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Physical Review A

Accessible information in combined and sequential quantum measurementson a binary-state signal

Masashi Ban, Kouichi Yamazaki, Osamu Hirota

Published Jan 1, 1997• 27 citations
Publisher

Abstract

The accessible information obtained in combined and sequential quantum measurements is calculated for a binary pure quantum state signal. It is shown that the accessible information obtained in the combined measurement is equal to that obtained in the sequential measurement. The result indicates that the conjecture by Peres and Wootters [Phys. Rev. Lett. 66, 1119 (1991)] is not valid, at least for a binary quantum state signal. Furthermore, the result is compared with that obtained by Brody and Meister [Phys. Rev. Lett. 76, 1 (1996)].

DOI
10.1103/physreva.55.22

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The accessible information obtained in combined and sequential quantum measurements is calculated for a binary pure quantum state signal. It is shown that the accessible information obtained in the combined measurement is equal to that obtained in the sequential measurement. The result indicates that the conjecture by Peres and Wootters [Phys. Rev. Lett. 66, 1119 (1991)] is not valid, at least for a binary quantum state signal. Furthermore, the result is compared with that obtained by Brody and Meister [Phys. Rev. Lett. 76, 1 (1996)].
To transmit classical information using a quantum system, the sender prepares the system in one of a set of possible states and sends it to the receiver. The receiver then makes a measurement on the system to obtain information about the senders choice of state. The amount of information which is accessible to the receiver depends upon the encoding and the measurement. Here we derive a bound on this information which generalizes the bound derived by Schumacher, Westmoreland and Wootters [Schumacher, Westmoreland and Wootters, Phys. Rev. Lett. 76, 3452 (1996)] to include inefficient measurements, and thus all quantum operations. This also allows us to obtain a generalization of a bound derived by Hall [Hall, Phys. Rev. A 55, 100 (1997)], and to show that the average reduction in the von Neumann entropy which accompanies a measurement is concave in the initial state, for all quantum operations.
Classical mixtures of quantum states often give rise to decoherence and are generally considered detrimental to quantum processing. However, in the framework of sequential measurement, such mixtures can be beneficial for state discrimination. We investigate the sequential discrimination of mixed states and compare the results with those of pure states under the condition of equal fidelity. It is found that the successful probability of the mixed-state protocol is superior to the pure one under the equal-fidelity condition. It is shown that the difference between the sequential discrimination of pure and mixed states is more reliable under the equal-fidelity condition than under single-shot discrimination, and this difference increases with the mixability of the initial mixed states. For scenarios in which classical communication is allowed, the optimal successful probability of pure-state discriminations is larger than that for mixed states on the contrary. We also show that the classical mixture of basic vectors from quantum decoherence has a subtle impact on the communication channel induced by the coincidence of the maximal mutual information and optimal successful probability of sequential discrimination for pure states.
As first shown by Popescu (1995 Phys. Rev. Lett. 74 2619), some quantum states only reveal their nonlocality when subjected to a sequence of measurements while giving rise to local correlations in standard Bell tests. Motivated by this manifestation of 'hidden nonlocality' we set out to develop a general framework for the study of nonlocality when sequences of measurements are performed. Similar to Gallego et al (2013 Phys. Rev. Lett. 109 070401) our approach is operational, i.e. the task is to identify the set of allowed operations in sequential correlation scenarios and define nonlocality as the resource that cannot be created by these operations. This leads to a characterization of sequential nonlocality that contains as particular cases standard nonlocality and hidden nonlocality.
We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states.
When a measurement is made on a quantum system in which classical information is encoded, the measurement reduces the observers' average Shannon entropy for the encoding ensemble. This reduction, being the mutual information, is always non-negative. For efficient measurements the state is also purified; that is, on average, the observers' von Neumann entropy for the state of the system is also reduced by a non-negative amount. Here we point out that by rewriting a bound derived by Hall [Phys. Rev. A 55, 100 (1997)], which is dual to the Holevo bound, one finds that for efficient measurements, the mutual information is bounded by the reduction in the von Neumann entropy. We also show that this result, which provides a physical interpretation for Hall's bound, may be derived directly from the Schumacher-Westmoreland-Wootters theorem [Phys. Rev. Lett. 76, 3452 (1996)]. We discuss these bounds, and their relationship to another bound, valid for efficient measurements on pure state ensembles, which involves the subentropy.