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Generalized Quantum Correlations in Discrete Variable Systems

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Quantum Correlations

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

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Abstract

This chapter deals with the development of our novel notion of generalized quantum correlations within discrete variable systems. In particular, we do four things: first, we discuss and motivate the need for a generalized definition of quantum correlations; second, we discuss the tool that we use to achieve our goal, that is, we review the concepts of local and global quantum coherence; third, we define nonclassicality from the perspective of quantum computation and coherence theories and establish an equivalence between the two. Finally, through a toy model, we define the notion of quantum correlations and provide analysis of its feasibility and operational significance.

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Notes

  1. 1.

    From now on, we refer to an orthonormal PVM only as a PVM.

  2. 2.

    Note that, by Definition 1.2.47, the dimensionality of a convex set is the maximum number of affinely independent points belonging to that set.

  3. 3.

    Note that, the actual number of points required for such a decomposition is \(d^2+1\). However, one point serves as the origin which is assumed to be the zero operator. Hence, we are left with only \(d^2\) pure states.

  4. 4.

    See also the discussion following Theorem 1.2.9.

  5. 5.

    Because it can create entanglement: consider, for example, \(\Lambda _{\mathrm{CNOT}}[ | + \rangle _{\mathrm{A}}\langle + |\otimes | 0 \rangle _{\mathrm{B}}\langle 0 |]= | \Phi ^+ \rangle _{\mathrm{AB}}\langle \Phi ^+ |\) where \(| + \rangle =(| 0 \rangle +| 1 \rangle )/\sqrt{2}\) and \(| \Phi ^+ \rangle =(| 0,0 \rangle +| 1,1 \rangle )/\sqrt{2}\).

  6. 6.

    Note that, the predictions of the theory do not need to be deterministic. For instance, the quantum theory is intrinsically nondeterministic.

  7. 7.

    This corresponds to the \(\mathrm{{BPP}}\) class of computational complexity.

  8. 8.

    Corollary 3.3.1 is a result that is implicitly known to the community of quantum information and quantum computation. However, the author cannot recall an explicit and rigorous statement of this result within literature to best of his knowledge.

  9. 9.

    This can also be justified via the fact that a (non)deterministic classical computer is able to (produce) simulate every incoherent state as its output, hence, a quantum computer must be able to do so.

  10. 10.

    The abbreviation stands for deterministic quantum computation with one clean qubit.

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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). Generalized Quantum Correlations in Discrete Variable Systems. In: Quantum Correlations. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-24120-9_3

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