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The Resource Theory of Entanglement

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Quantum Correlations

Part of the book series: Springer Theses ((Springer Theses))

Abstract

In this chapter, we discuss the phenomena of quantum entanglement within the framework of quantum resource theories. In particular, we study entanglement witnesses and their properties. This chapter incorporates two main results on the theory of entanglement, namely, ultrafine entanglement witnessing and measurement-device-independent quantification of entanglement. The first result presents a new approach for improving standard witnessing techniques by using additional information obtained within witnessing experiments and thus, cope with unexpected experimental imperfections. The second result illustrates the use of entanglement witnessing in construction of measures of entanglement that are independent of the specific description of measurement devices similar to a Bell scenario.

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Notes

  1. 1.

    By allowing the trace to decrease we allow for operations such as subselection. However, it is very common to consider trace-preserving completely positive maps (TPCP) for which \(\sum _i \hat{A}_i^\dag \hat{A_i} = \hat{\mathbbm {1}}\), which follows from a TnIPC map via a renormalization.

  2. 2.

    See Ref. [27] for the construction instructions of one such a basis.

  3. 3.

    Note that, in general, one may require multiple POVMs for such a decomposition. As an example, consider the CHSH test in which two POVMs corresponding to measurements in x and y directions are used. However, any finite number of POVMs can be combined to form a single POVM after a suitable renormalization.

  4. 4.

    This result also holds if multiple constraint are used.

  5. 5.

    Note that the former of these maps is ultimately equivalent to the latter one. That is, if \(f_{\mathrm{pr}}:\mathcal I\rightarrow \mathcal {P}\), then for every map \(h_{\mathrm{cl}}:\mathcal I\rightarrow \mathcal I\) there exists a map \(e_{\mathrm{cl}}:\mathcal {P}\rightarrow \mathcal {P}\) such that \(f_{\mathrm{pr}}\circ h_{\mathrm{cl}}=e_{\mathrm{cl}}\circ f_{\mathrm{pr}}\).

  6. 6.

    In this protocol, the indices \(\mathrm X_0\) and \(\mathrm X\) refer to the question input and shared state Hilbert spaces, respectively, for each party \(\mathrm A\) and \(\mathrm B\), while \(\tilde{{\mathrm X}}={\mathrm X}_0{\mathrm X}\) represents the joint Hilbert space for each party.

  7. 7.

    Without loss of generality we have assumed that the shared entangled state lives in a finite dimensional Hilbert space which is d-dimensional for both parties; see Theorem 2.2.2.

  8. 8.

    Note that, the parties can perform any two-outcome local measurements, however, it turns out that the one suggested by Charlie is one of the optimal measurements possible [44]. Moreover, for Alice and Bob to be able to choose their measurement strategy, Charlie has to inform them of the test operator \(\hat{L}\).

  9. 9.

    We follow the terminology used in Ref. [45] for consistency.

  10. 10.

    In fact, this can be replaced by a generic quantum state.

  11. 11.

    Recall that, \(\mathrm A\) (\(\mathrm B\)) and \(\mathrm{A}_0\) (\(\mathrm{B}_0\)) label Alice’s (Bob’s) input Hilbert spaces for shared state and quantum questions, respectively. The joint Hilbert space of Alice thus can be labelled by \({\tilde{\mathrm{A}}}={\mathrm{A_0A}}\), and similarly for Bob, \({\tilde{\mathrm{B}}}=\mathrm{B_0B}\).

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Authors and Affiliations

  1. Department of Physics, Swansea University, Swansea, UK

    Farid Shahandeh

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  1. Farid Shahandeh
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Correspondence to Farid Shahandeh .

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Shahandeh, F. (2019). The Resource Theory of Entanglement. In: Quantum Correlations. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-24120-9_2

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