Entanglement can increase asymptotic rates of zero-error classical communication over classical channels
It is known that the number of different classical messages which can be communicated with a single use of a classical channel with zero probability of decoding error can sometimes be increased by using entanglement shared between sender and receiver. It has been an open question to determine whether entanglement can ever increase the zero-error communication rates achievable in the limit of many channel uses. In this paper we show, by explicit examples, that entanglement can indeed increase asymptotic zero-error capacity, even to the extent that it is equal to the normal capacity of the channel. Interestingly, our examples are based on the exceptional simple root systems and .
A classical channel which is discrete and memoryless is fully described by its conditional probability distribution of producing output for a given input . The channel obtained by allowing one use of a channel and one use of is written as , reflecting the fact that its conditional probability matrix is the tensor (or Kronecker) product of those of the two constituent channels. Similarly, denotes uses of .
Let denote the maximum number of different messages which can be sent with a single use of with zero probability of a decoding error. The zero-error capacity of is
Two input symbols of a channel are said to be confusable if and for some output symbol . The confusability graph of a channel is a graph , whose vertices correspond to different input symbols of and two vertices are joined by an edge if the corresponding symbols are confusable. The confusability graph of is determined by those of and as follows: , where “” denotes the strong graph product.
Definition 2 (Strong graph product).
In general, the strong product of graphs is a graph , whose vertices are the -tuples and distinct vertices and are joined by an edge if they are entry-wise confusable, i.e., for each either or . Likewise, we define the strong power of graph by and .
An independent set of a graph is a subset of its vertices with no edges between them. The independence number of a graph is the maximum size of an independent set of . As Shannon observed , and depend only on the confusability graph of the channel: and where
is known as the Shannon capacity of the graph . Clearly, , since has an independent set of size . However, in general can be larger than . The simplest example is the -cycle for which but .
Computing the independence number of a graph is NP-hard, but conceptually simple. However, no algorithm is known to determine in general, although there is a celebrated upper bound due to Lovász . He defined an efficiently computable quantity called the Lovász number of which satisfies and . Because of these properties we also have .
Let denote the number of different messages which can be sent with a single use of with zero probability of a decoding error, when both parties share an arbitrary finite-dimensional entangled state on which each can perform arbitrary local measurements. The entanglement-assisted zero-error capacity of is
As in the unassisted case, the quantities and also depend only on the confusability graph of the channel . For this reason, we will talk about the assisted and unassisted zero-error capacities of graphs as well as of channels.
Whether graphs with exist, that is, whether entanglement can ever offer an advantage in terms of the rates achievable in the large block length limit was left as an open question. Clearly, the Lovász bound cannot be used to prove such a separation. Fortunately, there is another bound due to Haemers which is sometimes better than the Lovász bound [6, 7].
For let be a matrix with entries in any field . We say that fits if and whenever there is no edge between and . Then . In particular, .
Let be a maximal independent set in . If fits , then for all while the diagonal entries are non-zero. Hence, has full rank on a subspace of dimension and thus . As this is true for any that fits , we get .
Next, note that if fits and fits then fits , and . Hence, , which implies the desired result. ∎
The next section shows how Haemers bound applies to a particular graph to determine its unassisted zero-error capacity, and then provides an explicit entanglement-assisted protocol that achieves a higher rate. This shows that entanglement assistance can indeed increase the asymptotic zero-error rate, thus giving an affirmative answer to the previously open question.
The entanglement-assisted protocol is based on the fact that the graph in question can be constructed from the root system  , so in Section 3 we investigate constructions based on other irreducible root systems. Most notably we show that a construction based on provides another example with a larger gap in the capacities. In Section 4 we discuss open problems.
2 The zero-error capacities of the symplectic graph
Definition 5 (Symplectic space).
A non-degenerate symplectic space is a vector space (over a field ) equipped with a non-degenerate symplectic form, i.e., a bilinear map which is
skew-symmetric: for all , and
non-degenerate: if for all , then .
If has characteristic 2, we also require that for all (for other fields this is implied by the anti-symmetry property). On a -dimensional space, the canonical symplectic form is
where is the identity matrix. Any non-degenerate symplectic space with finite dimensional vector space is isomorphic to the canonical symplectic space .
Definition 6 (Symplectic graph).
Let be a finite field and let be a natural number. The vertices of the symplectic graph are the points of the projective space and there is an edge between if . In the case where , the points of the projective space are simply the non-zero elements of .
The symplectic graph is isomorphic to the graph whose vertices are all the -fold tensor products of Pauli matrices except for the identity, i.e., , and which has edges between commuting matrices.
The next two subsections prove that for channels with the confusability graph , the entanglement-assisted zero-error capacity is larger than the unassisted capacity. More precisely,
2.1 Capacity in the unassisted case
The fact that is a special case of a result in  which we prove explicitly here.
Theorem 9 (Peeters ).
For the upper bound we construct a matrix over which fits and which has rank and use Haemers’ bound (see Theorem 4). Let
be the -dimensional subspace of that consists of vectors which have an even number of entries equal to one. The restriction of the standard inner product on to the subspace is a non-degenerate symplectic form, so there is an isomorphism such that
Let be the all-ones vector in (note that for all ). For all let
Since if and only if and are not joined by an edge in , matrix fits . Since it is the Gram matrix of a set of -dimensional vectors (i.e. the entry at is the inner product of the vector and vector for some ordering of the set of vectors), its rank is at most . Therefore, by Haemers’ bound, .
For the matching lower bound, let for be the standard basis of , and let . Then so and . Therefore is an independent set of size in , so .
Hence, and the upper bound on the zero-error capacity is attained by a code of block length one. ∎
2.2 Entanglement-assisted capacity
In this section we will establish the entanglement-assisted capacity of . Our main tool is Theorem 11 together with some combinatorial results.
A -dimensional orthonormal representation of graph is a function that assigns unit vectors to the vertices of such that for each edge vectors and are orthogonal.
The following theorem appeared in  but for completeness we include the proof here.
Theorem 11 ().
If graph has an orthonormal representation in and its vertices can be partitioned into disjoint cliques each of size then .
Figure 1 describes a protocol that uses a rank- maximally entangled state to send one of messages with zero error by a single use of the channel, proving that . Removing edges from cannot decrease the Lovász number and in this way we can obtain the graph which is the strong product of the empty graph on vertices with the clique of size (i.e. the disjoint union of -cliques). This graph has Lovász number so . Since [4, 5], this provides the matching upper bound. ∎
The vertices of can be partitioned into cliques of size .
Such a partition of the symplectic graph is known as a symplectic spread, and is well-known to exist (see for example ). We give a simple construction from  below. Another proof in terms of commuting sets of Pauli operators is given in [13, 14].
Let , and identify the vertices of with the non-zero vectors in . Consider the following symplectic form on :
where is the finite field trace defined as . As is a non-degenerate inner product in , the form is also non-degenerate. Hence, the symplectic spaces and are isomorphic. We will describe the partition for the later space.
Denoting the multiplicative group (of order ) in by , the cells of a partition of the non-zero elements of are:
It is easy to check that these cells of size partition . Moreover, if and are in the same cell, then
Therefore each cell is a clique. ∎
The entire representation, grouped into 9 complete (unnormalised) orthogonal bases, is given in a table in Appendix A and it suffices to check that this has the desired properties to establish the result. Interestingly, it consists of vectors from the root system and it is possible to give a more insightful description and proof of the representation in relation to this. Such a proof is given in Appendix B.
2.3 The connection to
A deeper coincidence underlies the orthogonal representation of by roots of . The automorphism group of is the symplectic group , which is the group of linear maps on which preserve the symplectic form. This group is isomorphic to quotient , where is the Weyl group of .
3 Relationship to the normal capacity
Given a classical channel , its standard classical capacity cannot be increased by the use of entanglement or even arbitrary non-signalling correlations between the sender and receiver . The standard capacity and the assisted and unassisted zero-error capacities are related by
Given a graph which satisfies the premises of Theorem 11 (partitions into cliques of size and has an orthonormal representation in dimension ) and is also vertex-transitive, one can construct a channel whose normal capacity and are both equal to and are both achieved by a block-length one entanglement-assisted zero-error code.
Let be the vertices of and let be the set of all cliques of size in . Since is vertex-transitive, each vertex is contained in the same number of cliques from . Counting the number of pairs in the set in two ways we have
Let be the channel which on the input produces an output uniformly at random from the set . Using the analysis of Section 16 of ,
also partitions into cliques of size and, since it is a subgraph of , has an orthonormal representation in dimension . Therefore and, furthermore, this rate is achieved by the block-length one entanglement-assisted protocol of Figure 1. ∎
The symplectic graphs are all vertex transitive so, remarkably, the channel constructed in this way for has , even though there is a positive lower bound on the error probability for classical codes with any rate greater than (as well as an upper bound, both decaying exponentially with ) .
4 Graphs from and other root systems
We define the orthogonality graph of a root system as follows. The vectors of occur in antipodal pairs ; the vertices of the graph are the rays spanned by these antipodal pairs. Two vertices are adjacent if and only if their rays are orthogonal. The graph of Section 2 is precisely the orthogonality graph of . This raises the question of whether a channel whose confusability graph is the orthogonality graph of another irreducible root system can also exhibit a gap between the classical and entanglement-assisted zero-error capacities. We find that the orthogonality graph of provides a second example of such a gap, and furthermore, the ratio between the assisted and classical capacities is larger in this case.
The irreducible root systems consist of the infinite families , , , where , and the exceptional cases , , , , and (see ). We show that for all of the infinite families, and for , there is no gap between the independence number and the Lovász number , so for these graphs. However, the orthogonality graph of provides a second example of a gap between the classical and entanglement-assisted capacity. For , we show that while . It is interesting to note that the graph used in  is precisely the orthogonality graph of and while we know the entanglement-assisted capacity of this graph, we still do not know its unassisted capacity or whether it is smaller than the assisted one. We do not give either capacity for the graph of .
In what follows the name of the root system is also used as the name of the orthogonality graph and denotes the -th standard basis vector. We ignore correct normalization of the root vectors for simplicity, since it clearly doesn’t affect the orthogonality graph.
As pointed out in , is the graph whose vertices are the non-isotropic points in the ambient projective space of the polar space , with vertices adjacent if they are orthogonal with respect to the associated bilinear form. The ambient projective space of is . Since the bilinear form associated with is symplectic, it follows immediately that is an induced subgraph of the symplectic graph with . By Theorem 9,
On the other hand, let and identify the vertices of with the non-zero vectors of . Then we may choose the quadratic form of to be , where is the finite field trace. The polarization of this quadratic form is the symplectic form . With this choice, the vertices of , i.e., the non-isotropic vectors in , are those such that . Now, consider the partition of vertices into cliques given in Lemma 12, restricted to the vertices of :
Recall that . For each , there are exactly choices of such that . Therefore, is a partition of the vertices of into 15 cliques of size . By Theorem 11,
Root system ()
This graph is isomorphic to the Kneser graph . By a result of Lovász (Theorem 13 of ),
Root system ()
Note that the vertices induce a subgraph isomorphic to . Also note that and are adjacent and have the same neighbourhood (apart from themselves). It follows that is isomorphic to , the strong graph product of a Kneser graph and a complete graph on vertices. By Theorem 7 of ,
Since is an independent set of size , it follows that
Root system ()
To find the Lovász number we consider even and odd separately. When is odd, partition the vertices into sets , where
Each is a clique of size . When is even, partition the vertices into sets , where
Again each is a clique of size . In either case, we have partitioned the graph into cliques of size . By Theorem 11, .
Since is an independent set of size , we have
Root system ()
has the same orthogonality graph as .
By inspection has an independent set of size and can be partitioned into cliques of size . By Theorem 11,
We have shown that it is possible for entanglement to increase the asymptotic rate of zero-error classical communication over some classical channels. This is quite different from the situation for families of codes which only achieve arbitrarily small error rates asymptotically. The best rate that can be achieved by classical codes in this context is the Shannon capacity and entanglement cannot increase this rate. The entanglement-assisted capacity has a simple formula which reduces to the formula for the Shannon capacity when the channel is classical .
It is interesting to note that in every example of a graph with found to date, the entanglement-assisted capacity is attained by a code of block length one. This certainly is not true of the entanglement-assisted capacities of all graphs. In , an interesting observation of Arikan is reported: The graph consisting of a five cycle and one more isolated vertex has . Since no positive integer power of this quantity is an integer, the capacity is not attained by any finite length block code for this graph. Since the Lovász number of this graph is also (the Lovász number is additive for disjoint unions of graphs, and was calculated for cycles in ), the same observation is true for the entanglement-assisted capacity, which in this case is equal to the unassisted capacity.
Our result has an interesting interpretation in terms of Kochen-Specker (KS) proofs of non-contextuality. Such a proof specifies a set of complete, projective measurements, with some projectors in common, such that there is no way to consistently assign a truth value to each projector. An assignment is consistent if (a) precisely one projector in each measurement is “true” and (b) no two “true” projectors are orthogonal.
Ruuge  shows that the root systems and can be used to construct KS proofs using computer search to nullify the possibility of a consistent assignment. This is a corollary of our results, but our proof is analytic due to the novel application of the Haemers bound. In fact, the use of the Haemers bound provides a whole sequence of KS proofs which are increasingly strong in the following quantitative sense: For the set of measurements which are obtained by tensoring together of Alice’s 9 measurements, only can be assigned values in accordance with property (a) before property (b) must be violated.
Three main avenues for further research are apparent to us. First, is it possible to give a general algorithm to compute ? More specific related problems include determining whether can be arbitrarily large, and whether there are graphs where is strictly less than .
Secondly, we have already shown that there are some connections to multi-prover games and to non-contextuality, but we feel that a deeper understanding of these connections is possible and desirable. For example, the application of our result to KS proofs mentioned above suggests some stronger notion of non-contextuality in quantum mechanics.
Finally, our work on entanglement-assisted zero-error codes can be placed in the wider context of using entanglement to reduce decoding error in finite block length coding of classical information for classical channels (demonstrating this effect is even experimentally feasible ), and characterising this phenomenon presents an even wider set of questions.
We would like to thank Andrew Childs, Richard Cleve, David Roberson, Simone Severini, and Andreas Winter for useful discussions. This work was supported by NSERC, QuantumWorks, CIFAR, CFI, and ORF. Aidan Roy acknowledges support by a UW/Fields Institute Award.
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Appendix A The orthogonal representation of in full
Appendix B The proof of Lemma 13
We first give some basic definitions and facts about and its lattice. Let be the vectors given in Figure 2, known as simple roots of . Their inner products are encoded by Dynkin diagram shown in Figure 2 as follows:
All integer linear combinations of the simple roots form the lattice
For , let denote the lattice coordinates of . The inner product between two lattice vectors is
where is the set of edges in the Dynkin diagram. Note that the inner product is an even integer for all , .
The root system is the set vectors of norm in :111One can check that this agrees with the more common definition of as the orbit of under the reflection group , where and is the unit vector in direction .
In terms of lattice coordinates, the condition can be expressed as
The (defined in the figure above) are chosen so that . This extends to all lattice vectors by linearity of :
We can write where is defined by with the are treated as vectors in . It is easily checked that the kernel of is the set where (i.e. ). Therefore,
iff for some and .
If then is not a root.
If then for some , , so . The first two inner products are even integers and , so for some integer , and can’t be a root by definition. ∎
and iff .
iff for some , . We can rule out the case where because, if it were
since . Then, and cannot both be roots due to equation (30), which must necessarily hold modulo two. Therefore, and . Since is a root, the condition that is also a root
reduces to . By the Cauchy-Schwarz inequality, , or equivalently, , with equality iff is a scalar multiple of . Since inner products between lattice vectors are even integers, either and , or and . ∎
There are 126 roots in 63 antiparallel pairs. Let be a subset of with one root from each pair. We have just shown that both roots in a pair have the same image under , that these images are different for different pairs, and none are equal to 0. Therefore, the restriction of to the domain is a bijection between and whose inverse determines (by normalising the vectors in ) an orthonormal representation of thanks to the relationship (32).