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Physical Review Letters

Lower Bound for Mutual Information of a Quantum Channel

Horia Scutaru

Published Jul 31, 1995• 9 citations
Publisher

Abstract

We obtain a lower bound for the mutual information of a quantum transmission channel, which is perfectly analogous with Holevo's upper bound. The Uhlmann inequality for relative entropies is used, in a reverted order, for a completely positive mapping related with the mixed coherent states introduced by us seventeen years ago. Possible applications to quantum cryptography are discussed.

DOI
10.1103/physrevlett.75.773

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We obtain a lower bound for the mutual information of a quantum transmission channel, which is perfectly analogous with Holevo's upper bound. The Uhlmann inequality for relative entropies is used, in a reverted order, for a completely positive mapping related with the mixed coherent states introduced by us seventeen years ago. Possible applications to quantum cryptography are discussed.
While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measurement are obtained in a unified manner. In particular, it is shown that such bounds can all be stated as inequalities between mutual entropies. This approach based on channels gives rise to a unified picture of known and new bounds on the classical information (Holevo's, Shumacher-Westmoreland-Wootters', Hall's, Scutaru's bounds, a new upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.
We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, which we define similarly to the Holevo capacity but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.
While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measurement are obtained in a unified manner. In particular, it is shown that such bounds can all be stated as inequalities between mutual entropies. This approach based on channels gives rise to a unified picture of known and new bounds on the classical information (the bounds by Holevo, by Shumacher, Westmoreland and Wootters, by Hall, by Scutaru, a new upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.
Let X be an unbiased random bit, let Y be a qubit. whose mixed state depends on X, and let the qubit Z be the result of passing Y through a depolarizing channel, which replaces Y with a completely random qubit with probability p. We measure the quantum mutual information between X and Y by T(X; Y)=S(X)+S(Y)-S(X, Y), where S(...) denotes von Neumann's (1948) entropy. (Since X is a classical bit, the quantity T(X; Y) agrees with Holevo's (1973) bound /spl chi/(X; Y) to the classical mutual information between X and the outcome of any measurement of Y.) We show that T(X; Z) /spl les/ (1-p)/sup 2/T(X; Y). This generalizes an analogous bound for classical mutual information due to Evans and Schulman (1993), and provides a new proof of their result.
We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the [Formula: see text]-[Formula: see text]-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo–Rényi inequality. This result leads to a quantum bound for the [Formula: see text]-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.
A simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel is obtained. It implies that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion of existence of such states is given. All the obtained results may be reformulated as conditions for equality between the constrained Holevo capacity of a quantum channel and the input von Neumann entropy.