Flow of quantum correlations in noisy two-mode squeezed microwave states

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Flow of quantum correlations in noisy two-mode squeezed microwave states

M. Renger, S. Pogorzalek, F. Fesquet, K. Honasoge, F. Kronowetter, Q. Chen, Y. Nojiri, K. Inomata, Y. Nakamura, A. Marx, F. Deppe, R. Gross, and K. G. Fedorov

Phys. Rev. A 106, 052415 – Published 15 November 2022

Abstract

We study nonclassical correlations in propagating two-mode squeezed microwave states in the presence of noise. We focus on two different types of correlations, namely, quantum entanglement and quantum discord. Quantum discord has various intriguing fundamental properties which require experimental verification, such as the asymptotic robustness to environmental noise. Here, we experimentally investigate quantum discord in propagating two-mode squeezed microwave states generated via superconducting Josephson parametric amplifiers. By exploiting an asymmetric noise injection into these entangled states, we demonstrate the robustness of quantum discord against thermal noise while verifying the sudden death of entanglement. Furthermore, we investigate the difference between quantum discord and entanglement of formation, which can be directly related to the flow of locally inaccessible information between the environment and the bipartite subsystem. We observe a crossover behavior between quantum discord and entanglement for low noise photon numbers, which is a result of the tripartite nature of noise injection. We demonstrate that the difference between entanglement and quantum discord can be related to the security of certain quantum key distribution protocols.

Received 27 July 2022
Accepted 6 October 2022

DOI:https://doi.org/10.1103/PhysRevA.106.052415

Physics Subject Headings (PhySH)

Quantum circuits Quantum communication Quantum correlations in quantum information Quantum discord Quantum entanglement

Quantum Information, Science & Technology

Authors & Affiliations

M. Renger ^1,2,*, S. Pogorzalek^1,2, F. Fesquet^1,2, K. Honasoge^1,2, F. Kronowetter^1,2,3, Q. Chen^1,2, Y. Nojiri^1,2, K. Inomata^4,5, Y. Nakamura ^4,6, A. Marx¹, F. Deppe ^1,2,7, R. Gross ^1,2,7,†, and K. G. Fedorov^1,2,‡

¹Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
²Physik-Department, Technische Universität München, 85748 Garching, Germany
³Rohde & Schwarz GmbH & Co. KG, Mühldorfstraße 15, 81671 Munich, Germany
⁴RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan
⁵National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan
⁶Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
⁷Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 Munich, Germany

^*michael.renger@wmi.badw.de
^†rudolf.gross@wmi.badw.de
^‡kirill.fedorov@wmi.badw.de

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Vol. 106, Iss. 5 — November 2022

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Images

Figure 1
(a) Scheme of two-mode squeezing and noise injection. Nonlocal correlations are generated by superposing two orthogonally squeezed states on a symmetric beam splitter to generate a path-entangled frequency-degenerate TMS state. The asymmetric noise injection couples the environment to one of the TMS subsystems. (b) Equivalent experimental setup consisting of two JPAs for squeezed state generation, a microwave hybrid ring, and a directional coupler for injection of white Gaussian noise generated at room temperature by an arbitrary function generator (AFG). The two-mode signal is detected with a heterodyne receiver setup and digitally processed to extract the statistical signal moments and reconstruct the corresponding covariance matrix.
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Figure 2
Theoretical prediction of quantum correlations as a function of squeezing level $S$ and noise photon number $n$ . Panel (a) shows the quantum discord $D_{B}$ , thereby demonstrating the asymptotic robustness, $D_{B} > 0$ , for any finite level of noise and squeezing. Panel (b) shows the analytical lower bound $E_{F}$ for EOF. At $n = 1$ , as indicated by the vertical dashed line, we observe $E_{F} = 0$ , independent of $S$ , which demonstrates the sudden death of entanglement. In panel (c), $E_{F}$ and $D_{B}$ are plotted in the regime of $n ≪ 1$ , revealing a crossover region. Here, for low $n$ , EOF is larger than QD. For increasing $n$ , QD becomes larger than EOF. The crossover noise photon number $n_{c}$ is depicted by the solid line, dividing the blue (dark gray) and orange (light gray) plane.
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Figure 3
(a) Experimentally obtained values of quantum discord $D_{A}$ as a function of the injected noise photon number $n$ for various squeezing levels $S$ . Symbols indicate the measured data and lines are fits according to a realistic model, described in Appendix pp1-s3, which takes a finite JPA noise into account. The quantity $S_{e}$ denotes the experimentally determined squeezing level and $S_{t}$ is the corresponding squeezing level, obtained by fitting the data by the theory prediction. The JPA noise $n_{j}$ is extracted from the fit and is a function of gain. Although only shown for $D_{A}$ , the fitted values for $S_{e}$ and $n_{j}$ are the same for $D_{B}$ and $E_{F}$ . (b) Experimentally obtained values of quantum discord $D_{B}$ as a function of the injected noise photon number, $n$ , for various squeezing levels. The inset shows the same data in a log-log plot. (c) Experimental EOF (symbols) and corresponding fits (lines) for various squeezing levels. We observe the sudden death of entanglement at $n_{sd} ≃ 1$ . The inset shows that $n_{sd}$ is independent of $S$ , where $n_{sd}$ is obtained from the experimental data using cubic Hermite spline interpolation. Error bars are obtained from the statistical measurement error and are only plotted if the error exceeds the symbol size. (d) Zoom in of experimental results for $D_{B}$ and $E_{F}$ for low noise photon numbers $n$ and various squeezing levels $S_{e}$ . Solid (dashed) lines are the result from a cubic Hermite spline interpolation between the measured values for $D_{B}$ (EOF). Here, we observe the crossover behavior of QD and EOF, as predicted by the theory.
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Figure 4
(a) Experimentally determined crossover noise photon number, $n_{c}$ , as a function of the squeezing level for $D_{A}$ (blue squares) and $D_{B}$ (orange dots). Red triangles represent the arithmetic mean of $n_{c}$ for $D_{A}$ and $D_{B}$ , where we observe a minimum in the region of $5 dB$ . The error bars are determined from the experimental uncertainties of QD and EOF by randomized error sampling. (b) Theoretically predicted $n_{c}$ for $D_{A}$ (blue dashed line), $D_{B}$ (orange solid line) for the case of an idealized (lossless and noiseless, $β \to 0$ ) experiment. In the case of $D_{A}$ ( $D_{B}$ ), $n_{c}$ decreases (increases) monotonically with $S$ . We observe a minimum for $S_{\min} ≃ 5.73 dB$ for the red dotted-dashed curve which corresponds to the arithmetic mean of both discords, $D_{A}$ and $D_{B}$ . In the limit $S \to \infty, n_{c}$ converges to the same constant $n_{*} ≃ 0.26$ for $D_{A}$ and $D_{B}$ . The gray shaded region indicates the experimentally obtained squeezing levels, corresponding to panel (a). (c) Theoretical difference $Δ_{A}$ between $D_{A}$ and $E_{F}$ as a function of the noise photon number $n$ for various squeezing levels $S$ . (d) Theoretical difference $Δ_{B}$ between $D_{B}$ and $E_{F}$ as a function of the noise $n$ for various squeezing levels $S$ .
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Figure 5
(a) Schematic dependence of bipartite correlations between subsystems A, B, and E. Solid (dotted) black arrows indicate a monotonic increase (decrease) of bipartite correlations with a respective quantity. The dotted-dashed arrow, connecting A and E, indicates that correlations between these subsystems are not necessarily monotonic in $S$ or $n$ . The solid curved purple (dashed green) arrow indicates the LII flow $B \to A \to E$ ( $A \to B \to E$ ), described by $Δ_{A}$ ( $Δ_{B}$ ). (b) Theoretical secret key $K$ as a function of the resource state squeezing level $S$ and injected noise photon number $n_{q}$ in the detected quadrature. Here, we assume a continuous-variable QKD protocol between A and B [19], where the environment acts as an eavesdropper. The black dashed line indicates the threshold separating the areas of positive secret keys (secure), $K > 0$ , and negative keys (insecure), $K < 0$ . The orange (light gray) dashed line shows the corresponding $n_{c}$ for $D_{B}$ as a function of $S$ , which offers an intuitive explanation for the security of the QKD protocol on the language of the LII flow between the subsystems.
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