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Physica Scripta

POVM-based Tsallis coherence consumption in fixed rate inconclusive outcome discrimination for equidistant states

Xiaoyu Song, Hua Nan, Lingman Wang, Yicheng Hong

Published May 22, 2026• 0 citations
Publisher

Abstract

Abstract We investigate the coherence consumption in fixed-rate inconclusive outcome (FRIO) quantum state discrimination with an inconclusive rate ξ by extending both the set of states and the resource quantifier. We extend the set of states from symmetric states to the class of equidistant states and use Tsallis relative α -entropy to quantify positive operator-valued measure (POVM)-based coherence. Based on this extension, we derive the optimal success probability for the parametric separation of equiprobable equidistant states and provide an explicit construction of the separating POVM. Furthermore, we quantify the Tsallis relative α -entropy-based POVM coherence consumption required for three processes in FRIO discrimination: state separation, minimum-error discrimination of post-separation states, and FRIO protocol implementation. Notably, we demonstrate that the POVM-based Tsallis coherence consumed during the state separation stage is precisely equal to the Tsallis quantum discord of the system-ancilla state generated by the separation protocol. This establishes a direct quantitative link between the resource consumption of a generalized measurement and the creation of specific quantum correlations. Our results deepen the understanding of the interplay between quantum states, measurements, and resources.

DOI
10.1088/1402-4896/ae7213

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•Jan 1, 2015•3 citations
Abstract We investigate the coherence consumption in fixed-rate inconclusive outcome (FRIO) quantum state discrimination with an inconclusive rate ξ by extending both the set of states and the resource quantifier. We extend the set of states from symmetric states to the class of equidistant states and use Tsallis relative α -entropy to quantify positive operator-valued measure (POVM)-based coherence. Based on this extension, we derive the optimal success probability for the parametric separation of equiprobable equidistant states and provide an explicit construction of the separating POVM. Furthermore, we quantify the Tsallis relative α -entropy-based POVM coherence consumption required for three processes in FRIO discrimination: state separation, minimum-error discrimination of post-separation states, and FRIO protocol implementation. Notably, we demonstrate that the POVM-based Tsallis coherence consumed during the state separation stage is precisely equal to the Tsallis quantum discord of the system-ancilla state generated by the separation protocol. This establishes a direct quantitative link between the resource consumption of a generalized measurement and the creation of specific quantum correlations. Our results deepen the understanding of the interplay between quantum states, measurements, and resources.
The optimal measurement that discriminates nonorthogonal quantum states with fixed rates of inconclusive outcomes (FRIO) can be decomposed into an assisted separation of the inputs, yielding conclusive and inconclusive outputs, followed by a minimum-error (ME) measurement for the conclusive ones (standard FRIO) or both ones (concatenated FRIO). The implementation of these measurements is underpinned by quantum resources, and here we investigate coherence based on positive operator-valued measures (POVMs) as a resource for both strategies in discriminating equally probable symmetric states of arbitrary dimension. First, we show that the POVM coherence in the assisted separation stage decomposes into the coherence of the ancillary state and the quantum discord between the system and the ancilla, evidencing coherence as a more elementary resource than quantum correlations. Next, it is demonstrated that the POVM coherence for standard and concatenated FRIO decomposes into the POVM coherence measures for state separation and ME measurement, weighted by the probabilities of occurrence of each event. Due to the ME discrimination of inconclusive states, the coherence required for the concatenated scheme is shown to be greater than that of the standard one. We discuss other general aspects of our results by characterizing the POVM coherence in the discrimination of qutrit states, with respect to the distinguishability of the inputs and the inconclusive rate. Finally, by exploiting POVM-based coherence as a quantifier of cryptographic randomness gain, we discuss the standard and concatenated FRIO strategies from the perspective of generating random bits that are secret to an eavesdropper.
The problem of non-orthogonal state discrimination underlies crucial quantum information tasks, such as cryptography and computing protocols. Therefore, it is decisive to find optimal scenarios for discrimination among quantum states. We experimentally investigate the strategy for the optimal discrimination of two non-orthogonal states considering a fixed rate of inconclusive outcomes (FRIO). The main advantage of the FRIO strategy is to interpolate between unambiguous and minimum error discrimination by solely adjusting the rate of inconclusive outcomes. We present a versatile experimental scheme that performs the optimal FRIO measurement for any pair of generated non-orthogonal states with arbitrary a priori probabilities and any fixed rate of inconclusive outcomes. Considering different values of the free parameters in the FRIO protocol, we implement it upon qubit states encoded in the polarization mode of single photons generated in the spontaneous parametric down-conversion process. Moreover, we resort to a newfangled double-path Sagnac interferometer to perform a three-outcome non-projective measurement required for the discrimination task, showing excellent agreement with the theoretical prediction. This experiment provides a practical toolbox for a wide range of quantum state discrimination strategies using the FRIO scheme, which can significantly benefit quantum information applications and fundamental studies in quantum theory.
We studied the mutual information and quantum discord that Alice and Bob share when Bob implements a discrimination with a fixed rate of inconclusive outcomes (FRIO) onto two pure non-orthogonal quantum states, generated with arbitrary a priori probabilities. FRIO discrimination interpolates between minimum error (ME) and unambiguous state discrimination (UD). ME and UD are well known discrimination protocols with several applications in quantum information theory. FRIO discrimination provides a more general framework where the discrimination process together with its applications can be studied. In this setting, we compared the performance of optimum probability of discrimination, mutual information, and quantum discord. We found that the accessible information is obtained when Bob implements the ME strategy. The most (least) efficient discrimination scheme is ME (UD), from the point of view of correlations that are lost in the initial state and remain in the final state, after Bob's measurement.
In a general optimized measurement scheme for discriminating between nonorthogonal quantum states, the error rate is minimized under the constraint of a fixed rate of inconclusive outcomes (FRIO). This so-called optimal FRIO measurement encompasses the standard and well-known minimum-error and optimal unambiguous (or maximum-confidence) discrimination strategies as particular cases. Here, we experimentally demonstrate the optimal FRIO discrimination between N=2,3,5N=2,3,5N=2,3,5, and 7 equally likely symmetric states of a qubit encoded in photonic path modes. Our implementation consists of applying a probabilistic quantum map which increases the distinguishability between the inputs in a controlled way, followed by a minimum-error measurement on the successfully transformed outputs. The results obtained corroborate this two-step approach and, in our experimental scheme, can be straightforwardly extended to higher dimensions. The optimized measurement demonstrated here will be useful for quantum communication scenarios where the error rate and inconclusive rate must be kept below the levels provided by the respective standard strategies.
We study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure states occurring with arbitrary prior probabilities, and for the discrimination of NNN symmetric states, both pure and mixed. We also consider a case where the given density operators resolve the identity operator, and we specify the optimality conditions for partially symmetric states. Moreover, we show that from the complete solution for arbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement in another strategy where the error rate is fixed, and vice versa.
In our work we explore the field of quantum state discrimination and quantum cloning. Recently the problem of optimal state discrimination with a Fixed Rate of Inconclusive Outcomes (FRIO strategy) has been solved for two pure quantum states and a few other highly symmetric cases. An optical implementation to FRIO for pure states is provided. The physical implementation can be carried out with the use of a six-port interferometer constructed with optical fibers beam splitters, phase shifters and mirrors. The input states are composed of qubits which are realized as photons in the dual-rail representation. The non-unitary measurements are carried out at the output for the presence or absence of a photon. The setup optimally interpolates between minimum error and unambiguous state discrimination. We also extend the FRIO strategy to two mixed states, whose eigenvectors in their spectral representation form a Jordan basis. We derive the minimum error rate PE for a fixed inconclusive rate Q and, in particular, the optimal distribution of the total Q over the Jordan subspaces. As Q is varied between the two limits, 0 < Q < Qc, a structure with multiple thresholds, Q1(th) (=0) < Q2(th) < . . . < QN(th) < Qc, emerges. We also solve the problem of state separation of two known pure states in the general case where the known states have arbitrary prior probabilities. The solution emerges from a geometric formulation of the problem. This formulation also reveals a deeper connection between cloning and state discrimination. The results are then applied in designing a scheme for hybrid cloning which interpolates between approximate and probabilistic exact cloning. It is shown that state separation and hybrid cloning are generalized schemes to well established state discrimination and cloning strategies. The connection between cloning and state separation is derived in the asymptotic limit.