---
OA_place: publisher
OA_type: hybrid
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_id: '22813'
abstract:
- lang: eng
  text: "Consider a bipartite quantum system, where Alice and Bob jointly possess\r\na
    pure state |ψ\x02. Using local quantum operations on their respective subsystems,
    and\r\nunlimited classical communication, Alice and Bob may be able to transform
    |ψ\x02 into\r\nanother state |φ\x02. Famously, Nielsen’s theorem [28] provides
    a necessary and sufficient\r\nalgebraic criterion for such a transformation to
    be possible (namely, the entanglement\r\nspectrum of |φ\x02 should majorise the
    entanglement spectrum of |ψ\x02). In the paper where\r\nNielsen proved this theorem,
    he conjectured that in the limit of large dimensionality, for\r\nalmost all pairs
    of states |ψ\x02, |φ\x02 (according to the natural unitary invariant measure)
    such\r\na transformation is not possible. That is to say, typical pairs of quantum
    states |ψ\x02, |φ\x02\r\nare entangled in fundamentally different ways, that cannot
    be converted to each other via\r\nlocal operations and classical communication.ViaNielsen’s
    theorem, this conjecture can\r\nbe equivalently stated as a conjecture about majorisation
    of spectra of random matrices\r\nfrom the so-called trace-normalised complex Wishart–Laguerre
    ensemble. Concretely,\r\nlet X and Y be independent n × m random matrices whose
    entries are i.i.d. standard\r\ncomplex Gaussians; then Nielsen’s conjecture says
    that the probability that the spectrum\r\nof XX†/ tr(XX†) majorises the spectrum
    of YY†/ tr(YY†) tends to zero as both n and\r\nm grow large. We prove this conjecture,
    and we also confirm some related predictions\r\nof Cunden et al. [12]."
acknowledgement: Open access funding provided by Institute of Science and Technology
  (IST Austria). Vishesh Jain was partially supported by NSF CAREER award DMS-2237646.
  Matthew Kwan was supported by ERC Starting Grant “RANDSTRUCT” No. 101076777. Marcus
  Michelen was supported in part by NSF grants DMS-2137623 and DMS-2246624.
article_number: '214'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Vishesh
  full_name: Jain, Vishesh
  last_name: Jain
- first_name: Matthew Alan
  full_name: Kwan, Matthew Alan
  id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
  last_name: Kwan
  orcid: 0000-0002-4003-7567
- first_name: Marcus
  full_name: Michelen, Marcus
  last_name: Michelen
citation:
  ama: Jain V, Kwan MA, Michelen M. Entangled states are typically incomparable. <i>Communications
    in Mathematical Physics</i>. 2026;407(10). doi:<a href="https://doi.org/10.1007/s00220-026-05711-4">10.1007/s00220-026-05711-4</a>
  apa: Jain, V., Kwan, M. A., &#38; Michelen, M. (2026). Entangled states are typically
    incomparable. <i>Communications in Mathematical Physics</i>. Springer. <a href="https://doi.org/10.1007/s00220-026-05711-4">https://doi.org/10.1007/s00220-026-05711-4</a>
  chicago: Jain, Vishesh, Matthew Alan Kwan, and Marcus Michelen. “Entangled States
    Are Typically Incomparable.” <i>Communications in Mathematical Physics</i>. Springer,
    2026. <a href="https://doi.org/10.1007/s00220-026-05711-4">https://doi.org/10.1007/s00220-026-05711-4</a>.
  ieee: V. Jain, M. A. Kwan, and M. Michelen, “Entangled states are typically incomparable,”
    <i>Communications in Mathematical Physics</i>, vol. 407, no. 10. Springer, 2026.
  ista: Jain V, Kwan MA, Michelen M. 2026. Entangled states are typically incomparable.
    Communications in Mathematical Physics. 407(10), 214.
  mla: Jain, Vishesh, et al. “Entangled States Are Typically Incomparable.” <i>Communications
    in Mathematical Physics</i>, vol. 407, no. 10, 214, Springer, 2026, doi:<a href="https://doi.org/10.1007/s00220-026-05711-4">10.1007/s00220-026-05711-4</a>.
  short: V. Jain, M.A. Kwan, M. Michelen, Communications in Mathematical Physics 407
    (2026).
corr_author: '1'
das_tickbox: '1'
dataavailabilitystatement: No datasets were used in this research.
date_created: 2026-09-06T22:01:56Z
date_published: 2026-08-28T00:00:00Z
date_updated: 2026-09-09T12:35:24Z
day: '28'
ddc:
- '500'
department:
- _id: MaKw
doi: 10.1007/s00220-026-05711-4
external_id:
  arxiv:
  - '2406.03335'
  pmid:
  - '42668663'
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fulldoi: https://doi.org/10.1007/s00220-026-05711-4
has_accepted_license: '1'
intvolume: '       407'
issue: '10'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '08'
oa: 1
oa_version: Published Version
pmid: 1
project:
- _id: bd95085b-d553-11ed-ba76-e55d3349be45
  grant_number: '101076777'
  name: Randomness and structure in combinatorics
publication: Communications in Mathematical Physics
publication_identifier:
  eissn:
  - 1432-0916
  issn:
  - 0010-3616
publication_status: published
publisher: Springer
quality_controlled: '1'
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status: public
supplementarymaterial: no
title: Entangled states are typically incomparable
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type: journal_article
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volume: 407
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