Kuantum faz uzayında sürekli büyükleştirme

Kuantum faz uzayında sürekli büyükleştirme

Zaman Damgası: 24 Mayıs 2023 7:39
Kaynak Düğüm: 2674950

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

1Kuantum Bilgi ve İletişim Merkezi, École polytechnique de Bruxelles, CP 165/59, Université libre de Bruxelles, 1050 Brüksel, Belçika
2Wyant College of Optical Sciences, The University of Arizona, 1630 E. University Blvd., Tucson, AZ 85721, ABD
3DAMTP, Matematik Bilimleri Merkezi, Cambridge Üniversitesi, Cambridge CB3 0WA, Birleşik Krallık
4Fizik Bölümü, Danimarka Teknik Üniversitesi, 2800 Kongens Lyngby, Danimarka

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Özet

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.

Popüler özet

Belirsizlik ilkesi, kuantum fiziğinin en büyüleyici fenomenlerinden biridir. Bir parçacığın konumu ve momentumu gibi ölçülebilir nicelik çiftlerinin aynı anda doğru bir şekilde tahmin edilebilmesi doğal görünse de, kuantum fiziği aslında değişmeyen gözlemlenebilirler için bunu yasaklar. Heisenberg ve Kennard, belirsizliğini yakalamak için ölçülebilir herhangi bir miktarın varyansını kullanarak bunu kesinleştirdi. Yıllar sonra, Heisenberg'in belirsizlik ilkesi, belirsizliği ölçmek için uygun bir araç olarak entropiye dönülerek yeniden formüle edildi. Burada, faz uzayındaki kuantum değişkenlerinin belirsizliğini anlamak için daha güçlü bir bilgi-kuramsal paradigma, yani büyükleştirme teorisini tanıtıyoruz.

Bu matematiksel teori, bir asırdan fazla bir süre önce geliştirildi ve istatistikten fiziğe kadar çok sayıda bilim alanında kullanıldı. Dikkat çekici bir şekilde, kuantum fiziğine ancak nispeten yakın bir zamanda uygulandı ve burada kuantum dolaşıklığı keşfetmek için güçlü bir yaklaşım olduğu gösterildi. Bu nedenle, faz uzayındaki kuantum değişkenlerini, yani Wigner fonksiyonlarını tanımlayan sürekli yoğunlukları karakterize etmek için hiçbir zaman kullanılmadı. Bunun için uygun bir araç olarak sürekli büyükleştirme gösteriyoruz. Makalemizin ana itici gücü, bir bosonik modun (yani, harmonik osilatörün temel durumu) vakum durumunun Wigner fonksiyonunun, diğer herhangi bir Wigner fonksiyonunu sürekli olarak büyükleştirdiği ve onu büyükleştirme anlamında daha az belirsiz hale getirdiği ifadesiyle ilgilidir. .

Sonuçlarımızı kuantum optiği bağlamında açıklayıp tartışırken, bunlar herhangi bir kanonik çifte aktarılır ve bu nedenle fiziğin çeşitli alanlarında çıkarımları olmalıdır.

► BibTeX verileri

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Alıntılama

[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", Fiziksel İnceleme 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).

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