Hierarchy of correlation quantifiers comparable to negativity
1Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland
2International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk, 80-308 Gdańsk, Poland
3Centre of New Technologies, University of Warsaw, 02-097 Warsaw, Poland
Published: | 2022-02-16, volume 6, page 654 |
Eprint: | arXiv:2111.11887v2 |
Doi: | https://doi.org/10.22331/q-2022-02-16-654 |
Citation: | Quantum 6, 654 (2022). |
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Abstract
Quantum systems generally exhibit different kinds of correlations. In order to compare them on equal footing, one uses the so-called distance-based approach where different types of correlations are captured by the distance to different sets of states. However, these quantifiers are usually hard to compute as their definition involves optimization aiming to find the closest states within the set. On the other hand, negativity is one of the few computable entanglement monotones, but its comparison with other correlations required further justification. Here we place negativity as part of a family of correlation measures that has a distance-based construction. We introduce a suitable distance, discuss the emerging measures and their applications, and compare them to relative entropy-based correlation quantifiers. This work is a step towards correlation measures that are simultaneously comparable and computable.

Featured image: Geometrical illustration. Negativity of state $\rho$ is the trace distance from its partial transposition to the set of states. The closest state is known and given as $\sigma_N$ in the figure. The main claim of the paper is negativity can be computed by taking the partial transpose distance from $\rho$ to the set of PPT states (in the shadowed overlap of states and partially transposed states). This is proven when partial transposition of $\sigma_N$ is positive semi-definite, i.e. a state. In general, the closest PPT state in the partial transpose distance is unknown as depicted with the question mark. We emphasise that this is just an illustration as partial transposition of $\sigma_N$ can produce matrices with negative eigenvalues, i.e. outside the set of states.
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