Quantum relations in the general setting: composition and adjacency operators

Abstract: Quantum relations in the sense of Weaver are $M'$-bimodules, for a von Neumann algebra $M$, these generalising actual relations on a set $X$ when $M=\ell^\infty(X)$. Similarly, relations between two sets can be generalised as bimodules over the commutants of two algebras. We make an explicit study of this idea, developing some tools to check that constructions are well-defined. Motivation comes from Kornell's concept of a Quantum Set (for algebras which are sums of matrix algebras), and we find that $*$-homomorphisms correspond to certain quantum relations, extending unpublished work of Kornell. We find a functor from completely positive maps to quantum relations, related to the idea of taking a noisy communication channel and reducing it to its underlying relation''. As with Quantum Graphs, at least in finite-dimensions, quantum relations correspond toadjacency operators'', certain CP maps depending on a choices of faithful functional on the algebras. We develop some tools to deal with the non-Schur-idempotent case, and show links with our functor from CP maps, and work of Verdon. We explicitly compute the adjacency operator of a $*$-homomorphism.

• The paper establishes a representation-independent theory of quantum relations between von Neumann algebras and proves that coinjective relations correspond to normal *-homomorphisms, with quantum functions capturing unital maps.
• It constructs a functor from normal CP maps to quantum relations that recovers classical channel supports, preserves composition, and shows that finite-dimensional relations arise from CP maps while some cosurjective relations do not arise from UCP maps.
• It extends adjacency operators beyond Schur-idempotent maps so composition remains well behaved, characterizes symmetric CP-realisable relations through domination by a positive element, and identifies finite-dimensional and UCP-realisation limits.

Background and motivation

Quantum relations, introduced by Weaver, generalise ordinary relations on a set to the setting of a von Neumann algebra $M \subseteq B(H)$: a quantum relation is a weak$^*$-closed $M'$-bimodule $V \subseteq B(H)$, and quantum graphs are distinguished quantum relations over a single algebra. This paper by Daws develops the corresponding theory for relations between two von Neumann algebras $M$ and $N$, a notion implicit in Kornell's work on quantum sets and in unpublished material of Kornell, but not previously worked out in detail. The motivating examples are Kornell's quantum sets — von Neumann algebras that are $\ell^\infty$-direct sums of matrix algebras — for which a quantum relation is exactly an assignment of a subspace $R(X,Y) \subseteq B(X,Y)$ for each pair of atoms, recovering Kornell's category of quantum sets and relations.

Two equivalent formulations are established: a quantum relation from $M$ to $N$ is either an $^*$0-$^*$1-bimodule $^*$2, or a quantum relation on $^*$3 supported in the corner $^*$4. A substantial technical contribution of the early sections is a careful verification that all subsequent definitions — coinjectivity, cosurjectivity, the relation $^*$5 attached to a CP map, and the adjacency-operator correspondence — are independent of the faithful representations $^*$6, $^*$7 chosen. The key tool is a "base change" construction $^*$8 along a unital normal $^*$9-homomorphism, shown to commute with composition (so that the category of von Neumann algebras with quantum relation morphisms is well defined), and to preserve the corner condition.

Quantum functions and $M'$0-homomorphisms

The central structural result is a bijection between normal $M'$1-homomorphisms $M'$2 and coinjective quantum relations $M'$3 from $M'$4 to $M'$5, given by

$M'$6

A relation is coinjective when $M'$7 and cosurjective when $M'$8; a relation satisfying both is a quantum function. The proof proceeds by viewing $M'$9 as a self-dual Hilbert $V \subseteq B(H)$0-module over $V \subseteq B(H)$1 (self-duality following from separate weak$V \subseteq B(H)$2-continuity of the inner product) and applying the correspondence between self-dual Hilbert modules and normal $V \subseteq B(H)$3-homomorphisms. The paper refines this to a full characterisation: under the bijection, unital homomorphisms correspond to quantum functions, injective homomorphisms to surjective relations, $V \subseteq B(H)$4 is injective if and only if $V \subseteq B(H)$5 is central in $V \subseteq B(H)$6 with $V \subseteq B(H)$7, and for quantum functions surjectivity of $V \subseteq B(H)$8 matches injectivity of $V \subseteq B(H)$9. Moreover, $M$0 corresponds to an explicit partial inverse $M$1, constructed via the central projection cutting out $M$2, so that $M$3 when $M$4 is both injective and coinjective. This extends Kornell's finite-dimensional result to all von Neumann algebras and shows the category of quantum relations contains the category of normal $M$5-homomorphisms.

From channels to quantum relations

The paper then formalises the passage from a noisy communication channel to its underlying "relation". Classically, a channel with conditional distribution $M$6 yields the relation $M$7. In the quantum setting, given a normal CP map $M$8 with Stinespring dilation $M$9, one defines

$N$0

and proves this is a quantum relation from $N$1 to $N$2 independent of the dilation chosen: $N$3 is the weak$N$4-closure of $N$5 for any minimal dilation $N$6. In the classical case this recovers exactly the support relation of the channel. The construction is functorial: $N$7, generalising the corresponding fact for $N$8-homomorphisms, and $N$9 can be expressed as a pullback of the trivial relation $\ell^\infty$0 along $\ell^\infty$1, with pullbacks in turn realisable as $\ell^\infty$2.

Two results delineate the range of this construction. On the positive side, when $\ell^\infty$3 are finite-dimensional every quantum relation arises from some CP map. On the negative side — a notable explicit counterexample — not every cosurjective relation arises from a unital CP map: the paper exhibits $\ell^\infty$4 with $\ell^\infty$5 for which no UCP map $\ell^\infty$6 satisfies $\ell^\infty$7, the obstruction being that $\ell^\infty$8 admits no spanning family $\ell^\infty$9 with $R(X,Y) \subseteq B(X,Y)$0. The paper notes that the analogous phenomenon does not occur for quantum graphs, attributing this to the interplay of self-adjointness and reflexivity, and leaves the characterisation of UCP-realisable relations open.

Adjacency operators and the non-Schur-idempotent theory

In finite dimensions, quantum relations correspond to quantum adjacency operators: CP maps $R(X,Y) \subseteq B(X,Y)$1 that are Schur idempotent, equivalently projections $R(X,Y) \subseteq B(X,Y)$2, where $R(X,Y) \subseteq B(X,Y)$3 is a modularly twisted generalisation of the Choi matrix, $R(X,Y) \subseteq B(X,Y)$4. The paper extends this correspondence to operators that are not Schur idempotent: for arbitrary $R(X,Y) \subseteq B(X,Y)$5 with $R(X,Y) \subseteq B(X,Y)$6, the associated quantum relation is

$R(X,Y) \subseteq B(X,Y)$7

with the adjoint relation $R(X,Y) \subseteq B(X,Y)$8 corresponding to the KMS adjoint $R(X,Y) \subseteq B(X,Y)$9. This non-idempotent theory is necessary because composition of adjacency operators does not preserve Schur idempotency. The map $M$0 is shown to be a full, adjoint-preserving functor, and in the tracial case the relation attached to $M$1 coincides with $M$2. In general one obtains $M$3 for $M$4; the paper is candid that the appearance of the modular operators here is an artefact of the choices made, and that no alternative convention simultaneously makes $M$5 and identifies $M$6 with the KMS adjoint.

Two applications of this machinery are significant. First, the paper gives an operator-algebraic account of Verdon's result that every finite-dimensional quantum graph $M$7 (symmetric and reflexive) is the quantum confusability graph $M$8 of some UCP map $M$9, and then proves a new characterisation: a symmetric relation $N$0 arises from a (not necessarily unital) CP map if and only if there exists $N$1 in $N$2 such that every self-adjoint element of $N$3 is dominated by a scalar multiple of $N$4 — in which case $N$5 may be taken, and $N$6 may be taken finite-dimensional. Second, the paper explicitly computes the adjacency operator of a quantum function $N$7 in the non-tracial case. The answer is intricate: with $N$8 a $N$9-homomorphism, the Schur-idempotent $^*$00 has the form

$^*$01

where $^*$02 is constructed factor by factor from the central decomposition $^*$03 as the inverse of an explicitly defined positive invertible operator $^*$04 built from the matrix coefficients of $^*$05 and $^*$06. In the tracial case this collapses to the expected answer $^*$07, the map induced by $^*$08 on the GNS spaces.

Limitations and open questions

The paper is explicit that its adjacency-operator theory is finite-dimensional throughout: the identification of $^*$09 with $^*$10, the Markov trace, and the Choi-matrix bijection all rely on finite dimensionality, and the author notes that attempts to extend the Verdon-type arguments to general von Neumann algebras via conditional expectations $^*$11 fail because such expectations force $^*$12 to be a direct sum of matrix algebras. The ordering of relations translates only awkwardly to adjacency level: the inclusion order corresponds to the Schur-product order $^*$13, but no practical criterion for cosurjectivity of an adjacency operator is obtained. Several questions are left open: which cosurjective relations arise from UCP maps (beyond the counterexample showing not all do); a characterisation, in terms of the projection $^*$14, of when one can choose $^*$15 with $^*$16 — the paper shows Verdon's claimed invertibility condition on $^*$17 is necessary but not sufficient, with an explicit $^*$18 counterexample; and the computation of the adjacency operator for $^*$19 with $^*$20 merely CP, which the author observes is closely related to computing support projections of positive operators and appears intractable except by explicit calculation. The category-theoretic properties of the category of quantum relations between von Neumann algebras are likewise not pursued.

Conclusion

This paper consolidates and extends the theory of quantum relations between two von Neumann algebras, establishing well-definedness of the fundamental constructions, extending Kornell's bijection between quantum functions and $^*$21-homomorphisms to arbitrary von Neumann algebras, constructing a functor from CP maps to quantum relations that models the reduction of a channel to its support relation, and developing the non-Schur-idempotent adjacency-operator theory needed to make composition well behaved at the operator level. The explicit computation of the adjacency operator of a quantum function in the non-tracial case, and the characterisation of symmetric relations arising from CP maps via the domination condition on $^*$22, are the paper's most concrete new contributions, while the finite-dimensionality of the adjacency theory and the unresolved UCP-realisation question mark the boundaries of what is achieved.