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Square-Root Measurement for Ternary Coherent State Signal

Kentaro Kato

Published Jan 1, 2013• 3 citations
OpenAlex

Abstract

The closed-form expression of the square- root measurement for the ternary coherent state signal {|0⟩, |�⟩, |−�⟩} is derived. Further, the optimal distribution of the ternary coherent state signal that makes the square- root measurement Bayes-optimal is also derived. From a numerical analysis, it is shown that the average probability of error obtained from the square-root measurement and the corresponding optimal distribution provides a good lower bound of the minimax value for the ternary coherent state signal.

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The closed-form expression of the square- root measurement for the ternary coherent state signal {|0⟩, |�⟩, |−�⟩} is derived. Further, the optimal distribution of the ternary coherent state signal that makes the square- root measurement Bayes-optimal is also derived. From a numerical analysis, it is shown that the average probability of error obtained from the square-root measurement and the corresponding optimal distribution provides a good lower bound of the minimax value for the ternary coherent state signal.
The optimum quantum measurements that minimize the error probability for mixed‐state signals are considered. The square‐root measurement (SRM) is known to be the optimal measurement for many pure‐state signals, e.g., symmetric states, linear codes, etc. However, when the states are mixed, the SRM is no longer guaranteed to be optimal even if the states are symmetric. Here, we define the ‘pure’ square‐root measurement (pure‐SRM) which is the SRM for pure states and its optimality for mixed states. It is shown that the pure‐SRM is satisfied the necessary and sufficient conditions of the quantum Bayes strategy when the signals are mixed by transmission errors, and therefore the pure‐SRM is the optimal measurement.
To derive an optimal quantum detection that minimizes the average probability of detection error is an important problem in quantum information theory. Especially, a study of mixed-state signals is important not only for reliable communications but also for secure communications. To date, the Square-Root Measurement (SRM) has been received much attention because it is the optimal or asymptotically-optimal detection for quantum state signals. The main purpose of this correspondence is to consider the Pseudo-Classicality of the SRM.
This paper is concerned with the minimax strategy in quantum signal detection theory. First we show a numerical calculation method for finding a solution to the quantum minimax decision problem in the case that the average probability of decision errors is used as the quality function of a quantum communication system. To verify the numerical calculation method, ternary coherent state signal is considered in the absence of thermal noise. After that, the error probability of the quantum minimax receiver for the ternary coherent state signal in the pressure of thermal noise is computed by using this numerical calculation method.
In quantum discrimination, the value of the minimum error probability and the set of measurement operators which achieve this minimum are often difficult to derive. Here we present a comparison of the performance obtained by the optimal solution and by the available bounds, namely the square root measurement (SRM) and the Chernoff bound. Applied to some Gaussian states, namely to coherent states with thermal noise, it is shown that the SRM provides a much tighter bound with respect to the Chernoff bound, with a comparable numerical complexity.