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arXiv:0808.3844 · Physical Review A

Optimal state discrimination in general probabilistic theories

Gen Kimura, Takayuki Miyadera, Hideki Imai

Published Jun 9, 2009• 46 citations
arXivHTMLPDF

Abstract

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 $N$ 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.

DOI
10.1103/physreva.79.062306

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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.
General probabilistic theories provide the most general mathematical framework for the theory of probability in an operationally natural manner and generalize classical and quantum theories. In this article, we study state discrimination problems in general probabilistic theories using a Bayesian strategy. After reformulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that for two-state discrimination, this approach is indeed effective in arbitrary dimensional cases. Moreover, our method reveals an operational meaning of Gudder’s “intrinsic metric” by means of the optimal success probability, which turns out to be a generalization of the trace distance for quantum systems. As its by-product, an information disturbance theorem in general probabilistic theories is derived, generalizing its well known quantum version.
General Probabilistic Theories provide the most general mathematical framework for the theory of probability in an operationally natural manner, and generalize classical and quantum theories. In this article, we study state-discrimination problems in general probabilistic theories using a Bayesian strategy. After re-formulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that, for two-state discrimination, this approach is indeed effective in arbitrary dimensional cases. Moreover, our method reveals an operational meaning of Gudder’s “intrinsic metric ” by means of the optimal success probability, which turns out to be a generalization of the trace distance for quantum systems. As its by-product, an information-disturbance theorem in general probabilistic theories is derived, generalizing its well known quantum version.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
We consider optimal state discrimination in a general convex operational framework, so-called generalized probabilistic theories (GPTs), and present a general method of optimal discrimination by applying the complementarity problem from convex optimization. The method exploits the convex geometry of states but not other detailed conditions or relations of states and effects. We also show that properties in optimal quantum state discrimination are shared in GPTs in general: (i) no measurement sometimes gives optimal discrimination, and (ii) optimal measurement is not unique.
Using the convex optimization method and Helstrom family of ensembles introduced in Ref. [1], we have discussed optimal ambiguous discrimination in qubit systems. We have analyzed the problem of the optimal discrimination of N known quantum states and have obtained maximum success probability and optimal measurement for N known quantum states with equiprobable prior probabilities and equidistant from center of the Bloch ball, not all of which are on the one half of the Bloch ball and all of the conjugate states are pure. An exact solution has also been given for arbitrary three known quantum states. The given examples which use our method include: 1. Diagonal N mixed states; 2. N equiprobable states and equidistant from center of the Bloch ball which their corresponding Bloch vectors are inclined at the equal angle from z axis; 3. Three mirror-symmetric states; 4. States that have been prepared with equal prior probabilities on vertices of a Platonic solid. Keywords: minimum-error discrimination, success probability, measurement, POVM elements, Helstrom family of ensembles, convex optimization, conjugate states PACS Nos: 03.67.Hk, 03.65.Ta
A method to compute the optimal success probability of discrimination of NNN arbitrary quantum states is presented, based on the decomposition of any NNN-outcome measurement into sequences of nested two-outcome ones. In this way the optimization of the measurement operators can be carried out in successive steps, optimizing first the binary measurements at the deepest nesting level and then moving on to those at higher levels. We obtain an analytical expression for the maximum success probability after the first optimization step and examine its form for the specific case of N=3,4N=3,4N=3,4 states of a qubit. In this case, at variance with previous proposals, we are able to provide a compact expression for the success probability of any set of states, whose numerical optimization is straightforward; the results thus obtained highlight some lesser-known features of the discrimination problem.