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Kluwer Academic Publishers eBooks

Experimental Quantum State Discrimination

Stephen Mark Barnett, Roger B. M. Clarke, Viv M. Kendon, Erling Riis, Anthony Chefles, Masahide Sasaki

Published Feb 24, 2006• 1 citation
Publisher

Abstract

We present the results of generalized measurements of optical polarization designed to provide one of three or four distinct outcomes. This has allowed us to discriminate between non-orthogonal polarization states with an error probability that is close to the minimum allowed by quantum theory. The sets of states we have prepared and measured are (i) two non-orthogonal states of linear polarisation, (ii) the trine, consisting of three states of linear polarization separated by 60º and, (iii) the tetrad, consisting of four polarization states (two linear and two elliptical) arranged in a tetrahedron on the Poincaré sphere. This has enabled us to realize the generalized measurements required for (i) optimal unambiguous discrimination between two non-orthogonal polarization states, (ii) discrimination between the three trine states and four tetrad states with minimum probability of error and, (iii) maximizing the mutual information associated with our trine and tetrad measurements and showing that it exceeds the value assocated with the best possible von Neumann measurement. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/0-306-47114-0_11

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We present the results of generalized measurements of optical polarization designed to provide one of three or four distinct outcomes. This has allowed us to discriminate between non-orthogonal polarization states with an error probability that is close to the minimum allowed by quantum theory. The sets of states we have prepared and measured are (i) two non-orthogonal states of linear polarisation, (ii) the trine, consisting of three states of linear polarization separated by 60º and, (iii) the tetrad, consisting of four polarization states (two linear and two elliptical) arranged in a tetrahedron on the Poincaré sphere. This has enabled us to realize the generalized measurements required for (i) optimal unambiguous discrimination between two non-orthogonal polarization states, (ii) discrimination between the three trine states and four tetrad states with minimum probability of error and, (iii) maximizing the mutual information associated with our trine and tetrad measurements and showing that it exceeds the value assocated with the best possible von Neumann measurement. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We present the results of generalized measurements of optical polarization designed to provide one of three or four distinct outcomes. This has allowed us to discriminate between nonorthogonal polarization states with an error probability that is close to the minimum allowed by quantum theory. Employing these optimal detection strategies on sets of three (trine) or four (tetrad) equiprobable symmetric nonorthogonal polarization states, we obtain a mutual information that exceeds the maximum value attainable using conventional (von Neumann) polarization measurements.
Since the state of a quantum system is not an observable, it cannot be determined completely, in principle, if no a priori information is available. In this case, one needs to resort to various estimation strategies. The situation is completely different if a priori knowledge is available. Even unambigous discrimination is possible if the system is always prepared in a state that is selected from a known set of quantum states that are not orthogonal but linearly independent. This type of discrimination strategy has a higher probability of success by using generalised measurements than by using standard von Neumann measurements. We discuss here possible optical realizations, via generalized interferometers and single photon states, for the implementation of strategies that maximize the success rate. The implementations are based on exact analytical results. We also discuss possible applications for entanglement enhancement and quantum communication protocols.
It is shown that generalized measurements, required for optimally discriminating between non-orthogonal joint polarization states of two indistinguishable photons, can be realized with the help of polarization-dependent two-photon absorption and by means of sum-frequency generation. Optimization schemes are investigated with respect to minimizing the error probability in inferring the states, as well as with respect to maximizing the probability of success for unambiguous discrimination. Moreover, an implementation of error-minimizing discrimination between N symmetric single-photon states is studied. The latter can be used to extract information from the inconclusive results occurring in unambiguous discrimination between three symmetric two-photon polarization states.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
We present the first experimental demonstration of the maximum confidence measurement strategy for quantum state discrimination. Applying this strategy to an arbitrary set of states assigns to each input state a measurement outcome which, when realized, gives the highest possible confidence that the state was indeed present. The theoretically optimal measurement for discriminating between three equiprobable symmetric qubit states is implemented in a polarization-based free-space interferometer. The maximum confidence in the measurement result is 2/3. This is the first explicit demonstration that an improvement in the confidence over the optimal minimum error measurement is possible for linearly dependent states.
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.