This paper studies quantum variations of the Shannon capacity of graphs via the theory of asymptotic spectra. The Shannon capacity of a graph was introduced by Shannon in [Sha56] and is defined as
where denotes the independence number of and where denotes the -th strong graph product power of . (All concepts used in this introduction will be defined in Section 2.) The definition of this graph parameter is motivated by the study of classical noisy communication channels. One associates to a classical noisy communication channel the confusability graph
with vertices labeled by the input symbols of the channel, and edges given by pairs of input symbols that may be mapped to the same output by the channel with nonzero probability. The Shannon capacity then measures the amount of information that can be transmitted over the channel without error, asymptotically. Deciding whetheris NP-complete [Kar72], and Shannon capacity is not even known to be a computable function.
A natural approach to study the Shannon capacity is to construct graph parameters that are upper bounds on Shannon capacity. Shannon himself introduced an upper bound in [Sha56], which is known as the fractional packing number or Rosenfeld number. In the seminal work of Lovász [Lov79], the Lovász theta function was introduced to upper bound the Shannon capacity. Remarkably, the theta function can be written as a semidefinite program that is efficiently computable. Using the theta function, Lovász proved that , where is the -cycle graph. Lovász further conjectured that for every graph . This conjecture was shown to be false by Haemers. He introduced the Haemers bound as an upper bound on the Shannon capacity, and showed that for some Kneser graph and field [Hae79]
. For the odd cycle graphswith , it is still open whether . For example, the currently best lower bound on is [PS18], whereas .
Recently, a dual characterization of the Shannon capacity was found by Zuiddam in [Zui18b] via the theory of asymptotic spectra. This theory was developed by Strassen in [Str88], see also the exposition in [Zui18a, Chapter 1]. In the general theory we are given a commutative semiring with addition , multiplication , and a preorder on that satisfies the properties to be a “Strassen preorder”. For , the rank is defined as the minimum number such that , and the subrank is defined as the maximum number such that , where stands for the sum of times the element . The asymptotic rank of is defined as the regularization and the asymptotic subrank as the regularization . The asymptotic spectrum of with respect is the set of all -monotone semiring homomorphisms . Strassen proves that the asymptotic rank of equals the pointwise maximum over the asymptotic spectrum and the asymptotic subrank equals the pointwise minimum. Strassen also defines the asymptotic preorder on by if there exists a sequence such that and such that for all holds . He proves that if and only if for every in the asymptotic spectrum holds .
The theory of asymptotic spectra was originally motivated by the study of tensor rank and asymptotic tensor rank[Str86, Str87, Str88, Str91], which are the keys to understanding the arithmetic complexity of matrix multiplication (see, e.g., [BCS97]). Here we let be any family of isomorphism classes of tensors that is closed under direct sum and tensor product, and which contains the “diagonal tensors”. We let be the restriction preorder, which in quantum information theory language is the preorder corresponding to convertibility by stochastic local operations and classical communication (SLOCC). The restriction preorder is a Strassen preorder, the rank as defined above equals tensor rank, and the asymptotic rank as defined above equals asymptotic tensor rank. Recently, Christandl, Vrana and Zuiddam in [CVZ18] constructed for the first time an infinite family of elements in the asymptotic spectrum of tensors over the complex numbers. A study of tensors with respect to local operations and classical communication was carried out in [JV18].
Let us return to the study of graphs as in [Zui18b]. Here is any family of isomorphism classes of graphs that is closed under the disjoint union and the strong graph product, and which contains the -vertex empty graph for all . Let be the cohomomorphism preorder, which is defined by letting if there is a graph homomorphism from the complement of to the complement of . The cohomomorphism preorder is a Strassen preorder, the subrank of a graph equals the independence number, and the asymptotic subrank equals the Shannon capacity [Zui18b]. Known elements in the asymptotic spectrum of graphs are the Lovász theta function [Lov79], the fractional Haemers bounds over all fields [Bla13, BC18], the complement of the projective rank [MR16] and the fractional clique cover number (see [Sch03, Eq. ]. The fractional Haemers bounds provide an infinite family of elements in the asymptotic spectrum due to the separation result in [BC18]. We note that the dual characterization is nontrivial in the sense that the asymptotic tensor rank of tensors and the Shannon capacity of graphs are not multiplicative. We note that Fritz in [Fri17] developed a theory for commutative monoids analogous to Strassen’s theory of asymptotic spectra and that he applied this theory to graphs to obtain a dual characterization of Shannon capacity and of the asymptotic preorder in terms of -monotone monoid-homomorphisms.
Quantum Shannon capacity of graphs
We now turn to the quantum setting. We consider two quantum variants of graph homomorphism. The first variant is characterized by the existence of perfect quantum strategies for the graph homomorphism game [MR16], which is defined as follows. Two players Alice and Bob are given two graphs and . During the game, the referee sends to Alice some vertex and to Bob some vertex . Alice responds to the referee with a vertex and Bob respond to the referee with a vertex . Alice and Bob win this instance of the -homomorphism game, when their answer satisfy
Alice and Bob are not allowed to communicate with each other after having received their input from the referee, but they may together decide on a strategy beforehand. It is not hard to see that Alice and Bob can win the -homomorphism game with a classical strategy (i.e. not sharing entangled states) if and only if there is a graph homomorphism from to . We say that there is a quantum homomorphism from to , and write , if there exists a perfect quantum strategy for Alice and Bob to win the -homomorphism game. It is not hard to see that implies . The quantum cohomomorphism preorder is defined by letting if . The quantum independence number of is defined as the maximum number such that and the quantum Shannon capacity of is defined as its regularization.
Entanglement-assisted Shannon capacity of graphs
The second quantum variant of graph homomorphism comes from the study of entanglement-assisted zero-error capacity of classical channels, which is a quantum generalization of Shannon’s zero-error communication setting. In the zero-error communication model, Alice wants to transmit messages to Bob without error through some classical noisy channel. Shannon in [Sha56] showed that the maximum number of zero-error messages Alice can send to Bob equals the independence number of the confusability graph. In the entanglement-assisted setting, the maximum number of messages that can be sent with zero error turns out to be completely determined by the confusability graph as well. It is called the entanglement-assisted independence number of the confusability graph [Bei10]. Similarly, the entanglement-assisted Shannon capacity is its regularization. Based on this definition, one naturally defines an entanglement-assisted homomorphism between graphs, denoted by [CMR14]. Let the entanglement-assisted cohomomorphism preorder be defined by letting if . The entanglement-assisted homomorphism has applications in the study of the entanglement-assisted source-channel coding problem [BBL15, CMR14]. It is easy to see that the entanglement-assisted independence number of is the maximum number such that .
It is not hard to see that implies . Interestingly, the reverse direction holds if one can prove that the maximally entangled state is sufficient to achieve the entanglement-assisted homomorphism [MR16]. On the other hand, it is known that the entanglement-assisted Shannon capacity can be strictly larger than the Shannon capacity [LMM12, BBG13]. In fact, the lower bounds they proved for the entanglement-assisted Shannon capacity use the maximally entangled state, thus they are also lower bounds for the quantum Shannon capacity.
Entanglement-assisted Shannon capacity of non-commutative graphs
Finally, we consider the entanglement-assisted Shannon capacity of quantum channels. It turns out, analogous to the classical channel scenario, that the one-shot (entanglement-assisted) zero-error classical capacity of a quantum channel can be characterized in terms of the non-commutative graph associated with the channel [DSW13]. A non-commutative graph, or nc-graph for short, is a subspace
of the vector space ofcomplex matrices, satisfying and . Duan in [Dua09] and Cubitt, Chen and Harrow in [CCH11] have shown that every such subspace is indeed associated to a quantum channel. There is a natural preorder on nc-graphs such that the entanglement-assisted independence number , defined in [DSW13], equals the maximum number such that [Sta16], where is the non-commutative graph corresponding to the -message perfect classical channel. The entanglement-assisted Shannon capacity of nc-graphs is the regularization of .
Overview of our results
In this paper, we extend the study of the asymptotic spectrum of graphs to the quantum domain. We introduce three new asymptotic spectra:
the asymptotic spectrum of graphs with respect to the quantum cohomomorphism preorder
the asymptotic spectrum of graphs with respect to entanglement-assisted cohomomorphism preorder
the asymptotic spectrum of non-commutative graphs with respect to the entanglement-assisted cohomomorphism preorder.
We prove that the preorders in these scenarios are Strassen preorders. This allows us to apply Strassen’s spectral theorem to obtain a dual characterization of the corresponding Shannon capacities and asymptotic preorders in terms of the asymptotic spectra. We then perform a study of the relations among the three new asymptotic spectra and the asymptotic spectrum of graphs of [Zui18b]. The bounds on the several Shannon capacities that have appeared in the literature we fit into the corresponding asymptotic spectra, and we find a new element in the asymptotic spectrum of graphs with respect to quantum homomorphisms, namely the fractional Haemers bound over the complex numbers.
Organization of this paper
In Section 2 we cover the basic definitions of graph theory; the definition of the Lovász theta function, the fractional Haemers bounds, the projective rank and the fractional clique cover number; the theory of asymptotic spectra of Strassen; the known properties of the asymptotic spectrum of graphs; the definition of the quantum homomorphism; the definition of the entanglement-assisted homomorphism of graphs; and the definition of the (entanglement-assisted) Shannon capacity of non-commutative graphs. In Section 3 we study the quantum Shannon capacity and the entanglement-assisted Shannon capacity via the corresponding asymptotic spectra. In Section 4 we study the entanglement-assisted Shannon capacity of non-commutative graphs via the corresponding asymptotic spectrum.
2.1 Graphs, independence number, and Shannon capacity
In this paper we consider only finite simple graphs, so graph will mean finite simple graph. For a graph , we use to denote the vertex set of and to denote the edge set of . We write to denote an edge between vertex and . Since our graphs are simple, implies that . The complement of is the graph with and . (We emphasize that when we write we include the case that .) For , the complete graph is the graph with and . Thus is the empty graph and is the graph consisting of a single vertex and no edges. A graph homomorphism from to is a map , such that for all , implies . We write if there exists a graph homomorphism from to .
A clique of is a subset of , such that for any holds . The size of the largest clique of is called the clique number of and is denoted by . Equivalently,
An independent set of is a clique of . The size of the largest independent set of is called the independence number of and is denoted by . Equivalently,
Let and be graphs. The disjoint union is the graph with and . The strong graph product is the graph with
We use to denote . The Shannon capacity of [Sha56] is defined as
This limit exists and equals the supremum by Fekete’s lemma.
For any and any field , let be the space of matrices with coefficients in . Let be the identity matrix. Let . Then denotes the complex conjugate of . The element is called a projector if and , i.e. is Hermitian and idempotent.
2.2 Upper bounds on the Shannon capacity
Deciding whether is NP-hard [Kar72] and it is not known whether the Shannon capacity is a computable function. In the study of , the following graph parameters have been introduced that upper bound . They will play an important role in this paper.
Lovász theta function
An orthonormal representation of a graph is a collection of unit vectors indexed by the vertices of , such that non-adjacent vertices receive orthogonal vectors: for all . The celebrated Lovász theta function [Lov79], is defined as
where the minimization goes over unit vectors and orthonormal representations of . Lovász proved that
Fractional Haemers bound
A -representation of a graph over a field is a matrix of the form , such that for all and if . Let be the set of all -representation of over . The fractional Haemers bound [Bla13, BC18], as a fractional version of the Haemers bound [Hae79], is defined as
Bukh and Cox proved that
It is worth noting that whether the Haemers bound is computable remains unknown. Interestingly, for any field of nonzero characteristic and , there exists an explicit graph so that if is any field with a different characteristic, [BC18, Theorem 19].
A -representation of a graph is a collection of rank- projectors , such that if . The projective rank [MR16] is defined as
The complement of the projective rank, , is an upper bound on the Shannon capacity,
Fractional clique cover number
The chromatic number is the smallest number such that the vertices of can be colored with colors such that adjacent vertices receive different colors. The clique cover number is defined as the chromatic number of the complement, . Let be the lexicographic product of and . The fractional clique cover number is defined as
It is known that . It is also known that (e.g., [Sch03]).
Relationships between graph parameters
We know the following inequalities among the graph parameters that we have just defined:
The inequalities in (7) can be found in [Lov79, MR16]. The inequalities in (8) follow from the work in [BC18]. The inequalities in (9) are readily verified using Lemma 20, and the fact that the real fractional Haemers bound is at most the complement of the real projective rank , since we obtain the definition of from the definition of by requiring the -representations of to be positive semidefinite.
2.3 Asymptotic spectra and Strassen’s spectral theorem
A semiring is a set equipped with a binary addition operation , a binary multiplication operation , and elements , such that for all holds
A semiring is commutative if for all holds . For any natural number , let denote the sum of times the element .
A preorder on is a relation such that for any holds that , and that if and , then . A preorder on is a Strassen preorder if for all , holds
|(14)||in if and only if in|
|(15)||if , then and|
Let and be semirings. A semiring homomorphism from to is a map such that , for all , and . Let be the semiring of non-negative real numbers with the usual addition and multiplication operations. The asymptotic spectrum of the semiring with respect to the preorder is the set of -monotone semiring homomorphisms from to , i.e.
Let . The subrank of is defined as . The asymptotic subrank of is defined as
Since is supermultiplicative, Fekete’s lemma implies that the limit in (18) indeed exists and can be replaced by a supremum, i.e.
Strassen proved the following dual characterization of in terms of the asymptotic spectrum.
Let be a commutative semiring and let be a Strassen preorder on . For any such that and for some , holds
Besides asymptotic subrank, the asymptotic spectrum of a commutative semiring with respect to a Strassen preorder also characterizes the asymptotic rank and the asymptotic preorder associated to . The rank is defined as . The asymptotic rank is defined as , which equals by Fekete’s lemma. The dual characterization is that , assuming there is a with . The asymptotic preorder associated to is defined by if there is a sequence of natural numbers such that and such that for all holds . The dual characterization is that if and only if for all holds . See [Str88, Theorem 3.8, Cor. 2.6] and see also [Zui18a, Cor. 2.13, Theorem 2.12].
Finally, we mention that the asymptotic spectrum is well-behaved with respect to subsemirings. Let be a commutative semiring, let be a Strassen preorder on , and let be a subsemiring, which means that and that is closed under addition and multiplication. Then clearly the restriction of to is a Strassen preorder on . For any the restricted function is clearly an element of . The opposite is also true.
Let be a commutative semiring, let be a Strassen preorder on , and let be a subsemiring. For every element there is an element such that restricted to equals .
We note that the proof of Theorem 2 is nonconstructive.
2.4 Semiring of graphs and the dual characterization of Shannon capacity
Let be the set of isomorphism classes of (finite simple) graphs. The cohomomorphism preorder on is defined by if and only if , i.e. there is a graph homomorphism from the complement of to the complement of . Zuiddam proved in [Zui18b] that is a commutative semiring and that the cohomomorphism preorder is a Strassen preorder on . By definition, the asymptotic spectrum of graphs consists of all maps such that, for all , holds
Note that the subrank of a graph equals the independence number of , since equation (2) is exactly
By Theorem 1, the Shannon capacity is dually characterized as
The known elements belonging to the asymptotic spectrum of graphs are: the Lovász theta function [Lov79], the fractional Haemers bound over any field [BC18, Bla13], the complement of projective rank [MR16, CMR14] and the fractional clique cover number [Sch03]. Note that there are infinitely many elements in , due to the separation result in [BC18] of the fractional Haemers bound over different fields. In addition, we note that the fractional clique cover number is the pointwise largest element in . This is because the rank of a graph equals the clique cover number and the asymptotic clique cover number equals the fractional clique cover number, see [Zui18b].
2.5 Quantum variants of graph homomorphism
We present mathematical definitions of the two quantum variants of graph homomorphisms, arising from the theory of non-local games and from quantum zero-error information theory, respectively.
2.5.1 Quantum homomorphism
Definition 3 (Quantum homomorphism [Mr16]).
Let and be graphs. We say there is a quantum homomorphism from to , and write , if there exist and projectors for every and , such that the following two conditions hold:
|(25)||for every we have|
|(26)||if and , then .|
The first condition implies for all and . Namely, for a fixed and an arbitrary , implies . Since every is a projector, we have . We conclude that since projectors are also positive semidefinite.
For every collection of complex projectors satisfying the above two conditions, there exists a collection of real projectors which also satisfies the above two conditions. Namely, we take the collection of real matrices
where and denote the real part and the image part of , respectively. Noting that , we have
It is easy to see that implies . The opposite direction is not generally true [MR16]. The quantum cohomomorphism preorder on graphs is defined by if and only if , and the quantum independence number as . The quantum Shannon capacity is defined as .
2.5.2 Entanglement-assisted homomorphism
Definition 5 (Entanglement-assisted homomorphism [Cmr14]).
Let and be graphs. We say there is a quantum homomorphism from to , and write , if there exist and positive semidefinite matrices and , such that the following two conditions hold
|(27)||for every we have|
We note that the positive semidefinite matrix can be further restricted to be positive definite.
The entanglement-assisted cohomomorphism preorder is defined by if and only if . The entanglement-assisted independence number can be defined as . The entanglement-assisted Shannon capacity of is defined as .
2.6 (Entanglement-assisted) zero-error capacity of quantum channels
For related definitions in quantum information theory, we refer the reader to [NC10]. We use and to denote the (finite-dimensional) Hilbert spaces of the sender (Alice) and the receiver (Bob), respectively. Let be the space of linear operators from to . Let . The space is isomorphic to the matrix space with . Let be the set of all (mixed) quantum states, i.e. all trace- positive semidefinite operators in . A quantum state is pure if it has rank , i.e. if it can be written as for some unit vector . The support of a positive semidefinite matrix is the subspace ofcan be characterized by a completely positive and trace-preserving (CPTP) map. This is equivalent to saying that is of the form for some linear operators , called the Choi–Kraus operators associated to , satisfying . (The Choi–Kraus operators are not unique, but they are unique up to unitary transformations, see, e.g., [NC10].)
We focus on the setting in which Alice and Bob use a quantum channel to transmit classical zero-error messages. To transmit classical messages to Bob through the quantum channel , Alice prepares pairwise orthogonal states , where orthogonality is defined with respect to the Hilbert–Schmidt inner product . Bob needs to distinguish the output states perfectly, in order to obtain the messages without error. This is only possible when the output states are pairwise orthogonal. In this situation, without loss of generality, Alice may select the to be pure states for all . Note that if and only if for all , where are the Choi–Kraus operators of . We now see that the number of messages one can transmit through the channel is determined by the linear space of matrices . Duan, Severini and Winter called the linear space the non-commutative graph (nc-graph) of a quantum channel [DSW13]. The nc-graphs may be thought of as the quantum generalization of confusability graphs of classical channels, mentioned in Section 2.5.2. In this analogy, for an nc-graph the density operators are input symbols of the channel, and they are “non-adjacent” in the nc-graph if for all and in the support of and , respectively. As in the classical setting, “non-adjacent vertices” are nonconfusable.
Note that for every quantum channel , the associated nc-graph satisfies and , where is the identity operator in . It is shown in [Dua09, CCH11] that any subspace that satisfies and is associated to some quantum channel. From now, we define a non-commutative graph or nc-graph as a subspace satisfying and . We define the independence number as the maximum such that there exist pure states satisfying for all . The Shannon capacity is defined as , where denotes the tensor product of and . One verifies that if and are the nc-graphs of and respectively, then the tensor product is the nc-graph of the quantum channel . Then (the logarithm of) is exactly the classical zero-error capacity of quantum channels whose nc-graph is [DSW13].
In the quantum setting, it will be more natural to consider that Alice and Bob are allowed to share entanglement to assist the information transmission. To make use of the entanglement, say , Alice prepares quantum channels for encoding the classical messages. To send the th message, Alice applies to her part of , and sends the output state to Bob via the quantum channel . Bob needs to perfectly distinguish the output states for . The following lemma shows that the maximum number of classical message which can be sent via the quantum channel in the presence of entanglement can be also characterized by the nc-graph , as also mentioned in [DSW13, Sta16].
Lemma 7 (Implicitly in [Sta16]).
Let , , and as above. Let and be the Choi–Kraus operators of and , respectively, for . Then
is equivalent to
Let be the Schmidt decomposition of . We have
Thus, is equivalent to