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Computable and Operationally Meaningful Multipartite Entanglement Measures

Jacob L. Beckey, N. Gigena, Patrick J. Coles, and M. Cerezo
Phys. Rev. Lett. 127, 140501 – Published 27 September 2021
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  • Introduction.—
  • Concentratable entanglement.—
  • Efficient Computation.—
  • Properties of C|ψ⟩(s).—
  • Conclusion.—
  • ACKNOWLEDGMENTS
    • Supplemental Material
    References

    Abstract

    Multipartite entanglement is an essential resource for quantum communication, quantum computing, quantum sensing, and quantum networks. The utility of a quantum state |ψ⟩ for these applications is often directly related to the degree or type of entanglement present in |ψ⟩. Therefore, efficiently quantifying and characterizing multipartite entanglement is of paramount importance. In this work, we introduce a family of multipartite entanglement measures, called concentratable entanglements. Several well-known entanglement measures are recovered as special cases of our family of measures, and hence we provide a general framework for quantifying multipartite entanglement. We prove that the entire family does not increase, on average, under local operations and classical communications. We also provide an operational meaning for these measures in terms of probabilistic concentration of entanglement into Bell pairs. Finally, we show that these quantities can be efficiently estimated on a quantum computer by implementing a parallelized SWAP test, opening up a research direction for measuring multipartite entanglement on quantum devices.

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    • Received 4 May 2021
    • Accepted 12 August 2021

    DOI:https://doi.org/10.1103/PhysRevLett.127.140501

    © 2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum computationQuantum entanglement
    Quantum Information, Science & Technology

    Authors & Affiliations

    Jacob L. Beckey1,2,3,4, N. Gigena5, Patrick J. Coles1,4, and M. Cerezo1,4,6

    • 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
    • 2JILA, NIST and University of Colorado, Boulder, Colorado 80309, USA
    • 3Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
    • 4Quantum Science Center, Oak Ridge, Tennessee 37931, USA
    • 5Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
    • 6Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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    Issue

    Vol. 127, Iss. 14 — 1 October 2021

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    Images

    • Figure 1
      Figure 1

      Circuit for the n-qubit parallelized SWAP test. Given two copies of the quantum state |ψ⟩and n ancilla qubits, the n-qubit parallelized SWAP test consists of employing the kth ancilla to perform a controlled swap test on the kth qubit of each copy of |ψ⟩. Since the n SWAP test can be factorized, one can perform them in parallel, leading to a constant-depth circuit.

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    • Figure 2
      Figure 2

      Protocol for concentrating entanglement. Given two copies of |ψ⟩, one can employ the n-qubit parallelized SWAP test to prepare Bell pairs between qubits in the different copies of |ψ⟩. Specifically, measuring the kth control qubit in the state |1⟩ implies that the joint state of the kth qubit of each copy of |ψ⟩ is the Bell state |Φ−⟩=(1/2)(|01⟩−|10⟩). Hence, a single run of the n-qubit parallelized SWAP test has a probability p(z) of concentrating the multipartite entanglement in the copies of |ψ⟩ and producing w(z) Bell pairs.

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    • Figure 3
      Figure 3

      Comparison of the concentratable entanglements for the GHZ and W states. In the figure we show the difference C|GHZ⟩(s)−C|W⟩(s) versus the number of qubits n for different sets s with cardinalities c(s)=1, 2, n/2, n−1, n. In all cases we find C|GHZ⟩(s)>C|W⟩(s).

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