Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.03335 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911906172764160 |
|---|---|
| author | Jain, Vishesh Kwan, Matthew Michelen, Marcus |
| author_facet | Jain, Vishesh Kwan, Matthew Michelen, Marcus |
| contents | Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state $|ψ\rangle$. Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform $|ψ\rangle$ into another state $|ϕ\rangle$. 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 $|ϕ\rangle$ should majorise the local spectrum of $|ψ\rangle$).
In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states $|ψ\rangle, |ϕ\rangle$ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states $|ψ\rangle, |ϕ\rangle$ 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03335 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entangled states are typically incomparable Jain, Vishesh Kwan, Matthew Michelen, Marcus Quantum Physics Probability Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state $|ψ\rangle$. Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform $|ψ\rangle$ into another state $|ϕ\rangle$. 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 $|ϕ\rangle$ should majorise the local spectrum of $|ψ\rangle$). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states $|ψ\rangle, |ϕ\rangle$ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states $|ψ\rangle, |ϕ\rangle$ 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]. |
| title | Entangled states are typically incomparable |
| topic | Quantum Physics Probability |
| url | https://arxiv.org/abs/2406.03335 |