Quantum Correlations Theory
Entanglement is a purely quantum phenomenon without a classical counterpart.  It occurs when particles are generated or interact in such a way that the quantum state of each particle cannot be described separately. The study of quantum entanglement and the nature of the correlations allow us to explore the foundations of quantum mechanics. It belongs to the mainstream of quantum information theory, where entanglement is regarded as a resource. Entanglement measures and analysis methods have been developed since the beginning of the field. A review [1] presents almost the contemporary state-of-the-art. While bipartite entanglement is completely understood, cases of more particles still pose many interesting questions regarding the optimal analysis, structure of entanglement and relations between different kinds of entanglements. Moreover, many of the popular entanglement identifiers involve many measurements or even knowledge of the entire density matrix. In the case of complex systems, it is a significant barrier for experiments.

As our response to the problems described above, a powerful method for generating families of quadratic entanglement identifiers was shown in [2]. They enable a simple and practical method to reveal entanglement of all pure states and some mixed states by measuring only few correlations [3]. Since the method is adaptive, it does not require a priori knowledge of the state nor a shared reference frame between the possibly remote observers, and thus greatly simplifies the practical application. Recently, we have shown in [4] that pure state entanglement can be solely characterized by correlations between all involved particles and that it can be detected by measurements along random local directions.

The nature of multipartite quantum correlations was investigated in [5], where a task of  locally sharing pre-existing correlations was shown to be distinctive for quantum and classical resources.

The theory of entanglement, especially in the multipartite case, invokes a deep and fairly involved mathematical problems. In a series of papers, of which [6] and [7] present, respectively, fundamentals and the most advanced applications, we show how symplectic and algebraic geometry can be used for classification of multipartite entanglement in systems with arbitrary quantum statistics (distinguishable particles, bosons and fermions).

[1] Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
[2] P. Badziąg, C. Brukner, W. Laskowski, T. Paterek, M. Żukowski, Phys. Rev. Lett. 100, 140403 (2008).
[3] W. Laskowski, D. Richart, C. Schwemmer, T. Paterek, H. Weinfurter, Phys. Rev. Lett. 108, 240501 (2012).
[4] M. C. Tran, B. Dakic, F. Arnault, W. Laskowski, T. Paterek, Phys. Rev. A 92, 050301(R) (2015).
[5] M. Piani, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 100, 090502 (2008)
[6] A. Sawicki, A. Huckleberry, M. Kuś, Comm. Math. Phys. 305, 441–468 (2011).
[7] A. Sawicki, M. Oszmaniec, M. Kuś, Rev. Math. Phys. 26, 1450004 (2014).

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