Quantum graphs: different perspectives, homomorphisms and quantum automorphisms

We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional $C^*$-algebras $B$ equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on $L^2(B)$, the quantum adjacency matrices, or as certain operator bimodules over $B'$. We present a simple, purely algebraic approach to proving equivalence between these settings, thus recovering existing results in the tracial state setting. For non-tracial states, our approach naturally suggests a generalisation of the operator bimodule definition, which takes account of (some aspect of) the modular automorphism group of the state. Furthermore, we show that each such ``non-tracial'' quantum graphs corresponds to a ``tracial'' quantum graph which satisfies an extra symmetry condition. We study homomorphisms (or CP-morphisms) of quantum graphs arising from UCP maps, and the closely related examples of quantum graphs constructed from UCP maps. We show that these constructions satisfy automatic bimodule properties. We study quantum automorphisms of quantum graphs, give a definition of what it means for a compact quantum group to act on an operator bimodule, and prove an equivalence between this definition, and the usual notion defined using a quantum adjacency matrix. We strive to give a relatively self-contained, elementary, account, in the hope this will be of use to other researchers in the field.


Introduction
The notion of a Quantum (or sometimes non-commutative) Graph is a non-commutative generalisation of a classical (finite) graph which has been defined and explored in a number of different presentations. The initial definition arose from the study of quantum communication channels, [20], and treats them as operator systems. The series of papers [26,46,47] extend this definition to arbitrary von Neumann algebras (instead of just M n ), here being more motivated by a general notion of a quantum relation. A graphical approach, using string diagrams, was given in [32], showing that a notion of the adjacency matrix of quantum graph bijects with the quantum relation definition. Here again motivation comes from quantum computing considerations. Also here links with Quantum Groups arose, which has been further explored in [7]. Recent papers exploring quantum graphs are [25,27,31], while generalisations to the non-tracial case have been given in [7,31], and a more minimal set of axioms explored in [8,9,12]. Papers making use of quantum graphs include [6,11,12,21,22,23,39] to list but a few. This paper revisits the bijection between adjacency matrices and quantum relations, and develops a new, elementary, algebraic approach, eschewing graphical methods. We take the time to survey the area, presenting the different approaches, and then show that they are equivalent. Our different approach has the benefit of seamlessly allowing us to treat the non-tracial situation; we further can work with arbitrary faithful states, not δ-forms. We present a large number of new results. Once we have established our approach to equivalence, it becomes clear that in fact quantum graphs over a nontracial state correspond to "tracial" quantum graphs satisfying an extra symmetry, see Theorem 5. 26. We give a definition of the quantum automorphism group of a quantum graph which operates at the quantum relation level. We explore how quantum channels give rise to quantum graphs, showing how an analysis of the Stinespring dilation yields generalisations and improvements of results from [20,47].
Let us now quickly introduce the principle objects we shall study, and then outline the organisation of the paper.
For us a graph G is a (finite) vertex set V = V G together with a set of edges E = E G ⊆ V × V . Thus, in full generality, we allow directed graphs, and loops at vertices (but not multiple edges between vertices, and each loop, if present, is undirected). In this formalism, an undirected graph is when (x, y) ∈ E =⇒ (y, x) ∈ E; the graph has no loops when (x, x) ∈ E for all x ∈ V ; and the graph has a loop at every vertex when (x, x) ∈ E for all x ∈ V . We will typically only be interested in the case of no loops or a loop at every vertex, and there is an obvious bijection between the cases. Thus having a loop at every vertex can be viewed as simply a convention, and not some profound condition.
Let B be a finite-dimensional C * -algebra equipped with a faithful state ψ. We shall say that we are in the classical case when B is commutative, so B ∼ = C(V ) ∼ = ℓ ∞ (V ) for some finite set V and ψ corresponds to the counting measure on V . All the definitions of a quantum graph correspond to a genuine graph on vertex set V in the classical case. Briefly, the principle definitions of a quantum graph are: • a self-adjoint operator A on L 2 (B) satisfying certain conditions corresponding to, in the classical case, being {0, 1}-valued, being undirected, and having a loop at every vertex. See [32,25] with generalisations to non-tracial states in [7,8,31]; see Section 2.
• when B ⊆ B(H) for some Hilbert space H, a unital, self-adjoint subspace S of B(H) which is a bimodule over B ′ , the commutant of B, meaning that for a, b ∈ B ′ , x ∈ S also axb ∈ S. This definition is essentially independent of the choice of H. We call such S an operator bimodule. For B = M n this is [20,19], and in general, [26,46,47]; see Section 4.
• A completely positive map B → B satisfying similar axioms to the first case, explored at least in the case B = M n . See [12]; see Section 3.
We remark that examples of the second case, when B = B(H), arise naturally in the theory of quantum communication channels (this being the motivation of [19,20]). We explore what happens for general B (beyond B = M n ) in Section 6.2.
In the literature, it is common to restrict to the case when ψ is a δ-form, compare Remark 2.3 and Definition 2.8. However, this condition will play no role for us, excepting perhaps meaning that our definition of the empty quantum graph is slightly more complicated. See also Remark 9.11 which shows that seemingly we cannot just reduce to the δ-form case, in general.
When the state ψ is a trace (perhaps a tracial δ-form) these axiomatisations are well-known to be equivalent, see [32, Section VII] and [25,Section 1], the proofs using graphical techniques, and it is not clear what happens in the non-tracial case (but see [31] for some recent progress in adapting graphical ideas to the more general situation). In contrast, we present a simple, purely algebraic approach which allows us to give a quick, direct proof of the equivalence of the first two definitions. Our arguments are motivated by thinking about the problem using operator-algebraic techniques (e.g. working with commutants and projections, and using aspects of Tomita-Takesaki theory). Indeed, it has already been recognised that a key intermediate step between the first two axiomatisations is: • An (orthogonal) projection e ∈ B ⊗ B op satisfying certain symmetry axioms. Here B op is the algebra B with the opposite product. To see why such a projection e might arise, see Theorem 5.11 for example.
For example, [25] takes this as the principle definition. We also extend the "completely positive" approach suggest in [12] to arbitrary (finite-dimensional) C * -algebras. The technical tool here is a version of the Choi matrix correspondence, and we present a version which works for any B; while not technically hard, this might be of independent interest.
Our approach to these equivalences is Section 5. A major benefit of our approach is that, with relatively little work, it suggests an approach which works for any faithful state ψ, tracial or not.
Here we take the first definition (of operators on L 2 (B)) as primary, and show how to adapt the 2nd (operator bimodule) definition so that we still obtain a correspondence. This is an new definition, and a major benefit of the operator bimodule approach is that it is extremely easy to find non-trivial examples in this setting. We first introduce our key bijection(s) and begin to indicate how to pass from adjacency matrices to projections. This can be viewed as an algebraic version of the graphical methods, which are suitably "twisted" by the modular automorphism group, in the non-tracial setting. It is well-known that projections in a von Neumann algebra correspond to closed subspaces invariant for the commutant, and we quickly explore in Section 5.1 what happens with (non-self-adjoint) idempotents in place of projections. In Section 5.2 we state and prove our main results: Theorem 5.13 which works with just the "idempotency axiom", Theorem 5.16 which deals with general adjacency matrices, and Proposition 5.23 deals with the "loop at every vertex" axiom. Along the way we prove a number of auxiliary results: for example, the adjacency matrix commutes with the modular operator and conjugation, Proposition 5.20.
In Section 6.1 we see how standard examples present under these equivalences. In Section 6.2 we explore how quantum channels give rise to quantum graphs, extending known constructions to the case of general B. Here a careful study of Stinespring dilations are key to our results. In Section 6.3 we look at the condition of having a loop at every vertex (or that of having no loops) and produce bijections between these situations; we also look at graph complements. Here our main task is to clarify some comments in the literature, and to deal with the non-δ-form case. In Section 6.4 we quickly look at examples over B = M 2 and M 2 ⊕ M 2 .
In Section 7 we survey a number of different notions of "homomorphism" which have appeared in the literature, mostly concentrating upon [39,47]. By extending our careful study of Stinespring dilations from Section 6.2 we improve results of [47] by showing that "pushouts" and "pullbacks" are automatically bimodules over the appropriate algebra (this also avoids the need to use the techniques of [46]). This area is less settled that the main definitions, and in Section 7.1 we survey some other notions of "homomorphism", "subgraph" and so forth.
We finally turn our attention to automorphisms. In Section 8 we quickly look at automorphisms of quantum graphs: this is well-known in the adjacency matrix picture, but we also look at operator bimodules, and prove an equivalence. This also motivates the quantum case considered in Section 9. Again, for quantum adjacency matrices, this is well-known. However, we find some of the literature here a little hard to follow. For example, [25,Section 4] is readable, but only looks at the Hopf * -algebra case, while we work with the C * -algebra picture of compact quantum groups. We hope that this section will hence be helpful to the non-expert. After some set-up and technical calculation, in Section 9.2 we propose a definition of a coaction on an operator bimodule, and our main result, Theorem 9.18, shows that we then obtain exactly the same notion as that of a coaction on a quantum adjacency matrix. Here our result that every "generalised" quantum graph corresponds to a "tracial" quantum graph, see Remark 5.32, is vital in linking the two pictures of the quantum automorphism group. As operator bimodules are almost trivial to give examples of, we hope that this definition of a quantum automorphism group might aid, for example, in understanding which quantum automorphism groups can arise. It is notable that in our list of recent papers "using" quantum graphs, almost all use the Operator Bimodule approach, while papers looking at quantum automorphisms use the adjacency matrix approach. It is hence useful to have a comprehensive "bridge" between these different perspectives.
Let us end this introduction by saying that almost everything we do is finite-dimensional (but clearly motivated in many places by infinite-dimensional theory), an exception being results in Section 7. Of course, operator bimodules are an infinite-dimensional framework, [46], but it seems unclear what to expect of an "infinite-dimensional" adjacency matrix: many of the core ideas of Section 5 are finite-dimensional.

Notational conventions
We shall attempt to introduce terminology and technical background results, as needed, rather than collecting them up-front. However, let us fix some conventions here. The inner-product on a Hilbert space H will be written (·|·) and is assumed linear in the right variable. Write B(H), B 0 (H) for the algebras of bounded, respectively compact, operators on H. The identity on H is denoted by 1 H = 1, while the identity map on other spaces is denoted by id.
is an orthonormal basis for H then (e i ) n i=1 is an orthonormal basis for H, so with respect to these bases we have that B(H) ∼ = M n and B(H) ∼ = M n , and x → x ⊤ is simply taking the transpose of a matrix.
Throughout the paper, B will denote a finite-dimensional C * -algebra, and ψ will be a state on B, always assumed faithful, with further conditions imposed if necessary.

Acknowledgements
The author thanks Simon Schmidt and Christian Voigt for helpful conversations at an early stage of this project. The author thanks Larissa Kroell for a copy of [27].
The author is partially supported by EPSRC grant EP/T030992/1. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising.

Quantum adjacency matrices
We introduce the notion of a quantum adjacency matrix. We first briefly study finite-dimensional C * -algebras. Any finite-dimensional C * -algebra B has the form The addition and multiplication in B respects this direct sum decomposition, and so often it suffices to consider just a matrix algebra M n . We write e ij ∈ M n for the matrix units, so a typical matrix a = (a ij ) is a = i,j a ij e ij , and the trace satisfies Tr(e ij ) = δ i,j the Kronecker delta. We write a ⊤ for the transpose of a (compare with Section 1.1).
For ψ ∈ M * n there is Q ∈ M n with ψ(a) = Tr(Qa) for each a ∈ M n ; indeed, set Q ij = ψ(e ji ). Then ψ being positive corresponds to Q being positive. If ψ is positive, then ψ being faithful corresponds to Q being invertible, and ψ is a state exactly when Tr(Q) = 1. The same remarks hence apply to B, choosing Q in each matrix factor. Definition 2.1. Given B, ψ as throughout the paper, we shall denote by Q ∈ B the unique positive invertible element with ψ(a) = Tr(Qa) for each a ∈ B. Let L 2 (B) = L 2 (B, ψ) denote the GNS space for ψ.
Thus L 2 (B) is a finite-dimensional Hilbert space. Let Λ : B → L 2 (B) be the GNS map, which in this situation is a linear isomorphism. If necessary, we write π : B → B(L 2 (B)) for the GNS representation, but we shall often suppress this. Hence Thus also Λ ⊗ Λ is a linear isomorphism between B ⊗ B and L 2 (B) ⊗ L 2 (B). The multiplication map m : B ⊗ B → B may hence be regarded as a (bounded) operator L 2 (B) ⊗ L 2 (B) → L 2 (B). The identity 1 ∈ B induces a linear map C → B; λ → λ1. Hence η : λ → λΛ(1) and η * (ξ) = (Λ(1)|ξ) for ξ ∈ L 2 (B), so η * (Λ(a)) = ψ(a) for a ∈ B. Similarly, m(Λ(a) ⊗ Λ(b)) = Λ(ab). Remark 2.3. In the literature (for example, [7,8,25,31,32]) it is very common to assume that additionally ψ is a δ-form (see Definition 2.8) and in this case, we use the terminology that (B, ψ) is a Finite Quantum Space. However, being a δ-form will actually play no role in our arguments, and so it seems more sensible to not add this as a seemingly unnecessary hypothesis. To avoid confusion with the existing literature, we hence avoid the "Quantum Space" terminology. △ We can now proceed with the definition of a quantum adjacency matrix.
Definition 2.4. A quantum adjacency matrix is a self-adjoint operator A ∈ B(L 2 (B)) satisfying: By symmetry we also define variations of these axioms: We show later that when A is self-adjoint, axioms (2) and (4) (2) and (4). While the terminology is to call A a "matrix", it is of course merely a linear map, as L 2 (B) usually carries no canonical choice of basis, and in our approach, we shall not think of A as an actual matrix, except in the classical situation which motivates the more general theory.
Indeed, let us see what happens in the classical situation, where B = ℓ ∞ (V ) = C(V ) for a finite-set V , and ψ is given by the uniform measure on V . Let (δ v ) v∈V be the obvious basis of B consisting of idempotents, so ψ(δ v ) = 1 for each v, and m( . Also η is given by the vector Λ(1) = v Λ(δ v ), and η * Λ(δ v ) = 1 for each v ∈ V . Finally, we see that L 2 (B) ∼ = C V with orthonormal basis (e v ) v∈V where e v = Λ(δ v ). Hence A may be identified with a matrix (A u,v ). Indeed, given two operators A 1 , A 2 ∈ B(L 2 (B)), and so m( u,v ). Thus m(A 1 ⊗ A 2 )m * is the entry-wise, or Schur, or Hadamard, product of the matrices A 1 , A 2 . Thus axiom (1) corresponds to A being Schur-idempotent, equivalently, that A is 0, 1-valued.
Similarly, we calculate that Thus axiom (2) is equivalent to A ⊤ = A. In this situation, axiom (4) gives the same condition. Finally, and so axiom (3) (and here also (5)) is that A has 1 down the main diagonal.
Proposition 2.5. Let V be a finite set, and let B = C(V ). An operator A ∈ B(L 2 (B)) which satisfies axiom (1) corresponds to a directed graph structure on V . That A is self-adjoint, equivalently, that A satisfies axiom (2) or (4), corresponds to the graph being undirected. That A satisfies axiom (3) or (5) corresponds to the graph having a loop at every vertex.
Proof. A 0, 1-valued matrix corresponds to a directed graph structure on V by setting the edge set to be The results now follows from the discussion above.
Remark 2.6. We already see that there is a certain redundancy between requiring that A be self-adjoint and that A satisfy axiom (2) (and/or (4)). This is discussed a little more in [31] after Definition 2.19, in particular, [31,Lemma 2.22]. To an extent, this issue occurs even for general, non-commutative B, compare Remark 5. 19. △ If we prefer to have a loop at no vertex, then given the above discussion, it is natural to consider the axioms: Again, we show later that these axioms are equivalent, in the presence of the other axioms. We give some simple examples for quantum adjacency matrices.
Definition 2.7. The complete quantum graph is given by the adjacency matrix A direct computation shows that this A indeed satisfies all the axioms (1) through (5). We study the map mm * a little further. Suppose for the moment that B = M n . Given i, j set Tr(e kj y * )Tr(e ik x * ) = k Tr(Qx * e ik Q −1 )Tr(Qy * e kj Q −1 ). As x, y are arbitrary, this shows that Thus also where δ 2 = Tr(Q −1 ). As Q −1 is positive, δ 2 is indeed positive. In the general case, B is a direct sum of matrix factors, and Q = (Q k ) say. Then mm * respects the direct sum decomposition, and on each matrix factor is the identity times Tr(Q −1 k ), say mm * = (Tr(Q −1 k )1 k ). As the following appears extensively in the existing literature, we make a formal definition.
We now define the empty graph.
Definition 2.9. The empty quantum graph is given by the adjacency matrix A = (Tr(Q −1 A moment's thought reveals that if A ∈ B(L 2 (B)) respects the direct sum decomposition of L 2 (B) which is induced by that of B, then all of the axioms also respect the decomposition: that is, it suffices to check the axioms on each matrix block. With this observation, a direct calculation shows that the empty quantum graph adjacency matrix satisfies all the axioms (1) through (5); in this calculation it is helpful to show that (1 ⊗ η * m)(m * η ⊗ 1) = 1.
The notion of a quantum adjacency matrix was first considered in [32], in the special case that ψ is a trace and a δ-form. The paper [32] makes extensive use of graphical methods, and the axioms are stated purely in terms of graphical methods. These methods seem to make extensive use of the trace condition, but [31,Section 2] shows how these ideas can be adapted to the non-tracial (but still δ-form) setting. When ψ is merely a δ-form, the definition we have given is equivalent to that given in [7,Definition 3.4], except that we have chosen to normalise A differently. Again when ψ is a δ-form, [8,Definition 3.3] just considers axiom (1) (and without the requirement that A be self-adjoint). A different starting point is chosen in [25,Definition 1.8] which corresponds to the projections e ∈ B⊗B op which we consider in Section 5 below, again when ψ is a tracial δ-form, and again graphical methods are used to make links to adjacency matrices.

Adjacency super-operators
We slightly abuse notation and in this section consider m and m * as maps B ⊗ B → B and B → B ⊗ B respectively.
This terminology is our own; we chose "super-operator" from its use in Quantum Information Theory to refer to a linear map between C * -algebras rather than Hilbert spaces, as is the case here. Notice that the trace ψ enters the definition in the formation of the map m * .
We have stated this definition with the minimal axiom, m(A ⊗ A)m * = A, corresponding to axiom (1) in Definition 2.4. In the obvious way, any other axiom can be added.
This definition is make in [12,Definition 1.8], in the case when B = M n is a single matrix block. In [12,Section 1.2] it is briefly argued that for this B, this definition agrees with the other notions of a quantum graph. We shall show that this holds for all B, with respect to a trace. Indeed, we shall see that the super-operator A in Definition 3.1 agrees with the operator A in Definition 2.4, if we identify B and L 2 (B).
In the non-tracial situation, [9, Section 2] makes further study of the quantum adjacency superoperators. We make links between this and our approach in Remark 5.35.

Operator bimodules
The following definitions make perfect sense in the infinite-dimensional setting, and so we shall briefly work in more generality. We follow [46] and [26,Section 1].
We also call such an S an operator bimodule. As stated, this definitions depends on M ⊆ B(H) and not just on M . However, the definition is essentially independent of the choice of H, compare [46,Theorem 2.7]. This will also follow from Theorem 5.11.
Notice that as M ′ SM ′ ⊆ S, we have that S is reflexive if and only if 1 ∈ S. For the following, recall that an operator system is a unital, self-adjoint subspace of B(H), for some Hilbert space H.

Equivalence of definitions
We present a unified way to move between quantum adjacency matrices and the operator bimodule notion of a quantum graph in the tracial case, and also show how to obtain a correspondence in the general setting, by adjusting the definition of an operator bimodule to take account of (some aspect of) the functional ψ. We then also look at super-operators. We make some remarks about the existing literature.
Definition 5.1. Let B op be the opposite algebra to B, so B op is the same vector space as B, and has the same * -operation, but has the reversed product a op · b = ba. When it is clear from context, we shall already write the multiplication in B op just by juxtaposition.
As L 2 (B) is finite-dimensional, the linear span of the operators θ Λ(a),Λ(b) , as a, b ∈ B vary, gives all of B(L 2 (B)). Given s, t ∈ R, we define bijections from B(L 2 (B)) to B ⊗ B op by As θ ·,· is anti-linear in the first variable, and linear in the second, we see that Ψ is itself linear, and so extends by linearity to a bijection B(L 2 (B)) → B ⊗ B op . Similarly, Ψ ′ is linear. We work with B ⊗ B op to match the existing literature, which is natural for Ψ ′ , but we shall shortly see that when working with Ψ, it might have been slightly more natural to consider B op ⊗ B.
Definition 5.2. We write σ for the tensor swap map which is an anti- * -homomorphism.
, corresponding to e k , for k = 1, 2, we see that and so m(A 1 ⊗ A 2 )m * corresponds to as we work in B ⊗ B op . This shows (4).
We can immediately use this result to show one equivalence.
Proof. Let e = Ψ(A) (for some choice of s, t) so A = A * is equivalent to Then axiom (2) is equivalent to (σ −i(s+t) ⊗ σ i(t+s−1) )σ(e) = e. Taking the adjoint and applying the swap map σ shows that this is equivalent to where we also used (2). This shows that but using (2) again shows that which shows that axiom (4) holds. The converse is similar.
There is an analogous equivalence for Ψ ′ .

we have that
Proof. This could be proved in an entirely analogous way. However, we also observe that Ψ ′ t,s = σ•Ψ s,t , from which these claims follow immediately from Proposition 5.3.
The way to prove the equivalence between Definitions 2.4 and 4.4 requires two steps: • we use Ψ t,s or Ψ ′ t,s , for carefully chosen t, s, to associate A to an idempotent (ideally, an orthogonal projection) e; • associate idempotents (or projections) e to subspaces S ⊆ B(H).
We will shortly deal with the second point is detail, before then returning to the first point. However, the reader can already see that if ψ is a trace, then each σ z is trivial, and it is immediate that A = A * satisfying axioms (1) and (2) corresponds to e ∈ B ⊗ B op with σ(e * ) = e, e 2 = e and e = σ(e), or equivalently, e = e * = e 2 = σ(e), this using any choice of Ψ and Ψ ′ . In the non-tracial case, it is not possible to choose any s, t which gives such a direct bijection, though of course the benefit is that we obtain an interesting, different, theory in this case.
Remark 5.6. When ψ is a tracial δ-form, it was already shown in [32,Theorem 7.7] that a self-adjoint A satisfying axioms (1) and (2)  Elsewhere in the literature, compare for example [12,Lemma 1.6] or [25,Definition 1.12], the relation between A ∈ B(L 2 (B)) and e ∈ B ⊗ B op is given by e = (A ⊗ 1)m * η, and this again is Ψ ′ 0,0 . To make sense of the expression e = (A ⊗ 1)m * η, we identify, as linear spaces, B ⊗ B op with B ⊗ B, and in turn, with L 2 (B) ⊗ L 2 (B). By contrast, [27,Proposition 3.12] uses e = (1 ⊗ A)m * η (and a normalisation factor) which a calculation shows is e = Ψ 1,0 (A). We remark that unfortunately, [27,Lemma 3.11] is only true for a trace ψ, and so while results in [27] are stated for general δ-forms, the proofs only work for tracial δ-forms.
We shall shortly see that Ψ and Ψ ′ are essentially interchangeable, but it is helpful to have the choice between them for different arguments. △

From projections to subspaces
As H is finite-dimensional, every operator on B(H) is in the linear span of the rank-one operators, and so there is a linear isomorphism Once we identify B(H) with H ⊗ H, we find that That is, we obtain the natural action of B ⊗ B op on H ⊗ H. Elsewhere in the literature, links with completely bounded maps (and implicitly with the (extended) Haagerup tensor product) are made, compare [24], [23,Section 2.4] or [27,Lemma 3.9]. In contrast, our perspective in this section is that it is Hilbert Space techniques which are vitally important, for which H being finite-dimensional seems essential. △ An elementary result from von Neumann algebra theory is that invariant subspaces for a von Neumann algebra M biject with projections in the commutant M ′ . We would like to have a version of this result for (possibly) non-self-adjoint idempotents. Proof. Any ξ ∈ H is written as ξ = e(ξ) + (ξ − e(ξ)) ∈ H 0 + H 1 , and if ξ ∈ H 0 ∩ H 1 then ξ = e(ξ) = 0. Thus H = H 0 ⊕ H 1 as a direct sum of Banach spaces (that is, a not necessarily orthogonal direct sum). If e ∈ M then given n ∈ M ′ , for ξ ∈ H 0 we see that n(ξ) = ne(ξ) = en(ξ) so n(ξ) ∈ H 0 , while for ξ ∈ H 1 we have that en(ξ) = ne(ξ) = 0 so n(ξ) ∈ H 1 . Conversely, if H 0 , H 1 are M ′ -invariant, then to show that e ∈ M , by the bicommutant theorem, it suffices to show that ne = en for each n ∈ M ′ . Given ξ ∈ H write ξ = ξ 0 + ξ 1 ∈ H 0 ⊕ H 1 , so ne(ξ) = n(ξ 0 ) = en(ξ 0 ) as n(ξ 0 ) ∈ H 0 , while n(ξ) = n(ξ 0 ) + n(ξ 1 ) ∈ H 0 ⊕ H 1 so en(ξ) = en(ξ 0 ) because en(ξ 1 ) = 0. Hence ne(ξ) = en(ξ) as required.
We now collect some definitions which will allow us to talk about properties of subspaces of H ⊗ H which correspond to the properties of idempotents e ∈ B ⊗ B op which occurred in the previous section. We define an anti-linear map Let (e i ) be some orthogonal basis for H. As 1 = i θ e i ,e i we find that and that this is independent of the choice of (e i ). Proof. The bijection follows from Lemma 5.9 combined with Lemma 5.7. When e = a⊗b acting as a⊗b ⊤ on H ⊗H, so that e(η⊗ξ) = a(η)⊗b * (ξ), we see that σ(e * ) = b * ⊗a * , so that Thus in general σ(e * ) = J 0 eJ 0 , and so e = σ(e * ) if and only if eJ 0 = J 0 e. If this holds, then J 0 (V ) = J 0 e(H ⊗ H) = eJ 0 (H ⊗ H) = V , while also if ξ ∈ W then eJ 0 (ξ) = J 0 e(ξ) = 0 so J 0 (ξ) ∈ ker(e) = W and hence W = J 0 (W ) (as J 2 0 = 1). Conversely, if J 0 (V ) = V and J 0 (W ) = W , then given ξ ∈ H ⊗ H take the decomposition ξ = v + w ∈ V ⊕ W . By assumption, J 0 (ξ) has decomposition J 0 (v) + J 0 (w) ∈ V ⊕ W . Hence e(ξ) = v and eJ 0 (ξ) = J 0 (v), and so Hence σ(e * ) = e, and we have shown (1).

From adjacency matrices to subspaces
We now put together the results of the previous two sections to establish bijections between our two main definitions. We first consider just looking at the first axiom (1), m(A ⊗ A)m * = A.
Theorem 5.13. For any t ∈ R, consider Ψ = Ψ t,t or Ψ ′ = Ψ ′ t,t . These maps B(L 2 (B)) → B ⊗ B op give a bijection between: Proof. Proposition 5.3 shows that Ψ t,t gives a bijection between A and e such that A * corresponds to σ(e * ) and m(A ⊗ A)m * corresponding to e 2 . Similarly, Proposition 5.5 does the same for Ψ ′ t,t . Hence (1) and (2) are equivalent. We now use Theorem 5.11 to see that such e correspond to projections of . This is exactly condition (4) using H ⊗ H in place of B(H).
Remark 5.14. In particular, this applies to the bijection Ψ ′ 0,0 as used elsewhere in the literature (in the tracial case), see Remark 5.6. However, we notice that the condition we obtain is that e = e 2 = σ(e * ), and certainly not that e need be self-adjoint. Compare also Remark 5.19 below.
In light of just the use of axiom (1) in [8], we note that the same proof shows that we can remove that condition that A = A * if we remove the conditions e = σ(e * ), that V, W are J 0 -invariant, and the condition that V, W be self-adjoint. △ To make further progress, it really does seem necessary to take account of the modular automorphism group. First a lemma.
We now come to the first of two key theorems which establish a bijection between Definition 2.4 and a generalisation of Definition 4.4. Recall that Recall the meaning of J 0 from Definition 5.10.
Suppose that (2) holds. By Corollary 5.12, we know that e corresponds to a self-adjoint (2) and (3) are equivalent.
We immediately recover the known results in the tracial case, as here the modular automorphism group is trivial, and so the conditions involving σ z or Q become vacuous. Remark 5.17. As Ψ ′ = σΨ, and we have shown that e = σ(e), it should come as no surprise that working with Ψ or Ψ ′ are equivalent, because if A satisfies the first two axioms, then in fact Ψ(A) = Ψ ′ (A). △ Remark 5.18. The operator Q ∈ B appears in parts (3) and (4) only because Q implements (σ t ). In particular, if q is any positive invertible operator on H with σ t (a) = q it aq −it for a ∈ B, t ∈ R, then we can replace the condition in (3) with (q ⊗ (q −1 ) ⊤ )(V ) = V and that in (4) with qSq −1 = S. A special case of this observation occurs when H = L 2 (B) when we could use the operator ∇. △ Remark 5.19. As we hinted at above, there are really three natural axioms for A: that A is selfadjoint, the axiom (2) (or equivalently (4)) and the axiom (1). Axiom (1) corresponds to e 2 = e and hence to a subspace of B(H) (or more accurately, a choice of subspace and complement, not necessarily the orthogonal complement). That A = A * taken together with axiom (2) then yields that e = e * = σ(e) (and the condition of being invariant under the modular automorphism group) and these then correspond to an orthogonal decomposition of B(H), and the subspace S being self-adjoint. However, we wish to stress that taken individually, it is not clear what A * = A alone, or axiom (2) alone, correspond to, for a subspace of B(H). △ That (σ z ⊗ σ z )(e) = e implies commutation results for the adjacency matrix A.
Proposition 5.20. Let A ∈ B(L 2 (B)) be self-adjoint and satisfy axioms (1) and (2). Then A commutes with both the modular operator ∇, and the modular conjugation J.
as claimed. Similarly, as claimed.
For the following, recall the meaning of1 from Definition 5.10.
The following are equivalent, assuming the equivalent conditions of Theorem 5.16 already hold.
Using Ψ = Ψ 0,1/2 , we see that e acts as i a * i ⊗ σ i/2 (b i ) ⊤ and so from (3) and using that As the equivalent conditions of Theorem 5.16 hold, we know that The bijection between H ⊗ H and B(H) sends η ⊗ ξ to θ ξ,η and so Thus (3) and (4) are equivalent. For (5) consider that , we see that this condition is also equivalent to m(σ −i/2 ⊗ id)Ψ 0,1/2 (A) = 1. Hence (5) and (1) are equivalent.
Let us quickly adapt this to the case of the axioms (6) and (7).
For (4) we again simply follow the above proof.
We will address what axioms (6) and (7) mean for subspaces of B(H) below, see Proposition 6.12 and the discussion after.
Remark 5.24. We continue Remark 5.18. In the previous two propositions, the operator Q appears in conditions, and also in the formation of the tensors u 0 , u 1 . However, again this is only because Q implements the group (σ t ), and so any positive invertible operator q would suffice. In particular, if H = L 2 (B) then we can use ∇ in place of Q. △ As an immediate corollary, we can state the known correspondence between Definitions 2.4 and 4.4. Proof. Immediate from the correspondence between Theorem 5.16 parts (1) and (4), and Proposition 5.22 parts (1) and (4), as when ψ is a trace, Q = 1 and σ z = id for all z.

From adjacency matrices to subspaces
For general ψ, we only obtain a correspondence between Definitions 2.4 (that of a quantum adjacency matrix) and a generalisation of Definition 4.4 (the operator bimodule view of quantum graphs) taking account of the extra conditions of Theorem 5.16 and Proposition 5.22. In this section, we show how to use a further bijection to come in a full circle back to Definition 4.4. We also show some more direct links between adjacency matrices A and the associated subspaces S.
For the following, recall Remark 5.18. Proof. Firstly, we claim that for any z ∈ C, we have that q z xq −z ∈ B ′ for any x ∈ B ′ . Indeed, for b ∈ B, we have that q −z bq z = σ iz (b) ∈ B, so x commutes with this, and hence As b was arbitrary, this indeed shows that q z xq −z ∈ B ′ . Thus, if S is a B ′ -operator bimodule, then for x, y ∈ B ′ and s ∈ S, for s 0 = q 1/2 s ∈ S 0 , we see that then with x, y, s, s 0 as before, Now consider the condition that qSq −1 = S. By a functional calculus argument, it follows that also q 1/2 Sq −1/2 = S; compare with the proof of Theorem 5.16. Thus also S 0 = Sq 1/2 and so q 1/2 S 0 q −1/2 = q 1/2 S = S 0 . Hence qS 0 q −1 = S 0 . Furthermore, if also S is self-adjoint, then S * 0 = (q 1/2 S) * = Sq 1/2 = S 0 . Conversely, if qS 0 q −1 = S 0 then also q −1 S 0 q = S 0 , so also q −1/2 S 0 q 1/2 = S 0 . Thus In particular, we can always apply this result with q = Q, and in the special case when H = L 2 (B), with q = ∇. Let ψ 0 be a faithful trace on B. Combining Theorem 5.16 and Proposition 5.22 with the construction just given allows us to do the following. We start with a quantum adjacency matrix A, with respect to ψ, and form the subspace S. Then form S 0 , and using ψ 0 , reverse the process, to form a quantum adjacency matrix A 0 . Thus every quantum adjacency matrix for ψ corresponds (uniquely) to a quantum adjacency matrix A 0 for the trace ψ 0 .
We first of all see what the condition qS 0 q −1 = S 0 means for the operator A 0 . We continue to write (σ z ) for the modular automorphism group of ψ.
Lemma 5.27. Let B ⊆ B(H), let ψ 0 be a faithful trace on B, and let S 0 ⊆ B(H) be a quantum graph associated to the adjacency matrix A 0 ∈ B(L 2 (ψ 0 )). Furthermore, using that the GNS map Proof. As in the proof of Theorem 5.16, that By linearity, such a relationship holds in general, and so we conclude that Remark 5.28. Suppose that S, S 0 are linked as before. In terms of subspaces of H ⊗ H this means that V is associated to Thus e 0 is the orthogonal projection onto V 0 . We note that is an idempotent with image V 0 , here of course letting e act on H ⊗ H in the usual way. Then, in B ⊗ B op , we have that p = (σ −i/4 ⊗ σ i/4 )(e). So in general, p = e 0 as p is not self-adjoint.
There is a standard way to convert an idempotent into a projection; we follow [15, Proposition 3.2.10] for example, and just give a sketch. Setting a = 1 + (p − p * ) * (p − p * ), we see that a ≥ 1 so a is invertible. Then one can show that q = pp * a −1 is a projection with pq = q, qp = p so that q and p have the same image. Hence e 0 = q. This is a concrete formula, but it appears difficult to work with, as in particular, we have no expression for a −1 . Then , noting of course that really these are two different maps Ψ ′ 1/2,0 as they have different codomains. We do not see how to give a concrete formula for A in terms of A 0 .
However, Remark 5.32 below does give a direct link between A and S 0 . △ We seek a way to more directly link the adjacency matrix A with the subspace S ⊆ B(H) (and/or V ⊆ H ⊗ H). As H is essentially arbitrary (assuming B is faithfully represented on H) this seems hard, but with a special choice of H we can make progress.
We start by looking further at the opposite algebra B op . Define a faithful positive functional ψ op on B op by ψ op (a) = ψ(a) for a ∈ B op . Proof. We have that (Λ op (a)|Λ op (b)) = ψ op (ba * ) = ψ(ba * ), recalling that the product is reversed in it follows that w is unitary, and so w * Λ as required. By definition, we have that ψ op (a) = ψ(a) = Tr(Qa) = Tr(aQ) and so ψ op also has density Q.
, remembering that the multiplication in B op is reversed. Thus as claimed. Similarly, In the sequel, we shall often suppress w, and simply identify . As usual, we give B(L 2 (B)) the functional Tr, and this also has GNS space L 2 (B) ⊗ L 2 (B).
By linearity, this relationship holds for all f ∈ B ⊗ B op . As B ′ = JBJ, taking the linear span over x, y, we conclude that the image of e equals (B ′ ⊗ (B ′ ) op )(Λ ⊗ Λ op )(e), as claimed.
Proposition 5.31. Let B act on L 2 (B), and let A ∈ B(L 2 (B)) be self-adjoint and satisfy axioms (1) and (2). Let A be associated to the B ′ -operator bimodule S ⊆ B(L 2 (B)), and the By linearity, we see that Λ , and so by the previous lemma, S is the B ′ -operator bimodule generated by ∇ −1/2 A. The first claim follows. Further, as e is the projection onto V , the projection of ( , and so the second claim follows.
Remark 5.32. There is an interesting link with Theorem 5.26, which we apply with q = ∇. Thus S is associated to S 0 = ∇ 1/2 S which is a "tracial" quantum graph. As A ′ = ∇ −1/2 A, the proposition says that As x → ∇ 1/2 x∇ −1/2 is a bijection of B ′ , it follows that S 0 is the B ′ -operator bimodule generated by A. △

Adjacency super-operators to matrices
In this section we show that Definitions 3.1 and 2.4 are equivalent. This was shown in [12,Section 1.2] in the case when B = M n , using the Choi matrix of a completely positive map. We shall give an account of how to extend this formalism to general B, which might be of independent interest. As such, we proceed with a little more generality than strictly needed. We shall also use the language of Hilbert C * -modules, see [29] for example, but we shall not use any deep results from this theory. In this section, let C be an arbitrary C * -algebra, and let B, ψ be as usual. In the usual way, regard B * as a bimodule over B. In particular, for a ∈ B, we denote by ψa ∈ B * the functional b → ψ(ab). Every f ∈ B * is of the form ψa for some a ∈ B. This follows, as if not, by Hahn-Banach, there is a only when a = 0, again using that ψ is faithful.
As B is finite-dimensional, B(B, C) is spanned by finite-rank operators, that is, the span of operators of the form θ ψa,c for a ∈ B, c ∈ C, where θ ψa,c is the map B ∋ b → ψ(ab)c ∈ C. In fact, it will be profitable to again introduce the modular automorphism group, and consider the map Theorem 5.33. Ψ ′ restricts to a bijection between the completely positive maps B → C, and positive elements e ∈ C ⊗ B op .
The "usual" Choi matrix, compare [10, Proposition 1.5.12] associates a CP map A : M n → C with the positive operator (A(e ij )) ∈ M n (C). Let us see this in our generalisation. Set B = M n and ψ = Tr. We identify B op with M n using the matrix transpose, and so our bijection identifies To prove the theorem, we shall use the language of Hilbert C * -modules, in particular, the KSGNS construction, which we now briefly recall from [29,Chapter 5]. Given a completely positive map A : B → C, we consider the algebraic tensor product B ⊙ C, and define on this a (in general, degenerate) positive C-valued form by The separation completion is denoted by B ⊗ A C, and we continue to denote by b ⊗ c the equivalence class in B ⊗ A C defined by this simple tensor. Almost by definition, B ⊗ A C is a (right) Hilbert C * -module over C, and there is a * -homomorphism π : B → L(B ⊗ A C) given by π Finally, there is V ∈ L(C, B ⊗ A C) defined by V (c) = 1 ⊗ c, and then giving a canonical dilation for the CP map A.
Proof of Theorem 5.33. Suppose that A : B → C is completely positive, and set e = Ψ ′ (A). Perform the KSGNS construction to form E = B ⊗ A C, π and V . We wish to describe E in terms of e. For the moment suppose that A = θ ψa,c , so that e = c ⊗ σ −i/2 (a), and where the final product is formed in B op ⊗C and σ is again the swap map, and ψ op is as in Lemma 5.29. By linearity, this relationship holds for general A.
Let µ be a state on C, with GNS construction (L 2 (C, µ), Λ µ ). Given As this holds for all u, this shows that the image of σ(e) in B(L 2 (B op ) ⊗ L 2 (C, µ)) is positive. As µ was arbitrary, this shows that σ(e), and hence e, is positive.
We now consider the converse, assume that e = Ψ ′ (A) is positive, and aim to show that A is completely positive. We copy the KSGNS construction, but now define a sesquilinear form on B op ⊗ C by This form is clearly positive, so we may let E be the separation completion (as B is finite-dimensional, E is in fact a quotient of B ⊗ C).
We shall repeatedly move between B and B op , and it will be helpful to have notation to distinguish these. We shall write the product in B op by juxtaposition, and the product in B by a · b, for a, b ∈ B. We now show that π : We can now pull out the b * · b term, at the cost of a b 2 factor, and then reverse to calculation to conclude that Thus π is well-defined. A similar calculation shows that π is a * -homomorphism.
Without loss of generality, we may suppose that C is unital. Set u 0 = 1 ⊗ 1 ∈ E, so that We wish to link this to the map A. Suppose for the moment that e = c ⊗ a so that A(a ′ ) = ψ(σ i/2 (a) · a ′ )c, and hence By linearity, this relationship holds for all e, and hence A(b) = (u 0 |π(b)u 0 ) E for all b ∈ B. We conclude that A is completely positive.
We now apply this to our situation; the following is [12, Proposition 1.7] in the case B = M n .  (1) and (2).
Proof. We first note that the correspondence between A 0 and A means that if A 0 = θ ψa,b , for a, b ∈ B, then A 0 (c) = ψ(ac)b, and hence AΛ(c) = ψ(ac)  (1) and (2), showing (4). This argument can simply be reversed to show the converse. However, with reference to Proposition 5.3, neither Ψ 1/2,0 nor Ψ 1,0 seems to give particularly "nice" equivalent properties on e for the other axioms we might wish A (or A 0 ) to satisfy.

Constructions and Examples
In this section, we discuss some examples, explore how quantum channels (for us, UCP maps) give rise to quantum graphs, and also discuss various ways to construct new quantum graphs from existing ones.

Examples
We consider our basic examples, Definitions 2.7 and 2.9, and see what the different realisations from Section 5.2 are. The following is clear: for the complete quantum graph we obtain, for example, the "maximal" operator bimodule B(H).
The empty quantum graph given by A = (mm * ) −1 is much harder to work with, as ideally we would wish to express (mm * ) −1 as a span of operators of the form θ Λ(a), Λ(b) , and it is not immediate how to do this. However, recall from the discussion before Definition 2.9 that mm * is, in particular, an operator which is a central element in B.
For the following, recall u 0 from Proposition 5.22, which shows that axiom (3) is equivalent to u 0 ∈ V , and hence the characterisation of e as "minimal" is as expected.
For a ∈ B and y 1 , y 2 ∈ B ′ there are b 1 , b 2 ∈ B with y i = Jb i J, and . Thus S ⊆ Q −1/2 B ′ , but clearly given an arbitrary z, we may set b 1 = 1 and b * 2 = Q −1/2 σ −i/2 (z * ) to obtain the opposite inclusion. Thus Let S 0 be the tracial quantum graph associated to S, given by Theorem 5.26. So is the empty quantum graph from Definition 4.5, as we might expect.

From Quantum Channels
We now explore a class of examples (of quantum graphs in the sense of Definition 4.4) which come from quantum channels, see [39,20,19,14]. These examples are amongst the first which arose, and motivate much of the theory. We shall generalise the constructions to arbitrary finite-dimensional C * -algebras, and at the same time streamline them, using the Stinespring representation theorem directly, rather than passing via Kraus operators.
Quantum Channels are usually considered to be trace-preserving, completely positive maps between matrix algebras. From an operator algebraic perspective, this means they are most naturally considered to be maps between (pre)duals of algebras: namely trace-class operators. As such, we prefer to work with the Banach space adjoints, and instead look at unital, completely positive maps. We shall makes comments below where this leads to slightly different conventions. Remark 6.3. Classically we consider the following situation. Let X, Y be finite sets, and for each x ∈ X imagine we send x down a noisy communication channel, so that y ∈ Y may be received with probability p(y|x) ∈ [0, 1]. Thus y p(y|x) = 1 for each x. This gives a map ℓ 1 (X) → ℓ 1 (Y ), say sending a unit basis vector δ x to y p(y|x)δ y . The adjoint is a (completely) positive, unital map θ : ℓ ∞ (Y ) → ℓ ∞ (X) given by We say that x 1 , x 2 ∈ X are "confusable" if there is some y with both p(y|x 1 ) > 0 and p(y|x 2 ) > 0, that is, non-zero chance that y can be received if either of x 1 or x 2 is sent. This establishes a symmetric, reflexive relation on X, that is, a graph structure on X. The associated ℓ ∞ (X)-bimodule is We shall see shortly how to express this S using θ. △ We shall use a form of the Stinespring representation theorem, which follows [40, Theorem 3.6, Chapter IV]. For C * -algebras B, C with B ⊆ B(H), given a UCP map θ : C → B there is a Hilbert space K, a non-degenerate * -homomorphisms π : C → B(K), a normal unital * -homomorphism ρ : θ(C) ′ → π(C) ′ and an isometric linear operator V : As θ(C) ⊆ B ⊆ B(H), we shall always restrict ρ to B ′ . If we have the minimality condition that K = lin{π(c)V ξ : c ∈ C, ξ ∈ H}, then this construction is unique up to unitary conjugation in the obvious way. When (as is our case) B, H are finite-dimensional, a minimal K is also finite-dimensional. The following will be immediate from more general considerations below in Section 7, see Remark 7.6, but we give the proof, as it will in part motivate the more general constructions to be given later.
Theorem 6.4. Let C be some C * -algebra, let B as usual be a finite-dimensional C * -algebra with B ⊆ B(H) faithfully represented. Given a UCP map θ : C → B, construct V, ρ, π, K as above, and defineθ : π(C) ′ → B(H) byθ(y) = V * yV . Then S =θ(π(C) ′ ) is a quantum graph over B in the sense of Definition 4.4. This definition of S is independent of the choice of V, ρ, π, K (so in particular, K is not required to be minimal).
Proof. As V is an isometry, V * V = 1, and soθ is UCP, from which it immediately follows that S is a self-adjoint unital subspace of B(H). Given x 1 , x 2 ∈ B ′ , as ρ(x 1 ), ρ(x 2 ) ∈ π(C) ′ , we find that Thus S is a B ′ -operator bimodule.
Thus S = lin{E * k E l }, in agreement of [20, Section IV]. By contrast, [39] considers a different convention, which we discuss below after Proposition 6.12. △ When B = B(H), it is shown in [19, Lemma 2] that every quantum graph over B arises from some UCP map. We have been unable to decide if this remains true for general B.

The no-loops condition
In this section, we compare the "loops" condition m(A ⊗ 1)m * = 1 with the "no loops" condition m(A ⊗ 1)m * = 0. We also see what this means for the operator bimodule picture of quantum graphs. Into such considerations also comes the non-commutative generalisation of taking a graph complement, which we consider first.
Recall that, when we view B as a direct sum of matrix algebras, the operator mm * is a direct sum of scalar multiples of the identity (equivalently, mm * is in the centre of B) and is invertible, compare the discussion around Definition 2.9. We state the following for axioms (3) and (6), but we know that in the presence of the other axioms, these are equivalent to axioms (5) and (7). Proof. We know that (mm * ) −1 is itself a quantum adjacency matrix, see after Definition 2.9. As the axiom (2)  It hence remains to consider axiom (1). As B is a direct sum of n matrix algebras, also L 2 (B) is the direct sum of Hilbert spaces (H k ) n k=1 , and m, m * and hence (mm * ) −1 all respect this decomposition. Let ξ ∈ H k , so m * (ξ) ∈ H k ⊗ H k . There is a scalar δ k > 0 with (mm * ) −1 equal to δ 2 k on H k . Hence Conversely, let (2) hold. By Proposition 5.23, also m(1⊗A ′ )m * = 0. Using that A = A ′ +(mm * ) −1 an analogous calculation to the above shows that and so this equals A ′ (ξ) + δ 2 k ξ, for ξ ∈ H k . As k was arbitrary, m(A ⊗ A)m * = A, and so (1) holds.
The previous proposition shows that we can move between "all loops" and "no loops". We next look at the notion of a "complement", which classically would take an "all loops" graph to a "no loops" graph. We will discuss an intermediate process, which preserves "all loops", respectively, "no loops", below. Proposition 6.8. Let A ∈ B(L 2 (G)) be self-adjoint and define A ′′ = θ Λ(1),Λ(1) − A which is also self-adjoint. The following are equivalent:
Remark 6.11. The definition of the complement considered in Corollary 6.9 is noted in [7, Remark 3.6], and in somewhat more detail in [31, Propositions 2.26, 2.28], both in the case when ψ is a δ-form. Notice that implicit in the proof of [31, Propositions 2.26] is a consideration that m(A ⊗ 1)m * = 1 implies (in the presence of the other axioms) that also m(1 ⊗ A)m * = 1, which we also make essential use of. △ We now look at quantum graphs in the sense of Definition 4.4, that is, quantum relations over our algebra B ⊆ B(H) with additional properties. In the following, A is self-adjoint satisfying axioms (1) and (2), and satisfies other axiom(s) as appropriate. Proof. Using Theorem 5.16 we associate A with a projection e, and S is the image of e (once we identify B(H) with its GNS space H ⊗ H). Let (mm * ) −1 correspond to projection e min , which from Proposition 6.2 is the minimal projection fixing u 0 , and corresponds to the bimodule S min = B ′ Q −1/2 B ′ .
Consider (1), where A is assumed to satisfy axiom (3), so e(u 0 ) = u 0 by Proposition 5.22, and hence e min ≤ e. By linearity, A ′ corresponds to e − e min which has image the relative orthogonal complement of S min in S, namely S ∩ (B ′ Q −1/2 B ′ ) ⊥ , as claimed. Now consider (2). As θ Λ(1),Λ(1) corresponds to the projection 1 and the subspace B(H), it is immediate that S ′′ corresponds to S ⊥ . Then (3) and (4) follow in a similar way.
To be precise, x ∈ S ⊥ if and only if Tr(y * x) = 0 for each y ∈ S. It is easy to verify directly that when S is a B ′ -operator bimodule, so is S ⊥ , and that S is self-adjoint if and only if S ⊥ is. When ψ is a trace, the "all loops" condition corresponds to S being unital, and this is equivalent to Tr(x) = 0 for all x ∈ S ⊥ , or that S ⊥ is "trace-free". This trace-free condition was suggested in [39, Section III] from motivation from quantum information theory. Indeed, Remark 6.3 showed how to construct the "confusability graph" from a channel, but [39] argues that it is perhaps more useful to look at the "distinguishability graph" which indeed classically corresponds to S → S ⊥ at the level of operator bimodules.
Remark 6.13. This is one place where Theorem 5.26 does not interact as we might wish. Recall that we associate S to the "tracial" quantum graph S 0 = Q 1/2 S. A simple calculation shows that See Remark 6.15 below for an example suggesting that we cannot improve this.
However, we do note that if QSQ −1 = S then the same is true of S ⊥ , as given x ∈ S ⊥ , y ∈ S, we compute that Tr(y * QxQ −1 ) = Tr(Q −1 y * Qx) = Tr((QyQ −1 ) * x) = 0 as QyQ −1 ∈ S. So S → S ⊥ preserves all the properties of Theorem 5.16. △ Proof. From Lemma 5.15 we know that T (V ) = V , and so also (tT ) n (V ) = V for any t > 0 and n ∈ N. At T is self-adjoint, we can write T = i t i P i where {t i } are the eigenvalues of T , and each P i is the projection onto the eigenspace associated to t i . Thus the {P i } are mutually orthogonal, and i P i = 1. Suppose t 1 > t 2 > · · · , and let ξ ∈ V . Then
Thus P i (V ) ⊆ V for each i, and so V is the orthogonal direct sum of the subspaces P i (V ) (some of which may be {0}). The result follows.
Given Q, we can diagonalise, and so find a unitary u ∈ B and a (component-wise) diagonal Q d ∈ B with uQu * = Q d . If S satisfies the conditions of Theorem 5.16 for Q d , then T = u * Su satisfies the conditions for Q. As u ∈ B, it is easy to see that this procedure correctly interacts with Theorem 5.26; it also interacts smoothly with taking complements etc., Proposition 6.12. Finally, the notion of "equivalence" is compatible with the notion of automorphism we explore in Section 8, and with the obvious notion of "isomorphism" between two quantum graphs, as mentioned elsewhere in the literature. Thus, there is no loss of generality in working with Q d .
When B = M 2 , say acting on C 2 , we have that B ′ = C, and any Q is similar to a Powers density with 0 < q ≤ 1, compare [31, Section 3.3]. We will not consider the tracial case, so assume q = 1. With Lemma 6.14 in mind, the eigenspaces to consider are As S should also be self-adjoint, there are just four quantum graphs over M 2 with Q d SQ −1 d = S and Q −1/2 d ∈ S (this from Proposition 5.22), namely Here S 1 is the empty quantum graph, S 3 the complete quantum graph, and S 2A , S 2B lie strictly between these, and are incomparable. A simple computation shows that taking the complement, S → S c from Proposition 6.12, interchanges S 1 , S 3 and S 2A , S 2B . If we apply Theorem 5.26, then (S 1 ) 0 is the span of the identity, (S 2A ) 0 is the span of the identity and the off-diagonal matrices, and S 2B , S 3 remain unchanged. Thus, in this special case, Theorem 5.26 interacts as we might hope with the complement.
We identify B(H) with 2 × 2 matrices of operators in B(C 2 ) ∼ = M 2 . Set Finally let T = lin{x, x * }, so T is a self-adjoint, B ′ -bimodule. Then from which it follows that QT Q −1 = T . By the final part of Remark 6.13, S 0 = T ⊥ also satisfies the required conditions, and so by Theorem 5.26, S = Q −1/2 S 0 satisfies all the conditions of Theorem 5.16. Towards a contradiction, suppose that (S c ) 0 = (S 0 ) ⊥ + B ′ . By Remark 6.13, always (S c ) 0 = Q(S 0 ) ⊥ + B ′ , so we obtain that (S 0 ) ⊥ + B ′ = Q(S 0 ) ⊥ + B ′ , that is, T + B ′ = QT + B ′ , that is, QT ⊆ T + B ′ . However, given the form of Qx computed above, it is clear that Qx ∈ T + B ′ , giving the required contradiction. △

Homomorphisms
In this section, we look at various notions of "homomorphism" between quantum graph. We note that there have been notions of a "quantum homomorphism" between classical graphs (see [34,Section 4] or [32, Theorem 4.14, Remark 5.5] for example) which can further be extended to quantum homomorphisms between quantum graphs. These notions are linked to the quantum automorphisms we discuss below, and are discussed in detail in [25,Section 4.4], [31,Section 2.3], for example. In contrast, in this section we look at what might be thought of as "classical homomorphisms" between (possibly) quantum graphs. Such notions seem to be less firmly established than other definitions made in this paper, so we compare and contrast them, without offering a definitive definition. We start with two ideas from Weaver in [47]; again we work with quantum graphs in the sense of Definition 4.4. For exactly what we mean by "Kraus form" see the discussion below. It is not at all clear that the following definition is independent of the choice of Kraus form, but this follows from [47], or our, different, argument to follow.
Again here we have translated from trace-preserving CP maps to UCP maps, equivalently, have "reversed the arrows", compare with Remark 6.3 above. We notice that the pullback is always unital if S 2 is unital, but the pushfoward may not be, in accordance with what happens in the classical situation. While we stated this for quantum graphs, obviously the definition works for any quantum relation. The reader who would like some motivation can see Remark 7.11 below.
The paper [47] develops an "intrinsic" way to characterise quantum graphs (making use of results from [46]), and then shows that the pushfoward and pullback only depend on θ (and not the Kraus form chosen) when recast using this intrinsic characterisation of S 1 and S 2 . We shall show how to use Stinespring dilations (and a duality argument) to give a direct argument. A biproduct of our construction is that in the above definition, it suffices to take the linear span of the sets given: they are automatically operator bimodules over the relevant commutants.
Let us recall how to construct the Kraus form of a normal CP map between arbitrary von Neumann algebras. This is surely known, but we are not aware of a canonical reference. Let M, N be von Neumann algebras acting on Hilbert spaces H M , H N respectively. We start by observing that the usual Stinespring dilation of a normal CP map θ : M → N between von Neumann algebras, say V, ρ, π, K automatically has that π is normal. Consider now an arbitrary dilation (so we do not assume minimality). By the structure theorem for normal * -homomorphisms between von Neumann algebras (for example, [40, Theorem 5.5, Chapter IV]) there is an index set I and an isometry W : and there is a family of operators  2. there is a normal unital * -homomorphism ρ : Proof. Suppose first that θ maps into N . Let V 0 , π 0 , K 0 , ρ 0 be the minimal Stinespring dilation for θ. We may define u : it follows that u is well-defined, an isometry, and by minimality, extends by linearity and continuity to all of K 0 . By construction, uπ 0 (x) = π(x)u for x ∈ M , and uV 0 = V . Define ρ(x ′ ) = uρ 0 (x ′ )u * which is a normal * -homomorphism, as u * u = 1. It follows that Conversely, if we have such a ρ then for As there is a bijection between dilations V, π, K with K = H M ⊗ ℓ 2 (I) and π(x) = x ⊗ 1, and Kraus forms θ(x) = i∈I b * i xb i , this lemma can be translated into a (slightly cumbersome) condition on a Kraus form.
Weaver shows in [46,Theorem 2.7] that the definition of a Quantum Relation, Definition 4.1, is independent of the ambient Hilbert space H upon which our von Neumann algebra M acts. We will study the ideas of this proof, and separate them out into a number of useful constructions. Let  Proof. The only non-obvious thing to check is that every quantum relation on π(M ) is of the form S π . Let T be a weak * -closed operator bimodule for π(M ) ′ = M ′⊗ B(H). For x ∈ T , consider which is seen to be an M ′ -bimodule. By linearity, and that T is weak * -closed, it follows that y ⊗ a ∈ T for each y ∈ S, a ∈ B(H). Furthermore, we can approximate x ∈ T in the weak * -topology by multiplying on the left and right by 1 ⊗ θ for suitable finite-rank operators θ, and so x is in S⊗B(H). Thus T = S⊗B(H) = S π , as required.
where π(x) = x ⊗ 1 as before. Notice that the definition of S θ generalises the definition of S π . Lemma 7.4. The definition of S θ is independent of the choice of u, H, and S θ is a quantum relation on θ(M ). The map S → S θ is order preserving, and preserves the properties of S being self-adjoint, respectively, S being unital.
To show that S θ is weak * -closed, we used the Krein-Smulian Theorem, [13, Theorem 12.1] for example, which tells us that it suffices to show that the (norm closed) unit ball of S θ is weak * -closed. Towards this, let (u * α i u) be a bounded net in S θ converging weak * to x ∈ B(K). There seems to be no reason why (α i ) is a bounded net, but for each i, set β i = uu * α i uu * ∈ B(H M ⊗H). As uu * ∈ M ′⊗ B(H), it follows that each β i ∈ S⊗B(H), and u * β i u = u * α i u. Of course, β i ≤ u * α i u , and so (β i ) is a bounded net. By moving to a subnet if necessary, we may suppose that β i → β ∈ S⊗B(H), as S⊗B(H) is weak * -closed. Then u * β i → u * βu and so x = u * βu ∈ S θ , as required.
The remaining claims are clear.
The following shows how to compute the pullback from a Stinespring dilation directly (not using the Kraus form) and shows that even the Kraus form definition is already an N ′ -operator bimodule. We prove this for possibly infinite-dimensional operator bimodules, as this is not much more work.
Theorem 7.5. Let θ : M → N be a normal CP map with Kraus form θ(x) = i∈I b * i xb i . Let S ⊆ B(H M ) be a quantum relation over M , and as before, define the pullback of S along θ to be ← − S , the weak * -closed N ′ -operator bimodule generated by elements b * i xb j for x ∈ S, i, j ∈ I. Then: 1. already the weak * -closed subspace generated by elements b * i xb j for x ∈ S, i, j ∈ I is an N ′ -operator bimodule; 2. if V, π, K is any Stinespring dilation of θ, then ← − S is the weak * -closure of V * S π V , where S π is as defined above.
Proof. We first show how to express ← − S using a special Stinespring dilation. For the moment, suppose that V, π, K = H M ⊗ ℓ 2 (I) comes from the Kraus representation, so V ξ = i b i ξ ⊗ δ i and π(x) = x⊗ 1.
and so ← − S is also equal to the weak * -closure of lin{V * (x ⊗ a)V : x ∈ S, a ∈ B(ℓ 2 (I))}, that is, equals the weak * -closure of V * S π V . Now let ρ : N ′ → π(M ) ′ = M ′⊗ B(ℓ 2 (I)) be as in Lemma 7.2. Then, for the operator bimodule property of S π . Thus V * S π V is an N ′ -operator bimodule, and thus so is its weak * -closure. We have hence shown claim (1). Now let V 0 , π 0 , K 0 be the minimal Stinespring dilation for θ. There is an isometry u : K 0 → K with uV 0 = V and π(x)u = uπ 0 (x) for x ∈ M , compare the proof of Lemma 7.2. Notice then that π 0 (x) = u * (x ⊗ 1)u for each x ∈ M , so by definition, S π 0 = u * S π u, and hence V * S π V = V * 0 u * S π uV = V * 0 S π 0 V 0 . Suppose now that V, π, K is any Stinespring dilation for θ. Again, there is an isometry u : K 0 → K with uV 0 = V and π(x)u = uπ 0 (x) for x ∈ M . For a suitable Hilbert space K ′ , we can find an isometry v : Remark 7.6. Let us compare this with the quantum graph we constructed in Theorem 6.4. There we started with θ : C → B a UCP map and constructed a quantum graph over B which is S = V * π(C) ′ V . Here V, π, K is a Stinespring dilation of θ. As this is independent of the choice, we can choose K = H C ⊗ K 0 for a suitable K 0 , and π(x) = x ⊗ 1, so that π(C) ′ = C ′⊗ B(K 0 ) = T π , where T = C ′ the trivial quantum graph over C. Thus S is actually just the pullback of the trivial quantum graph C ′ for the UCP map θ. △ Corollary 7.7. Let θ : B 2 → B 1 be a UCP map, and let S be a quantum graph over B 2 . Then ← − S is a quantum graph over B 1 . If V, π, K is any Stinespring dilation of θ, then Proof. In the finite-dimensional setting, there is no need to take weak * -closure, and so ← − S = V * S π V , for any quantum relation S. If S is self-adjoint, clearly also V * S π V is. As θ is assumed to be unital, V * V = 1, and so if S is unital, so is S π , and hence so is V * S π V .
While this result gives a rather pleasing form for the pullback at the level of operator bimodules, we have not been able to formulate a simple result for the associated projections and quantum adjacency matrices.
The pushforward seemingly does not make sense in the infinite-dimensional case, but in our usual finite-dimensional case, we can use a duality argument. The following should be compared to a more general construction which holds in the infinite-dimensional case under some conditions, see [1, Proposition 3.1] for example.
Thenθ exists, and is completely positive.θ is UCP if and only if θ is trace preserving: Proof. As ψ 2 is faithful, any functional f on B 2 arises from a unique b 0 ∈ B 2 as f (b) = ψ 2 (b 0 b) for b ∈ B 2 , compare with the discussion at the start of Section 5.4. Applying this to the functional b → ψ 1 (aθ(b)) gives the existence and uniqueness ofθ(a). Assuming that θ is positive, for a ≥ 0, This shows thatθ(a) ≥ 0, soθ is positive. Consider nowθ ⊗ id : B 1 ⊗ M n → B 2 ⊗ M n . Giving M n the trace Tr and hence B i ⊗ M n the faithful trace ψ i ⊗ Tr, it follows readily thatθ ⊗ id and θ ⊗ id are related in the same way, and hence the previous argument shows thatθ ⊗ id is positive when θ ⊗ id is. It follows that θ CP implies that θ is CP.
Corollary 7.10. Let θ : B 2 → B 1 be a CP map, and let S be a quantum graph over B 1 . Then − → S is a self-adjoint quantum relation over B 1 , and − → S is equal to lin{b i xb * j : x ∈ S} for any Kraus form Proof. Formθ : B 1 → B 2 which is CP (but maybe not unital) and has Kraus formθ(x) = i b i xb * i . Hence, the pullback of S alongθ is exactly the pushforward of S along θ, and so the claims for − → S follow from Theorem 7.5.
We have not discussed quantum graphs which arise from quantum adjacency matrices with respect to a non-tracial state. By Theorem 5.26, this is equivalent to studying quantum graphs S with the additional property that QSQ −1 = S, but it is not very clear what condition we might place on a CP map θ to ensure this. Further, we have not discussed links with quantum adjacency matrices A nor projections e, because it seems somewhat hard to give concrete formula for, say, ← − e associated to ← − S .

Further notions
We give a quick summary of some of the other suggestions for what a "homomorphism" for a quantum graph should be, and related ideas. In [47, Section 6] given a quantum graph S ⊆ B(H) over B ⊆ B(H) and a projection p ∈ B, the restriction S p is pSp the quantum graph over pBp ⊆ B(p(H)). The map θ : B → pBp, a → pap is UCP with Stinespring dilation π = id, K = H, V = p, and so pSp is simply the pullback of S along this UCP map.
In [39,Definition 7], given quantum graphs S i ⊆ B(H i ) over B(H i ) for i = 1, 2, a homomorphism S 1 → S 2 is defined to be a UCP map θ : B(H 2 ) → B(H 1 ) with Kraus form θ(x) = i b * i xb i , with the property that b i S 1 b * j ⊆ S 2 for each i, j. This is equivalent to − → S 1 ⊆ S 2 , which agrees with Weaver's notion of a CP morphism, [47,Definition 7.5], once we recall that we always work with UCP maps, not TPCP map.
We should note that here Stahkle works with trace-free quantum graphs, see the remarks after Proposition 6.12. If we look instead at the quantum graphs S ⊥ i , then the condition becomes b * i S ⊥ 2 b j ⊆ S ⊥ 1 for each i, j. That is, However, it seems to the author that these notions make sense in general, regardless of whether we work with unital, trace-free, or more general quantum relations.
Remark 7.11. We follow [39] and motivate this definition from the classical situation. Consider a classical graph G with associated S G = lin{e u,v : (u, v) ∈ E(G)} over C(V (G)). Similarly for a graph For the moment, let f : V (G) → V (H) be any map. We obtain a unital * -homomorphism θ : C(V (H)) → C(V (G)) given by θ(a)(u) = a(f (u)) for a ∈ C(V (H)), u ∈ V (G). Then θ is in particular a UCP map, and we can construct a Stinespring dilation as follows.

The Kraus form is obtained by setting
if u is the ith vertex which maps to f (u), and 0 otherwise. For (u, v) ∈ E(G), so that e u,v ∈ S G , we see that or 0, depending on i, j (and that there is some choice of i, j which gives e f (u),f (v) ). Hence and so − → S G ⊆ S H if and only if f is a homomorphism. Thus V * (a ⊗ 1)V = θ(a) for each a, and as the dilation has the correct form, it follows that A calculation shows that then ← − S H is the linear span of "blocks" of the form {e u,v : f (u) = x, f (v) = y} as x, y vary. This indeed corresponds to a "pullback" of H along f to a graph on G, where we expand each edge in H to a clique in G. △ Stahkle shows further in [39] that given any CP morphism C(V (H)) → C(V (G)), there is some graph theoretic homomorphism G → H. However, this construction is not a bijection, but rather an existence result (which motivates Weaver to use the new terminology of "CP morphism"). Nevertheless, for many graph theory questions, we are merely interested in the existence of any homomorphism between two graphs, and for this, the notion of a CP morphism exactly extends this to quantum graphs (which is indeed Stahkle's motivation).
[20] (see Section II) and [39] (see after Proposition 9) give tentative definitions for what subgraphs or induced subgraphs should be. Gromada in [25, Section 2.2] looks at two notions of a "quantum subgraph", which are defined using the formalism of projections e ∈ B ⊗ B op . Firstly, e 1 is a subgraph made by removing edges of e 2 if e 1 ≤ e 2 , which is clearly equivalent to containment of the related operator bimodules. When we have q : B 1 → B 2 a surjective * -homomorphism, then e q the induced subgraph of e along q is given by e q = (q ⊗ q)(e). It seems tricky to give a simple description of the associated operator bimodules. Finally, [25,Section 2.4] looks at more general notions of "quotient", which requires the consideration of a more general object, a weighted quantum graph.

Automorphisms of Quantum graphs
As a warm-up to looking at quantum automorphisms, we look at (classical) automorphisms of quantum graphs. Let us quickly discuss some well-known results about automorphisms of B. By Aut(B, ψ) we mean the automorphisms of B, the bijective * -homomorphisms B → B, which leave ψ invariant. As B is finite-dimensional, it is easy to understand automorphisms of B. We give a quick proof of the following well-known result, for completeness. Proof. It is well known that a * -isomorphism M m → M m is of the form stated, so we just shown the first part of the claim. For projections e, f we have by definition that e ≤ f if and only if ef = e, and so an automorphism preserves this partial ordering. Clearly also θ maps the centre of B to itself. Thus θ gives a bijection when restricted to the set of minimal central projections. The minimal central projections of B are exactly the units of the matrix blocks. Fix some matrix block M n k and let the unit be e. Let (e ij ) n k i,j=1 be the matrix units of this block, so ee ij = e ij e = e ij for each i, j. If θ(e) is the unit of the matrix block M n l , then we see that θ(e ij ) ≤ θ(e) for each i, j, and so each θ(e ij ) is in this block, as required.
As ψ is given by the positive invertible Q ∈ B, we can work out which automorphisms are in Aut(B, ψ). Proof. We first show that θ ∈ Aut(B, ψ) commutes with (σ t ). We use the uniqueness property of (σ t ), see for example [41,Theorem 1.2,Chapter VIII], which says that ψ • σ t = σ t for each t ∈ R, and for every a, b ∈ B there is a continuous function F = F a,b on {z ∈ C : 0 ≤ Im(z) ≤ 1} which is holomorphic on the interior, and with F (t) = ψ(σ t (a)b) and F (t + i) = ψ(bσ t (a)). These properties determine (σ t ) uniquely.
Shortly A will denote a quantum group, and so in this section we shall always write A G for a quantum adjacency matrix. In the following, we are deliberately a little vague as to which exact axioms we need A G to satisfy. Definition 8.4. Let A G ∈ B(L 2 (B)) be a quantum adjacency matrix. An automorphism of A G is θ ∈ Aut(B, ψ) such that A G andθ commute. Remark 8.5. Let B = C(V ) and let A = A G be the adjacency matrix of a classical (perhaps directed) graph, so A ij = 1 exactly when there is an edge from j to i. Let us denote that there is an edge from j to i by j → i. Any automorphism θ is given by some σ ∈ Sym(V ), and notice that the uniform measure on V is automatically preserved. Then L 2 (B) ∼ = C V andθ is also induced by σ, in that θ(δ i ) = δ σ(i) . As ThusθA = Aθ if and only if, for each i, the sets {j : i → σ −1 (j)} and {j : σ(i) → j} agree. This is equivalent to the statement that i → k if and only if σ(i) → σ(k), which we observe by setting k = σ −1 (j). That is, σ is an automorphism of the (directed) graph. △ As a final remark, we notice that it is not really necessary to work with L 2 (B) here. Instead, we could interpret the quantum adjacency matrix A G as a linear map on B (compare Section 5.4), and then require that A G and θ commute; this is [25,Definition 2.4] for example.
So far, we have been discussing well-known results. To our knowledge, however, there is little or no discussion in the literature around what an automorphism of an operator bimodule might be. We shall follow Theorem 5.16. Let A G ∈ B(L 2 (B)) be a quantum adjacency matrix, satisfying axioms (1) and (2). Then A G corresponds to a projection e ∈ B ⊗ B op with e = σ(e) and with e invariant under (σ t ⊗ σ t ). Notice that when θ is an automorphism of B, also θ is an automorphism of B op . Proposition 8.6. We have that A Gθ =θA G if and only if (θ ⊗ θ)(e) = e.
Proof. It follows that e ′ = (θ ⊗ θ)(e) satisfies the conditions of Theorem 5.16 and so corresponds to A ′ G ∈ B(L 2 (B)) which satisfies axioms (1) and (2). We will show that A ′ G =θA Gθ * , and so A Gθ =θA G if and only if A ′ G = A G if and only if e = e ′ , as claimed. Proof. As θ B restricts to an automorphism of B ′ , it follows that B θ is a self-adjoint B ′ -bimodule. From Lemma 8.2, we know that θ B (Q) = Q, and so QSQ −1 = S implies that QS θ Q −1 = S θ . Let S corresponds to V ⊆ H ⊗ H, and similarly S θ correspond to V θ . As noted above, V θ = (u θ ⊗ u θ )(V ). Let e ∈ B ⊗ B op be the projection onto V , so that and so e θ = (u θ ⊗ u θ )e(u θ ⊗ u θ ) * is the orthogonal projection onto V θ . However, then e θ = (u θ ⊗ u θ )e(u * θ ⊗ u ⊤ θ ) = (θ ⊗ θ)(e), as claimed. (Indeed, note here that, acting on H, we have that Thus we might define an automorphism of S to be such a θ B with θ B (S) = S, equivalently, with (θ ⊗θ)(e) = e. However, there are a number of problems here. Firstly, different θ B might give the same θ, and hence the same automorphism of S. Secondly, we would wish there to be a bijection between automorphisms of a quantum adjacency matrix A G and automorphisms of the associated S, but it is not clear that every suitable θ arises from some θ B . Finally, when studying operator bimodules S, there is no real dependence on H, but it does not appear to be the case that any θ B on B(H), say restricting to θ on B, with give rise to some automorphism of B(K) whenever B ⊆ B(K).
With a special choice of H, we can say something. Set H = L 2 (B) and given θ ∈ Aut(B, ψ) formθ as in Definition 8.3. We now define θ B (x) =θxθ * for x ∈ B(L 2 (B)), an automorphism which restricts to θ on B.
Proposition 8.8. Let A G ∈ B(L 2 (B)) be a quantum adjacency matrix, satisfying axioms (1) and (2). Then θ ∈ Aut(B, ψ) is an automorphism of A G if and only if θ B (S) = S, where S is the B ′ -operator bimodule associated with A G .
Proof. This follows immediately from Proposition 8.6 and Lemma 8.7 by thinking about the projection associated to A G and S.

Quantum automorphisms of Quantum graphs
We now look at how (compact) quantum groups (co)act on (quantum) graphs. For compact quantum groups acting on classical graphs, this goes back to work of Banica in [2]. In the subsequent years, there has been much activity, with the quantum automorphisms of some small graphs computed, and much interest in the question of which graphs have a non-commutative quantum automorphism group. In this direction, let us just mention one recent paper [35] which constructs a finite graph which has a non-commutative, yet finite-dimensional, quantum automorphism group. More recently quantum automorphisms of quantum graphs have been defined and studied, [32, Section V, Part C] and more completely, [7,Section 3], compare [25,Section 4], [31,Section 2.3].
We find some of the literature in this area to be quite hard to follow, and so we shall both motivate the definitions from the classical situation, and develop certain aspects of compact quantum group theory to allow us to make broader definitions than available elsewhere. We will assume the reader is familiar with the basics of compact quantum group theory, for example, [33,42].
We start again from the classical situation, which while well-known, we feel provides interesting motivation. Let G = Aut(B, ψ) and let A = C(G) the group algebra, with coproduct The action of G on B gives a coaction of C(G) on B by Let us check that the coaction equation holds.
showing that indeed (α ⊗ id)α = (id ⊗∆)α. Notice that with these conventions, we are considering right coactions. One can also verify the Podlés density condition, that lin{(1 ⊗ a)α(b) : b ∈ B, a ∈ C(G)} is dense in B ⊗ C(G). Even in this setting, verifying this condition does not appear to be totally trivial to do by direct calculation, and we point the reader to [37,Section 2] for an interesting discussion.
That G is a group of automorphisms which preserve ψ means that (ψ ⊗ id)α(b) = ψ(b)1 for each b ∈ B. If we regard a quantum adjacency matrix A G as acting on B, then that G is a group of automorphisms of A G is equivalent to θ(A G (b)) = A G (θ(b)) for each b ∈ B, that is, (A G ⊗ id)α = αA G . If we, as is usual, regard A G as acting on L 2 (B), then we have to work harder to interpret this at the coaction level, see below.
The condition of preserving ψ was first considered by Wang in [45] (in fact, only the case ψ = Tr are worked out in [45,Section 6], but as remarked there, the same ideas work in general). We write QAut(B, ψ) for the maximal compact quantum group coacting on B and preserving ψ. This means that if (A, ∆) is any compact quantum group coacting on B and preserving ψ, there there is a unique * -homomorphism QAut(B, ψ) → A intertwining the coproducts and the coactions. That such a maximal quantum group exists is shown in [45,Theorem 6.1], where it is also show that QAut(B) does not exist: there is no maximal compact quantum group coacting on B, unless B is commutative.
Let us think more about coactions. The second condition is often called the Podleś density condition after Podleś's early work in this area. Let O A be the dense Hopf * -algebra of A formed from the coefficients of irreducible unitary corepresentations of A. There is a * -algebra O α ⊆ B (the Podleś subalgebra or algebraic core of B) which is dense, and such that α restricts to a * -homomorphism O α → O α ⊗ O A . In fact, this restriction is a Hopf * -algebraic coaction; in particular, it is injective. A modern reference for these results is [16].
For us, B is finite-dimensional, so O B = B, and hence α actually maps into B ⊗ O A . A modern approach to constructing QAut(B, ψ) makes use of the following observation. Let (e i ) n i=1 be an algebraic basis of B such that (Λ(e i )) is an orthonormal basis of L 2 (B). For any linear map α : B → B ⊗ A there is a matrix v = v ij ∈ M n (A) with α(e i ) = j e j ⊗ v ji for each i. When α is a coaction, our observations show that already v ij ∈ O A for each i, j.
We follow here ideas of Banica, see [3,Section 4], or [4, Section 1] for the case when ψ is a trace.
, here using the common leg-numbering notation. Define hom(v, w) to be the space of linear maps t : M n → M m with (t ⊗ 1)v = w(t ⊗ 1), an equation we can interpret simply using matrix multiplication. These definitions should of course be familiar to the reader who knows about corepresentations of compact quantum groups.
With respect to the basis (Λ(e i )), write the multiplication map m : B ⊗ B → B as a n × n 2 matrix, and write the unit map η : C → B as a n × 1 matrix. Also write the projection η • η * as a n × n matrix. The following proposition makes links between properties of a linear map α : B → B ⊗ A, and properties of v.
Proposition 9.2. Let A be a compact quantum group. Under this bijection between α and v, the following hold:

α is a unital map if and only if
3. assuming α is unital, α is ψ-preserving if and only if η • η * ∈ hom(v, v);

α satisfies the coaction equation if and only if v is a corepresentation.
Assume now that α is multiplicative and unital. If v is unitary, then α is a * -map. If α is a * -map, satisfies the coaction equation, and ψ-preserving, then v is unitary.
Proof. Most of this follows by simple direct calculation, and is rather similar to [4,Lemma 1.2] or [25,Lemma 4.5] (which both just considers the case when ψ is a trace). However, we prove the final claim, to check that it works for a non-trace, and to indicate that this step does not follow from direct calculation, but rather from quite a lot of theory.
Suppose that α is a unital * -homomorphism and ψ-preserving. Then consider, for i, j, As α preserves ψ, this is equal to ψ(e * i e j )1 = δ i,j 1 and so v * v = 1. As we know that actually v is a matrix in M n (O A ), Hopf * -algebra theory would show that v is invertible, as it has a left inverse. An alternative way to show this is to check that v is non-degenerate and then to appeal to [48,Proposition 3.2]. (We note here that we use "non-degenerate" in the sense of [48, page 630], which is weaker than the usual Hopf * -algebra definition.) Regardless, v is invertible, so necessarily also vv * = 1, so v is unitary.
Remark 9.3. The condition that η • η * ∈ hom(v, v) is often omitted, which is perhaps surprising. In fact, it is automatic in the presence of the other properties. Indeed, η ∈ hom(1, v) means that If v is assumed unitary, this is implies that η • η * ∈ hom(v, v).

△
It is now easy to see how to define the quantum automorphism group of B which preserves ψ, written as QAut(B, ψ). We define QAut(B, ψ) to be the universal C * -algebra generated by the coefficients of an n × n unitary matrix v = (v ij ) with the conditions The coproduct on QAut(B, ψ) is defined to satisfy ∆(v ij ) = n k=1 v ik ⊗ v kj , and exists by universality. We define α by α(e i ) = n j=1 e j ⊗ v ji , and the proposition shows that α is a coaction. As QAut(B, ψ) is generated by elements of the matrix v, which is by definition a (unitary) corepresentation, in fact QAut(B, ψ) is a compact matrix quantum group. One point in this definition which is far from obvious to the author is why the axioms of a compact quantum group hold, equivalently, why the transpose of v is an invertible matrix (an alternative axiom in the case of a compact matrix quantum group). We find it helpful to look at [5, Section 3] which explains links with the Tannaka-Krein reconstruction theorem. Also not considered in this construction is the Podleś density condition for α, but this follows easily once we know that v t is invertible. In fact, as we shall need the following later, we shall prove this by a more direct argument.
using that α is ψ-invariant. A similar calculation holds with arbitrary tensors, which shows that U is well-defined, and a co-isometry. The density condition on α (and that A acts non-degenerately on K) shows that U is unitary. Then We remark that we could instead have worked in the language of Hilbert C * -modules, which would remove the need to consider K, at the cost of more technicalities. The following now shows how α commuting with A G makes sense at the level of U .
We shall call U the unitary implementation of α, compare with the much more general construction of [43]. Recall from Proposition 9.2 that there is a bijection between coactions α and certain (unitary) matrices v ∈ M n (A). The following clarifies the relation between v and the unitary U . Proof. We simply calculate that for a ∈ A, Thus the matrix of U equals (v ij ), acting on L 2 (B) ⊗ K, as claimed.
In particular, U is a corepresentation (which also follows by direct calculation). The following definition should now be clear, given our motivation above from the case of classical groups, and Lemma 9.5. Definition 9.7. Let (A, ∆) be a compact quantum group, and let α : B → B ⊗ A be a coaction which preserves ψ. Let A G ∈ B(L 2 (B)) be a quantum adjacency matrix associated to the linear map A 0 on B. Then α is action on A G when αA 0 = (A 0 ⊗ id)α, equivalently, when A G and v commute, where v is the unitary corepresentation given by α.
Hence, in agreement with intuition, as soon as α is a ψ-preserving coaction on B, it is automatically an action on both the empty and complete quantum graphs. △ We next show that there exists a universal compact quantum group which acts on A G , and which we can then be justified in calling the quantum automorphism group of A G . We shall compare this definition to others in the literature in Remark 9.10. Proposition 9.9. Consider QAut(B, ψ) as constructed above from a generating unitary corepresentation v. Let A be the quotient formed by imposing the extra conditions that A G v = vA G . Then ∆ drops to A making (A, ∆) into a quantum subgroup of QAut(B, ψ), which is the quantum automorphism group of A G : that is, universal amoungst all compact quantum groups acting on A G .
Proof. Let J be the ideal generated by the relations coming from A G v = vA G , that is, J is the ideal generated by the element If we apply ∆ to one of these elements, we obtain n k,l=1 In the first line we add in and subtract the same element, but notice that the element in the brackets is manifestly in J. In the second step we re-index, and the re-arrange in the third step so both brackets are now clearly members of J. This calculation and a simple continuity argument shows that ∆ gives a well-defined map A → A ⊗ A, where A = QAut(B, ψ)/J. By definition, A coacts on A G . Now let A 0 be some other compact quantum group which coacts on A G . Then, by Proposition 9.2 for any orthonormal basis of L 2 (B) there is a unitary corepresentation v ∈ M n (A 0 ) which defines the coaction. If necessary replacing A 0 by the C * -subalgebra generated by the elements of v, we may suppose that A 0 is a compact matrix quantum group generated by v.
Consider now QAut(B, ψ) with generating corepresentation w say. As A 0 coacts in a ψ-preserving way, v satisfies all the relations which are satisfied by w, and so by universality, there is a *homomorphism θ : QAut(B, ψ) → A 0 which intertwines the coproducts and which sends w to v. Hence θ is a surjection. As A G v = vA G it follows that θ(J) = {0}, and so θ drops to a morphism from A, which is necessarily uniquely defined. This has hence demonstrated the required universal property for A.
We shall write QAut(A G ) for the resulting compact quantum group.  (QAut(B, ψ)), where the extra relations come from requiring that ρA 0 = (A 0 ⊗ id)ρ. As in Lemma 9.5, this is equivalent to requiring that u and A G commute. Thus this construction does agree with Proposition 9.9.
Gromada in [25, Section 4] makes much the same definition, but with a slightly different presentation of the relations. The links with Tannaka-Krein reconstruction results are made explicit. However, note that all definitions are made at the level of Hopf * -algebras, and so, for example, the definition of a coaction [25,Definition 4.2] differs from ours (but is equivalent). △ Remark 9.11. As alluded to a number of times, much of the existing work on quantum graphs restricts to the case when ψ is a δ-form (tracial, or not). There is some motivation for this if we consider QAut(B, ψ), as a result attributed to Brannan, [17,Proposition 20], describes QAut(B, ψ) in terms of a free product of a decomposition of B into the coarsest direct sum B = B i where ψ restricts to a δ-form on each B i . Thus for many questions, it suffices to look at each factor B i separately.
We might wonder if QAut(A G ) factors similarly. This does not appear to be the case, as there appears to be no reason why A G should restrict to each factor B i . To find a counter-example, let B = M 2 ⊕ M 2 where we equip each M 2 factor with differently normalised traces, say ψ 1 , ψ 2 with ψ = ψ 1 ⊕ ψ 2 . Then ψ is not a δ-form, but each ψ i is.
Let B act on H = C 2 ⊕ C 2 so that B ′ ∼ = ℓ ∞ 2 , say the span of e 1 , e 2 where e i is the projection onto the ith C 2 component. As usual, we identify B(H) with M 2 (B(C 2 )) ∼ = M 2 (M 2 ). Let for some non-zero t : C 2 → C 2 . Then S is an operator system, and so as B has a tracial state, S corresponds to a quantum adjacency matrix A G say, and to a projection in B ⊗ B op , say e G . Towards a contradiction, suppose that A G commutes with the projections onto each factor of B. As A G is self-adjoint, it follows that A G can be written as a sum of rank-one operators of the form θ Λ(a),Λ(b) where a, b both come from the same factor of B. It follows that e G is the span of tensors b ⊗ a * where a, b both come from the same factor of B. Thus the projection of B(L 2 (B)) onto S is a sum of maps of the form θ → bθa * where a, b both come from the same factor of B. However, b 0 0 0 x 0 a * 0 0 0 = 0 bt 0 0 a * 0 0 0 = 0, and similarly if b, a come from the 2nd factor. This is a contradiction, as the projection onto S should preserve x 0 . △ Examples of quantum automorphism groups of quantum graphs are considered in [25], and in particular in [31,Section 4], where all quantum graphs over B = M 2 are considered.

Quantum automorphisms of operator bimodules
As B ′ -operator bimodules S satisfying the conditions of Theorem 5.16 biject with quantum adjacency matrices A G , we immediately obtain a notion of an action of a quantum group on S. Of course, it would be more satisfying to have a definition which only used S.
If we consider what happens for (non-quantum) automorphisms of A G , say associated to a projection e G and B ′ -bimodule S, as in Section 8, then we found a link between automorphisms θ of A G , the condition that (θ ⊗ θ)(e G ) = e G , and θ B (S) = S, at least when S ⊆ B(L 2 (B)). Unfortunately, as is well-known, for quantum groups, forming the analogue of the "diagonal action" θ ⊗ θ is essentially impossible. We hence focus on quantum automorphisms of S, but guided by the commutative situation, we shall only consider the situation when B is acting on L 2 (B).
Proof. From Lemma 9.4, we know that X −1 = X, and so P is positive and invertible, with P −1 = XX t . By definition, x −2 0 = (P t ) −1 = XX * , and we calculate and so x −2 0 = XX * represents the operator ∇ −1 , so x 0 agrees with ∇ 1/2 . As U = XU X so that U t = X * U * X t , we see that U t P U = X * U * X t X * XXU X = X * U * (XX) t (XX)U X = X * U * U X = X * X = P.
Form now L 2 (B(H),ψ) with GNS mapΛ, so (Λ(θ ξ,η )|Λ(θ ξ ′ ,η ′ )) =ψ(θ η,ξ θ ξ ′ ,η ′ ) = (η|η ′ )Tr(P t θ ξ ′ ,ξ ) = (η|η ′ )(x 0 (ξ ′ )|x 0 (ξ)) = (η|η ′ )(x 0 (ξ)|x 0 (ξ ′ )) H . x → x ⊤ which is an anti- * -homomorphism. Similarly, so is the unitary antipode R : A → A. Following common usage in quantum group theory, for Y ∈ B(H) ⊗ A we write Y ⊤⊗R for the image of U under the tensor product of x → x ⊤ and R. One can check that U ⊤⊗R is also a unitary corepresentation, termed the contragradient corepresentation, compare [38,Section 3.3.3]. With respect to the orthonormal basis (Λ(e i )) of H, we have that x → x ⊤ is the transpose map on matrices, and so U ⊤⊗R has matrix (R(U ji )) i,j . Proof. As in Section 9.1, let A ⊆ B(K), and then for 1 ≤ s, t ≤ n and ν ∈ K, Compare this to We hence need to show that (x −1 0 U * x 0 ) tj = R(U tj ) for each t, j. To show this, we use some compact quantum group theory. Let S be the antipode on O A , which has decomposition S = Rτ −i/2 where (τ t ) is the scaling group, and τ −i/2 is the analytic extension. Recall the definition of the operator ρ U from [33, Section 1.4], and from before [33,Definition 1.4.5] recall that ρ U is the unique (up to a positive scalar) operator with ((ρ t U ) 1/2 ⊗ 1)U ((ρ t U ) −1/2 ⊗ 1) is unitary. From Lemma 9.4, we know that U = XU X −1 , and so ρ 1/2 U is |X| t , up to a scalar. As P = X * X, we see that |X| t = (P t ) 1/2 = x 0 . From after [33, Remark 1.7.7] we have that (id ⊗τ i/2 )(U ) = (ρ This previous result should be compared with [44, Corollary 2.6.3] (which itself follows quickly from [43,Proposition 4.2]). We do not invoke this directly, as we will need to use the exact form of the unitary w, and it seems about as much work to check that w really is the intertwining unitary as it does to give the above lemma (which is also of independent interest).
As κ restricts to a bijection B → B ′ , and κ(A G ) = JA * G J = A G because A G is self-adjoint, and using Proposition 5.20, it follows that S ′ 0 is the B-operator bimodule generated by A G . It now follows that wΛ( However, for u ∈ B ⊗ B op , with reference to Lemma 5.29, we see that Theorem 9.18. Let α be a coaction of a compact quantum group (A, ∆) on B, which preserves ψ, and has unitary implementation U . Let A G be a quantum adjacency matrix, and let S 0 be the B ′ -operator bimodule generated by A G . The following are equivalent: Proof. We have already seen that (2) and (3) are equivalent. Suppose that α U (S ′ 0 ) ⊆ S ′ 0 ⊗ A, and let A ⊆ B(K). By definition of the unitary implementatioñ U , given ξ =Λ(s ′ 0 ) ∈Λ(S ′ 0 ) and τ ∈ K, we see that Let f be the orthogonal projection of L 2 (B(H),ψ) ontoΛ(S ′ 0 ), so this calculation shows thatŨ (f ⊗ 1) = (f ⊗ 1)Ũ (f ⊗ 1). A non-trivial result about compact quantum groups (see the proof of [30, Proposition 6.2] for example) shows that thenŨ (f ⊗ 1) = (f ⊗ 1)Ũ .
Given the discussion before, we might have hoped instead to find that U A G = A G U was equivalent to α(κ(S)) ⊆ κ(S) ⊗ A, but it just seems that this is not so. However, working with S 0 and κ(S 0 ) instead of S and κ(S) is not so unnatural in light of Remark 5.32, because S 0 is exactly the "tracial" quantum graph which arises from S using Theorem 5.26. The dependence on the state ψ arises from the fact that we always assume that (A, ∆) coacts on B in a way which preserves ψ.
In the tracial case we have a more direct result.