ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 3, pp. 239–264.
Pleiades Publishing, Inc., 2016.
Original Russian Text
M.E. Shirokov, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 3, pp. 45–72.
Estimates for Discontinuity Jumps of Information
Characteristics of Quantum Systems and Channels
M. E. Shirokov
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
Received February 24, 2016
Abstract—Quantitative analysis of discontinuity of information characteristics of quantum
states and channels is presented. Estimates for discontinuity jump (loss) of the von Neumann
entropy for a given converging sequence of states are obtained. It is shown, in particular,
that for any sequence the loss of entropy is upper bounded by the loss of mean energy (with
the coeﬃcient characterizing the Hamiltonian of a system). Then we prove that discontinuity
jumps of basic measures of classical and quantum correlations in composite quantum systems
are upper bounded by the loss of one of the marginal entropies (with a corresponding coeﬃ-
cient). Quantitative discontinuity analysis of the output entropy of a quantum operation and
of basic information characteristics of a quantum channel considered as functions of a pair
(channel, input state) is presented.
One of the main diﬃculties in study of inﬁnite-dimensional quantum systems consists in discon-
tinuity of basic characteristics of quantum states and channels (such as the von Neumann entropy,
conditional entropy, quantum mutual information, entanglement measures, etc.). This shows the
necessity to ﬁnd conditions for local continuity of such characteristic. The ﬁrst results in this direc-
tion seems to be Simon’s convergence theorems for the von Neumann entropy [1, Appendix]. Since
then, many diﬀerent continuity conditions for the entropy and other basic information characteris-
tics have been found (see [2–5] and the references therein).
In this paper we present quantitative analysis of discontinuity of several important characteristics
of quantum states and channels starting with the von Neumann entropy.
In Section 3 we consider general estimates for discontinuity jumps of the von Neumann entropy
(Propositions 1 and 3 and their corollaries). We also consider relations between discontinuity jumps
of the entropy and majorization (Proposition 2). Then we focus on estimating discontinuity of the
entropy on the set of states with bounded mean energy, i.e., states ρ satisfying the inequality
Tr Hρ ≤ E, (1)
where H is the Hamiltonian of a system. It is known that the entropy is continuous on this set if
(and only if) Tr e
is ﬁnite for all λ>0 . Explicit continuity bounds for the entropy on this
set were recently obtained by Winter . We analyze discontinuity jumps (losses) of the entropy
on the set determined by inequality (1) in the case of logarithmic growth of the eigenvalues of H,
< +∞ for some λ>0. (2)
The research was carried out at the expense of the Russian Science Foundation, project no. 14-21-00162.