Problems of Information Transmission, Vol. 41, No. 2, 2005, pp. 76–90. Translated from Problemy Peredachi Informatsii, No. 2, 2005, pp. 9–25.
Original Russian Text Copyright
2005 by Datta, Holevo, Suhov.
On a Suﬃcient Condition for Additivity
in Quantum Information Theory
, and Yu. M. Suhov
Statistical Laboratory, Centre for Mathematical Science, University of Cambridge, UK
Steklov Mathematical Institute, Moscow
Received September 6, 2004
Abstract—In this paper we give a suﬃcient condition for additivity of the minimum output
entropy for a pair of given channels and an analytic veriﬁcation of this condition for speciﬁc
quantum channels breaking a closely related multiplicativity property [1,2]. This yields validity
of the additivity conjecture for these channels, a result obtained by a diﬀerent method in .
Our proof relies heavily upon certain concavity properties of the output entropy, which are of
A number of important issues in quantum information theory would be greatly clariﬁed if some
resources and channel parameters were proved to be additive. However, the proof of additivity
of such characteristics as the minimum output entropy for quantum memoryless channels and
their classical capacity remains in general an open problem, see, e.g., . Recently, Shor 
provided a new insight into how several additivity-type properties are related to each other by
proving, in particular, global equivalence of the above properties (i.e., of their validity for all chan-
In this paper we give a suﬃcient condition for additivity of the minimum output entropy for a
pair of given channels and an analytic veriﬁcation of this condition for speciﬁc quantum channels
breaking a closely related multiplicativity property  (see  for preliminary results). For every
channel from this family, the additivity of the Holevo capacity (in what follows, χ-capacity) and
that of the minimum output entropy are equivalent. This gives an alternative proof of the results
of , where the additivity was established by a diﬀerent method. The key observation that ensures
the additivity in our case is that the output entropy of a product channel exhibits speciﬁc concavity
properties as a function of the Schmidt coeﬃcients of the input pure state.
A similar condition was independently found to be suﬃcient for the additivity of the χ-capacity
of an arbitrary pair of quantum channels in a recent paper , where it was used for a numerical
check of the additivity for a newly-found qubit channel, which requires four input states to achieve
The same mechanism might be responsible for the additivity in other interesting
This work was initiated when the second author was an overseas visiting scholar at St John’s College, Cam-
bridge. Afterwards he was supported by the Russian Scientiﬁc School Program, project no. 1758.2003.1.
The ﬁrst and third authors worked in association with the CMI, University of Cambridge – MIT.
It is known that the χ-capacity of a channel in a d-dimensional space is attained on an ensemble of at
states. Thus, this estimate was shown to be tight for qubit channels (d =2).
2005 Pleiades Publishing, Inc.