We compare several instances of pure-state Belavkin weighted square-root measurements from the standpoint of minimum-error discrimination of quantum states. The quadratically weighted measurement is proven superior to the so-called ``pretty good measurement'' (PGM) in a number of respects: (1) Holevo's quadratic weighting unconditionally outperforms the PGM in the case of two-state ensembles, with equality only in trivial cases. (2) A converse of a theorem of Holevo is proven, showing that a weighted measurement is asymptotically optimal only if it is quadratically weighted. Counterexamples for three states are constructed. The cube-weighted measurement of Ballester, Wehner, and Winter is also considered. Sufficient optimality conditions for various weights are compared.
We compare several instances of pure-state Belavkin weighted square-root measurements from the standpoint of minimum-error discrimination of quantum states. The quadratically weighted measurement is proven superior to the so-called ``pretty good measurement'' (PGM) in a number of respects: (1) Holevo's quadratic weighting unconditionally outperforms the PGM in the case of two-state ensembles, with equality only in trivial cases. (2) A converse of a theorem of Holevo is proven, showing that a weighted measurement is asymptotically optimal only if it is quadratically weighted. Counterexamples for three states are constructed. The cube-weighted measurement of Ballester, Wehner, and Winter is also considered. Sufficient optimality conditions for various weights are compared.
We prove a concise factor-of-2 estimate for the failure rate of optimally distinguishing an arbitrary ensemble of mixed quantum states, generalizing work of Holevo [Theor. Probab. Appl. 23, 411 (1978)] and Curlander [Ph.D. Thesis, MIT, 1979]. A modification to the minimal principle of Cocha and Poor [Proceedings of the 6th International Conference on Quantum Communication, Measurement, and Computing (Rinton, Princeton, NJ, 2003)] is used to derive a suboptimal measurement which has an error rate within a factor of 2 of the optimal by construction. This measurement is quadratically weighted and has appeared as the first iterate of a sequence of measurements proposed by Ježek et al. [Phys. Rev. A 65, 060301 (2002)]. Unlike the so-called pretty good measurement, it coincides with Holevo’s asymptotically optimal measurement in the case of nonequiprobable pure states. A quadratically weighted version of the measurement bound by Barnum and Knill [J. Math. Phys. 43, 2097 (2002)] is proven. Bounds on the distinguishability of syndromes in the sense of Schumacher and Westmoreland [Phys. Rev. A 56, 131 (1997)] appear as a corollary. An appendix relates our bounds to the trace-Jensen inequality.
We prove a concise factor-of-2 estimate for the failure rate of optimally distinguishing an arbitrary ensemble of mixed quantum states, generalizing work of Holevo [Theor. Probab. Appl. 23, 411 (1978)] and Curlander [Ph.D. Thesis, MIT, 1979]. A modification to the minimal principle of Cocha and Poor [Proceedings of the 6th International Conference on Quantum Communication, Measurement, and Computing (Rinton, Princeton, NJ, 2003)] is used to derive a suboptimal measurement which has an error rate within a factor of 2 of the optimal by construction. This measurement is quadratically weighted and has appeared as the first iterate of a sequence of measurements proposed by Ježek et al. [Phys. Rev. A 65, 060301 (2002)]. Unlike the so-called pretty good measurement, it coincides with Holevo’s asymptotically optimal measurement in the case of nonequiprobable pure states. A quadratically weighted version of the measurement bound by Barnum and Knill [J. Math. Phys. 43, 2097 (2002)] is proven. Bounds on the distinguishability of syndromes in the sense of Schumacher and Westmoreland [Phys. Rev. A 56, 131 (1997)] appear as a corollary. An appendix relates our bounds to the trace-Jensen inequality.
We derive a new lower bound on the success probability of the Pretty Good Measurement (PGM) for worst-case quantum state discrimination among m pure states. Our bound is strictly tighter than the previously known Gram-matrix-based bound for m≥4. The proof adapts techniques from Barnum and Knill's analysis of the average-case PGM, applied here to the worst-case scenario. By comparing the PGM to the sequential measurement algorithm, we obtain a guarantee showing that, in the low-fidelity regime, the PGM's success probability decreases quadratically with respect to the maximum pairwise overlap, rather than linearly as in earlier bounds.
We investigate the problem of computing optimal quantum measurements in both minimal measuring and minimax strategies. A Belavkin weighted square-root measurement (BWSRM) with appropriate weights can represent the measurement that maximizes the correct probability for any given prior probabilities of quantum states. Using this fact, we propose methods for computing optimal solutions by optimizing the weights of the BWSRM. First, we explain the conditions for the BWSRM to be optimal. In particular, we argue that if a BWSRM with certain weights is a minimax measurement, then the minimax probabilities can be immediately obtained. Next, we propose an extension of the iterative algorithm developed by Je\ifmmode \check{z}\else \v{z}\fi{}ek et al. [Phys. Rev. A 65, 060301 (2002)] for maximizing the correct probability. We prove that, for a linearly independent pure state set, Je\ifmmode \check{z}\else \v{z}\fi{}ek et al.'s algorithm converges to an optimal measurement. We also propose an iterative algorithm for a minimax solution and prove that, for a pure state set, our algorithm monotonically decreases the difference between estimated and true minimax values. Finally the performance of our algorithms is evaluated through numerical experiments.
Given a quantum pure state chosen from a set with some a priori probabilities, what is the optimal measurement needed to correctly guess the given state? We show that a good choice is the family of square-root or ``pretty good'' measurements, as each measurement in the family is optimal for at least one discrimination problem with the same quantum states but possibly different a priori probabilities. Furthermore, the map from measurement to discrimination problems can be explicitly described. In fact, for linearly independent states, every pair of discrimination problem and optimal measurement can be explicitly generated this way.
The Barnum-Knill theorem states that the optimal success probability in the multiple state discrimination task is not more than the square root of the success probability when the pretty good or square-root measurement is used for this task. An assumption of the theorem is that the underlying ensemble consists of finitely many quantum states over a finite-dimensional quantum system. Motivated in part by the fact that the success probability is not a relevant metric for continuous ensembles, in this paper we provide a generalization of the notion of pretty good measurement and the Barnum-Knill theorem for general quantum ensembles, including those described by a continuous parameter space and an infinite-dimensional Hilbert space. To achieve this, we also design a general metric of performance for quantum measurements that generalizes the success probability, namely, the expected gain of the measurement with respect to a positive score function. A notable consequence of the main result is that, in a Bayesian estimation task, the mean square error of the generalized pretty good measurement does not exceed twice the optimal mean square error.