Continuous majorization in quantum phase space

Zacharie Van Herstraeten1,2, Michael G. Jabbour1,3,4, and Nicolas J. Cerf1

1Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165/59, Université libre de Bruxelles, 1050 Brussels, Belgium
2Wyant College of Optical Sciences, The University of Arizona, 1630 E. University Blvd., Tucson, AZ 85721, USA
3DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
4Department of Physics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark

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We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson's theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.

The uncertainty principle is one of the most fascinating phenomena in quantum physics. While it may seem natural that pairs of measurable quantities, such as the position and momentum of a particle, could accurately be predicted simultaneously, quantum physics actually forbids this for non-commuting observables. Heisenberg and Kennard made this precise by employing the variance of any measurable quantity in order to capture its uncertainty. Years later, Heisenberg’s uncertainty principle was reformulated by turning to entropy as a proper means to quantify uncertainty. Here, we introduce yet a stronger information-theoretical paradigm for understanding the uncertainty of quantum variables in phase space, namely the theory of majorization.

This mathematical theory has been developed more than a century ago and has been used in numerous fields of science, ranging from statistics to physics. Remarkably, it has been applied to quantum physics only relatively recently, where it was shown to be a powerful approach for exploring quantum entanglement. As such, it has never been exploited to characterize the continuous densities that describe quantum variables in phase space, that is, Wigner functions. We show continuous majorization to be a fitting tool for this. The main thrust of our paper concerns the statement that the Wigner function of the vacuum state of a bosonic mode (i.e., the ground state of the harmonic oscillator) continuous-majorizes any other Wigner function, making it the less uncertain in the sense of majorization.

While we expose and discuss our results in the context of quantum optics, they carry over to any canonical pair and should therefore have implications in various areas of physics.

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Cited by

[1] Jan de Boer, Victor Godet, Jani Kastikainen, and Esko Keski-Vakkuri, "Quantum information geometry of driven CFTs", arXiv:2306.00099, (2023).

[2] Nuno Costa Dias and João Nuno Prata, "On a recent conjecture by Z. Van Herstraeten and N.J. Cerf for the quantum Wigner entropy", arXiv:2303.10531, (2023).

[3] Zacharie Van Herstraeten and Nicolas J. Cerf, "Quantum Wigner entropy", Physical Review A 104 4, 042211 (2021).

[4] Martin Gärttner, Tobias Haas, and Johannes Noll, "Detecting continuous variable entanglement in phase space with the $Q$-distribution", arXiv:2211.17165, (2022).

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