The problem addressed is to design a detector which is maximally sensitive to specific quantum states. Here we concentrate on quantum state detection using the worst-case a posteriori probability of detection as the design criterion. This objective is equivalent to asking the question: if the detector declares that a specific state is present, what is the probability of that state actually being present? We show that maximizing this worst-case probability (maximizing the smallest possible value of this probability) is a quasiconvex optimization over the matrices of the POVM (positive operator valued measure) which characterize the measurement apparatus. We also show that with a given POVM, the optimization is quasiconvex in the matrix which characterizes the Kraus operator sum representation (OSR) in a fixed basis. We use Lagrange Duality Theory to establish the optimality conditions for both deterministic and randomized detection. We also examine the special case of detecting a single pure state. Numerical aspects of using convex optimization for quantum state detection are also discussed.
The problem addressed is to design a detector which is maximally sensitive to specific quantum states. Here we concentrate on quantum state detection using the worst-case a posteriori probability of detection as the design criterion. This objective is equivalent to asking the question: if the detector declares that a specific state is present, what is the probability of that state actually being present? We show that maximizing this worst-case probability (maximizing the smallest possible value of this probability) is a quasiconvex optimization over the matrices of the POVM (positive operator valued measure) which characterize the measurement apparatus. We also show that with a given POVM, the optimization is quasiconvex in the matrix which characterizes the Kraus operator sum representation (OSR) in a fixed basis. We use Lagrange Duality Theory to establish the optimality conditions for both deterministic and randomized detection. We also examine the special case of detecting a single pure state. Numerical aspects of using convex optimization for quantum state detection are also discussed.
Quantum detectors provide information about the microscopic properties of quantum systems by establishing correlations between those properties and a set of macroscopically distinct events that we observe. The question of how much information a quantum detector can extract from a system is therefore of fundamental significance. In this paper, we address this question within a precise framework: given a measurement apparatus implementing a specific POVM measurement, what is the optimal performance achievable with it for a specific information readout task and what is the optimal way to encode information in the quantum system in order to achieve this performance? We consider some of the most common information transmission tasks—the Bayes cost problem, unambiguous message discrimination and the maximal mutual information. We provide general solutions to the Bayesian and unambiguous discrimination problems. We also show that the maximal mutual information is equal to the classical capacity of the quantum-to-classical channel describing the measurement, and study its properties in certain special cases. For a group covariant measurement, we show that the problem is equivalent to the problem of accessible information of a group covariant ensemble of states. We give analytical proofs of optimality in some relevant cases. The framework presented here provides a natural way to characterize generalized quantum measurements in terms of their information readout capabilities.
We address the problem of distinguishing among a finite collection of quantum states, when the states are not entirely known. For completely specified states, necessary and sufficient conditions on a quantum measurement minimizing the probability of a detection error have been derived. In this work, we assume that each of the states in our collection is a mixture of a known state and an unknown state. We investigate two criteria for optimality. The first is minimization of the worst-case probability of a detection error. For the second we assume a probability distribution on the unknown states, and minimize the expected probability of a detection error. We find that under both criteria, the optimal detectors are equivalent to the optimal detectors of an effective ensemble. In the worst case, the effective ensemble is comprised of the known states with altered prior probabilities, and in the average case it is made up of altered states with the original prior probabilities.
We consider the problem of designing an optimal quantum detector with a fixed rate of inconclusive results that maximizes the probability of correct detection, when distinguishing between a collection of mixed quantum states. We show that the design of the optimal detector can be formulated as a semidefinite programming problem, and derive a set of necessary and sufficient conditions for an optimal measurement. We then develop a sufficient condition for the scaled inverse measurement to maximize the probability of correct detection for the case in which the rate of inconclusive results exceeds a certain threshold. Using this condition we derive the optimal measurement for linearly independent pure-state sets, and for mixed-state sets with a broad class of symmetries. Specifically, we consider geometrically uniform (GU) state sets and compound geometrically uniform (CGU) state sets with generators that satisfy a certain constraint. We then show that the optimal measurements corresponding to GU and CGU state sets with arbitrary generators are also GU and CGU, respectively, with generators that can be computed very efficiently in polynomial time within any desired accuracy by solving a reduced size semidefinite programming problem.
We consider the problem of constructing measurements optimized to distinguish between a collection of possibly nonorthogonal quantum states. We consider a collection of pure states and seek a positive operator-valued measure (POVM) consisting of rank-one operators with measurement vectors closest in squared norm to the given states. We compare our results to previous measurements suggested by Peres and Wootters (1991) and Hausladen et al. (1996), where we refer to the latter as the square-root measurement (SRM). We obtain a new characterization of the SRM, and prove that it is optimal in a least-squares sense. In addition, we show that for a geometrically uniform state set the SRM minimizes the probability of a detection error. This generalizes a similar result of Ban et al. (see Int. J. Theor. Phys., vol.36, p.1269-88, 1997).
For discrimination between two signaling states of a qubit, the optimal detector minimizing the probability of error is applied to the situation where detection has to be performed from a noisy qubit affected by an arbitrary quantum noise separately characterized. With no noise, any pair of orthogonal pure quantum states is optimal for signaling as it enables error-free detection. In the presence of noise, detection errors are in general inevitable, and the pairs of signaling states best resistant to such noise are investigated. With an arbitrary quantum noise, modeled as a channel affecting the qubit, and when minimum-error detection is performed from the output, a characterization of the optimal input signaling pairs and of their best detection performance is obtained through a simple maximization of a quadratic scalar criterion in three constrained real variables. This general characterization enables to establish that such optimal signaling pairs are always made of two orthogonal pure quantum states, but that they must be specifically selected to match the noise properties and prior probabilities. The maximization is explicitly solved for several generic quantum noise processes relevant to the qubit, such as the squeezed generalized amplitude damping noise which describes interaction with a thermal bath representing a decohering environment and which includes as special cases both the generalized and the regular amplitude damping noise processes, and such as general Pauli noise processes which include for instance the bit-flip noise and the depolarizing noise. Also, examined is the situation of one imposed (pure or mixed) signaling state, for which the other associated signaling state optimal for noisy detection is determined as a pure state, yet not necessarily orthogonal.
We consider the problem of designing an optimal quantum detector that distinguishes unambiguously between a collection of mixed quantum states. Using arguments of duality in vector space optimization, we derive necessary and sufficient conditions for an optimal measurement that minimizes the probability of an inconclusive result. We show that the previous optimal measurements that were derived for certain special cases satisfy these optimality conditions. We then consider state sets with strong symmetry properties, and show that the optimal measurement operators for distinguishing between these states share the same symmetries, and can be computed very efficiently by solving a reduced size semidefinite program.