Entanglement can increase asymptotic rates of zero-error classical communication over classical channels

0.88 MB·English

by  Debbie Leung

Entanglement can increase asymptotic rates of zero-error classical
communication over classical channels
Debbie Leung, Laura Mancinska, William Matthews, Maris Ozols, Aidan Roy
Institute of Quantum Computing, University of Waterloo,
200 University Ave. West, Waterloo, ON N2L 3G1, Canada
January 6, 2011
1
1
0
2 Abstract

n
It is known that the number of different classical messages which can be communicated
a
withasingle use ofaclassicalchannelwithzeroprobabilityofdecodingerrorcansometimes
J
beincreasedbyusingentanglementsharedbetweensenderandreceiver. Ithasbeenanopen
5
questiontodeterminewhetherentanglementcaneverincreasethezero-errorcommunication

] rates achievable in the limit of many channel uses. In this paper we show, by explicit
h
examples, thatentanglementcanindeedincreaseasymptoticzero-errorcapacity, eventothe
p
extentthatitisequaltothenormal capacityofthechannel. Interestingly, ourexamplesare
-
t based on the exceptional simple root systems E and E .
n 7 8
a
u
q 1 Introduction
[

2 A classical channel which is discrete and memoryless is fully described by its conditional
N
v probabilitydistribution (y x)ofproducingoutputy foragiveninputx. Thechannelobtained
5 N |
by allowing one use of a channel and one use of is written as , reflecting the fact
9 1 2 1 2
N N N ⊗N
1 that its conditional probability matrix is the tensor (or Kronecker) product of those of the two
1 constituent channels. Similarly, ⊗n denotes n uses of .
. N N
9
0 Definition 1. Let M ( ) denote the maximum number of different messages which can be sent
0
0 N
with a single use of with zero probability of a decoding error. The zero-error capacity of
1 N
: is
v N 1
i C0( ) := lim logM0( ⊗n). (1)
X N n→∞ n N
r
Two input symbols x ,x of a channel are said to be confusable if (y x ) > 0 and
a 1 2 1
N N |
(y x ) > 0 for some output symbol y. The confusability graph of a channel is a graph
2
N | N
G( ), whose vertices correspond to different input symbols of and two vertices are joined
N N
by an edge if the corresponding symbols are confusable. The confusability graph of
1 2
N ⊗N
is determined by those of and as follows: G( ) = G( )(cid:2)G( ), where “(cid:2)”
1 2 1 2 1 2
N N N ⊗N N N
denotes the strong graph product.
Definition 2 (Strong graph product). In general, the strong product of graphs G ,...,G is
1 n
a graph G (cid:2) (cid:2)G , whose vertices are the n-tuples V(G ) V(G ) and distinct vertices
1 n 1 n
··· ×···×
(a ,...,a ) and (b ,...,b ) are joined by an edge if they are entry-wise confusable, i.e., for each
1 n 1 n
j 1,...,n either a b E(G ) or a = b . Likewise, we define the strong power of graph
j j j j j
G∈by{G(cid:2)1 :=}G and G(cid:2)n :∈= G(cid:2)G(cid:2)(n−1).
1
An independent set of a graph is a subset of its vertices with no edges between them. The
independence number α(G) of a graph G is the maximum size of an independent set of G.
As Shannon observed [1], M and C depend only on the confusability graph of the channel:
0 0
M ( ) = α(G( )) and C ( ) = logΘ(G( )) where
0 0
N N N N
Θ(G) := lim n α(G(cid:2)n) (2)
n→∞
(cid:113)
is known as the Shannon capacity of the graph G. Clearly, Θ(G) α(G), since G(cid:2)n has an
≥
independent set of size α(G)n. However, in general Θ(G) can be larger than α(G). The simplest
example is the 5-cycle C for which α(C ) = 2 but Θ(C ) = √5.
5 5 5
Computing the independence number of a graph is NP-hard, but conceptually simple. How-
ever, no algorithm is known to determine Θ(G) in general, although there is a celebrated upper
bound due to Lov´asz [2]. He defined an efficiently computable quantity ϑ(G) called the Lov´asz
number of G which satisfies ϑ(G) α(G) and ϑ(G (cid:2) G ) = ϑ(G )ϑ(G ). Because of these
1 2 1 2
≥
properties we also have ϑ(G) Θ(G).
≥
Definition 3. Let ME( ) denote the number of different messages which can be sent with a
0 N
single use of with zero probability of a decoding error, when both parties share an arbitrary
N
finite-dimensionalentangledstateonwhicheachcanperformarbitrarylocalmeasurements. The
entanglement-assisted zero-error capacity of is
N
1
CE( ) := lim logME( ⊗n). (3)
0 N n→∞ n 0 N
As in the unassisted case, the quantities ME and CE also depend only on the confusability
0 0
graphofthechannel[3]. Forthisreason,wewilltalkabouttheassistedandunassistedzero-error
capacities of graphs as well as of channels.
In [3] it was shown that graphs G exist with ME(G) > M (G). Shortly afterwards it was
0 0
shown [4, 5] that the Lov´asz bound also applies to the entanglement-assisted quantities, so
ME(G) ϑ(G) and hence CE(G) logϑ(G).
0 ≤ 0 ≤
Whether graphs with CE(G) > C (G) exist, that is, whether entanglement can ever offer
0 0
an advantage in terms of the rates achievable in the large block length limit was left as an open
question. Clearly, the Lov´asz bound cannot be used to prove such a separation. Fortunately,
there is another bound due to Haemers which is sometimes better than the Lov´asz bound [6, 7].
Theorem 4 (Haemers [6, 7]). For u,v V(G) let M be a matrix with entries in any field K.
uv
∈
We say that M fits G if M = 0 and M = 0 whenever there is no edge between u and v.
uu uv
(cid:54)
Then Θ(G) R(G) := min rank(M) : M fits G . In particular, C (G) logR(G).
0
≤ { } ≤
Proof. Let S be a maximal independent set in G. If M fits G, then M = 0 for all u = v S
uv
(cid:54) ∈
while the diagonal entries are non-zero. Hence, M has full rank on a subspace of dimension S
| |
and thus rank(M) S = α(G). As this is true for any M that fits G, we get R(G) α(G).
≥ | | ≥
Next, note that if M fits G and M fits G then M M fits G (cid:2)G , and rank(A B) =
1 1 2 2 1 2 1 2
⊗ ⊗
rank(A)rank(B). Hence, R(G (cid:2)G ) R(G )R(G ), which implies the desired result.
1 2 1 2
≤
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 [8] E , so in Section 3 we investigate constructions based on
7
other irreducible root systems. Most notably we show that a construction based on E provides
8
another example with a larger gap in the capacities. In Section 4 we discuss open problems.
2
2 The zero-error capacities of the symplectic graph sp(6, )
F
2
Definition 5 (Symplectic space). A non-degenerate symplectic space (V,S) is a vector
space V (over a field K) equipped with a non-degenerate symplectic form, i.e., a bilinear
map S : V V K which is
× →
skew-symmetric: S(u,v) = S(v,u) for all u,v V, and
• − ∈
non-degenerate: if S(u,v) = 0 for all v V, then u = 0.
• ∈
If K has characteristic 2, we also require that S(u,u) = 0 for all u V (for other fields
∈
this is implied by the anti-symmetry property). On a 2m-dimensional space, the canonical
symplectic form is
0 11
σ(u,v) := uT m v. (4)
11 0
m
(cid:18) − (cid:19)
where 11 is the m m identity matrix. Any non-degenerate symplectic space with finite
m
×
dimensional vector space V is isomorphic to the canonical symplectic space (V,σ).
Definition 6 (Symplectic graph). Let K be a finite field and let m be a natural number. The
vertices of the symplectic graph sp(2m,K) are the points of the projective space PK2m and
there is an edge between u,v PK2m if σ(u,v) = 0. In the case where K = F2, the points of
∈
the projective space are simply the 22m 1 non-zero elements of F2m.
− 2
Remark 7. The symplectic graph sp(2m,F2) is isomorphic to the graph whose vertices are all
them-foldtensorproductsofPaulimatricesexceptfortheidentity, i.e., 11,X,Y,Z ⊗m 1⊗m ,
{ } \{ }
and which has edges between commuting matrices.
The next two subsections prove that for channels with the confusability graph sp(6,F2), the
entanglement-assisted zero-error capacity is larger than the unassisted capacity. More precisely,
Theorem 8. C0 sp(6,F2) = log7 while C0E sp(6,F2) = log9.
(cid:0) (cid:1) (cid:0) (cid:1)
2.1 Capacity in the unassisted case
The fact that C0 sp(6,F2) = log7 is a special case of a result in [9] which we prove explicitly
here.
(cid:0) (cid:1)
Theorem 9 (Peeters [9]). C0 sp(2m,F2) = log(2m+1).
Proof. For the upper bound w(cid:0)e construct(cid:1)a matrix over F2 which fits sp(2m,F2) and which has
rank 2m+1 and use Haemers’ bound (see Theorem 4). Let
Um := {v ∈ F22m+1 : (cid:104)v,v(cid:105) = 0} (5)
be the 2m-dimensional subspace of F2m+1 that consists of vectors which have an even number of
2
entriesequaltoone. Therestrictionofthestandardinnerproduct , onF2m+1 tothesubspace
(cid:104)· ·(cid:105) 2
Um is a non-degenerate symplectic form, so there is an isomorphism T : (F22m,σ) → (Um,(cid:104)·,·(cid:105))
such that
u,v F2m : σ(u,v) = T(u),T(v) . (6)
∀ ∈ 2 (cid:104) (cid:105)
Let 1 be the all-ones vector in F22m+1 (note that (cid:104)1,v(cid:105) = 0 for all v ∈ Um). For all u,v ∈ F22m let
M := 1+T(u),1+T(v)
uv
(cid:104) (cid:105)
= 1,1 + 1,T(v) + T(u),1 + T(u),T(v)
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
= 1+σ(u,v).
3
Since σ(u,v) = 1 if and only if u and v are not joined by an edge in sp(2m,F2), matrix M fits
sp(2m,F2). Since it is the Gram matrix of a set of (2m+1)-dimensional vectors (i.e. the entry
at i,j is the inner product of the vector i and vector j for some ordering of the set of vectors),
its rank is at most 2m+1. Therefore, by Haemers’ bound, C0 sp(2m,F2) log(2m+1).
≤
For the matching lower bound, let ei for 1 ≤ i ≤ 2m+1(cid:0)be the stan(cid:1)dard basis of F22m+1,
and let f := e + 1. Then f ,f = 1 δ so f U and σ T−1(f ),T−1(f ) = 1 δ .
i i i j ij i m i j ij
(cid:104) (cid:105) − ∈ −
Therefore T−1(fi) : i = 1,...,2m+1 is an independent set of size 2m+1 in sp(2m,F2), so
{ } (cid:0) (cid:1)
α sp(2m,F2) 2m+1.
≥
Hence, C0 sp(2m,F2) = logα sp(2m,F2) = log(2m + 1) and the upper bound on the
(cid:0) (cid:1)
zero-error capacity is attained by a code of block length one.
(cid:0) (cid:1) (cid:0) (cid:1)
2.2 Entanglement-assisted capacity
In this section we will establish the entanglement-assisted capacity of sp(6,F2). Our main tool
is Theorem 11 together with some combinatorial results.
Definition 10. A d-dimensional orthonormal representation of graph G = (V,E) is a
function φ : V Cd that assigns unit vectors to the vertices of G such that for each edge
→
uv E vectors φ(u) and φ(v) are orthogonal.
∈
The following theorem appeared in [10] but for completeness we include the proof here.
Theorem 11 ([10]). If graph G has an orthonormal representation in Cd and its vertices can
be partitioned into k disjoint cliques each of size d then CE(G) = logk.
0
Proof. Figure 1 describes a protocol that uses a rank-d maximally entangled state to send one
of k messages with zero error by a single use of the channel, proving that CE(G) k. Removing
0 ≥
edgesfromGcannotdecreasetheLov´asznumberandinthiswaywecanobtainthegraphwhich
is the strong product of the empty graph on k vertices with the clique of size d (i.e. the disjoint
union of k d-cliques). This graph has Lov´asz number k so ϑ(G) k. Since CE(G) logϑ(G)
≤ 0 ≤
[4, 5], this provides the matching upper bound.
Lemma 12. The 22m 1 vertices of sp(2m,F2) can be partitioned into 2m +1 cliques of size
−
2m 1.
−
Proof. Such a partition of the symplectic graph is known as a symplectic spread, and is well-
known to exist (see for example [11]). We give a simple construction from [12] below. Another
proof in terms of commuting sets of Pauli operators is given in [13, 14].
LetN = 2m,andidentifytheverticesofsp(2m,F2)withthenon-zerovectorsinF2N. Consider
the following symplectic form on F2 :
N
σ (w,x),(y,z) = Tr(wz+xy), (7)
N
where Tr : FN F2 is the finite fi(cid:0)eld trace defi(cid:1)ned as Tr(a) := a+a2 +a22 +...+a2m−1. As
→
x,y N := Tr(xy) is a non-degenerate inner product in FN, the form σN is also non-degenerate.
(cid:104) (cid:105)
Hence,thesymplecticspaces(Fm2 ,σ)and(F2N,σN)areisomorphic. Wewilldescribethepartition
for the later space.
Denoting the multiplicative group (of order N −1) in FN by F×N := FN \{0}, the cells of a
partition of the non-zero elements of F2 are:
N
πa = {(x,ax) : x ∈ F×N} (a ∈ FN),
πN+1 = {(0,x) : x ∈ F×N}.
4
SupposeG( )partitionsintokcliquesofsized. Labelthevertices(inputsymbols)bypairs(c,i)wherec [k]identifies
N ∈
thecliqueandi [d]thevertexwithintheclique. FurthersupposethatG( )hasanorthogonalrepresentationmapping
∈ N
(c,i)to|ψ(c,i)￿i.e. ￿ψ(c,i)|ψ(c￿,i￿)￿=0if(c,i)and(c￿,i￿)areconfusable. Tosendoneofkmessageswithzero-error:
1 d q i y (2)Alicemakestheinputlabelled(q,j).
i i
√d | ￿A⊗| ￿B
i=1
￿ (3) The output y tells Bob that the
inputwasinthemutuallyconfusableset
ALICE j q i y Sy:={(c,i):Pr(y|input=(c,i))>0}.
Therefore, he knows his system is in
N
one of the set of orthogonal states
BOB (1)Tosendmessageq∈[k]Alice {su|ψre(ci,ni)￿a∗b:a(csi,si)in∈clSuyd}inagndthehseecsatnatemsetao-
measures in the orthogonal ba-
determine(q,j)withcertainty.
sis ψ(q,i) : i [d] obtaining q i y
{| ￿ ∈ }
the outcome j. Conditioned on
knowledgeof(q,j),Bob’sstateis
|ψ(q,j)￿∗(thecomplexconjugate
of ψ(q,j) ).
| ￿
q i y
Figure 1: If the confusability graph of can partitioned into k cliques of size d then C ( ) k
0
N N ≤
by the Lov´asz bound. If it also has a d-dimensional orthonormal representation then this rate
can be achieved by the entanglement-assisted zero-error code (of block length one) described in
this figure.
It is easy to check that these N +1 cells of size N 1 partition F2 . Moreover, if (x,ax) and
− N
(y,ay) are in the same cell, then
σ (x,ax),(y,ay) = Tr(xay+axy) = Tr(0) = 0. (8)
Therefore each cell is a cl(cid:0)ique. (cid:1)
Lemma 13 ([15, 16]). sp(6,F2) has an orthonormal representation in R7.
Theentirerepresentation, groupedinto9complete(unnormalised)orthogonalbases, isgiven
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 E and it is possible to give
7
a more insightful description and proof of the representation in relation to this. Such a proof is
given in Appendix B.
Since the 63 vertices of sp(6,F2) partition into 9 cliques each of size 7 (Lemma 12), it follows
by Theorem 11 that C0E sp(6,F2) = log9, whereas we already established that C0 sp(6,F2) =
log(2 3+1) = log7. This concludes the proof of Theorem 8.
× (cid:0) (cid:1) (cid:0) (cid:1)
2.3 The connection to E
7
A deeper coincidence underlies the orthogonal representation of sp(6,F2) by roots of E7. The
automorphism group of sp(2m,F2) is the symplectic group Sp(2m,F2), which is the group of
linear maps on F2m which preserve the symplectic form. This group is isomorphic to quotient
2
W(E )/ 11 , where W(E ) is the Weyl group of E .
7 7 7
{± }
3 Relationship to the normal capacity
Given a classical channel , its standard classical capacity C( ) cannot be increased by the
N N
use of entanglement or even arbitrary non-signalling correlations between the sender and re-
5
ceiver [17]. The standard capacity and the assisted and unassisted zero-error capacities are
related by
C ( ) CE( ) C( ). (9)
0 N ≤ 0 N ≤ N
Theorem 14. Given a graph G which satisfies the premises of Theorem 11 (partitions into
k cliques of size d and has an orthonormal representation in dimension d) and is also vertex-
transitive, one can construct a channel whose normal capacity C( ) and CE( ) are both
N N 0 N
equal to CE(G) and are both achieved by a block-length one entanglement-assisted zero-error
0
code.
Proof. Let X be the vertices of G and let Y be the set of all cliques of size d in G. Since G is
vertex-transitive, each vertex is contained in the same number m of cliques from Y. Counting
the number of pairs in the set (x,y) X Y : x y in two ways we have
{ ∈ × ∈ }
(x,y) X Y : x y = Y d = X m. (10)
{ ∈ × ∈ } | | | |
Let be the channel whic(cid:12)h on the input x X pro(cid:12)duces an output uniformly at random from
N (cid:12) ∈ (cid:12)
the set y : x y Y. Using the analysis of Section 16 of [18],
{ ∈ } ⊆
Y X
C( ) = log | | = log | | = logk. (11)
N m d
G( ) also partitions into k cliques of size d and, since it is a subgraph of G, has an orthonormal
N
representation in dimension d. Therefore CE( ) = logk and, furthermore, this rate is achieved
0 N
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 sp(6,F2) has C = C0E = log9, even though there is a positive lower bound on the
error probability for classical codes with any rate greater than log7 (as well as an upper bound,
both decaying exponentially with n) [19].
4 Graphs from E and other root systems
8
We define the orthogonality graph of a root system R as follows. The vectors of R occur in
antipodal pairs v, v ; the vertices V(R) of the graph are the R /2 rays spanned by these
{ − } | |
antipodal pairs. Two vertices are adjacent if and only if their rays are orthogonal. The graph of
Section2ispreciselytheorthogonalitygraphofE . Thisraisesthequestionofwhetherachannel
7
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 E provides a second example of such a gap, and furthermore,
8
the ratio between the assisted and classical capacities is larger in this case.
The irreducible root systems consist of the infinite families An, Bn, Cn, Dn where n N,
∈
and the exceptional cases E , E , E , F , and G (see [20]). We show that for all of the infinite
6 7 8 4 2
families, and for G , there is no gap between the independence number α and the Lov´asz
2
number ϑ, so CE = C for these graphs. However, the orthogonality graph of E provides a
0 0 8
second example of a gap between the classical and entanglement-assisted capacity. For E , we
8
show that C log9 while CE = log15. It is interesting to note that the graph used in [3] is
0 ≤ 0
precisely the orthogonality graph of F and while we know the entanglement-assisted capacity of
4
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 E .
6
In what follows the name of the root system is also used as the name of the orthogonality
graph and e denotes the i-th standard basis vector. We ignore correct normalization of the root
i
vectors for simplicity, since it clearly doesn’t affect the orthogonality graph.
6
Root system E
8
V(E ) = e e : 1 i < j 8 (x ,...,x ) : x = 1, 8 x = 1 (12)
8 { i± j ≤ ≤ }∪ 1 8 i ± i=1 i
Aspointedoutin[21], E8 isthegraphwhosever(cid:8)ticesarethenon-isotrop(cid:81)icpointsin(cid:9)theambient
projective space of the polar space O+(8,F2), with vertices adjacent if they are orthogonal with
respect to the associated bilinear form. The ambient projective space of O+(2m,F2) is PF22m.
Since the bilinear form associated with O+(2m,F2) is symplectic, it follows immediately that
E8 is an induced subgraph of the symplectic graph sp(2m,F2) with m = 4. By Theorem 9,
C0(E8) C0 sp(8,F2) = log9. (13)
≤
(cid:0) (cid:1)
On the other hand, let N = 16 and identify the vertices of sp(8,F2) with the non-zero
vectors of F2N. Then we may choose the quadratic form of O+(8,F2) to be (x,y) (cid:55)→ Tr(xy),
where Tr : FN F2 is the finite field trace. The polarization of this quadratic form is the
→
symplectic form σ (w,x),(y,z) = Tr(wz +xy). With this choice, the vertices of E , i.e., the
8
non-isotropic vecto(cid:0)rs in sp(8,F2(cid:1)), are those (x,y) ∈ F2N such that Tr(xy) = 1. Now, consider
the partition of vertices into cliques given in Lemma 12, restricted to the vertices of E :
8
πa = {(x,ax) : x ∈ F×N,Tr(ax2) = 1}, (a ∈ F×N). (14)
Recall that Tr(ax2) = Tr(a2x4) = ... = Tr(a8x). For each a F×, there are exactly 8 choices
of x ∈ F×N such that Tr(a8x) = 1. Therefore, {πa : a ∈ F×N} is∈a pNartition of the vertices of E8
into 15 cliques of size 8. By Theorem 11,
CE(E ) = log15. (15)
0 8
Root system A (n 1)
n
≥
V(A ) = e e : 1 i < j n+1 . (16)
n i j
{ − ≤ ≤ }
This graph is isomorphic to the Kneser graph KG . By a result of Lov´asz (Theorem 13
n+1,2
of [2]),
α(A ) = ϑ(A ) = Θ(A ) = n. (17)
n n n
Root system D (n 4)
n
≥
V(D ) = e e : 1 i < j n . (18)
n i j
{ ± ≤ ≤ }
Note that the vertices ei ej : 1 i < j n induce a subgraph isomorphic to An−1 ∼= KGn,2.
{ − ≤ ≤ }
Also note that e +e and e e are adjacent and have the same neighbourhood (apart from
i j i j
−
themselves). It follows that D is isomorphic to KG (cid:2)K , the strong graph product of a
n n,2 2
Kneser graph and a complete graph on 2 vertices. By Theorem 7 of [2],
Θ(D ) = Θ(KG )Θ(K ) = n 1. (19)
n n,2 2
−
Since e e : 2 j n is an independent set of size n 1, it follows that
1 j
{ − ≤ ≤ } −
α(D ) = ϑ(D ) = Θ(D ) = n 1. (20)
n n n
−
7
Root system B (n 2)
n
≥
V(B ) = e e : 1 i < j n e : 1 i n . (21)
n i j i
{ ± ≤ ≤ }∪{ ≤ ≤ }
To find the Lov´asz number we consider even and odd n separately. When n is odd, partition
the vertices into sets π ,...,π , where
1 n
π = e e : i < j,i+j 2k (mod n) e . (22)
k i j k
{ ± ≡ }∪{ }
Each π is a clique of size n. When n is even, partition the vertices into sets π ,...,π , where
k 1 n
π = e e : i < j,i+j 2k (mod n 1) e e (k n 1);
k i j k n
{ ± ≡ − }∪{ ± } ≤ −
π = e ,...,e .
n 1 n
{ }
Again each π is a clique of size n. In either case, we have partitioned the graph into n cliques
k
of size n. By Theorem 11, Θ(B ) = n.
n
Since e e : 2 j n e is an independent set of size n, we have
1 j 1
{ − ≤ ≤ }∪{ }
α(B ) = ϑ(B ) = Θ(B ) = n. (23)
n n n
Root system C (n 2)
n
≥
V(C ) = e e : 1 i < j n 2e : 1 i n . (24)
n i j i
{ ± ≤ ≤ }∪{ ≤ ≤ }
C has the same orthogonality graph as B .
n n
Root system G
2
V(G ) = (1, 1,0),(1,0, 1),(0,1, 1),(1,1, 2),(1, 2,1),( 2,1,1) . (25)
2
{ − − − − − − }
By inspection G has an independent set of size 3 and can be partitioned into 3 cliques of size
2
2. By Theorem 11,
α(G ) = ϑ(G ) = Θ(G ) = 3. (26)
2 2 2
5 Conclusion
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
[22].
ItisinterestingtonotethatineveryexampleofagraphwithM (G) > ME(G)foundtodate,
0 0
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 [23], an interesting observation
of Arikan is reported: The graph consisting of a five cycle and one more isolated vertex has
Θ = √5+1. 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 Lov´asz number of this graph
is also √5+1 (the Lov´asz number is additive for disjoint unions of graphs, and was calculated
for cycles in [2]), the same observation is true for the entanglement-assisted capacity, which in
this case is equal to the unassisted capacity.
8
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 [24] shows that the root systems E and E can be used to construct KS proofs using
7 8
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 9n measurements which are obtained
by tensoring together n of Alice’s 9 measurements, only 7n 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 CE? More specific related problems include determining whether
0
CE(G)/C (G) can be arbitrarily large, and whether there are graphs where CE(G) is strictly
0 0 0
less than logϑ(G).
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 con-
text of using entanglement to reduce decoding error in finite block length coding of classical
informationforclassicalchannels(demonstratingthiseffectisevenexperimentallyfeasible[25]),
and characterising this phenomenon presents an even wider set of questions.
Acknowledgements
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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