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An Introduction to Entanglement Theory

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Quantum Information and Coherence

Part of the book series: Scottish Graduate Series ((SGS))

Abstract

We review the theory of entanglement measures, concentrating mostly on the finite dimensional two-party case. We begin with a non technical introduction, followed by topics such as: single-copy and asymptotic entanglement manipulation; the entanglement of formation; the entanglement cost; the distillable entanglement; the relative entropic measures; the squashed entanglement; log-negativity; the robustness monotones; relationship between entanglement and many-body physics. We conclude with a short introduction to the problem of quantitative entanglement verification via experimental data.

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Notes

  1. 1.

    If we can transport a qubit without any decoherence, then any entanglement shared by that qubit will also be distributed perfectly. Conversely, if we can distribute entangled states perfectly then with a small amount of classical communication we may use teleportation [10] to perfectly transmit quantum states.

  2. 2.

    The class of PPT operations was proposed by Rains [16, 39], and is defined as the set of completely positive operations \(\varPhi \) such that \(\Gamma _B \circ \varPhi \circ \Gamma _B\) is also completely positive, where \(\Gamma _B\) corresponds to transposition of all of Bob’s particles, including ancillas. One can also consider transposition only of those particles belonging to Bob that undergo the operation \(\varPhi \), and this leads to a different definition which may be useful in other contexts. It is also irrelevant whether the transposition is taken over Alice or Bob, and so one may simply assert that \(\Gamma \circ \varPhi \circ \Gamma \) must be completely positive, where \(\Gamma \) is the transposition of one party. It can be shown that the PPT operations are precisely those operations that preserve the set of PPT states (see later for a definition of the PPT criterion). Hence the set of non-PPT operations includes any operation that creates a free (non-bound) entangled state out of one that is PPT. Hence PPT operations correspond to some notion of locality, and in contrast to separable operations it is relatively easy to check whether a quantum operation is PPT [16].

  3. 3.

    That this is true can be proven as follows. Consider a general bipartite state \(|\psi \rangle = \sum a_{ij}|i\rangle |j\rangle \). The amplitudes \(a_{ij}\) can be considered as the matrix elements of a matrix \(A\). This matrix hence completely represents the state (as long as a local basis is specified). If we perform the local unitary transformation \(U \otimes V|\psi \rangle \) then the matrix \(A\) gets transformed as \(A \rightarrow UAV^T\). It is a well established result of matrix analysis—the singular value decomposition [31]—that any matrix \(A\) can be diagonalised into the form \(A_{ij}=\lambda _{i}\delta _{ij}\) by a suitable choice of (\(U,V\)), even if \(A\) is not square. The coefficients \(\lambda _i\) are the so-called singular values of \(A\), and correspond to the Schmidt coefficients.

  4. 4.

    This is reminiscent of Shannon compression in classical information theory—where the compression process loses all imperfections in the limit of infinite block sizes as long as the compression rate is below a threshold [34].

  5. 5.

    As mentioned earlier under a class of operations that is slightly larger than LOCC the relative entropy of entanglement actually equals \(E_C\) and \(E_D\) under this class of operations and is in this setting the unique measure of entanglement [61–63].

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Acknowledgments

This work was supported by an Alexander von Humboldt Professorship and a Starter grant at the University of Strathclyde.

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Ulm, Albert Einstein Allee 11, 89069, Ulm, Germany

    Martin B. Plenio

  2. Department of Mathematical Sciences, Brunel University, Brunel University, Uxbridge, UB8 3PH, UK

    Shashank S. Virmani

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  1. Martin B. Plenio
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Correspondence to Martin B. Plenio .

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  1. Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, United Kingdom

    Erika Andersson

  2. Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, United Kingdom

    Patrik Öhberg

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Plenio, M.B., Virmani, S.S. (2014). An Introduction to Entanglement Theory. In: Andersson, E., Öhberg, P. (eds) Quantum Information and Coherence. Scottish Graduate Series. Springer, Cham. https://doi.org/10.1007/978-3-319-04063-9_8

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