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arXiv:2204.06689 · Scientific Reports

Experimental quantum state discrimination using the optimal fixed rate of inconclusive outcomes strategy

Santiago Gómez, Esteban S. Gómez, O. Jiménez, A. Delgado, Stephen Patrick Walborn, G. Lima

Published Oct 15, 2022• 15 citations
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Abstract

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.

DOI
10.1038/s41598-022-22314-w

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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.
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 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 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.
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.
We present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between unambiguous and minimum error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.
We investigate the optimum strategy for state discrimination of two pure states when the probability of inconclusive answers is fixed. By varying a given rate of inconclusive probabilities, the strategy optimally interpolates between Unambiguous and Minimum-Error discrimination, the two standard approaches to quantum state discrimination. We derive the explicit expressions of all the probabilities using an ancillary system and introducing unitary transformations which act on the input states and produce the output measured on the ancilla. By exploring physically accessible transformation acting on the input states, we present an optical scheme to implement the state discrimination.