Kontinuirana majorizacija v kvantnem faznem prostoru

Kontinuirana majorizacija v kvantnem faznem prostoru

Časovni žig: 24. maj 2023 7:39
Izvorno vozlišče: 2674950

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

1Center za kvantne informacije in komunikacijo, École polytechnique de Bruxelles, CP 165/59, Université libre de Bruxelles, 1050 Bruselj, Belgija
2Wyant College of Optical Sciences, Univerza v Arizoni, 1630 E. University Blvd., Tucson, AZ 85721, ZDA
3DAMTP, Center za matematične znanosti, Univerza v Cambridgeu, Cambridge CB3 0WA, Združeno kraljestvo
4Oddelek za fiziko, Tehnična univerza na Danskem, 2800 Kongens Lyngby, Danska

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Minimalizem

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.

Priljubljen povzetek

Načelo negotovosti je eden najbolj fascinantnih pojavov v kvantni fiziki. Čeprav se morda zdi naravno, da je mogoče pare merljivih količin, kot sta položaj in zagon delca, natančno predvideti hkrati, kvantna fizika to dejansko prepoveduje za nekomutirajoče opazovalce. Heisenberg in Kennard sta to natančno določila z uporabo variance katere koli merljive količine, da bi zajela njeno negotovost. Leta kasneje je bilo Heisenbergovo načelo negotovosti preoblikovano tako, da se je obrnil na entropijo kot ustrezno sredstvo za kvantificiranje negotovosti. Tu uvajamo še močnejšo informacijsko-teoretično paradigmo za razumevanje negotovosti kvantnih spremenljivk v faznem prostoru, in sicer teorijo majorizacije.

Ta matematična teorija je bila razvita pred več kot stoletjem in se uporablja na številnih področjih znanosti, od statistike do fizike. Zanimivo je, da so ga v kvantni fiziki uporabili šele relativno nedavno, kjer se je izkazalo, da je močan pristop za raziskovanje kvantne prepletenosti. Kot taka ni bila nikoli izkoriščena za karakterizacijo zveznih gostot, ki opisujejo kvantne spremenljivke v faznem prostoru, to je Wignerjevih funkcij. Kažemo, da je stalna majorizacija primerno orodje za to. Glavna usmeritev našega prispevka se nanaša na izjavo, da Wignerjeva funkcija vakuumskega stanja bozonskega načina (tj. osnovnega stanja harmoničnega oscilatorja) zvezno majorizira katero koli drugo Wignerjevo funkcijo, zaradi česar je manj negotova v smislu majorizacije. .

Čeprav izpostavljamo in razpravljamo o naših rezultatih v kontekstu kvantne optike, se ti prenašajo na kateri koli kanonični par in bi zato morali imeti posledice na različnih področjih fizike.

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Navedel

[1] 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).

[2] Zacharie Van Herstraeten and Nicolas J. Cerf, "Quantum Wigner entropy", Fizični pregled A 104 4, 042211 (2021).

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

Zgornji citati so iz SAO / NASA ADS (zadnjič posodobljeno 2023-05-24 23:55:18). Seznam je morda nepopoln, saj vsi založniki ne dajejo ustreznih in popolnih podatkov o citiranju.

On Storitev, ki jo citira Crossref ni bilo najdenih podatkov o navajanju del (zadnji poskus 2023-05-24 23:55:17).

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