Entangled states are typically incomparable

• URL: http://arxiv.org/abs/2406.03335v1
• Date: Wed, 5 Jun 2024 14:49:53 GMT
• Title: Entangled states are typically incomparable
• Authors: Vishesh Jain, Matthew Kwan, Marcus Michelen,
• Abstract summary: In a bipartite quantum system, Alice and Bob may be able to transform $|psirangle$ into another state $|phirangle$. Via Nielsen's theorem, this conjecture can be equivalently stated as a conjecture about majorisation of spectra of random matrices. We prove this conjecture, and we also confirm some related predictions of Cunden, Facchi, Florio and Gramegna.
• Score: 3.9735602856280137
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state $|\psi\rangle$. Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform $|\psi\rangle$ into another state $|\phi\rangle$. Famously, Nielsen's theorem [Phys. Rev. Lett., 1999] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the local spectrum of $|\phi\rangle$ should majorise the local spectrum of $|\psi\rangle$). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states $|\psi\rangle, |\phi\rangle$ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states $|\psi\rangle, |\phi\rangle$ are entangled in fundamentally different ways, that cannot be converted to each other via local operations and classical communication. Via Nielsen's theorem, this conjecture can be equivalently stated as a conjecture about majorisation of spectra of random matrices from the so-called trace-normalised complex Wishart-Laguerre ensemble. Concretely, let $X$ and $Y$ be independent $n \times m$ random matrices whose entries are i.i.d. standard complex Gaussians; then Nielsen's conjecture says that the probability that the spectrum of $X X^\dagger / \operatorname{tr}(X X^\dagger)$ majorises the spectrum of $Y Y^\dagger / \operatorname{tr}(Y Y^\dagger)$ tends to zero as both $n$ and $m$ grow large. We prove this conjecture, and we also confirm some related predictions of Cunden, Facchi, Florio and Gramegna [J. Phys. A., 2020; Phys. Rev. A., 2021].

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