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arXiv:quant-ph/0208024

Complementarity, measurement and information in interference experiments

Giuseppe Bimonte, R. Musto

Published Aug 5, 2002• 0 citations
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Abstract

Different criteria (Shannon's entropy, Bayes' average cost, Durr's normalized rms spread) have been introduced to measure the "which-way" information present in interference experiments where, due to non-orthogonality of the detector states, the path determination is incomplete. For each of these criteria, we determine the optimal measurement to be carried on the detectors, in order to read out the maximum which-way information. We show that, while in two-beam experiments, the optimal measurement is always provided by an observable involving the detector only, in multibeam experiments, with equally populated beams and two-state detectors, this is the case only for the Durr criterion, as the other two require the introduction of an ancillary quantum system, as part of the read-out apparatus.

DOI
10.48550/arxiv.quant-ph/0208024

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Different criteria (Shannon's entropy, Bayes' average cost, Durr's normalized rms spread) have been introduced to measure the "which-way" information present in interference experiments where, due to non-orthogonality of the detector states, the path determination is incomplete. For each of these criteria, we determine the optimal measurement to be carried on the detectors, in order to read out the maximum which-way information. We show that, while in two-beam experiments, the optimal measurement is always provided by an observable involving the detector only, in multibeam experiments, with equally populated beams and two-state detectors, this is the case only for the Durr criterion, as the other two require the introduction of an ancillary quantum system, as part of the read-out apparatus.
Different criteria (Shannon's entropy, Bayes' average cost, Durr's normalized rms spread) have been introduced to measure the information present in interference experiments where, due to non-orthogonality of the detector states, the path determination is incomplete. For each of these criteria, we determine the optimal measurement to be carried on the detectors, in order to read out the maximum which-way information. We show that, while in two-beam experiments, the optimal measurement is always provided by an observable involving the detector only, in multibeam experiments, with equally populated beams and two-state detectors, this is the case only for the Durr criterion, as the other two require the introduction of an ancillary quantum system, as part of the read-out apparatus.
We critically analyse the problem of formulating duality between fringe visibility and which-way information, in multibeam interference experiments. We show that the traditional notion of visibility is incompatible with any intuitive idea of complementarity, but for the two-beam case. We derive a number of new inequalities, not present in the two-beam case, one of them coinciding with a recently proposed multibeam generalization of the inequality found by Greenberger and YaSin. We show, by an explicit procedure of optimization in a three-beam case, that suggested generalizations of Englert's inequality do not convey, differently from the two-beam case, the idea of complementarity, according to which an increase of visibility is at the cost of a loss in path information, and vice versa.
Abstract In interferometers, the more information about a quantum’s path that is available in an ancillary quantum system (AQS), the less visibility the interference has. By use of Shannon entropy, we try to compare the amount of which-phase information with the amount of which-way information stored in the AQS of two-path interferometers with symmetric beam merging. We show that the former is less than or equal to the latter if the bipartite system of the single quantum and the AQS is initially prepared in a pure state and the interaction between the two parts is unitary. The equality holds when symmetry exists. No which-way information is obtained by the measurement that we use for extracting the which-phase information and vice versa. In order to verify the results experimentally, we propose assembling a new single-photon interferometer.
Quantum detectors provide information about the microscopic properties of quantum systems by establishing correlations between those properties and a set of macroscopically distinct events that we observe. The question of how much information a quantum detector can extract from a system is therefore of fundamental significance. In this paper, we address this question within a precise framework: given a measurement apparatus implementing a specific POVM measurement, what is the optimal performance achievable with it for a specific information readout task and what is the optimal way to encode information in the quantum system in order to achieve this performance? We consider some of the most common information transmission tasks—the Bayes cost problem, unambiguous message discrimination and the maximal mutual information. We provide general solutions to the Bayesian and unambiguous discrimination problems. We also show that the maximal mutual information is equal to the classical capacity of the quantum-to-classical channel describing the measurement, and study its properties in certain special cases. For a group covariant measurement, we show that the problem is equivalent to the problem of accessible information of a group covariant ensemble of states. We give analytical proofs of optimality in some relevant cases. The framework presented here provides a natural way to characterize generalized quantum measurements in terms of their information readout capabilities.
The generally accepted view in quantum theory is that information about which way the quantum system traveled and interference visibility are complementary. In all which-way experiments, however, an intervention takes place in the interference process in order to determine which way the quantum system took. This intervention can imply the tagging of a which-way marker to a quantum system or, for instance, blocking off one of the paths in a Mach-Zehnder interferometer so that one indirectly knows that the quantum system took the other (open) path. It is, however, this intervention that destroys the interference. In this paper a novel two-slit which-way interference experiment will be discussed and proposed for implementation that provides maximum which-way information without intervening in the interference process so that simultaneously maximum interference visibility remains preserved. This, in fact, implies an uncoupling of which-way information from interference and consequently also entails violating the duality relation P^2+V^2<1. Basically, the purpose of the proposed experiment and of this paper is to scrutinize this duality relation. The experiment makes use of a super-focused laser beam that is launched into only one of the two slits of the two-slit interference experiment.
We propose a scheme to realize the famous two-slit gedanken experiment using photons. Measurements using optical parametric amplifiers (OPAs) are proposed to infer the probabilities that the input photon has taken a particular path in a Mach-Zehnder interferometer, without destroying the photon. Using information theory and Bayes' theorem, we find that the visibility of ensemble averaged interference fringes is inversely related to the which-path information that is made available by the OPAs. This is in agreement with the complementarity principle. A method is found to erase partially the which-path information before it is recorded; then by making conditional measurements the fringe visibility can be partially restored.