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arXiv:quant-ph/0007063 · Physical Review A

Experimental demonstration of optimal unambiguous state discrimination

Roger B. M. Clarke, Anthony Chefles, Stephen Mark Barnett, Erling Riis

Published Mar 21, 2001• 148 citations
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Abstract

We present a full demonstration of unambiguous state discrimination between nonorthogonal quantum states. Using a free-space interferometer we have realized the optimum quantum measurement scheme for two nonorthogonal states of light, known as the Ivanovic-Dieks-Peres (IDP) measurement. We have gained access to all three possible outcomes of this measurement. All aspects of this generalized measurement scheme, including its superiority over a standard von Neumann measurement, have been demonstrated within 1.5% of the IDP predictions.

DOI
10.1103/physreva.63.040305

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We present a full demonstration of unambiguous state discrimination between nonorthogonal quantum states. Using a free-space interferometer we have realized the optimum quantum measurement scheme for two nonorthogonal states of light, known as the Ivanovic-Dieks-Peres (IDP) measurement. We have gained access to all three possible outcomes of this measurement. All aspects of this generalized measurement scheme, including its superiority over a standard von Neumann measurement, have been demonstrated within 1.5% of the IDP predictions.
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.
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.
It is known that unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a variant of that problem. Instead of discriminating among all of the different states, we shall only discriminate between two subsets of them. In particular, for the case of three nonorthogonal states, ${|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉},$ we show that the optimal strategy to distinguish $|{\ensuremath{\psi}}_{1}〉$ from the set ${|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉}$ has a higher success rate than if we wish to discriminate among all three states. Somewhat surprisingly, for unambiguous discrimination the subsets need not be linearly independent. A fully analytical solution is presented, and we also show how to construct generalized interferometers (multiport) which provide an optical implementation of the optimal strategy.
Summary form only given. It is not possible to discriminate infallibly between non-orthogonal states. Attempts to do so will inevitably result in incorrect or inconclusive results, or both. One limiting case is the Helstrom measurement, which always gives a result, but not necessarily the correct one. The other is the Ivanovic-Dieks-Peres case, which sometimes gives an inconclusive result, but when it provides a discrimination between the states it is always correct. Here we present an experimental demonstration of measurement of this second type. The results were in excellent agreement with the theoretical predictions and demonstrated significant advantages over previous experimental techniques. We used a free-space interferometer constructed from four polarising beam splitters to perform this task. Two nonorthogonal linear polarisations of heavily attenuated pulses of mode-locked laser light formed the states to be discriminated.
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.