Experimental Investigation of Uncertainty Relations for Non-Hermitian Operators

Xinzhi Zhao, Xinglei Yu, Wenting Zhou, Chengjie Zhang, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo

Phys. Rev. Lett. 132, 070203 – Published 15 February 2024

Abstract

Uncertainty relations for Hermitian operators have been confirmed through many experiments. However, previous experiments have only tested the special case of non-Hermitian operators, i.e., uncertainty relations for unitary operators. In this study, we explore uncertainty relations for general non-Hermitian operators, which include Hermitian and unitary operators as special cases. We perform experiments with both real and complex non-Hermitian operators for qubit states, and confirm the validity of the uncertainty relations within the experimental error. Our results provide experimental evidence of uncertainty relations for non-Hermitian operators. Furthermore, our methods for realizing and measuring non-Hermitian operators are valuable in characterizing open-system dynamics and enhancing parameter estimation.

Received 22 June 2023
Accepted 17 January 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.070203

Physics Subject Headings (PhySH)

Quantum foundations Quantum optics Uncertainty relation

Non-Hermitian systems

Quantum Information, Science & Technology

Authors & Affiliations

Xinzhi Zhao¹, Xinglei Yu¹, Wenting Zhou ¹, Chengjie Zhang ^1,2,*, Jin-Shi Xu ^3,4,2,†, Chuan-Feng Li ^3,4,2,‡, and Guang-Can Guo^3,4,2

¹School of Physical Science and Technology, Ningbo University, Ningbo 315211, China
²Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
³CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
⁴CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China

^*chengjie.zhang@gmail.com
^†jsxu@ustc.edu.cn
^‡cfli@ustc.edu.cn

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Vol. 132, Iss. 7 — 16 February 2024

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Images

Figure 1
An experimental setup for testing the uncertainty relations between two non-Hermitian operators $A$ and $B$ in two scenarios: case (a) both $A$ and $B$ are real, and case (b) both $A$ and $B$ are complex. Pairs of single photons are generated via SPDC using a type-II PPKTP crystal. A wave plate group consisting of a half wave plate (HWP), a quarter wave plate (QWP), a polarization beam splitter (PBS) and a HWP are used to prepare initial input states encoded in the polarization of signal photons. We use a $50 : 50$ non-polarizing beam splitter (BS) to construct a shifted Sagnac ring interferometer, with small interferometers using two beam displacer (BD) crystals embedded in both transmission (red) and reflection (orange) paths. Corresponding to the real case and the complex case, the non-Hermitian operator $A$ is realized by a small interferometer with HWPs and QWPs as shown in (a) and (b). The detailed construction of $A$ and $B$ can be found in Supplemental Material [52]. The flat window on one path is tilted to act as a phase shifter (PS), while the fixed flat window on the other path keeps the path length difference within the coherent length. Two single photon detectors (SPDs) detect signal and idle photons, and coincidence counts are measured by a time-correlated single photon counting system. Dichroic mirror (DM); Dichroic polarization beam splitter (DPBS); Dichroic half wave plate (DHWP).
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Figure 2
Interferogram recorded by a coincidence counter. Non-Hermitian and identity operators are applied to each arm of the interference ring as follows: (a) Transmission, $I$ ; reflection, $I$ . (b) Transmission, $B$ ; reflection, $I$ .
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Figure 3
Verification of the uncertainty relation for non-Hermitian operators in the form of Eq. (5), where both $A$ and $B$ are real. Using fixed non-Hermitian operators $A$ and $B$ , we prepare different initial states. The left hand side (lhs) and right hand side (rhs) of Eq. (5) are red dots and blue triangles measured from experiments, respectively. The solid brown line represents the theoretical curve of Eq. (5).
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Figure 4
Verification of the uncertainty relation for non-Hermitian operators in the form of Eq. (2). Ensure that non-Hermitian operators on both arms of Sagnac ring interferometer are fixed and input different initial states. The blue dot represents the experimental value of lhs in Eq. (2), and the brown solid line represents the theocratical value of lhs in Eq. (2). The maximum uncertainty state corresponds to a unit value indicated by a gray dashed line.
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