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Quantum Information and Measurement (QIM)

Determination of a weak value in an interferometric experiment and connection with non-Hermitian operators

Mathilde Remy, Mirko Cormann, Branko Kolarić, Yves Caudano

Published Jan 1, 2017• 0 citations
Publisher

Abstract

We develop a new method to measure a weak value by an interferometric experiment without a weakness condition. We demonstrate a link between this experience and the measurement of the average of a non-Hermitian operator.
DOI
10.1364/qim.2017.qt6a.28

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Quantum theory allows direct measurement of the average of a non-Hermitian operator using the weak value of the positive semidefinite part of the non-Hermitian operator. Here, we experimentally demonstrate the measurement of the weak value and the average value of non-Hermitian operator by an interferometric technique. Our scheme is unique as we can directly obtain the weak value from the interference visibility and the phase shift in a Mach-Zehnder interferometer without using any weak measurement or postselection. Both the experiments discussed here were performed with laser sources, but the results would be the same with average statistics of single-photon experiments. Thus, the present experiment opens up the possibility of measuring the weak value and the average value of a non-Hermitian operator without weak interaction and postselection, which can have several technological applications.
In quantum theory, a physical observable is represented by a Hermitian operator as it admits real eigenvalues. This stems from the fact that any measuring apparatus that is supposed to measure a physical observable will always yield a real number. However, the reality of an eigenvalue of some operator does not mean that it is necessarily Hermitian. There are examples of non-Hermitian operators that may admit real eigenvalues under some symmetry conditions. In general, given a non-Hermitian operator, its average value in a quantum state is a complex number and there are only very limited methods available to measure it. Following standard quantum mechanics, we provide an experimentally feasible protocol to measure the expectation value of any non-Hermitian operator via weak measurements. The average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive-semidefinite part of the non-Hermitian operator. We also prove an uncertainty relation for any two non-Hermitian operators and show that the fidelity of a quantum state under a quantum channel can be measured using the average of the corresponding Kraus operators. The importance of our method is shown in testing the stronger uncertainty relation, verifying the Ramanujan formula, and measuring the product of noncommuting projectors.
Time averaging of weak values using the quantum transition path time probability distribution enables us to establish a general uncertainty relation for the weak values of two not necessarily Hermitian operators. This new relation is a weak value analog of the Schr\"odinger strong value uncertainty relation. It leads to the conclusion that it is possible to determine with high accuracy the simultaneous mean weak values of noncommuting operators by judicious choice of the pre- and postselected states even when the postselected state is not an eigenfunction of one of the respective operators. When the time fluctuations of the two weak values are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision even though their corresponding operators do not commute. To exemplify these properties we consider specific weak value uncertainty relations for the time-energy, coordinate-momentum, and coordinate--kinetic-energy pairs. In addition we analyze spin operators and the Stern-Gerlach experiment in weak and strong inhomogeneous magnetic fields. This classic case leads to anomalous spin values when the field is weak. The weak value uncertainty relation implies that anomalous spin values are associated with large variances so that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in scattering experiments.