Violating the Shannon capacity of metric graphs with entanglement

Abstract

The Shannon capacity of a graph G is the maximum asymptotic rate at which messages can be sent with zero probability of error through a noisy channel with confusability graph G. This extensively studied graph parameter disregards the fact that on atomic scales, nature behaves in line with quantum mechanics. Entanglement, arguably the most counterintuitive feature of the theory, turns out to be a useful resource for communication across noisy channels. Recently [Leung D, Mančinska L, Matthews W, Ozols M, Roy A (2012) Commun Math Phys 311:97–111], two examples of graphs were presented whose Shannon capacity is strictly less than the capacity attainable if the sender and receiver have entangled quantum systems. Here, we give natural, possibly infinite, families of graphs for which the entanglement-assisted capacity exceeds the Shannon capacity.

A sender transmits a message to a receiver. The main problem that information theory addresses is that noise could make the sender’s announcement ambiguous. To analyze this problem, one models a noisy communication channel by an input alphabet , an output alphabet ℛ, and a set of conditional probabilities for the probability that the receiver gets the letter when the sender transmits the letter . Two input letters x and y can then be confused with one another if they can lead to the same signal on the receiver’s end of the channel, that is, if there is an output letter a such that both the probability of the receiver getting a when the sender sent x and the probability of the receiver getting a when the sender sent y are nonzero. To cope with noise, the communicating parties could agree that the sender uses only input letters that lead to distinct signals on the receiver’s end, in which case the sender restricts to some set such that for every pair of distinct inputs and , at least one of the probabilities and vanishes. In a celebrated paper, Shannon (1) initiated the study of the zero-error capacity, the maximum rate of error-free communication with sequential uses of a memoryless channel. A channel is memoryless if using it does not change its behavior on later messages. By encoding messages into words consisting of multiple input symbols that are transmitted in sequence, a channel can sometimes be used more efficiently than if nonconfusable symbols are simply concatenated. Shannon demonstrated this with a famous example of a channel with five inputs and outputs. At most, two symbols can be sent perfectly with one use of the channel. Instead of the expected four, five messages can be sent perfectly with two uses of the channel (Fig. 1i).

Fig. 1.

(i) Channel with five inputs (numbers) and five outputs (letters), where an input symbol x is connected to an output symbol a if . Notice that none of the five pairs can be confused with one another, as either the first or the second two symbols are nonconfusable. (ii) Confusability graph of the channel, the five cycle .

Associated with a noisy channel is its confusability graph , with as vertex set V the input alphabet of the channel and edge set E consisting of those pairs of inputs that can be confused (Fig. 1ii). The largest number of messages that can be sent through a channel with zero probability of error equals , the maximum cardinality of an independent set in the graph G. The graph has as vertex set , the set of all n-tuples of elements from V. Two different vertices and are adjacent in if and only if they can be confused (i.e., if for every , either or is adjacent to in G). The independence number thus gives the maximum number of pairwise nonconfusable messages corresponding to n-letter codewords that are sent by using the channel n times in sequence. The Shannon capacity of the confusability graph,

gives the zero-error capacity of a channel.

Inevitably, devices used for information processing are subject to the laws of physics, and on atomic scales quantum mechanics is currently the most accurate model of nature. Thirty years before Shannon’s paper was published, Einstein et al. (2) pointed out an anomaly of quantum mechanics that allows spatially separated parties to establish peculiar correlations: entanglement. Later, Bell (3) proved that local measurements on a pair of spatially separated, entangled quantum systems can give rise to joint probability distributions that violate certain inequalities (now called Bell inequalities) satisfied by any distribution that may arise in classical mechanics. Experimental results of Aspect et al. (4) give strong evidence that nature indeed allows distant physical systems to be correlated in such nonclassical ways. Motivated by these important discoveries, one defines the zero-error entanglement-assisted capacity of a classical channel to be the maximum rate at which messages can be transmitted when the sender and receiver share a pair of entangled quantum systems (the precise model is described below).

Analogous to the classical setting, Cubitt et al. (5) defined the graph parameter (a quantum variant of the independence number) and proved that it equals the maximum number of pairwise nonconfusable messages that can be sent with a single use of a noisy channel and shared entanglement. They found examples of graphs for which , showing that the use of entanglement can increase the “one-shot” zero-error capacity of a channel [more examples were found recently by Mančinska et al. (6)]. This result was surprising because entanglement cannot increase the standard capacity of a (classical) discrete memoryless channel, as shown by Bennett et al. (7). Of course, the next question was if the entanglement-assisted capacity

could be strictly greater than the Shannon capacity.

In contrast with the combinatorial nature of the Shannon capacity, the entanglement-assisted capacity can sometimes be lower-bounded using geometric constructions. An orthonormal representation of a graph is a map that sends each of the vertices of the graph to a vector on a Euclidean unit sphere, such that adjacent vertices are sent to orthogonal vectors.* It turns out that if a graph G has a d-dimensional orthonormal representation, then , and therefore , is at least the number of disjoint d cliques in G (see Proposition 4).

Using the above argument, Leung et al. (8) recently found the first two examples of graphs whose entanglement-assisted capacity exceeds the Shannon capacity. Their graphs are based on the exceptional root systems and , giving a graph on 63 vertices such that and a graph on 157 vertices such that . They also show that their construction fails on any of the (four) infinite families of root systems, in the sense that it yields graphs whose Shannon and entangled capacities are equal.

The above-described lower-bound technique perhaps makes the orthogonality graph—whose vertices are the binary strings of length n, and whose edges are all pairs with Hamming distance (for n even)—the most natural candidate to separate the two capacities. This graph lies at the heart of many constructions that show a separation between some classical quantity and its quantum analog, such as in computational/communication complexity (†, 9), in Bell-inequality violations (10–12), and in the one-shot zero-error capacity of a noisy channel [where one compares to ].

By using the basis for bits, one directly obtains an n-dimensional orthonormal representation for the orthogonality graph because two strings have a Hamming distance (meaning they are adjacent) if and only if they are orthogonal. A Hadamard matrix is a square matrix with entries in such that its rows are mutually orthogonal. If a Hadamard matrix of size n exists, its rows thus give a clique of size n in the orthogonality graph. The fact that this graph is vertex transitive then implies that it has at least disjoint cliques (see Lemma 6 and Proposition 8 below), giving the same lower bound on its entanglement-assisted capacity. It is well known that Hadamard matrices exist when n is a power of 2; the famous Hadamard conjecture states that they exist whenever n is a multiple of 4. Although this conjecture remains unproven, it is widely believed to be true.

An indication that the orthogonality graph might exhibit a separation between the Shannon and entangled capacities is given by a well known result of Frankl and Rödl (13) showing that if n is a large enough multiple of 4, then the independence number is less than for some independent of n. However, despite effort from the quantum-information community, it remains unknown if this graph gives such a separation. Our main result shows that under certain conditions a separation holds for a “quarter” of the orthogonality graph.

Our Results

In this paper we present two natural, possibly infinite, families of basic graphs whose entanglement-assisted capacity exceeds the Shannon capacity. The graphs are defined as follows. Let be the graph with as vertex set all binary strings of odd length n and even Hamming weight, and as edge set the pairs with Hamming distance . Let be the subgraph of induced by the strings of Hamming weight . We prove the following.

Theorem 1.

Let p be an odd prime such that there exists a Hadamard matrix of size . Then, for and G either or , we have

Notice that the graph is a subgraph of the orthogonality graph on , induced by the -bit strings with even Hamming weight and first coordinate equal to 0. Based on constructions of Hadamard matrices by Scarpis (14) and Paley (15), Theorem 1 holds for any prime p such that for some odd prime q and positive integer k. The first three examples of such pairs for are , and . Examples for and can readily be generated with little computing power. The subset of strings in the vertex set of that have zeros on the last coordinates is an independent set of size , showing that the Shannon capacity of is exponential in n. Because is an induced subgraph of , the same holds for the latter graph.

We observe that the results of ref. 8 imply that the sequence of graphs has a capacity ratio that grows as roughly , where is the number of vertices of the graph G (the graph gives better dependence on the number of vertices than does). Our results show that the family of graphs gives a slightly higher ratio of roughly .

Theorem 1 follows directly from the following lemmas, which give lower and upper bounds on the entanglement-assisted capacity and Shannon capacity, respectively.

Lemma 1.

Let n be a positive integer such that there exists a Hadamard matrix of size . Then, for G either or , we have .

Lemma 2.

Let p be an odd prime and let . Then, for G either or , we have

Aside from relying on a few basic facts of graph theory and the theory of finite fields, our proofs of these lemmas are straightforward and self-contained. To obtain the asymptotic bound of Theorem 1, we upper-bound the binomial sum of Lemma 2 by the well-known estimate , where is the binary entropy function, and use the bound .

Entanglement-Assisted Capacity of a Graph

In this section we give the formal definition of the entanglement-assisted capacity of a graph. Let G be a finite simple undirected graph, with vertex set and edge set . The strong graph product of two graphs G and H has as vertex set all pairs . Two vertices and are adjacent in if and only if x and are adjacent in G or equal, and y and are adjacent in H or equal. For example, if and , then the four pairs , , , and form a 4 clique in .^‡ We denote by the n-fold strong graph product of G with itself. Recall that the Shannon capacity of G is defined as .

The entanglement-assisted capacity of a graph is defined as follows.

Definition 1. Entangled capacity of a graph:

Let be a graph. Define as the largest natural number M such that there exists a Hilbert space ℋ, a trace-1 positive semidefinite operator ρ on ℋ, and for every and , a positive semidefinite operator on ℋ satisfying

• 1. for every ;
• 2. for every and ;
• 3. if and .

The entanglement-assisted capacity of G is defined by

The parameter satisfies and is a generalization of the independence number. To see this, restrict in Definition 1 the space ℋ to be one-dimensional and add the restrictions and . Say that a vertex gets label i if . Condition 1 says that exactly one vertex gets label i, Condition 2 says that each vertex gets at most one label, and Condition 3 says that no two adjacent vertices belong to the privileged subset of labeled vertices. Hence, the system gives an independent set of size M, namely the set . Because relaxes this characterization of , it follows that .

By using tensor products of the operators ρ and , it is not hard to see that is nondecreasing with k. It follows that and (by Fekete’s Lemma) that .

Entanglement-Assisted Communication

In this section we describe the model of zero-error entanglement-assisted communication over classical channels. Readers who are familiar with this model or want to move on to the proof of the main result can safely skip this section. We start with some basic definitions of quantum-information theory. For more details we refer to Nielsen and Chuang (16).

Shared Entangled States.

A state is a positive-semidefinite matrix whose trace equals 1. We identify a matrix of size with a linear operator on in the obvious way. A state should be thought of as describing the configuration of a quantum system: an abstract physical object, or a collection of objects, on which one can perform experiments. Associated with a quantum system Q is a complex Euclidean vector space , for some dimension d. The possible configurations of Q are the states on .

Suppose the sender and receiver hold quantum systems S and R, respectively. Associated with the sender’s system is a space , and associated with the receiver’s system is a space . Then, by definition, the possible configurations of the joint system are the states on . If the system is in the state ρ, then the sender and receiver are said to share the state ρ. A state on is entangled if it is not a convex combination of states of the form , where is a state on χ and a state on .

Measurements.

Let be a finite set and let Q be a quantum system with associated vector space . A measurement on the system Q with outcomes in is a system of positive semidefinite matrices on , , which satisfies

where denotes the identity on .

Let be a measurement on the sender’s quantum system S. The numbers

define a probability distribution on . This follows easily from the properties of the matrices and ρ and the fact that for positive semidefinite matrices A and B, we have .

The partial trace function over of a matrix M on is defined by

where are the canonical basis vectors for . This function yields an matrix (i.e., a linear operator on the space ). It is not hard to see that the matrices

are each, in fact, states on the space associated with the receiver’s system R.

The postulates of quantum mechanics dictate that if the sender performs the measurement defined by the matrices on her system S, then the following two things happen:

• 1.She obtains outcome with probability ,
• 2.The receiver’s system R is left in the state on .

For some finite set ℛ, the receiver can perform a measurement on R, and he will obtain outcome “a” with probability . The joint probability of the sender and receiver obtaining outcomes x and a, respectively, is then given by . If the state ρ is not entangled, then this probability distribution is classical. Entanglement is thus necessary to obtain nonclassical quantum distributions.

Entanglement-Assisted Communication.

To send messages across a noisy channel defined by input alphabet , output alphabet ℛ, and conditional probability distribution P, the sender and receiver can use shared entanglement as follows. Let ρ be a state shared between the sender and receiver. Let M be a positive integer and for every let be a measurement on the sender’s system S with outcomes in . For every , let be a measurement on the receiver’s system R with outcomes in . Suppose that for every and such that , we have

To communicate the index i, the sender can then perform the ith measurement on her system and send her outcome through the channel. The receiver gets a message satisfying . The above discussion shows that if the receiver then performs the measurement labeled by a, he obtains outcome i with probability 1.

The link between this model and Definition 1 is given by the following theorem. Let us denote by the maximum number M such that a state ρ and matrices with the above property exists.

Theorem 2 [Cubitt et al. (5)].

Let be a noisy channel and let G be its confusability graph. Then, .

Preliminaries

Notation.

We use the following notation:

• •For strings , let denote their Hamming distance.
• •For vectors , let denote their Euclidean inner product.
• •For a prime number p, we write for a finite field consisting of p elements.
• •For vectors , let denote their inner product over .
• •For a field , we denote by the ring of n-variate polynomials with coefficients in .

Some Basic Graph Theory.

Let G be a graph. A permutation of the vertices is an automorphism of G if for every , the pair is an edge if and only if is an edge. Let denote the group of automorphisms of G. For , the set is the orbit of x and the set is the stabilizer of x.

Definition 2:

A graph G is vertex transitive if for every vertex , we have .

Lemma 3. Orbit-Stabilizer Theorem (17).

Let G be a graph and . Then

Corollary 3.

Let G be a vertex transitive graph and . Then, there are exactly automorphisms of G that map x to y.

Proof:

Because G is vertex transitive, there exists an automorphism such that . Consider the set of automorphisms . Clearly for every . We claim that contains all automorphisms that map x to y. To see this, notice that for any such that , we have and hence . Because implies that , we have . The claim follows because, by the Orbit-Stabilizer Theorem, we have , and by vertex transitivity of G, we have .

Lower Bounds on the Entanglement-Assisted Capacity

In this section we lower-bound the entanglement-assisted capacity of the graphs and . We start by dealing with the graph . The graph will be treated afterward in a similar manner.

To prove the lower bounds, we use a straightforward general method which was also used before in refs. 5 and 8. Recall that a (real) d-dimensional orthonormal representation of a graph G is a mapping satisfying and for every .

Proposition 4.

If a graph G has an orthonormal representation and has M disjoint d cliques, then .

Proof:

Let be a label set for the disjoint cliques. Let , where I is the d × d identity matrix. For every and let if x belongs to the ith clique and let be the zero matrix otherwise. Clearly these matrices are positive semidefinite, and it is easy to check that they satisfy the conditions of Definition 1 using the fact that for every d clique , the set is a complete orthonormal basis for . This gives .

The lower bounds on the entanglement-assisted capacity given in Lemma 1 follow immediately from the following two lemmas and Proposition 4.

Lemma 4.

Let n be an odd integer. Then, the graph has an n-dimensional orthonormal representation.

Lemma 5.

Let n be such that there exists a Hadamard matrix of size . Then, the graph has at least disjoint cliques of size n.

We proceed by proving these lemmas.

Proof of Lemma 4:

Associate with every vertex a sign vector given by . Let 1 denote the n-dimensional all-ones vector. Note that for every , we have , as the Hamming weight of x is . Moreover, for every we have , which follows from the fact that .

Now consider the -dimensional unit vectors (i.e., the column vector with a 1 appended to it, normalized). These vectors satisfy

• 1.
  For every we have

  
• 2.For every we have

The first item shows that f forms an orthonormal representation of G. The second item says that the vectors lie on a single n-dimensional hyperplane (orthogonal to the all-ones vector). Hence these vectors span a space of dimension at most n. It follows that there is an n-dimensional orthonormal representation of .

To prove Lemma 5, we need to find a large number of disjoint n cliques in . We achieve this by first finding just one n clique. Using the fact that is vertex transitive, we show that the existence of a single clique implies the existence of many disjoint cliques. More explicitly, one can produce many pairwise disjoint n cliques by simultaneously permuting the coordinates of the strings in this one clique. Notice that this permutation operation leaves both the Hamming weights and the Hamming distances invariant. A suitable choice of such permutations give pairwise disjoint cliques from any single clique, as whether or not a set of n-bit strings forms a clique in depends only on their Hamming weights and Hamming distances.

The following proposition tells us when we can find a single n clique in .

Proposition 5.

Let n be such that there exists a Hadamard matrix of size . Then, there exists an n clique in .

Proof:

Let M be an Hadamard matrix. We may assume that the first row and column of M contain only 's, because multiplying all entries in a row (or column) by gives again a Hadamard matrix. Because each of the last n rows of M is orthogonal to the first row, it has exactly entries equal to . Moreover, because each pair from the last n rows of M is orthogonal, the two rows differ in exactly coordinates.

Let C be the matrix obtained by removing the first row and column from M. Then, each row of C has exactly entries equal to , and every pair of rows from C differs in exactly coordinates. Hence, the rows of C are a clique in .

Next, we lower-bound the number of disjoint n cliques of size n in . We use the following lemma and proposition.

Lemma 6.

Let G be a vertex transitive graph that has a d clique as an induced subgraph. Then, G has at least vertex-disjoint induced d cliques.

Proof:

Let be a union of k disjoint d cliques, with k maximal. Because G is vertex transitive, Corollary 3 implies that for every pair of vertices there are exactly automorphisms mapping u to v. It follows that at most

automorphisms map a vertex in to a vertex in W.

On the other hand, by maximality of k, is nonempty for every automorphism σ. It follows that , and hence .

Proposition 6.

For every n, the graph is vertex transitive.

Proof:

Consider the group of permutations on . For every define the map by . As leaves the Hamming weight invariant, we have . Moreover, . Hence, . Finally, for every we have , and we are done.

Proof of Lemma 5:

The result follows by combining Propositions 5 and 6 and Lemma 6.

We deal with the graphs in the same way as we did with the graphs . We directly obtain the result of Lemma 1 for these graphs by combining the following two lemmas with Proposition 4.

Lemma 7.

Let n be an odd integer. Then, has an orthonormal representation of dimension .

Lemma 8.

Let n be such that there exists a Hadamard graph of size n. Then, the graph has at least disjoint cliques of size .

Proof of Lemma 7:

Associate with every vertex the vector

Then, the unit vectors form an -dimensional orthonormal representation of .

To prove Lemma 8 we proceed as in the proof of Lemma 5: We first find a single -clique in . Then, we prove that is vertex transitive and use Lemma 6.

Proposition 7.

Let n be such that there exists a Hadamard matrix of size . Then, there exists an clique in .

Proof:

Let n be an n clique in the graph . Then, because each of the vertices in C has Hamming weight , the union of C and the all-zeros string gives an clique in . The result now follows from Proposition 5.

Proposition 8.

For every n, the graph is vertex transitive.

Proof:

Recall that consists of the strings of even Hamming weight. For every define the linear bijection by . As leaves the parity of Hamming weight invariant, we have . Moreover, . Hence, . For every we have , and we are done.

Proof of Lemma 8:

The result follows by combining Propositions 7 and 8 and Lemma 6.

Upper Bounds on the Shannon Capacity

In this section we upper-bound the Shannon capacity of the graphs and . We recall that has as vertex set all binary strings of odd length n and Hamming weight , and as edge set the pairs of vertices with Hamming distance . The graph has as vertex set all binary strings of odd length n and even Hamming weight, and as edge set the pairs of vertices with Hamming distance . The proof of the upper bounds in Lemmas 2 is based on a general method of Haemers (18) and an algebraic lemma of Frankl and Wilson (19).

Lemma 9 [Haemers (18)].

Let be a graph. Let F be a field. Let be a matrix such that for every we have and for every nonadjacent pair we have . Then, .

Proof:

Say that a matrix fits G if it satisfies the conditions stated in the lemma. Let be a maximum-sized independent set and let A be a matrix that fits G. Then, the principal submatrix of A defined by S has rank . Hence, we have . The result follows because fits and .

We say that a polynomial is multilinear if its degree in each variable is at most 1.

Lemma 10 [Frankl–Wilson (19)].

Let p be an odd prime, let r be a natural number, and let . Let be a set of vectors over . Then, for every there exists a multilinear polynomial satisfying

• 1.;
• 2.For every such that , we have ,
• 3..

Proof:

For every vector let be the polynomial defined by

Because , every satisfies . By Wilson’s Theorem [for example Lidl and Niederreiter (20)], it follows that . If we have because in this case we have . In particular, we have . Because is a linear function, we have .

Define the multilinear polynomial by expanding in the monomial basis and changing the powers of in the monomial to 0 if is even and to 1 if is odd. Then, is multilinear and agrees with everywhere on and satisfies .

We now show how these two lemmas can be combined to give Lemma 2, which states that for p an odd prime and , and G either or , we have .

Proof of Lemma 2:

Let be either or . For every let be the corresponding sign vector in . Because and is isomorphic to the ring of integers mod p, we have for every

Let be distinct vertices such that is a nonedge. We claim that . It is not hard to see that if two strings have even Hamming weight, then their Hamming distance is also even. Hence, we have . Moreover, because p is odd, the only possible multiple of p that can attain is . Because edges are formed by pairs with Hamming distance , we have . This implies that , and the claim follows from Eq. 1.

Set and let . Then, Lemma 10 gives a multilinear polynomial for every , satisfying

• 1.;
• 2.For every such that and , we have ;
• 3..

The set ℳ of multilinear monomials in n variables of degree at most forms a basis for the space of multilinear polynomials of degree at most . For every vertex , define vectors as follows. For monomial let be the coefficient of m in the expansion of the polynomial in the basis ℳ, and let be the value obtained by evaluation the monomial m at . Then, for every we have .

Consider now the matrix defined by . Because the vectors and have dimension , we have . Additionally, it follows from the properties of the polynomials that the matrix A satisfies for every and for every nonadjacent pair .

The claim now follows from Lemma 9 and the fact that

This completes the proof.