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Quantum Correlations in Multipartite Quantum Systems

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Lectures on General Quantum Correlations and their Applications

Part of the book series: Quantum Science and Technology ((QST))

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Abstract

We review some concepts and properties of quantum correlations, in particular multipartite measures, geometric measures and monogamy relations. We also discuss the relation between classical and total correlations.

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Notes

  1. 1.

    There is a vast literature studying this gap and looking for more general types of Bell inequality which may close the gap; see [3].

  2. 2.

    Actually \(S(\rho _A|\rho _B)\) is always positive in classical setting, but can be negative for entangled states and took it a long time to understand this negativity; see [5].

  3. 3.

    We should stress that it is a natural requirement that correlation measurements should not increase under local operations: one should no be able to increase their correlation with someone far away only acting on their own system.

  4. 4.

    Note that the relative entropy can also be understood as a distance measure, even though technically it is not a genuine distance since it is not symmetric.

  5. 5.

    There are many different ways to define a norm for a matrix (or operator). One should first consider the p-norms of a vector \(\vec {v}\) given by \(||\vec {v}||=(\sum _i |v_i|^ p)^ {1/p}\) with \(v_i\) being the components of \(\vec {v}\) in some basis and \(p\ge 1\). For \(p=2\) we have the Euclidean norm. One can then define the induced norm for the matrix as the maximum norm the matrix can induce in a unit vector: \(||X||=\sup _{|\vec {v}|=1} ||X\vec {v}||\). Then given a vector p-norm vector we get a operator p-norm. Another possibility is to consider an m x n matrix as a mn vector and use an vector norm. These are usually called “entrywise” norms. A third possibility, the Schatten norms, is to apply the vector p-norm to the singular values of the matrix (the singular values are the square root of the eingenvalues of \(X^\dagger X\)). For \(p=2\) we have the Hilbert-Schmidt, also called Frobenius, norm which is equivalent to the \(p=2\) entrywise norm mentioned before. For \(p=1\) we have the trace norm and for \(p=\infty \) we have the spectral norm which is equivalent to the induced \(p=2\) norm and also called operator norm and given by the largest singular value.

  6. 6.

    This follows directly from the fact that the trace distance itself has a interpretation in terms of distinguishability: Suppose Alice prepares a quantum system in state \(\rho \) with probability 1/2 and in state \(\sigma \) with probability 1/2. She then gives the system to Bob, who performs a POVM measurement to distinguish the two states. It can be shown that Bob’s probability of correctly identifying which state Alice prepared is \(1/2+1/2||\rho -\sigma ||_1\) (see sec. 9.2 of [22]).

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Acknowledgements

I would like to thanks Marcelo Sarandy for many discussion about discord and quantum correlation and Ernesto Galvão for carefully reading the first draft. I also acknowledge financial support from the Brazilian agencies CNPq, CAPES, FAPERJ, and the Brazilian National Institute of Science and Technology for Quantum Information (INCT-IQ).

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Authors and Affiliations

  1. Instituto de Física, Universidade Federal Fluminense, Niterói, RJ, 24210-346, Brazil

    Thiago R. de Oliveira

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  1. Thiago R. de Oliveira
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Correspondence to Thiago R. de Oliveira .

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Editors and Affiliations

  1. São Paulo State University, São Paulo, Brazil

    Felipe Fernandes Fanchini

  2. São Carlos Institute of Physics, University of São Paulo, São Paulo, Brazil

    Diogo de Oliveira Soares Pinto

  3. School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom

    Gerardo Adesso

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de Oliveira, T.R. (2017). Quantum Correlations in Multipartite Quantum Systems. In: Fanchini, F., Soares Pinto, D., Adesso, G. (eds) Lectures on General Quantum Correlations and their Applications. Quantum Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-53412-1_5

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