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arXiv:2112.10014 · Quantum Science and Technology

Enhanced discrimination of high-dimensional quantum states by concatenated optimal measurement strategies

M. A. Solís-Prosser, O. Jiménez, A. Delgado, Leonardo Neves

Published Nov 9, 2021• 13 citations
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Abstract

Abstract The impossibility of deterministic and error-free discrimination among nonorthogonal quantum states lies at the core of quantum theory and constitutes a primitive for secure quantum communication. Demanding determinism leads to errors, while demanding certainty leads to some inconclusiveness. One of the most fundamental strategies developed for this task is the optimal unambiguous measurement. It encompasses conclusive results, which allow for error-free state retrodictions with the maximum success probability, and inconclusive results, which are discarded for not allowing perfect identifications. Interestingly, in high-dimensional Hilbert spaces the inconclusive results may contain valuable information about the input states. Here, we theoretically describe and experimentally demonstrate the discrimination of nonorthogonal states from both conclusive and inconclusive results in the optimal unambiguous strategy, by concatenating a minimum-error measurement at its inconclusive space. Our implementation comprises four- and nine-dimensional spatially encoded photonic states. By accessing the inconclusive space to retrieve the information that is wasted in the conventional protocol, we achieve significant increases of up to a factor of 2.07 and 3.73, respectively, in the overall probabilities of correct retrodictions. The concept of concatenated optimal measurements demonstrated here can be extended to other strategies and will enable one to explore the full potential of high-dimensional nonorthogonal states for quantum communication with larger alphabets.

DOI
10.1088/2058-9565/ac37c4

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Abstract The impossibility of deterministic and error-free discrimination among nonorthogonal quantum states lies at the core of quantum theory and constitutes a primitive for secure quantum communication. Demanding determinism leads to errors, while demanding certainty leads to some inconclusiveness. One of the most fundamental strategies developed for this task is the optimal unambiguous measurement. It encompasses conclusive results, which allow for error-free state retrodictions with the maximum success probability, and inconclusive results, which are discarded for not allowing perfect identifications. Interestingly, in high-dimensional Hilbert spaces the inconclusive results may contain valuable information about the input states. Here, we theoretically describe and experimentally demonstrate the discrimination of nonorthogonal states from both conclusive and inconclusive results in the optimal unambiguous strategy, by concatenating a minimum-error measurement at its inconclusive space. Our implementation comprises four- and nine-dimensional spatially encoded photonic states. By accessing the inconclusive space to retrieve the information that is wasted in the conventional protocol, we achieve significant increases of up to a factor of 2.07 and 3.73, respectively, in the overall probabilities of correct retrodictions. The concept of concatenated optimal measurements demonstrated here can be extended to other strategies and will enable one to explore the full potential of high-dimensional nonorthogonal states for quantum communication with larger alphabets.
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.
Abstract Quantum state discrimination is a central problem in quantum measurement theory, with applications spanning from quantum communication to computation. Typical measurement paradigms for state discrimination involve a minimum probability of error or unambiguous discrimination with a minimum probability of inconclusive results. Alternatively, an optimal inconclusive measurement, a non-projective measurement, achieves minimal error for a given inconclusive probability. This more general measurement encompasses the standard measurement paradigms for state discrimination and provides a much more powerful tool for quantum information and communication. Here, we experimentally demonstrate the optimal inconclusive measurement for the discrimination of binary coherent states using linear optics and single-photon detection. Our demonstration uses coherent displacement operations based on interference, single-photon detection, and fast feedback to prepare the optimal feedback policy for the optimal non-projective quantum measurement with high fidelity. This generalized measurement allows us to transition among standard measurement paradigms in an optimal way from minimum error to unambiguous measurements for binary coherent states. As a particular case, we use this general measurement to implement the optimal minimum error measurement for phase-coherent states, which is the optimal modulation for communications under the average power constraint. Moreover, we propose a hybrid measurement that leverages the binary optimal inconclusive measurement in conjunction with sequential, unambiguous state elimination to realize higher dimensional inconclusive measurements of coherent states.
Quantum mechanics allows for the realization of optimized measurements based on photon counting for the discrimination of nonorthogonal coherent states able to surpass the conventional limits of detection, such as the homodyne and heterodyne limits. Such measurements have a large potential for increasing sensitivities and information transfer in communications and for information processing. In this talk I will describe our current work in the problem of generalized measurements for coherent state discrimination. We implement an optimal inconclusive measurement for binary coherent states [1], a non-projective measurement that allows for achieving the lowest probability of error for a given rate of inconclusive results. This measurement encompasses standard measurement paradigms for state discrimination, specifically minimum error and unambiguous discrimination, and allows to transition between them in an optimal way. [1] M. T. DiMario, F. E. Becerra, Npj Quantum Inf. 8 (1), 1-8.
Research in nonorthogonal-state discrimination has given rise to two conventional optimal strategies: unambiguous discrimination (UD) and minimum error discrimination. We explore the experimentally relevant range of measurement strategies between the two, where the rate of inconclusive results is minimized for a bounded-error rate. We first provide some constraints on the problem that apply to generalized measurements [positive-operator-valued measurements (POVMs)]. We then provide the theory for the optimal projective measurement in this range. Through analytical and numerical results we investigate this family of projective, bounded-error strategies and compare it to the POVM family as well as to experimental implementation of UD using POVMs. We also discuss a possible application of these bounded-error strategies to quantum key distribution.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.
Quantum mechanics forbids perfect discrimination among nonorthogonal states through a single shot measurement. To optimize this task, many strategies were devised that later became fundamental tools for quantum information processing. Here, we address the pioneering minimum-error (ME) measurement and give the first experimental demonstration of its application for discriminating nonorthogonal states in high dimensions. Our scheme is designed to distinguish symmetric pure states encoded in the transverse spatial modes of an optical field; the optimal measurement is performed by a projection onto the Fourier transform basis of these modes. For dimensions ranging from D=2 to D=21 and nearly 14 000 states tested, the deviations of the experimental results from the theoretical values range from 0.3% to 3.6% (getting below 2% for the vast majority), thus showing the excellent performance of our scheme. This ME measurement is a building block for high-dimensional implementations of many quantum communication protocols, including probabilistic state discrimination, dense coding with nonmaximal entanglement, and cryptographic schemes.