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arXiv:1001.3583

Another look at the optimal Bayes cost in the binary decision problem

Bernhard K. Meister

Published Jan 20, 2010• 0 citations
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Abstract

The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability for the state discrimination is known to be given by the Helstrom bound. A new strategy is introduced here whereby the square well is compressed isoenergetically, modifying the wave-functions. The new contracted chamber is then probed using the conventional optimal strategy, and the error probability is calculated. It is shown that in some cases the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.

DOI
10.48550/arxiv.1001.3583

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The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability for the state discrimination is known to be given by the Helstrom bound. A new strategy is introduced here whereby the square well is compressed isoenergetically, modifying the wave-functions. The new contracted chamber is then probed using the conventional optimal strategy, and the error probability is calculated. It is shown that in some cases the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.
The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting an impenetrable barrier in the middle of the square well, which is either a nodal or non-nodal point of the wave function. The energy required to insert the barrier is dependent on the initial state. This enables the experimenter to gain additional information beyond the standard probing of the state envisaged by Helstrom and to improve the distinguishability of the states. It is shown that under some conditions the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.
The problem of quantum state discrimination between two wave functions on a ring is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting either adiabatically or instantaneously an impenetrable barrier. The insertion point, independent of the shape of the initial wave function, becomes a node. The resulting modified wave functions can be, if the initial functions are judiciously chosen, distinguished with a smaller error probability, and as a consequence the Helstrom bound can be violated under idealised conditions.
Quantum state discrimination between two wave functions on a ring is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting instantaneously two impenetrable barriers dividing the ring into two chambers. In the process, the candidate wave functions, as the insertion points become nodes, get entangled with the barriers and can, if judiciously chosen, be distinguished with smaller error probability. As a consequence, the Helstrom bound under idealised conditions can be violated.
For the optimal success probability under minimum-error discrimination between $r\ensuremath{\ge}2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the known general bounds, lower and upper. We also present the example where the values of the new general lower and upper bounds on the optimal success probability are tighter than the values of most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to r>2r>2r>2 the form of the Helstrom bound. For r=2r=2r=2, each of our new bounds, lower and upper, reduces to the Helstrom bound.
We investigate a state discrimination problem in operationally the most general framework to use a probability, including both classical, quantum theories, and more. In this wide framework, introducing closely related family of ensembles (which we call a Helstrom family of ensembles) with the problem, we provide a geometrical method to find an optimal measurement for state discrimination by means of Bayesian strategy. We illustrate our method in two-level quantum systems and in a probabilistic model with square-state space to reproduce, e.g., the optimal success probabilities for binary state discrimination and NNN numbers of symmetric quantum states. The existences of families of ensembles in binary cases are shown both in classical and quantum theories in any generic cases.