Hierarchy of correlation quantifiers comparable to negativity

Ray Ganardi1,2, Marek Miller3, Tomasz Paterek1, and Marek Żukowski2

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

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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.

Classical bits (bits in our present computers) can be correlated in only one way because there is only one question one can ask a bit: "Are you zero or one?" The correlations are present if the value of one bit tells something new about a value of another bit and these are therefore called "classical correlations." Quite differently, quantum bits can be asked infinitely many more questions and can be prepared in states which give a deterministic, non-random, yes answer to just ONE of them. This leads to many different types of correlations between quantum objects, including entanglement or total correlations in addition to the classical correlations. Currently, they are measured in many ways that are not necessarily comparable. This is because the definition of the measure is often tied to a specific task. For example, negativity shows that in case of some quantum states, a change of sign of one of the parameters that mathematically describe them leads to a prediction of nonsensical negative probabilities. The special thing about negativity is that, unlike many other entanglement measures, we can compute it for any quantum state. This makes negativity a popular choice in practical situations. In this work, we develop a way to measure different types of correlations that are comparable to negativity. This is done as a step towards developing a framework in which different types of correlations can be meaningfully compared and efficiently computed.

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[1] Mirko Consiglio, Tony J. G. Apollaro, and Marcin Wieśniak, "Variational approach to the quantum separability problem", Physical Review A 106 6, 062413 (2022).

[2] Ray Ganardi, Ekta Panwar, Mahasweta Pandit, Bianka Woloncewicz, and Tomasz Paterek, "Quantitative Nonclassicality of Mediated Interactions", PRX Quantum 5 1, 010318 (2024).

[3] Mikhail A. Yurischev and Saeed Haddadi, "Local quantum Fisher information and local quantum uncertainty for general X states", Physics Letters A 476, 128868 (2023).

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