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Detecting Gaussian entanglement via extractable work

Matteo Brunelli, Marco G. Genoni, Marco Barbieri, and Mauro Paternostro
Phys. Rev. A 96, 062311 – Published 11 December 2017
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  • INTRODUCTION
  • INFORMATION-TO-WORK CONVERSION IN A…
  • EXTRACTING WORK FROM CORRELATED SZILARD…
  • WORK EXTRACTION FROM BIPARTITE GAUSSIAN…
  • MEASUREMENT ON BOTH PARTIES
  • WORK EXTRACTION BEYOND THE GAUSSIAN…
  • WORK EXTRACTION FROM TRIPARTITE GAUSSIAN…
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We show how the presence of entanglement in a bipartite Gaussian state can be detected by the amount of work extracted by a continuous-variable Szilard-like device, where the bipartite state serves as the working medium of the engine. We provide an expression for the work extracted in such a process and specialize it to the case of Gaussian states. The extractable work provides a sufficient condition to witness entanglement in generic two-mode states, becoming also necessary for squeezed thermal states. We extend the protocol to tripartite Gaussian states and show that the full structure of inseparability classes cannot be discriminated based on the extractable work. This suggests that bipartite entanglement is the fundamental resource underpinning work extraction.

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    • Received 10 May 2017

    DOI:https://doi.org/10.1103/PhysRevA.96.062311

    ©2017 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum correlations in quantum informationQuantum entanglementQuantum information processing with continuous variables
    Quantum Information, Science & Technology

    Authors & Affiliations

    Matteo Brunelli1, Marco G. Genoni2, Marco Barbieri3, and Mauro Paternostro1

    • 1Centre for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen's University, Belfast BT7 1NN, United Kingdom
    • 2Quantum Technology Laboratory, Dipartimento di Fisica, UniversitÀ Degli Studi Di Milano, 20133 Milano, Italy
    • 3Dipartimento di Scienze, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy

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    Issue

    Vol. 96, Iss. 6 — December 2017

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    Images

    • Figure 1
      Figure 1

      (a) Gaussian demons Alice (orange) and Bob (purple) share a bipartite Gaussian state of modes â and b̂ and want to know whether the state is entangled (yellow line) or separable (gray line). In order to do so, they check how much work Alice can extract from a heat bath when only local Gaussian measurements are allowed. In the first strategy (b) Bob performs a Gaussian measurement π̂b and Alice extracts mechanical work by letting her conditional state σaπb expand (from orange to red), e.g., pushing the demon's board. As a result of the protocol Alice extracts an amount of work W. In the second approach (c), both demons perform a measurement and the work is extracted from the classical register of the results.

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    • Figure 2
      Figure 2

      Extractable work W (in units of kBT) against a for randomly generated states. Each point corresponds to a state obtained by a uniform sampling of the parameters a and c. Points corresponding to entangled (separable) states are marked in yellow (gray). (a) Homodyne detection and (b) heterodyne detection. The red dashed curve represents the maximum amount of extractable work Wmax, while the black solid curve stands for the work at the separability threshold Wsep(k), k=0,1. (c) Extractable work against the parameter c for different Gaussian measurements and a=3. Solid, dashed, and dot-dashed curves refer to λ=0, 5, and 1, respectively. The vertical dashed line refers to the value csep=a−1/2, while the horizontal ones refer to the corresponding values of Wsep(k), k=0,1.

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    • Figure 3
      Figure 3

      Extractable work W (in units of kBT) against local energies a and b for randomly generated STSs. Each point corresponds to a state obtained by a uniform sampling of the parameters a, b and c. Points corresponding to entangled (separable) states are marked in yellow (gray). (a) Homodyne detection (λ=0) and (b) heterodyne detection (λ=1). Maximum work Wmax(k) (red) and separable work Wsep(k) (black), k=0,1 correspond to red and gray surfaces, respectively. In the right column, sections of both plots are shown.

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    • Figure 4
      Figure 4

      Extractable work W (in units of kBT) for a STS against the parameter a. Random generated states are constrained to have b≤bmax where we set bmax=3. Points corresponding to entangled (separable) states are marked in yellow (gray) and we performed homodyne detection. The black solid curve is given by Wsep(0) evaluated at b=bmax, while the red dashed one is given by Wmax(0) evaluated at b=a.

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    • Figure 5
      Figure 5

      Extractable work W¯(λ) (in units of kBT) averaged over the detection angle ϕ against the local energies a and b. Points are obtained by random sampling. Detection strength has been fixed to a generic value λ=3. Points corresponding to entangled (separable) states are marked in yellow (gray). Maximum and separable work Wmax(λ) and W¯sep(λ) correspond to red dashed and black solid curves, respectively.

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    • Figure 6
      Figure 6

      Extractable work W (in units of kBT) for symmetric STS against the parameter c and for fixed a=3. The red dashed curve is for W(1,1), while the black solid one is for W¯(0,0). We also show a comparison with work extracted via single heterodyne detection W(1) (light red thin dashed curve) and homodyne detection W(0) (gray thin curve). The vertical dashed line refers to the value csep=a−1/2.

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    • Figure 7
      Figure 7

      Extractable work W (in units of kBT) against local energy a for a fully symmetric mixed state σabcS when either heterodyne detection (red dot-dashed curve) or homodyne detection (black solid curve) is performed on both Bob's and Charlie's sides. In the case of homodyne detection the work has been averaged over the two angular variable. The black dashed line corresponds to the work extracted from a pure symmetric tripartite state σabcP.

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    • Figure 8
      Figure 8

      Extractable work W (in units of kBT) against local energy a for randomly generated tripartite states σabcM. Points corresponding to fully inseparable states are marked in yellow, 1-biseparable in red, 2-biseparable in purple, and 3-biseparable in gray. (a)–(c) Homodyne detection (λ=0) on Bob's and Charlie's sides and (d)–(f) heterodyne detection (λ=1). The maximum work Wmax corresponds to the black curve.

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