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arXiv:1403.3830 · Physical Review Letters

Discriminating Single-Photon States Unambiguously in High Dimensions

Megan Agnew, Eliot Bolduc, Kevin J. Resch, Sonja Franke‐Arnold, Jonathan Leach

Published Jul 7, 2014• 31 citations
arXivHTMLPDF

Abstract

The ability to uniquely identify a quantum state is integral to quantum science, but for nonorthogonal states, quantum mechanics precludes deterministic, error-free discrimination. However, using the nondeterministic protocol of unambiguous state discrimination enables the error-free differentiation of states, at the cost of a lower frequency of success. We discriminate experimentally between nonorthogonal, high-dimensional states encoded in single photons; our results range from dimension d=2 to d=14. We quantify the performance of our method by comparing the total measured error rate to the theoretical rate predicted by minimum-error state discrimination. For the chosen states, we find a lower error rate by more than 1 standard deviation for dimensions up to d=12. This method will find immediate application in high-dimensional implementations of quantum information protocols, such as quantum cryptography.

DOI
10.1103/physrevlett.113.020501

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The ability to uniquely identify a quantum state is integral to quantum science, but for nonorthogonal states, quantum mechanics precludes deterministic, error-free discrimination. However, using the nondeterministic protocol of unambiguous state discrimination enables the error-free differentiation of states, at the cost of a lower frequency of success. We discriminate experimentally between nonorthogonal, high-dimensional states encoded in single photons; our results range from dimension d=2 to d=14. We quantify the performance of our method by comparing the total measured error rate to the theoretical rate predicted by minimum-error state discrimination. For the chosen states, we find a lower error rate by more than 1 standard deviation for dimensions up to d=12. This method will find immediate application in high-dimensional implementations of quantum information protocols, such as quantum cryptography.
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.
Identification of nonorthogonal quantum states without error is crucial for various applications in quantum information technology, as well as the foundations of quantum physics. Theoretical studies have proposed measurements that maximize the success probability of unambiguously discriminating quantum states. However, these methods are not always experimentally feasible, which has led most demonstrations to focus on equiprobable symmetric states. Here, we establish a projective measurement scheme that optimally discriminates multiple asymmetric qudit states. We experimentally demonstrate this optimal projective measurement using a photonic orbital angular momentum state, where asymmetric qudit states are encoded in the Laguerre-Gaussian modes of a heralded single-photon state. Our results have broad applications in high-dimensional quantum state-based quantum information processing, including quantum key distribution and quantum sensing.
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.
An important task for quantum-information processing is optimal discrimination between two nonorthogonal quantum states, which until now has been realized only optically. Here, we present and compare experimental realizations of optimal quantum measurements for distinguishing between two nonorthogonal quantum states encoded in a single (14)N nuclear spin at a nitrogen-vacancy defect in diamond. Implemented measurement schemes are the minimum-error measurement (known as Helstrom measurement), unambiguous state discrimination using a standard projective measurement, and optimal unambiguous state discrimination [known as Ivanovic-Dieks-Peres (IDP) measurement], which utilizes a three-dimensional Hilbert space. This allows us to benchmark the IDP measurement against the standard projective measurements. Measurement efficiencies are found to be above 80% for all schemes and reach a value of 90% for the IDP measurement.
The quest for the realization of effective quantum state discrimination strategies is of great interest for quantum information technology, as well as for fundamental studies. Therefore, it is crucial to develop new and more efficient methods to implement discrimination protocols for quantum states. Among the others, single photon implementations are more advisable, because of their inherent security advantage in quantum communication scenarios. In this work, we present the experimental realization of a protocol employing a time-multiplexing strategy to optimally discriminate among eight non-orthogonal states, encoded in the four-dimensional Hilbert space spanning both the polarization degree of freedom and photon energy. The experiment, built on a custom-designed bulk optics analyser setup and single photons generated by a nearly deterministic solid-state source, represents a benchmarking example of minimum error discrimination with actual quantum states, requiring only linear optics and two photodetectors to be realized. Our work paves the way for more complex applications and delivers a novel approach towards high-dimensional quantum encoding and decoding operations.
Quantum mechanics forbids deterministic discrimination among nonorthogonal states. Nonetheless, the capability to distinguish nonorthogonal states unambiguously is an important primitive in quantum information processing. In this work, we experimentally implement generalized measurements in an optical system and demonstrate the first optimal unambiguous discrimination between three nonorthogonal states, with a success rate of 55%, to be compared with the 25% maximum achievable using projective measurements. Furthermore, we present the first realization of unambiguous discrimination between a pure state and a nonorthogonal mixed state.