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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Design of POVMs with linear optics and continuous measurement

Masahiro Takeoka, Masahide Sasaki, Peter van Loock

Published Jan 11, 2005• 4 citations
Publisher

Abstract

We investigate the possibility of implementing a given measurement using linear optics and continuous measurement. In particular, we revisit the so-called Dolinar receiver, a quasi-physical model attaining the minimum error discrimination of binary coherent states, to give an alternative derivation for the optimal discrimination scheme. Our approach is rather simple and can be applied to various kinds of measurements such as projection measurements in the regime of photonic-qubit signals.

DOI
10.1117/12.576708

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We investigate the possibility of implementing a given measurement using linear optics and continuous measurement. In particular, we revisit the so-called Dolinar receiver, a quasi-physical model attaining the minimum error discrimination of binary coherent states, to give an alternative derivation for the optimal discrimination scheme. Our approach is rather simple and can be applied to various kinds of measurements such as projection measurements in the regime of photonic-qubit signals.
We investigate the possibility of implementing a given projection measurement using linear optics and arbitrarily fast feedforward based on the continuous detection of photons. In particular, we systematically derive the so-called Dolinar scheme that achieves the minimum-error discrimination of binary coherent states. Moreover, we show that the Dolinar-type approach can also be applied to projection measurements in the regime of photonic-qubit signals. Our results demonstrate that for implementing a projection measurement with linear optics, in principle, unit success probability may be approached even without the use of expensive entangled auxiliary states, as they are needed in all known (near-)deterministic linear-optics proposals.
We discuss the problem of implementing generalized measurements [positive operator-valued measures (POVMs)] with linear optics, either based upon a static linear array or including conditional dynamics. In our approach, a given POVM shall be identified as a solution to an optimization problem for a chosen cost function. We formulate a general principle: the implementation is only possible if a linear-optics circuit exists for which the quantum mechanical optimum (minimum) is still attainable after dephasing the corresponding quantum states. The general principle enables us, for instance, to derive a set of necessary conditions for the linear-optics implementation of the POVM that realizes the quantum mechanically optimal unambiguous discrimination of two pure nonorthogonal states. This extends our previous results on projection measurements and the exact discrimination of orthogonal states.
We consider the problem of discrimination between two pure quantum states. It is well known that the optimal measurement under both the error-probability and log-loss criteria is a projection, while under an ``erasure-distortion'' criterion it is a three-outcome positive operator-valued measure (POVM). These results were derived separately. We present a unified approach which finds the optimal measurement under any distortion measure that satisfies a convexity relation with respect to the Bhattacharyya distance. Namely, whenever the measure is relatively convex (resp. concave), the measurement is the projection (resp. three-outcome POVM) above. The three above-mentioned results are obtained as special cases of this simple derivation. As for further measures for which our result applies, we prove that Renyi entropies of order 111 and above (resp. 1/21/21/2 and below) are relatively convex (resp. concave). A special setting of great practical interest, is the discrimination between two coherent-light waveforms. In a remarkable work by Dolinar it was shown that a simple detector consisting of a photon counter and a feedback-controlled local oscillator obtains the quantum-optimal error probability. Later it was shown that the same detector (with the same local signal) is also optimal in the log-loss sense. By applying a similar convexity approach, we obtain in a unified manner the optimal signal for a variety of criteria.