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AIP conference proceedings

Quantum measurement with post-selection for two mixed states

N. R. Kenbaev, Dmitry Anatolevich Kronberg

Published Jan 1, 2021• 0 citations
Publisher

Abstract

In this paper, we will discuss the possibility of increasing the Holevo value when measuring with post-selection. By measurement with post-selection we mean that some of the outcomes of the measurement are discarded, and then we work with the resulting states. Using the example of two mixed states, it will be shown that mutual information after performing post-selection is greater than their initial Holevo value. An increase in the Holevo value follows from the monotonicity property of the quantum relative entropy.

DOI
10.1063/5.0054964

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In this paper, we will discuss the possibility of increasing the Holevo value when measuring with post-selection. By measurement with post-selection we mean that some of the outcomes of the measurement are discarded, and then we work with the resulting states. Using the example of two mixed states, it will be shown that mutual information after performing post-selection is greater than their initial Holevo value. An increase in the Holevo value follows from the monotonicity property of the quantum relative entropy.
Given an arbitrary measurement over a system of interest, the outcome of a posterior measurement can be used for improving the statistical estimation of the system state after the former measurement. Here, we realize an informational-entropic study of this kind of (Bayesian) retrodicted quantum measurement formulated in the context of quantum state smoothing. We show that the (average) entropy of the system state after the retrodicted measurement (smoothed state) is bounded from below and above by the entropies of the first measurement when performed in a selective and nonselective standard predictive way, respectively. For bipartite systems the same property is also valid for each subsystem. Their mutual information, in the case of a former single projective measurement, is also bounded in a similar way. The corresponding inequalities provide a kind of retrodicted extension of Holevo bound for quantum communication channels. These results quantify how much information gain is obtained through retrodicted quantum measurements in quantum state smoothing. While an entropic reduction is always granted, in bipartite systems mutual information may be degraded. Relevant physical examples confirm these features.
Abstract The discrimination of quantum states is a central problem in quantum information science and technology. Meanwhile, partial post-selection has emerged as a valuable tool for quantum state engineering. In this work, we bring these two areas together and ask whether partial measurements can enhance the discrimination performance between two unknown and non-orthogonal pure states. Our framework is general: the two unknown states interact with the same environment—set in a pure state—via an arbitrary unitary transformation. A measurement is then performed on one of the output modes (i.e. a partial measurement), modeled by an arbitrary positive operator-valued measure (POVM). We then allow classical communication to inform the unmeasured mode of the outcome of the partial measurement, which is subsequently measured by a POVM that is optimal in the sense that the discrimination probability of error is minimized. The two POVMs act locally and classical information is exchanged between the two modes, representing a single-round (feed-forward) form of local operations with classical communication. Under these considerations, we first show that, as expected, the minimum error probability, averaged over all possible conditional states, cannot be reduced below the minimum error probability of discriminating the original input states. Then, we devise a generic setup produces specific examples where the conditional discrimination can achieve strictly lower error probabilities than the original optimal measurement, illustrating that while post-selection does not improve the average performance, it can enable better discrimination in certain post-selected ensembles.
We studied in this paper the bipartite quantum systems in finite Hilbert space. We applied the measurement on the first subsystem only and studied how this measurement is affecting the entropy of the second subsystem. The analytic work carried out in this paper shows that the post measurement state is factorizable when measurement is applied on the first subsystem. This result shows that the disorder of the system is reduced and hence the entropy of the whole system is reduced as expected. However, it is also shown in this work that the entropy of the second subsystem of the post-measurement state could be greater than that of the original state. Increasing entropy after measurement means that the disorder in the second subsystem is increased. The information we gain from measuring the first subsystem reduces our information about the second subsystem rather than increasing it or at least keeping it the same as expected. This means that more factors may be required in order to calculate the probability of finding the second subsystem in a certain state.
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.
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.