Bigalois extensions and the graph isomorphism game
We study the graph isomorphism game that arises in quantum information theory from the perspective of bigalois extensions of compact quantum groups. We show that every algebraic quantum isomorphism between a pair of (quantum) graphs and arises as a quotient of a certain measured bigalois extension for the quantum automorphism groups and of the graphs and . In particular, this implies that the quantum groups and are monoidally equivalent. We also establish a converse to this result, which says that every compact quantum group monoidally equivalent to is of the form for a suitably chosen quantum graph that is quantum isomorphic to . As an application of these results, we deduce that the -algebraic, C-algebraic, and quantum commuting (qc) notions of a quantum isomorphism between classical graphs and all coincide. Using the notion of equivalence for non-local games, we deduce the same result for other synchronous non-local games, including the synBCS game and certain related graph homomorphism games.
Key words: quantum automorphism group, graph, non-local game, entanglement, monoidal equivalence, bigalois extension, representation category
MSC 2010: 20G42; 46L52; 16T20
Finite input-output games have received considerable attention in the quantum information theory literature as tools for investigating the structure of quantum correlations. The latter are meant to model the following setup (in one of several ways, depending on the specific chosen model ):
The perennial experimenters Alice and Bob, share an entangled quantum state, each performs one of quantum experiments and returns one of outputs. The respective quantum correlation is then defined as the collection of conditional probabilities of obtaining a given pair of outputs for a given pair of inputs, .
To recast this in game theoretic terms one typically proceeds as follows. Alice (A) and Bob (B) are regarded as cooperating players trying to supply “correct” answers to a referee (R) who communicates with A/B via sets of questions (inputs) . Alice and Bob then reply with answers from their respective sets of outputs . The game rules, which are known to A, B, and R, are embodied by a function , such that A and B win a round of the game if their respective replies and to the questions and satisfy , and they lose the round otherwise. Prior to the round, A and B can cooperate to develop a winning strategy, but are not allowed to communicate once the game begins. Cooperation consists of a pre-arranged strategy, which can be deterministic or random. We identify probabilistic strategies with the collection of conditional probability densities that they produce.
A probabilistic strategy is called perfect or winning, if the probability that they give an incorrect pair of answers is 0, i.e., . The key point is that the set of such conditional probability densities that can be obtained via quantum experiments in an entangled state is larger than the set of densities that can be obtained from classical shared randomness. For this reason winning quantum strategies can often be shown to exist when classical ones do not exist.
Our starting point in this context is the graph isomorphism game introduced in . Given two finite graphs and this game has inputs that are the disjoint union of the vertices of and the vertices of and outputs that are the same set. The referee sends A, B each a vertex and receives respective answers . Winning conditions require that
and belong to different graphs;
ditto for and ;
the “relatedness” of the inputs is reflected by that of the outputs: , or are distinct and connected by an edge, or distinct and disconnected if and only if the same holds for the vertices.
It turns out that the game has a perfect deterministic strategy if and only if the two graphs are isomorphic. This observation motivates the authors of  to introduce a notion of quantum isomorphism between finite graphs, relating to the existence of less constrained, random strategies for the graph isomorphism game. Whether or not two finite graphs and are quantum-isomorphic is governed by a -algebra , a non-commutative counterpart to the function algebra of the space of isomorphisms . More precisely, the algebra of continuous functions on the space of isomorphisms can be recovered as the abelianization of .
One is thus prompted to consider “quantum spaces” in the sense of non-commutative geometry: -algebras or C-algebras, thought of as function algebras on the otherwise non-existent spaces. In the same spirit we will work with quantum graphs (finite-dimensional C-algebras equipped with some additional structure mimicking an “adjacency matrix” § 3.1) and quantum groups i.e. non-commutative -algebras with enough structure to resemble algebras of representative functions on compact groups (§ 2.5).
As is the case classically, every quantum graph has a quantum automorphism group . The recent papers [18, 19, 20] uncover further remarkable connections between graph isomorphism games and quantum automorphism groups. Moreover, while  focuses on classical graphs, [19, 20] consider a more general categorical quantum mechanical framework which leads naturally to the notion of a quantum graphs and the generalization of the graph isomorphism game to that framework.
In particular,  obtains a characterization of (finite-dimensional) quantum isomorphic quantum graphs , in terms of simple dagger Frobenius monoids in the category of finite dimensional representations of the Hopf -algebra of the corresponding quantum automorphism group . On the other hand,  uses ideas from quantum group theory to establish the equivalence between the existence of C-quantum isomorphisms for graphs and the existence of perfect strategies for the isomorphism game within the so-called quantum commuting framework.
We regard the game -algebra as indicated above, as a non-commutative analogue of the space of isomorphisms . In particular, we say that and are algebraically quantum isomorphic, and simply write , if . Classically, the space of isomorphisms between two graphs is a principal homogeneous bundle over the automorphism groups of both graphs. In other words, if denotes the space of graph isomorphisms and (resp. ) denotes the automorphism group of (resp. ), then the canonical left/right actions are free and transitive. One of our main results is the quantum analogue of this remark; it is Theorem 4.33 below, and can be paraphrased as follows:
Let and be two quantum graphs. If the quantum isomorphism space is non-trivial then it is a quantum principal bi-bundle (bigalois extension) over the quantum automorphism groups and of and respectively.
This (non-commutative) bundle-theoretic perspective on has advantages: Although the construction of is purely algebraic and does not assume the existence of any C-representations of this object, we use the above result to show that this algebra always admits a faithful invariant state whenever it is non-zero (cf. Theorem 4.35), leading to connections with the notion of monoidal equivalence between quantum automorphism groups. Loosely speaking, we say that two compact quantum groups are monoidally equivalent if their categories of finite-dimensional unitary representations are equivalent as rigid C-tensor categories. Our main result here is an amalgamation of Theorem 4.35 and Theorem 4.39, and says:
Let be a quantum graph and its quantum automorphism group. Then the following hold:
If is another quantum graph such that , then admits a faithful state and is monoidally equivalent to the quantum automorphism group . If both and are moreover classical graphs, then admits a faithful tracial state.
Conversely, for any compact quantum group monoidally equivalent to , one can construct from this monoidal equivalence a quantum graph , an isomorphism of quantum groups , and an algebraic quantum isomorphism .
Recasting all of the above in the context of the (classical) graph isomorphism game, our results show that the condition is sufficient to ensure the existence of perfect quantum strategies for this game (Corollary 4.36 and Theorem 4.37):
Two classical graphs and are algebraically quantum isomorphic if and only if the graph isomorphism game has a perfect quantum-commuting -strategy.
We mention that a weaker version of the above theorem (that assumed the existence of a non-zero C-algebra representation of ) was recently proved in .
Using a notion of -equivalence for non-local games, we also use the above result to deduce the existence of perfect -strategies from purely algebraic data for synchronous binary constraint system (syncBCS) games and certain related graph homomorphism games (Corollary 5.52 and Corollary 5.53). We find all these results very striking because for generic synchronous games (e.g. the graph homomorphism game) the -algebra governing the game may be non-zero even if this algebra has no C-representations (and hence no perfect quantum strategies) .
We note also that the above results could be recast in a much broader context – rather than quantum graphs, we could consider arbitrary finite quantum structures: quantum sets (i.e. finite-dimensional -algebras with a fixed state) equipped with arbitrary tensors. Input-output isomorphism games could then be constructed as in the case of graphs, and the discussion replicated in that general framework. Our focus on (quantum) graphs is motivated by the contingent fact that the latter have received considerable interest in the literature.
2.1 Some notation
If is a natural number, we sometimes write for the ordered set . All vector spaces considered here are over the complex field. We use the standard leg numbering notation for linear operators on tensor products of vector spaces. For example, if are vector spaces and is a linear map, then is the linear map which acts as on the first and third leg of the triple tensor product, and as the identity on the second leg. We also typically denote the identity map on a vector space by .
2.2 Games and strategies
We lay out some definitions and a few basic properties of games and strategies. We will primarily be concerned with the graph isomorphism game, the graph homomorphism game and two versions of a game based on solving systems of linear equations over the binary field.
By a two-person finite input-output game we mean a tuple where are finite sets and
is a function that represents the rules of the game, sometimes called the predicate. The sets and represent the inputs that the players Alice and Bob can receive, and the sets and , represent the outputs that Alice and Bob can produce, respectively. A referee selects a pair , gives Alice and Bob , and they then produce outputs (answers), and , respectively. They win the game if and loose otherwise. Alice and Bob are allowed to know the sets and the function and cooperate before the game to produce a strategy for providing outputs, but while producing outputs, Alice and Bob only know their own inputs and are not allowed to know the other person’s input. Each time that they are given an input and produce an output is referred to as a round of the game.
We call such a game synchronous provided that: (i) Alice and Bob have the same input sets and the same output sets, which we denote by and , respectively, and (ii) satisfies:
that is, whenever Alice and Bob receive the same inputs then they must produce the same outputs. To simplify notation we write a synchronous game as .
A deterministic strategy for a game is a pair of functions, and such that if Alice and Bob receive inputs then they produce outputs . A deterministic strategy wins every round of the game if and only if
Such a strategy is called a perfect deterministic strategy. It is not hard to see that for a synchronous game, any perfect deterministic strategy must satisfy, .
On the other hand, a strategy for a game is called random if it can happen that for different rounds of the game, when Alice and Bob receive the input pair they may produce different output pairs. A random strategy thus yields a conditional probability density , which represents the probability that, given inputs , Alice and Bob produce outputs . Thus, and for each
In this paper we identify random strategies with their conditional probability densities, so that a random strategy will simply be a conditional probability density .
A random strategy is called perfect if
Thus, for each round of the game, a perfect strategy gives a winning output with probability 1.
Given a particular set of conditional probability densities, one can ask if the game not only has a perfect random strategy, but has one that belongs to a particular set of densities. The different kinds of probability densities that are studied in this context generally fall into two types: There are the local (loc) densities, also called the classical densities, which arise from ordinary random variables defined on probability spaces, and then there are the quantum densities that arise from the random outcomes of, especially, entangled quantum experiments. However, there are several different mathematical models for describing the densities obtained from quantum experiments. These models lead to sets of conditional probability densities know variously as the quantum (q), quantum spatial (qs) (or sometimes quantum tensor), quantum approximate (qa), and quantum commuting (qc) models.
Rather than go into a long explanation of the definitions of each of these sets, which is done many other places, we refer the reader to , for their definitions and merely summarize some of their basic relations below. Given inputs and outputs, we denote the set of conditional probability densities that belong to each of these sets by , where can be or . The following containments are known:
Moreover, for , it is known that . While for , we have by , and for , we have . The most famous question is whether or not , since this is known to be equivalent to Connes’ embedding conjecture .
We shall say that a game has a perfect t-strategy provided that it has a perfect random strategy that belongs to one of these sets, where can be either or . Moreover, we work with even broader classes of strategies we term C and (the latter being the broadest, i.e. weakest class; see Definition 2.2).
2.3 The -algebra of a synchronous game
In  a -algebra was affiliated with the graph homomorphism game, , whose representation theory determined whether or not a perfect t-strategy existed (see Section 2.4 for definitions). Later in  and [16, 17] these ideas were extended to any synchronous game. We begin by recalling the -algebra of a synchronous game and summarizing these results. This -algebra is defined by generators and relations arising from the rule function of the game.
Let be a synchronous game and assume that the cardinality of is while the cardinality of is . We will often identify with and with . We let denote the free product of copies of the cyclic group of order and let denote the complex -algebra of the group. We regard the group algebra as a -algebra, where for each group element we have .
For each we have a unitary generator such that . If we set then the eigenvalues of each is the set . The “orthogonal projection” onto the eigenspace corresponding to is given by
and these satisfy
The set is another set of generators for .
We let denote the 2-sided -ideal in generated by the set
and refer to it as the ideal of the game . We define the -algebra of to be the quotient
Note that since , in the quotient we will have that .
It is not hard to see that if we, alternatively, started with the free -algebra generated by and formed the quotient by the two-sided ideal generated by:
such that ,
then we obtain the same -algebra. We are not asserting that this algebra is non-zero. In fact, it can be the case that the identity belongs to the ideal, in which case the algebra is zero.
Let be a synchronous game.
has a perfect deterministic strategy if and only if has a perfect loc-strategy if and only if there exists a unital -homomorphism from to .
has a perfect q-strategy if and only if has a perfect qs-strategy if and only if there exists a unital -homomorphism from to for some non-zero finite dimensional Hilbert space.
has a perfect qa-strategy if and only if there exists a unital -homomorphism of into an ultrapower of the hyperfinite -factor,
has a perfect qc-strategy if and only if there exists a unital C-algebra with a faithful trace and a unital -homomorphism .
This theorem motivates the following definitions.
Let be a synchronous game. We say that has a perfect A-strategy provided is non-zero, and we say that has a perfect C-strategy provided that there is a unital -homomorphism from into for some non-zero Hilbert space .
2.4 Graphs and related games
A graph is specified by a vertex set and an edge set , satisfying and . Given two graphs and , a graph homomorphism from X to Y is a function with the property that . We write to indicate that there exists a graph homomorphisms from to . Graph homomorphisms encapsulate many familiar graph theoretic parameters. If we let denote the complete graph on vertices, i.e., the graph where every pair of vertices is connected by an edge, then
the chromatic number of is ,
the clique number of is ,
the independence number of is,
where denotes the graph complement of , i.e., the graph whose edge set is the complement of ’s.
The graph homomorphism game from to , which we shall denote by , is a synchronous game with inputs and outputs Alice and Bob win a round provided that whenever they receive inputs that are an edge in , then their outputs are an edge in and that whenever Alice and Bob receive the same vertex in they produce the same vertex in . This is also a synchronous game.
Note that a perfect deterministic strategy for the graph homomorphism game from to is a function that is a graph homomorphism. In particular, a perfect deterministic strategy exists if and only if . Similarly, we say that there is a t-homomorphism from to and write if and only if there exists a perfect t-strategy for the graph homomorphism game from to for , , etc.
2.4.1 The graph isomorphism game
Two graphs and are isomorphic if and only if there exists a one-to-one onto function such that is an edge in if and only if is an edge in . We write to indicate that and are isomorphic. If we let denote the adjacency matrix of and analogously for , then it is well-known and easy to check that if and only if there is a permutation matrix such that .
The graph isomorphism game, Iso(X,Y) between and is a game with the property that two graphs are isomorphic if and only if there exists a perfect deterministic strategy for . It was introduced by Atserias et al. .
The easiest way to describe the rules for this game is in terms of the relation between a pair of vertices. Formally, the relation on a graph is a function with
We remark that the matrix is known as the Seidel adjacency matrix of the graph.
The rules for this game can be stated loosely as requiring that to win, outputs must come from different graphs than inputs, outputs must have the same relations as inputs, and whenever one player’s output is the same as the other player’s input, then the same must hold for the other player. This final rule makes a deterministic strategy be a function and its inverse, instead of just a pair of functions. The input set and output set for this game is the disjoint union of with and
satisfies if and only if the following conditions are met:
belongs to a different graph than and belongs to a different graph than ,
if and are both vertices of the same graph, then .
if and are from different graphs and , then ,
if and are from different graphs and , then .
Now it is not hard to see that this game is synchronous and it has a perfect deterministic strategy if and only if . Indeed, if it has a perfect deterministic strategy, then there must be a function and the rules force and . Denoting the restrictions of to and by and . The fact that tells us that is one-to-one and preserves the edge relationships, since is also one-to-one, and so both and define graph isomorphisms. However, note that the rules of the game do not require that and be mutual inverses.
We will write if and only if this game has a perfect t-strategy for .
The following result characterizes .
Let and be graphs on vertices. Then is generated by self-adjoint idempotents satisfying:
for and , ,
Recall that for any game, we will have generators, with , , and for . So (2) and (6) are automatically met.
To see (1), note that if , then for all . Hence for and fixed we have that
The case that is identical.
Note that (4) follows from (1).
To see (3), note that
Now unless , so we have that A similar calculation shows that . Hence, .
Now (5) follows from (3) and (4). Similarly, (7) follows from (3) and (6).
Finally to see (8), we have that
since unless . Similarly, one shows that is equal to this latter sum and (8) follows.
A nice compact way to represent the above relations is to consider the matrix . Then by (2) every entry is a self-adjoint idempotent, while (4) and (5) imply that is the identity matrix, i.e., that is a unitary. We also, by (6) and (7), have that entries in each row and column are pairwise “orthogonal”, i.e., have pairwise 0 product. Such a matrix will be referred to as a quantum permutation over the -algebra .
Equation (8) implies that where and denote the adjacency matrices of the graphs, and is the unit of the algebra. Thus, Proposition 2.3 can be summarized as saying that is the -algebra generated by subject to the relations that is a quantum permutation with . We have that if and only if a non-trivial -algebra exists satisfying these relations.
if and only if there exist a and projections such that is a unitary in and
if and only if there exist projections such that is a unitary and
if and only if there exists projections in some C-algebra with a trace such that is a unitary and ,
if and only if there exists projections on a Hilbert space such that is a unitary and .
Also, if there exists a unital -homomorphism from , then will be a permutation matrix, satisfying , which is the classical notion of isomorphism for graphs.
Note that we have the following obvious implications.
Moreover, it is known that the first two implications are not reversible [1, 17]. The question of whether the third implication holds is still open. The question whether the implications hold for generic and had remained open for quite some time. Only very recently the implication was obtained in . One of our main results is that the implication holds. In other words, if and only if admits a tracial state. This is somehow surprising, because the same conclusion cannot be made for the algebras .
2.5 Compact quantum groups
We begin by recalling that a Hopf algebra is a quadruple where is a unital associative algebra with multiplication map , and , , are unital algebra morphisms satisfying
The maps given above are called the comultiplication, counit, and antipode, respectively. We typically just refer to a Hopf algebra with the symbol if the other structure maps are understood and there is no danger of confusion. A Hopf -algebra is a Hopf algebra where is a -algebra and the comultiplication and counit are -homomorphisms.
A compact quantum group (CQG) is a Hopf -algebra for which there exists a C-norm on making the comultiplication continuous with respect to the minimal C-tensor norm (in short, we say that is -compatible).
The motivating example of a CQG is given by the Hopf -algebra of representative functions on a compact group . Here, is the map , , and , where is the unit. Here the C-norm on is the uniform norm coming from , and it is relatively easy to see that this is the unique C-norm making the comultiplication -continuous.
Motivated by the above example, it is customary to use the symbol to to denote an arbitrary CQG and write for the Hopf -algebra associated to . Here we are viewing as a non-commutative algebra of “representative functions” on some “quantum space” , which comes equipped with a group structure.
Another example of a CQG is given by the Pontryagin dual of a discrete group . Here , , , and for each . In this case, one can in general choose from a variety of -compatible C-norms. The two most common ones are the maximal C-norm on and the reduced C-norm, the latter being induced by the left regular representation of on .
A few “purely quantum” examples follow.
Wang and Van Daele’s universal unitary quantum group  associated to a matrix is given by
together with Hopf--algebra maps , and .
The term universal in the above definition will be made precise in the paragraph following Remark 2.11. For the time being, now we suffice it to say that the quantum groups play the analogous universal role for compact matrix quantum groups that the ordinary compact matrix groups: Any compact matrix group arises as a closed quantum subgroup of some . That is, there exists a surjective Hopf -algebra morphism .
The quantum permutation group on points  is given by underlying Hopf -algebra which is the universal -algebra generated by the entries of an magic unitary: a matrix
consisting of self-adjoint projections summing up to across all rows and columns, and satisfying the orthogonality relations and . The Hopf -algebra maps are defined exactly as for .
Every compact quantum group comes equipped with a unique Haar state, which is a faithful state satisfying the left and right invariance conditions
The norm on induced by the GNS construction with respect to is always a -compatible norm, and it is the minimal such C-norm. We denote by , the corresponding C-algebra (the reduced C-algebra of ). The universal C-algebra of , , is the enveloping C-algebra of . By universality, this C-norm is also -compatible.
Often in the literature compact quantum groups are defined in terms of a pair where is a unital C-algebra and is a co-associative unital -homomorphism such that and are linearly dense in . One then obtains the Hopf -algebra as a certain dense -subalgebra (spanned by coefficients of unitary representations of , which we describe below).
Let be a CQG and a finite dimensional Hilbert space. A representation of on is an invertible element such that
A representation of is called unitary if is unitary. Note that if we fix an orthonormal basis for , then a representation corresponds to an invertible matrix such that
For any CQG, we always have the trivial representation on given by . Given two representations and , we can always form the direct sum , tensor product , and conjugate representation given by (if ). A morphism between and is a linear map such that . The Banach space of all morphisms between and is denoted by . If and are unitary representations, then . We say that two representations and are equivalent if there exists an invertible element . We say that is irreducible if .
The fundamental theorem on finite dimensional representations of CQGs is stated as follows.
Theorem 2.10 ()
Let be a CQG. Every finite dimensional representation of is equivalent to a unitary representation, and every finite dimensional unitary representation of is equivalent to a direct sum of irreducible representations. Moreover, is linearly spanned by the matrix elements of irreducible unitary representations of
In the language of Hopf algebras, a (unitary) representation of is typically called a (unitary) comodule over . These notions obviously make sense for general Hopf -algebras.
We end this section by recalling that a matrix Hopf -algebra is a Hopf -algebra that is generated by the coefficients of some corepresentation of . A useful fact in this regard from  is that if a matrix Hopf -algebra is generated by a corepresentation that is equivalent to a unitary one, then is the Hopf -algebra of some compact quantum group . In this case, we call a compact matrix quantum group and we call a fundamental representation of . By replacing with an equivalent unitary representation , note that is still generated by the matrix elements of , and is a unitarizable representation. Hence there exists some so that is a unitary representation. This means that there is a surjective morphism of Hopf -algebras defined by ,where is the fundamental representation of given in its definition. In particular, is a so-called closed quantum subgroup of (written ).
3 Quantum sets, graphs and their quantum automorphism groups
The examples of CQGs that feature in this paper are the quantum automorphism groups of certain finite structures, such as sets, graphs, and their quantizations. In order to describe these objects, we first quantize the notion of a (measured) finite set, then proceed to quantum graphs. All of the definitions that follow are quite standard in the operator algebra literature [28, 2, 3, 11]. The idea of a quantum set or a quantum graph also appears in [19, 20] using the language of special symmetric dagger Frobenius algebras.
3.1 Quantum sets and graphs
A (finite, measured) quantum set is a pair , where is a finite dimensional C-algebra and is a faithful state.
We write for , and refer to this value as the cardinality or size of .
The reason for our choice of notation is that when is commutative, Gelfand theory tells us that we are really just talking about a finite set (the spectrum of ) equipped with a probability measure defined for each .
Let be a quantum set. Let and be the multiplication and unit maps, respectively. In what follows, we will generally only be interested in a special class of finite quantum sets – namely those that are measured by a -form , which we now define:
Definition 3.13 ()
Let . A state is called a -form  if
where the adjoint is taken with respect to the Hilbert space structure on coming from the GNS construction with respect to .
For purposes of distinguishing between the Hilbert and C-structures on , we denote this Hilbert space by .
The most basic examples of -forms are given by the uniform measure on the -point set and the canonical normalized trace on . In the first case, a simple calculation shows that , where is the standard basis of projections for , and so we have . In the second case, one can show that , where are the matrix units for . So in this case we have . More generally, if we have a multimatrix decomposition and is a faithful state (so and ), then is a -form if and only if for all . In particular, admits a unique tracial -form with given by .