On Quantum Steering and Wigner Negativity

Mattia Walschaers

Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4 place Jussieu, F-75252 Paris, France

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Quantum correlations and Wigner negativity are two important signatures of nonclassicality in continuous-variable quantum systems. In this work, we investigate how both are intertwined in the context of the conditional generation of Wigner negativity. It was previously shown that when Alice and Bob share a Gaussian state, Bob can perform some measurement on his system to create Wigner negativity on Alice's side if and only if there is Gaussian steering from Alice to Bob. In this work, we attempt to generalise these findings to a much broader class of scenarios on which Alice and Bob share a non-Gaussian state. We show that if Alice can initially steer Bob's system with Wigner-positive measurements, Bob can remotely create Wigner negativity in Alice's subsystem. Even though this shows that quantum steering is sufficient, we also show that quantum correlations are in general not necessary for the conditional generation of Wigner negativity.

Quantum correlations lie at the basis of many of the most illustrious of all quantum phenomena. One such phenomenon is known as quantum steering. As ever so often, this aspect of quantum correlations is best explained using two separated labs, one controlled by Alice, the other by Bob. We say that Alice can “steer” Bob, if her measurement settings and outcomes can be used by Bob to make estimate parameters with extremely high precision in his lab. So precise that it would be impossible to achieve with classical physics. Using only the information that Alice communicates, Bob can now perform measurements in his lab and test quantum steering without making any further assumptions about what is happening in Alice’s lab.

In this work, it is shown that one can also look at quantum steering from a different point of view. We show that whenever Alice can steer Bob, Bob’s measurements can induce a property known as Wigner negativity in Alice’s lab. The presence of Wigner negativity means that the measurements done in Alice’s lab cannot all be consistently described by classical physics. This makes it a very important quantum effect, which has even been shown to be necessary for building quantum computers.
If Alice and Bob share a very specific type of quantum state, known as a Gaussian state, it was proven that if Alice cannot steer Bob, Bob can never create Wigner negativity, no matter how hard he tries. However, we show that this is not the case when Alice and Bob share a more general quantum state. We found that there are even quantum states without any quantum correlations which still allow Bob’s measurements to create Wigner negativity is Alice’s lab.

This work completes our understanding of the link between quantum steering and Wigner negativity. On the one hand, it shows that quantum steering is an extremely useful resource to remotely prepare Wigner negativity. On the other hand, it also shows that the ability to remotely prepare Wigner negativity is not enough to conclude the presence of any type of quantum correlation.

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Cited by

[1] Zi-wei Zhan, Bo Lan, Jian Wang, and Xue-xiang Xu, "Coupled three-mode squeezed vacuum: Gaussian steering and remote generation of Wigner negativity", Physical Review A 108 1, 012436 (2023).

[2] Carlos L. Benavides-Riveros, "Orbital-free quasidensity functional theory", Physical Review Research 6 1, 013060 (2024).

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