ISSN 0032-9460, Problems of Information Transmission, 2014, Vol. 50, No. 3, pp. 232–246.
Pleiades Publishing, Inc., 2014.
Original Russian Text
M.E. Shirokov, T.V. Shulman, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 3, pp. 35–50.
On Superactivation of One-Shot Quantum
Zero-Error Capacity and the Related Property
of Quantum Measurements
M. E. Shirokov
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
University of Copenhagen, Denmark
Received March 3, 2014
Abstract—We give a detailed description of a low-dimensional quantum channel (input dimen-
sion 4, Choi rank 3) demonstrating the symmetric form of superactivation of one-shot quantum
zero-error capacity. This property means appearance of a noiseless (perfectly reversible) sub-
channel in the tensor square of a channel having no noiseless subchannels. Then we describe
a quantum channel with an arbitrary given level of symmetric superactivation (including the
inﬁnite value). We also show that superactivation of one-shot quantum zero-error capacity of
a channel can be reformulated in terms of quantum measurement theory as appearance of an
indistinguishable subspace for the tensor product of two observables having no indistinguishable
Quantum channels, i.e., completely positive trace-preserving linear maps between spaces of trace-
class operators (spaces of matrices in the ﬁnite-dimensional case), can be considered as noncommu-
tative analogs of classical communication channels. There exist various protocols for transmission
of classical and quantum information over quantum channels, determined by used resources, secu-
rity requirements, etc. For each such protocol and for a given quantum channel there is an ultimate
rate of errorless (or asymptotically errorless) information transmission. Similarly to the classical
case, it is called the capacity of this channel (corresponding to this protocol) [1,2].
The superactivation phenomenon means that a particular capacity of the tensor product of two
quantum channels may be positive despite that the same capacity of each of these channels is zero.
This phenomenon, which has no classical counterpart, has been discovered by G. Smith and J. Yard
in 2008 for the case of quantum capacity .
Later it was shown that superactivation may hold for diﬀerent quantum channel capacities, in
particular, for (one-shot and asymptotic) classical and quantum zero-error capacities [4, 6]. The
eﬀect of (so-called) extreme superactivation of zero-error capacities was also discovered .
This paper is devoted to superactivation of one-shot quantum zero-error capacity. This capacity
(Φ) (strictly deﬁned in Section 2) characterizes the possibility of zero-error transmission of
Supported in part by the Scientiﬁc Program “Mathematical Theory of Control and Dynamic Systems” of
the Russian Academy of Sciences and by the Russian Foundation for Basic Research, project nos. 2-01-
00319-a and 13-01-00295-a).
Supported in part by the Danish Research Council through the Centre for Symmetry and Deformation at
the University of Copenhagen.