Abstract
Non-classical correlations such as entanglement and quantum discord are essential concepts in the context of quantum technologies nowadays. A plethora of both theoretical and experimental advances on this sort of correlation can be found in the literature. However, it is worth pointing out that new insights on their quantification, detection and application, to name a few, continue to be very welcome. In this chapter, we present a discussion on the quantification of correlations in quantum states. For doing so, we discuss how to quantify the so-called quantum discord in the light of a recently proposed measure inspired by the resource theory for coherence and operationally well-defined in the context of parameter estimation. On the other hand, we also comment on the problem of identifying genuine multipartite correlations in many-body systems. In this case, we discuss a proposal to quantify genuine total (classical plus quantum) correlations at different order \(2\le k\le N\).
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Notes
- 1.
A system with two parts (subsystems) is usually referred to as a bipartite system. Along the chapter, we make use of the term multipartite to refer to many-body systems (generally with more than two parts).
- 2.
The SIO is assumed to have the Kraus representation \(\mathcal {E}(\rho )=\sum _kK_k\rho K_k^{\dagger }\) with \(K_k^{\dagger }=\sum _ic_{k,i}^*\left| i \rangle \langle f_k(i) \right| \), and \(f_k(i)\) any permutation of the set \(\{0,1,\dots ,d-1\}\). It turns out that \(\left\langle l \right| K_k\rho K_k^{\dagger }\left| l \right\rangle =\sum _{j,i}c_{k,j}c_{k,i}^*\left\langle j \right| \rho \left| i \right\rangle \) with vanishing off-diagonal elements as \(K_k^{\dagger }\left| j \right\rangle =\sum _ic_{k,i}^*\left| i \right\rangle \langle f_k(i) | j \rangle \) has only one nonzero term for a given k.
- 3.
In the context of quantum resource theories, the free operations for quantum discord is still subject to debate. The accepted set so far is the set of commutativity-preserving operations (CPO) [5]; \([\mathcal {E}(\rho ),\mathcal {E}(\sigma )]=0\), for any CPO \(\mathcal {E}\) and any density operators such that \([\rho ,\sigma ]=0\). In [6], we introduced the extension ECPO as operations defined as follows: for any extension \(\rho _{SA}\) of \(\rho _S\) throughout an ancillary system A; there exists an operation \(\mathcal {F}_{SA}(\rho _{SA})\) such that \(\mathcal {E}_{S}=\text {Tr}_{A}[\mathcal {F}_{SA}]\). We say that \(\mathcal {F}_{SA}(\rho _{SA})\) is free for any free \(\mathcal {E}_{S}(\rho _S)\).
- 4.
A genuine multipartite correlation is understood as a joint correlation that cannot be reduced to properties of the composing parts. For instance, a genuine tripartite correlation cannot be described in terms of either two-partite correlations or single-part properties.
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Acknowledgements
I thank all my colleagues and friends who have been part of the development of these projects. In particular, I would like to thank Davide Girolami for his hospitality and all the knowledge shared with me. I acknowledge partial support from University of Córdoba.
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Susa-Quintero, C.E. (2023). Quantification of Correlations in Quantum States. In: Pandey, R., Srivastava, N., Singh, N.K., Tyagi, K. (eds) Quantum Computing: A Shift from Bits to Qubits. Studies in Computational Intelligence, vol 1085. Springer, Singapore. https://doi.org/10.1007/978-981-19-9530-9_1
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