The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven, so upper bounds such as the Lovász theta function play an important role in zero-error communication. In this paper, we show that the Lovász theta function is an upper bound on the zero-error capacity even in the presence of entanglement between the sender and receiver

Beigi, Salman

English

Caltech Authors - Main

RAPID COMMUNICATIONS
PHYSICAL REVIEW A 82, 010303(R) (2010)
Entanglement-assisted zero-error capacity is upper-bounded by the Lova´sz ϑ function
Salman Beigi
Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA
(Received 29 March 2010; published 26 July 2010)
The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph
and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven,
so upper bounds such as the Lova´sz theta function play an important role in zero-error communication. In this
paper, we show that the Lova´sz theta function is an upper bound on the zero-error capacity even in the presence
of entanglement between the sender and receiver.
DOI: 10.1103/PhysRevA.82.010303 PACS number(s): 03.67.Ac, 03.67.Bg, 89.70.Kn
I. INTRODUCTION
To provide an arbitrary small amount of error for transmit-
ting information via a communication channel, the number of
channel uses or equivalently the length of code words should
tend to infinity, which results in inefficient encoding and
decoding processes. To overcome this problem Shannon [1]
defined the zero-error channel capacity as the maximum rate
of information that can be sent through the channel with no
error. Later it was shown by Shannon et al. [2] that even the
rate at which the probability of error in the usual definition of
channel capacity decreases is related to the zero-error capacity,
turning this quantity into an important notion in information
theory [3].
To compute the zero-error capacity, the exact distribution of
the output of the channel under a certain input is not important;
what matters is whether the probability of receiving an output
is zero or not. Indeed, to encode m messages into code words of
length one and transmitting them with zero error we should find
m inputs of the channels which are not confusable after passing
through the channel. Thus, we can forget about the output set
of the channel and only consider its confusability graph; that
is, a graph on the input set of the channel with two vertices
being connected if with nonzero probability the output of the
channel is the same under those two inputs. Then encoding m
messages into code words of length one is equivalent to finding
an independent set of size m in the graph, and the one-shot
zero-error capacity of the channel is equal to the logarithm
of the independence number of the graph. Moreover, multiple
uses of the channel correspond to tensor product of the graph
(defined below) with itself, and then the zero-error capacity
is given in terms of the independence number of the tensor
powers of the graph.
It is known that computing the independence number of
graphs is an NP -complete problem; thus estimating the zero-
error capacity is a much harder problem. Even for small graphs
such as the cycle of length seven the capacity is unknown.
Shannon [1] computed the capacity of all graphs up to four
vertices, but the capacity of the cycle of length five remained an
open problem until Lova´sz [4] introduced an upper bound on
the zero-error capacity. Via semidefinite programming (SDP)
relaxation he found an upper bound on the independence
number of graphs. Then using the primal-dual framework of
SDPs he showed that the upper bound is multiplicative, and
concluded that this is indeed an upper bound on the zero-error
capacity.
The entanglement-assisted zero-error capacity of classical
channels was recently studied in [5]; that is, what happens if the
sender and receiver can share an entangled state? The authors
provided a graph for which the entanglement-assisted one-shot
capacity is greater than the independence number of the graph.
This example is evidence that unlike the usual capacity we may
increase the zero-error capacity by allowing entanglement as a
resource. This example, however, deals only with the one-shot
capacity, and computing the entanglement-assisted zero-error
capacity seems even harder than the graph capacity.
In this paper, we show that the Lova´sz upper bound
on the zero-error capacity is also an upper bound on the
entanglement-assisted zero-error capacity.
II. ENTANGLEMENT-ASSISTED ZERO-ERROR
CAPACITY
A discrete memoryless (classical) channel consists of a
finite input set X and an output set Y and the set of probabilities
p(x|y) for every x ∈ X, y ∈ Y , meaning that under the input
x the output of the channel is y with probability p(x|y). The
confusability graph of this channel is a graph G on the vertex
set X in which two vertices x,x ′ ∈ X are adjacent if there
exists y ∈ Y such that p(x|y) and p(x ′|y) are both nonzero
(see Fig. 1 for an example). Inputs x1, . . . ,xm can encode
m messages with zero error if they are not confusable after
passing through the channel, and then the one-shot zero-error
capacity of the channel is equal to log2 α(G) where α(G)
denotes the independence number of G (i.e., the maximum
number of vertices no two of which are adjacent). The graph
corresponding to k uses of the channel is given by the k-fold
tensor product of G with itself; that is, G⊗k is a graph on
the vertex set Xk and there is an edge between two vertices
(x1, . . . , xk) and (x ′1, . . . , x ′k) if for each 1  i  k, xi and x ′i
are either equal or adjacent in G. As a result, the zero-error
capacity is given by log2[lim α(G⊗k)1/k] as k → ∞.
In the presence of entanglement, the sender (Alice) and
receiver (Bob) can share an entangled state. To send a message,
Alice may apply some local measurement (which depends on
the message) in order to decide about the input of the channel.
Then Bob after receiving the output applies a measurement
(which depends on the output) to find Alice’s message.
Being a zero-error protocol, Bob’s residual states for different
messages, but a fixed output y ∈ Y , should be distinguishable.
However, note that Bob’s residual state depends on Alice’s
1050-2947/2010/82(1)/010303(4) 010303-1 ©2010 The American Physical Society
RAPID COMMUNICATIONS
SALMAN BEIGI PHYSICAL REVIEW A 82, 010303(R) (2010)
1
1
1
0.4
0.6
0.7
0.3
x1
x2
x3
x4
x5
y1
y2
y3
x1
x2
x3
x4x5
FIG. 1. The left graph shows a channel with input set {x1, . . . , x5}
and output set {y1,y2,y3}. If, for example, the input of the channel is
x3 the output is y1 or y2 with probabilities 0.4 and 0.6 respectively.
If the receiver sees y2 as the output, the input is either x3 or x4, so
(x3,x4) is a confusable pair. The right graph depicts the confusability
graph of the channel. The independence number of this graph is 2,
so log2 2 = 1 bit of information, by using code words {x3,x5}, can
be sent through this channel with no error. Moreover, it is not hard
to convince oneself that the independence of the kth tensor power of
this graph (the graph corresponding to k use of the channel) is equal
to 2k , and then the zero-error capacity of this channel is 1.
measurement and not directly ony which is not known to Alice.
So elements of Y do not appear in the orthogonality relations
that the residual states should satisfy. As a result, to compute
the entanglement-assisted zero-error capacity we can again
ignore the output set Y and only consider the confusability
graph G (see [5] for more details).
This observation allows us to assume that the channel has
the following special form: the output set of the channel is
the edge set of the graph, and under the input x we receive
one of the edges connected to x as the output (say) all with
equal probability. (For an isolated vertex x we also put x in
the output set.) Although this assumption is not crucial and the
proof can be given for an arbitrary channel, it simplifies the
presentation.
Assume that the one-shot entanglement-assisted zero-error
capacity of the channel is log2 m. Then the encoding and
decoding processes are as follows. Alice and Bob share a
bipartite state |ψ〉; to transmit the message 1  i  m, Alice
measures her part of |ψ〉 using the positive operator-valued
measure (POVM) {P ix : x ∈ X}, and sends the outcome of the
measurement x ∈ X through the channel. The output of the
channel is an edge xx ′ connected to x. Then Bob measures
the other part of |ψ〉 using the POVM {Qxx ′j : 1  j  m}
and considers the outcome of this measurement as Alice’s
message. Since this protocol has no error, the outcome of
Bob’s measurement must coincide with Alice’s message i.
That is, for every edge xx ′,
〈ψ |P ix ⊗ Qxx
′
j |ψ〉 = δij , (2.1)
where δij denotes the Kronecker delta function. In case x is an
isolated vertex, Bob’s measurement is of the form {Qxj : 1 
j  m} and we should have 〈ψ |P ix ⊗ Qxj |ψ〉 = δij .
As a result, the one-shot entanglement-assisted zero-error
capacity of the channel is equal to log2 α∗(G) where α∗(G)
is the maximum number m for which there exist POVMs
{P ix : x ∈ X} and {Qxx
′
j : 1  j  m} satisfying (2.1). Also,
the entanglement-assisted zero-error capacity is equal to
log2[lim α∗(G⊗k)1/k] as k → ∞.
III. THE LOV ´ASZ THETA FUNCTION
There are several characterizations of the Lova´sz theta
function [4]. The following definition is appropriate for us.
ϑ(G) = max tr(BJ ),
B  0,
trB = 1,
Bxx ′ = 0, for every edge xx ′,
where B is an |X| × |X| matrix and J is the matrix with
all entries equal to one. Considering an independent set of
size α(G), and letting Bxx ′ be equal to 1/[α(G)] for every
x,x ′ in the independent set and 0 otherwise, we find that
α(G)  ϑ(G). Lova´sz showed that ϑ(G) is multiplicative
[ϑ(G ⊗ H ) = ϑ(G)ϑ(H )] and concluded α(G⊗k)1/k  ϑ(G)
and that log2 ϑ(G) is an upper bound on the zero-error capacity
of G.
Lova´sz using this upper bound computed the capacity of
C5, the cycle of length five. If we let the vertices of C5 be
x1, . . . , x5, {x1x1,x2x3,x3x5,x4x2,x5x4} is an independent set
in C5 ⊗ C5. Then the capacity of C5 is at least log2
√
5. On the
other hand, ϑ(C5) =
√
5 [4]. Therefore, the capacity of C5 is
equal to log2
√
5.
IV. MAIN RESULT
Theorem 1. ϑ(G)  α∗(G), and since ϑ(G) is multiplica-
tive, log2 ϑ(G) is an upper bound on the entanglement-assisted
zero-error capacity of a classical channel with the confusability
graph G.
Let us first explain the idea behind the proof of this theorem
and then go through the details. Assume that α∗(G) = m and
for every 1  i  m and x ∈ X there exists a vectorwix in some
inner product vector space satisfying the following conditions:
(1) For every i, ∑x wix = w where w is a vector of length
one,
(2) 〈wix,wix ′ 〉 = 0 for every x 
= x ′,
(3) 〈wix,wjx ′ 〉 = 0 for every edge xx ′ of G,
(4) and 〈wix,wjx〉 = 0 for every i 
= j .
Then we can define the |X| × |X| matrix B by Bxx ′ =
〈wx,wx ′ 〉 where wx =
∑
i w
i
x . In this case, B is a positive
semidefinite matrix and one can show that Bxx ′ = 0 for
every edge xx ′, trB = m and tr(BJ ) = m2 (later we will
prove them all). Then by the definition of ϑ(G) we have
ϑ(G)  tr(BJ )/trB = m = α∗(G). So our main problem is
to define the vectors wix satisfying the above properties. In
the case where the shared state between Alice and Bob is
the maximally entangled state, these vectors are basically the
residual states of Alice after her measurement, but the general
case needs more work because we should define some twisted
inner product.
Assume that m = α∗(G) and there exist a bipartite state
|ψ〉 and POVM measurements {P ix : x ∈ X} for 1  i  m,
and {Qxx ′j : 1  j  m} for every edge xx ′, satisfying (2.1).
Due to the normalization of the POVMs we have
∑
x∈X
P ix = I,
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ENTANGLEMENT-ASSISTED ZERO-ERROR CAPACITY IS . . . PHYSICAL REVIEW A 82, 010303(R) (2010)
and
m∑
j=1
Qxx
′
j = I.
Without loss of generality, we assume that the POVMs
{P ix : x ∈ X} are projective measurements (P ixP ix ′ = δx,x ′P ix )
because every POVM measurement can be written as a
projective measurement on an extended space. Also, we may
assume that the local spaces of Alice and Bob are isomorphic
and of dimension d. Let |〉 be the maximally entangled state
on these two spaces. Then there exists a matrix S such that
|ψ〉 = I ⊗ S|〉,
and due to the normalization of |ψ〉 we have trS†S = d.
For every 1  j  m and edge xx ′ define Rxx ′j =
(S†Qxx ′j S)T, where T denotes the transpose operation with
respect to the orthonormal basis in which the maximally
entangled state |〉 is defined. Then for every edge xx ′ we have
Rxx
′
1 + · · · + Rxx
′
m = M,
where M = (S†S)T. Observe that
δij = 〈ψ |P ix ⊗ Qxx
′
j |ψ〉
= 〈|P ix ⊗ S†Qxx
′
j S|〉
= 1
d
tr
(
P ixR
xx ′
j
)
.
Since both P ix and Rxx
′
j are positive semidefinite we conclude
that
P ixR
xx ′
j = 0, (4.1)
for every edge xx ′ and every i 
= j .
Consider the vector space of complex d × d matrices
equipped with the bilinear form,
〈A,B〉 = 1
d
tr(A†BM).
Note that, for every nonzero matrix A, 〈A,A〉 is non-negative
because M is positive semidefinite. So 〈·,·〉 is a non-negative
form.1 There are some orthogonality relations among matrices
P ix with respect to this bilinear form.
(a) 〈P ix ,P ix ′ 〉 = 0 for every x 
= x ′. This is simply because
P ixP
i
x ′ = 0.
(b) 〈P ix ,P jx ′ 〉 = 0 for every edge xx ′. From (a) we may
assume that i 
= j . Then we have
〈
P ix ,P
j
x ′
〉 = 1
d
tr
(
P ixP
j
x ′M
) = 1
d
m∑
k=1
tr
(
P ixP
j
x ′R
xx ′
k
) = 0,
since for every k, by (4.1), either P ixRxx
′
k = 0 or P jx ′Rxx
′
k = 0.
(c) 〈P ix ,P jx 〉 = 0 for every i 
= j . Let x ′ be an adjacent
vertex of x. As before for every k either P ixRxx
′
k = 0 or
P
j
x R
xx ′
k = 0 and orthogonality follows. Similar equations can
be written if x is an isolated vertex.
1It may not be positive because M can have zero eigenvalues.
Now for every vertex x of G let Vx =
∑m
i=1 P
i
x and define
the matrix B by Bxx ′ = 〈Vx,Vx ′ 〉. Since 〈·,·〉 is a non-negative
form, B is positive semidefinite. Moreover, by (b) for every
edge xx ′ we have
Bxx ′ = 〈Vx,Vx ′ 〉 =
m∑
i,j=1
〈
P ix ,P
j
x ′
〉 = 0.
As a result, tr(BJ )/tr(B)  ϑ(G). So the only remaining part
is to show tr(BJ )/tr(B) = m. Using (c) we have
trB =
∑
x
〈Vx,Vx〉 =
m∑
i,j=1
∑
x
〈
P ix ,P
j
x
〉 =
m∑
i=1
∑
x
〈
P ix ,P
i
x
〉
= 1
d
m∑
i=1
∑
x
tr
(
P ixM
) = 1
d
m∑
i=1
tr(M) = m,
where in the second line we use (P ix )2 = P ix . Also by
∑
x P
i
x =
I we obtain
tr(BJ ) = ∑x,x ′ 〈Vx,Vx ′ 〉 =
∑m
i,j=1
∑
x,x ′
〈
P ix ,P
j
x ′
〉
= ∑mi,j 〈I,I 〉 = m2.
We are done.
V. DISCUSSION
The proof of Theorem 1 indeed gives the following stronger
statement. Let β(G) be the maximum number m such that
there exist vectors wix , 1  i  m, in an inner product vector
space satisfying conditions (1)–(4) of the previous section.
Then α∗(G)  β(G)  ϑ(G). It is a very interesting question
whether α∗(G) and β(G) are always equal or not. The vectors
constructed in the proof of Theorem 1 come from positive
semidefinite matrices (POVM elements), but there is no such
constraint in the definition of β(G), and this point seems to
be the main difference between α∗(G) and β(G). Another
interesting problem is the relation between β(G) and ϑ(G); for
example, do we have β(G) = ϑ(G)? To prove this equality,
given a matrix B in the definition of ϑ(G), if Bxx ′ = 〈wx,wx ′ 〉,
we should be able to decompose each wx into ϑ(G) vectors
wx =
∑
i w
i
x satisfying (1)–(4). In general, computing β(G)
seems easier than α∗(G) and we hope this quantity furthers
research in this direction.
Examples of [5] which show a gap between α(G) and
α∗(G) are defined based on the Kochen-Specker sets. In
these examples one can easily check that α∗(G) and ϑ(G)
coincide, so according to Theorem 1 the entanglement-assisted
zero-error capacity of these channels is equal to log2 ϑ(G).
Thus computing the usual zero-error capacity of these graphs
and comparing it with the Lova´sz theta function would clarify
the role of entanglement in zero-error communication. Also,
finding instances of graphs G with α(G) < α(G)∗ < ϑ(G) is
of interest.
Finally, as noted in [5] entanglement-assisted zero-error
communication protocols are related to pseudotelepathy
games. So techniques of this paper might be useful for studying
such games.
Note added. Recently, we found that in an independent
work Runyao Duan, Simone Severini, and Andreas Winter
[6] have defined a generalization of the Lova´sz bound for
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RAPID COMMUNICATIONS
SALMAN BEIGI PHYSICAL REVIEW A 82, 010303(R) (2010)
quantum channels, and have shown that it is an upper bound
on the entanglement-assisted zero-error capacity of quantum
channels. For classical channels this bound coincides with
the Lova´sz bound and then their work gives our result by
a different approach. Besides being shorter, an advantage of
our proof is that it gives the intuition of how to construct
an entanglement-assisted communication protocol from the
optimal vectors in the Lova´sz bound.
ACKNOWLEDGMENTS
The author is grateful to the unknown referee whose
comments helped to improve the presentation of the paper.
This work has been supported in part by National Science
Foundation under Grant No. PHY-0803371 and by National
Security Agency/Army Research Office under Grant No.
W911NF-09-1-0442.
[1] C. E. Shannon, IRE Trans. Inform. Theory 2, 8 (1956).
[2] C. E. Shannon, R. G. Gallager, and E. R. Berlekamp, Inform.
Contr. 10, 65 (1967).
[3] J. Korner and A. Orlitsky, IEEE Trans. Inf. Theory 44, 2207
(1998).
[4] L. Lova´sz, IEEE Trans. Inf. Theory 25, 1 (1979).
[5] T. S. Cubitt, D. Leung, W. Matthews, and A. Winter, e-print
arXiv:0911.5300.
[6] R. Duan, S. Severini, and A. Winter, e-print
arXiv:1002.2514.
010303-4


Entanglement-assisted zero-error capacity is upper-bounded by the Lovász ϑ function

RAPID COMMUNICATIONS
PHYSICAL REVIEW A 82, 010303(R) (2010)
Entanglement-assisted zero-error capacity is upper-bounded by the Lovász ϑ function
Salman Beigi
Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA
(Received 29 March 2010; published 26 July 2010)
The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph
and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven,
so upper bounds such as the Lovász theta function play an important role in zero-error communication. In this
paper, we show that the Lovász theta function is an upper bound on the zero-error capacity even in the presence
of entanglement between the sender and receiver.
DOI: 10.1103/PhysRevA.82.010303 PACS number(s): 03.67.Ac, 03.67.Bg, 89.70.Kn
I. INTRODUCTION
To provide an arbitrary small amount of error for transmit-
ting information via a communication channel, the number of
channel uses or equivalently the length of code words should
tend to infinity, which results in inefficient encoding and
decoding processes. To overcome this problem Shannon [1]
defined the zero-error channel capacity as the maximum rate
of information that can be sent through the channel with no
error. Later it was shown by Shannon et al. [2] that even the
rate at which the probability of error in the usual definition of
channel capacity decreases is related to the zero-error capacity,
turning this quantity into an important notion in information
theory [3].
To compute the zero-error capacity, the exact distribution of
the output of the channel under a certain input is not important;
what matters is whether the probability of receiving an output
is zero or not. Indeed, to encode m messages into code words of
length one and transmitting them with zero error we should find
m inputs of the channels which are not confusable after passing
through the channel. Thus, we can forget about the output set
of the channel and only consider its confusability graph; that
is, a graph on the input set of the channel with two vertices
being connected if with nonzero probability the output of the
channel is the same under those two inputs. Then encoding m
messages into code words of length one is equivalent to finding
an independent set of size m in the graph, and the one-shot
zero-error capacity of the channel is equal to the logarithm
of the independence number of the graph. Moreover, multiple
uses of the channel correspond to tensor product of the graph
(defined below) with itself, and then the zero-error capacity
is given in terms of the independence number of the tensor
powers of the graph.
It is known that computing the independence number of
graphs is an NP -complete problem; thus estimating the zero-
error capacity is a much harder problem. Even for small graphs
such as the cycle of length seven the capacity is unknown.
Shannon [1] computed the capacity of all graphs up to four
vertices, but the capacity of the cycle of length five remained an
open problem until Lovász [4] introduced an upper bound on
the zero-error capacity. Via semidefinite programming (SDP)
relaxation he found an upper bound on the independence
number of graphs. Then using the primal-dual framework of
SDPs he showed that the upper bound is multiplicative, and
concluded that this is indeed an upper bound on the zero-error
capacity.
The entanglement-assisted zero-error capacity of classical
channels was recently studied in [5]; that is, what happens if the
sender and receiver can share an entangled state? The authors
provided a graph for which the entanglement-assisted one-shot
capacity is greater than the independence number of the graph.
This example is evidence that unlike the usual capacity we may
increase the zero-error capacity by allowing entanglement as a
resource. This example, however, deals only with the one-shot
capacity, and computing the entanglement-assisted zero-error
capacity seems even harder than the graph capacity.
In this paper, we show that the Lovász upper bound
on the zero-error capacity is also an upper bound on the
entanglement-assisted zero-error capacity.
II. ENTANGLEMENT-ASSISTED ZERO-ERROR
CAPACITY
A discrete memoryless (classical) channel consists of a
finite input set X and an output set Y and the set of probabilities
p(x|y) for every x ∈ X, y ∈ Y , meaning that under the input
x the output of the channel is y with probability p(x|y). The
confusability graph of this channel is a graph G on the vertex
set X in which two vertices x,x ′ ∈ X are adjacent if there
exists y ∈ Y such that p(x|y) and p(x ′|y) are both nonzero
(see Fig. 1 for an example). Inputs x1, . . . ,xm can encode
m messages with zero error if they are not confusable after
passing through the channel, and then the one-shot zero-error
capacity of the channel is equal to log2 α(G) where α(G)
denotes the independence number of G (i.e., the maximum
number of vertices no two of which are adjacent). The graph
corresponding to k uses of the channel is given by the k-fold
tensor product of G with itself; that is, G⊗k is a graph on
the vertex set Xk and there is an edge between two vertices
(x1, . . . , xk) and (x ′1, . . . , x
′
k) if for each 1  i  k, xi and x ′i
are either equal or adjacent in G. As a result, the zero-error
capacity is given by log2[lim α(G
⊗k)1/k] as k → ∞.
In the presence of entanglement, the sender (Alice) and
receiver (Bob) can share an entangled state. To send a message,
Alice may apply some local measurement (which depends on
the message) in order to decide about the input of the channel.
Then Bob after receiving the output applies a measurement
(which depends on the output) to find Alice’s message.
Being a zero-error protocol, Bob’s residual states for different
messages, but a fixed output y ∈ Y , should be distinguishable.
However, note that Bob’s residual state depends on Alice’s
1050-2947/2010/82(1)/010303(4) 010303-1 ©2010 The American Physical Society
RAPID COMMUNICATIONS
SALMAN BEIGI PHYSICAL REVIEW A 82, 010303(R) (2010)
1
1
1
0.4
0.6
0.7
0.3
x1
x2
x3
x4
x5
y1
y2
y3
x1
x2
x3
x4x5
FIG. 1. The left graph shows a channel with input set {x1, . . . , x5}
and output set {y1,y2,y3}. If, for example, the input of the channel is
x3 the output is y1 or y2 with probabilities 0.4 and 0.6 respectively.
If the receiver sees y2 as the output, the input is either x3 or x4, so
(x3,x4) is a confusable pair. The right graph depicts the confusability
graph of the channel. The independence number of this graph is 2,
so log2 2 = 1 bit of information, by using code words {x3,x5}, can
be sent through this channel with no error. Moreover, it is not hard
to convince oneself that the independence of the kth tensor power of
this graph (the graph corresponding to k use of the channel) is equal
to 2k , and then the zero-error capacity of this channel is 1.
measurement and not directly on y which is not known to Alice.
So elements of Y do not appear in the orthogonality relations
that the residual states should satisfy. As a result, to compute
the entanglement-assisted zero-error capacity we can again
ignore the output set Y and only consider the confusability
graph G (see [5] for more details).
This observation allows us to assume that the channel has
the following special form: the output set of the channel is
the edge set of the graph, and under the input x we receive
one of the edges connected to x as the output (say) all with
equal probability. (For an isolated vertex x we also put x in
the output set.) Although this assumption is not crucial and the
proof can be given for an arbitrary channel, it simplifies the
presentation.
Assume that the one-shot entanglement-assisted zero-error
capacity of the channel is log2 m. Then the encoding and
decoding processes are as follows. Alice and Bob share a
bipartite state |ψ〉; to transmit the message 1  i  m, Alice
measures her part of |ψ〉 using the positive operator-valued
measure (POVM) {P ix : x ∈ X}, and sends the outcome of the
measurement x ∈ X through the channel. The output of the
channel is an edge xx ′ connected to x. Then Bob measures
the other part of |ψ〉 using the POVM {Qxx ′j : 1  j  m}
and considers the outcome of this measurement as Alice’s
message. Since this protocol has no error, the outcome of
Bob’s measurement must coincide with Alice’s message i.
That is, for every edge xx ′,
〈ψ |P ix ⊗ Qxx
′
j |ψ〉 = δij , (2.1)
where δij denotes the Kronecker delta function. In case x is an
isolated vertex, Bob’s measurement is of the form {Qxj : 1 
j  m} and we should have 〈ψ |P ix ⊗ Qxj |ψ〉 = δij .
As a result, the one-shot entanglement-assisted zero-error
capacity of the channel is equal to log2 α
∗(G) where α∗(G)
is the maximum number m for which there exist POVMs
{P ix : x ∈ X} and {Qxx
′
j : 1  j  m} satisfying (2.1). Also,
the entanglement-assisted zero-error capacity is equal to
log2[lim α
∗(G⊗k)1/k] as k → ∞.
III. THE LOVÁSZ THETA FUNCTION
There are several characterizations of the Lovász theta
function [4]. The following definition is appropriate for us.
ϑ(G) = max tr(BJ ),
B  0,
trB = 1,
Bxx ′ = 0, for every edge xx ′,
where B is an |X| × |X| matrix and J is the matrix with
all entries equal to one. Considering an independent set of
size α(G), and letting Bxx ′ be equal to 1/[α(G)] for every
x,x ′ in the independent set and 0 otherwise, we find that
α(G)  ϑ(G). Lovász showed that ϑ(G) is multiplicative
[ϑ(G ⊗ H ) = ϑ(G)ϑ(H )] and concluded α(G⊗k)1/k  ϑ(G)
and that log2 ϑ(G) is an upper bound on the zero-error capacity
of G.
Lovász using this upper bound computed the capacity of
C5, the cycle of length five. If we let the vertices of C5 be
x1, . . . , x5, {x1x1,x2x3,x3x5,x4x2,x5x4} is an independent set
in C5 ⊗ C5. Then the capacity of C5 is at least log2
√
5. On the
other hand, ϑ(C5) =
√
5 [4]. Therefore, the capacity of C5 is
equal to log2
√
5.
IV. MAIN RESULT
Theorem 1. ϑ(G)  α∗(G), and since ϑ(G) is multiplica-
tive, log2 ϑ(G) is an upper bound on the entanglement-assisted
zero-error capacity of a classical channel with the confusability
graph G.
Let us first explain the idea behind the proof of this theorem
and then go through the details. Assume that α∗(G) = m and
for every 1  i  m and x ∈ X there exists a vector wix in some
inner product vector space satisfying the following conditions:
(1) For every i,
∑
x w
i
x = w where w is a vector of length
one,
(2) 〈wix,wix ′ 〉 = 0 for every x 
= x ′,
(3) 〈wix,wjx ′ 〉 = 0 for every edge xx ′ of G,
(4) and 〈wix,wjx〉 = 0 for every i 
= j .
Then we can define the |X| × |X| matrix B by Bxx ′ =
〈wx,wx ′ 〉 where wx =
∑
i w
i
x . In this case, B is a positive
semidefinite matrix and one can show that Bxx ′ = 0 for
every edge xx ′, trB = m and tr(BJ ) = m2 (later we will
prove them all). Then by the definition of ϑ(G) we have
ϑ(G)  tr(BJ )/trB = m = α∗(G). So our main problem is
to define the vectors wix satisfying the above properties. In
the case where the shared state between Alice and Bob is
the maximally entangled state, these vectors are basically the
residual states of Alice after her measurement, but the general
case needs more work because we should define some twisted
inner product.
Assume that m = α∗(G) and there exist a bipartite state
|ψ〉 and POVM measurements {P ix : x ∈ X} for 1  i  m,
and {Qxx ′j : 1  j  m} for every edge xx ′, satisfying (2.1).
Due to the normalization of the POVMs we have
∑
x∈X
P ix = I,
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and
m∑
j=1
Qxx
′
j = I.
Without loss of generality, we assume that the POVMs
{P ix : x ∈ X} are projective measurements (P ixP ix ′ = δx,x ′P ix )
because every POVM measurement can be written as a
projective measurement on an extended space. Also, we may
assume that the local spaces of Alice and Bob are isomorphic
and of dimension d. Let |〉 be the maximally entangled state
on these two spaces. Then there exists a matrix S such that
|ψ〉 = I ⊗ S|〉,
and due to the normalization of |ψ〉 we have trS†S = d.
For every 1  j  m and edge xx ′ define Rxx ′j =
(S†Qxx
′
j S)
T, where T denotes the transpose operation with
respect to the orthonormal basis in which the maximally
entangled state |〉 is defined. Then for every edge xx ′ we have
Rxx
′
1 + · · · + Rxx
′
m = M,
where M = (S†S)T. Observe that
δij = 〈ψ |P ix ⊗ Qxx
′
j |ψ〉
= 〈|P ix ⊗ S†Qxx
′
j S|〉
= 1
d
tr
(
P ixR
xx ′
j
)
.
Since both P ix and R
xx ′
j are positive semidefinite we conclude
that
P ixR
xx ′
j = 0, (4.1)
for every edge xx ′ and every i 
= j .
Consider the vector space of complex d × d matrices
equipped with the bilinear form,
〈A,B〉 = 1
d
tr(A†BM).
Note that, for every nonzero matrix A, 〈A,A〉 is non-negative
because M is positive semidefinite. So 〈·,·〉 is a non-negative
form.1 There are some orthogonality relations among matrices
P ix with respect to this bilinear form.
(a) 〈P ix ,P ix ′ 〉 = 0 for every x 
= x ′. This is simply because
P ixP
i
x ′ = 0.
(b) 〈P ix ,P jx ′ 〉 = 0 for every edge xx ′. From (a) we may
assume that i 
= j . Then we have
〈
P ix ,P
j
x ′
〉 = 1
d
tr
(
P ixP
j
x ′M
) = 1
d
m∑
k=1
tr
(
P ixP
j
x ′R
xx ′
k
) = 0,
since for every k, by (4.1), either P ixR
xx ′
k = 0 or P jx ′Rxx
′
k = 0.
(c) 〈P ix ,P jx 〉 = 0 for every i 
= j . Let x ′ be an adjacent
vertex of x. As before for every k either P ixR
xx ′
k = 0 or
P
j
x R
xx ′
k = 0 and orthogonality follows. Similar equations can
be written if x is an isolated vertex.
1It may not be positive because M can have zero eigenvalues.
Now for every vertex x of G let Vx =
∑m
i=1 P
i
x and define
the matrix B by Bxx ′ = 〈Vx,Vx ′ 〉. Since 〈·,·〉 is a non-negative
form, B is positive semidefinite. Moreover, by (b) for every
edge xx ′ we have
Bxx ′ = 〈Vx,Vx ′ 〉 =
m∑
i,j=1
〈
P ix ,P
j
x ′
〉 = 0.
As a result, tr(BJ )/tr(B)  ϑ(G). So the only remaining part
is to show tr(BJ )/tr(B) = m. Using (c) we have
trB =
∑
x
〈Vx,Vx〉 =
m∑
i,j=1
∑
x
〈
P ix ,P
j
x
〉 =
m∑
i=1
∑
x
〈
P ix ,P
i
x
〉
= 1
d
m∑
i=1
∑
x
tr
(
P ixM
) = 1
d
m∑
i=1
tr(M) = m,
where in the second line we use (P ix )
2 = P ix . Also by
∑
x P
i
x =
I we obtain
tr(BJ ) = ∑x,x ′ 〈Vx,Vx ′ 〉 =
∑m
i,j=1
∑
x,x ′
〈
P ix ,P
j
x ′
〉
= ∑mi,j 〈I,I 〉 = m2.
We are done.
V. DISCUSSION
The proof of Theorem 1 indeed gives the following stronger
statement. Let β(G) be the maximum number m such that
there exist vectors wix , 1  i  m, in an inner product vector
space satisfying conditions (1)–(4) of the previous section.
Then α∗(G)  β(G)  ϑ(G). It is a very interesting question
whether α∗(G) and β(G) are always equal or not. The vectors
constructed in the proof of Theorem 1 come from positive
semidefinite matrices (POVM elements), but there is no such
constraint in the definition of β(G), and this point seems to
be the main difference between α∗(G) and β(G). Another
interesting problem is the relation between β(G) and ϑ(G); for
example, do we have β(G) = ϑ(G)? To prove this equality,
given a matrix B in the definition of ϑ(G), if Bxx ′ = 〈wx,wx ′ 〉,
we should be able to decompose each wx into ϑ(G) vectors
wx =
∑
i w
i
x satisfying (1)–(4). In general, computing β(G)
seems easier than α∗(G) and we hope this quantity furthers
research in this direction.
Examples of [5] which show a gap between α(G) and
α∗(G) are defined based on the Kochen-Specker sets. In
these examples one can easily check that α∗(G) and ϑ(G)
coincide, so according to Theorem 1 the entanglement-assisted
zero-error capacity of these channels is equal to log2 ϑ(G).
Thus computing the usual zero-error capacity of these graphs
and comparing it with the Lovász theta function would clarify
the role of entanglement in zero-error communication. Also,
finding instances of graphs G with α(G) < α(G)∗ < ϑ(G) is
of interest.
Finally, as noted in [5] entanglement-assisted zero-error
communication protocols are related to pseudotelepathy
games. So techniques of this paper might be useful for studying
such games.
Note added. Recently, we found that in an independent
work Runyao Duan, Simone Severini, and Andreas Winter
[6] have defined a generalization of the Lovász bound for
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RAPID COMMUNICATIONS
SALMAN BEIGI PHYSICAL REVIEW A 82, 010303(R) (2010)
quantum channels, and have shown that it is an upper bound
on the entanglement-assisted zero-error capacity of quantum
channels. For classical channels this bound coincides with
the Lovász bound and then their work gives our result by
a different approach. Besides being shorter, an advantage of
our proof is that it gives the intuition of how to construct
an entanglement-assisted communication protocol from the
optimal vectors in the Lovász bound.
ACKNOWLEDGMENTS
The author is grateful to the unknown referee whose
comments helped to improve the presentation of the paper.
This work has been supported in part by National Science
Foundation under Grant No. PHY-0803371 and by National
Security Agency/Army Research Office under Grant No.
W911NF-09-1-0442.
[1] C. E. Shannon, IRE Trans. Inform. Theory 2, 8 (1956).
[2] C. E. Shannon, R. G. Gallager, and E. R. Berlekamp, Inform.
Contr. 10, 65 (1967).
[3] J. Korner and A. Orlitsky, IEEE Trans. Inf. Theory 44, 2207
(1998).
[4] L. Lovász, IEEE Trans. Inf. Theory 25, 1 (1979).
[5] T. S. Cubitt, D. Leung, W. Matthews, and A. Winter, e-print
arXiv:0911.5300.
[6] R. Duan, S. Severini, and A. Winter, e-print
arXiv:1002.2514.
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Entanglement-assisted zero-error capacity is upper-bounded by the Lovász ϑ function

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