1 Introduction
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 whether
is NPcomplete [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 graphs
with , 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 monoidhomomorphisms.
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.
Entanglementassisted Shannon capacity of graphs
The second quantum variant of graph homomorphism comes from the study of entanglementassisted zeroerror capacity of classical channels, which is a quantum generalization of Shannon’s zeroerror communication setting. In the zeroerror 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 zeroerror messages Alice can send to Bob equals the independence number of the confusability graph. In the entanglementassisted 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 entanglementassisted independence number of the confusability graph [Bei10]. Similarly, the entanglementassisted Shannon capacity is its regularization. Based on this definition, one naturally defines an entanglementassisted homomorphism between graphs, denoted by [CMR14]. Let the entanglementassisted cohomomorphism preorder be defined by letting if . The entanglementassisted homomorphism has applications in the study of the entanglementassisted sourcechannel coding problem [BBL15, CMR14]. It is easy to see that the entanglementassisted 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 entanglementassisted homomorphism [MR16]. On the other hand, it is known that the entanglementassisted Shannon capacity can be strictly larger than the Shannon capacity [LMM12, BBG13]. In fact, the lower bounds they proved for the entanglementassisted Shannon capacity use the maximally entangled state, thus they are also lower bounds for the quantum Shannon capacity.
Entanglementassisted Shannon capacity of noncommutative graphs
Finally, we consider the entanglementassisted Shannon capacity of quantum channels. It turns out, analogous to the classical channel scenario, that the oneshot (entanglementassisted) zeroerror classical capacity of a quantum channel can be characterized in terms of the noncommutative graph associated with the channel [DSW13]. A noncommutative graph, or ncgraph for short, is a subspace
of the vector space of
complex 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 ncgraphs such that the entanglementassisted independence number , defined in [DSW13], equals the maximum number such that [Sta16], where is the noncommutative graph corresponding to the message perfect classical channel. The entanglementassisted Shannon capacity of ncgraphs 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 entanglementassisted cohomomorphism preorder

the asymptotic spectrum of noncommutative graphs with respect to the entanglementassisted 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 entanglementassisted homomorphism of graphs; and the definition of the (entanglementassisted) Shannon capacity of noncommutative graphs. In Section 3 we study the quantum Shannon capacity and the entanglementassisted Shannon capacity via the corresponding asymptotic spectra. In Section 4 we study the entanglementassisted Shannon capacity of noncommutative graphs via the corresponding asymptotic spectrum.
2 Preliminaries
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,
(1) 
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,
(2) 
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
(3) 
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 NPhard [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 nonadjacent vertices receive orthogonal vectors: for all . The celebrated Lovász theta function [Lov79], is defined as
(4) 
where the minimization goes over unit vectors and orthonormal representations of . Lovász proved that
Equation (4) is a semidefinite program that is efficiently computable. We also point out that there are several useful alternative characterizations of in the literature, see [Lov79].
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
(5) 
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].
Projective rank
A representation of a graph is a collection of rank projectors , such that if . The projective rank [MR16] is defined as
(6) 
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]).
We note that the fractional clique cover number can be written as a linear program (of large size), which is the dual linear program of the fractional packing number (see, e.g.,
[Sch03] or [ADR17, Eq. (A.16)]). Thus these two graph parameters coincide.Relationships between graph parameters
We know the following inequalities among the graph parameters that we have just defined:
(7)  
(8)  
(9) 
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
We present some fundamental abstract concepts and theorems from Strassen’s theory of asymptotic spectra. For a detailed description, we refer the reader to [Str88, Zui18a].
A semiring is a set equipped with a binary addition operation , a binary multiplication operation , and elements , such that for all holds
(10)  
(11)  
(12)  
(13) 
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  
(16) 
Let and be semirings. A semiring homomorphism from to is a map such that , for all , and . Let be the semiring of nonnegative 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.
(17) 
Let . The subrank of is defined as . The asymptotic subrank of is defined as
(18) 
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.
Theorem 1 ([Str88, Theorem 3.8], see also [Zui18a, Cor. 2.14]).
Let be a commutative semiring and let be a Strassen preorder on . For any such that and for some , holds
(19) 
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 wellbehaved 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.
Theorem 2 ([Str88, Cor. 2.7], see also [Zui18a, Cor. 2.17]).
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
(20)  
(21)  
(22)  
(23) 
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
(24) 
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 nonlocal games and from quantum zeroerror 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 . 
Remark 4.

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
Moreover, it is easy to verify that satisfies the conditions in Definition 3 [MR16].
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 Entanglementassisted homomorphism
Definition 5 (Entanglementassisted 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  
(28) 
Remark 6.
We note that the positive semidefinite matrix can be further restricted to be positive definite.
The entanglementassisted cohomomorphism preorder is defined by if and only if . The entanglementassisted independence number can be defined as . The entanglementassisted Shannon capacity of is defined as .
2.6 (Entanglementassisted) zeroerror capacity of quantum channels
For related definitions in quantum information theory, we refer the reader to [NC10]. We use and to denote the (finitedimensional) 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 of
spanned by the eigenvectors with positive eigenvalues. A
quantum channel can be characterized by a completely positive and tracepreserving (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 zeroerror 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 noncommutative graph (ncgraph) of a quantum channel [DSW13]. The ncgraphs 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 ncgraph the density operators are input symbols of the channel, and they are “nonadjacent” in the ncgraph if for all and in the support of and , respectively. As in the classical setting, “nonadjacent vertices” are nonconfusable.
Note that for every quantum channel , the associated ncgraph 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 noncommutative graph or ncgraph 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 ncgraphs of and respectively, then the tensor product is the ncgraph of the quantum channel . Then (the logarithm of) is exactly the classical zeroerror capacity of quantum channels whose ncgraph 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 ncgraph , 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
Proof.
Let be the Schmidt decomposition of . We have
Thus, is equivalent to for all possible . ∎
We call