and j is replaced by the condition that V ⊆ i,j K * i WK j . The second condition is implied by the first (multiply the first condition on the left by K * i and on the right by K j , then sum over i and j and invoke the identity K * i K i = I m ). Classically, the first version demands that if the point x is related to the point y then x ′ must be related to y ′ for any x ′ and y ′ such that the transition probabilities x → x ′ and y → y ′ are both nonzero, while the second version asks only that there be at least one such pair (x ′ , y ′ ).