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arXiv:1807.09014 · Physical Review A

Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer

Gaurav Nirala, Surya Narayan Sahoo, Arun Kumar Pati, Urbasi Sinha

Published Feb 13, 2019• 28 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.99.022111

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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.
Weak values provide a powerful framework for characterizing quantum systems. Their experimental extraction conventionally relies on weak conditioned von Neumann measurements, involving weak interactions and meter states that increase experimental complexity and often limit measurement efficiency. Here we introduce a method to fully characterize path weak-values in a generalized Mach-Zehnder interferometer employing neither meter states nor weak interactions. We experimentally demonstrate the technique in matter-wave interferometry. We identify anomalous weak values and, equivalently, negative quasiprobability distributions, which reflect the nonclassical behavior of the quantum system. The approach relies uniquely on intensity measurements at the output ports of the interferometer combined with controlled relative phase shifts between the paths. The absence of meter states enables considerable simplification of the setup and shorter measurement times, while preserving full access to weak values with comparable or increased accuracy. The scheme is directly applicable to a broad class of experiments involving two-level quantum systems.
The statistical properties of the detection events constituting the interference fringes at the output of an interferometer are well-known. Nevertheless, there is still no unified view of what is happening to a quantum system inside an interferometer. Strong measurements of path operators destroy the interference effect. In weak measurements, an observable is weakly coupled to a pointer system and the resulting weak values quantify the observable by minimally disturbing the system. Previous which-way experiments with weak measurements could extract either the real or imaginary part of a single weak value with each ensemble. Here, we present the simultaneous full complex quantification of two path weak values with a single ensemble in a Mach-Zehnder neutron interferometer. Magnetic fields, oscillating with different frequencies, change the energy state in each interferometer path. The time-dependent phase between the energy states distinctly marks each path. The resulting beating intensity modulation at the interferometer output gives both path weak values. For the present experiment, the weak values' absolute value and phase directly describe the observed amplitude and phase of the intensity modulation.
The theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.