Entangled states are typically incomparable

article Entangled states are typically incomparable published yes Vishesh Jain author Matthew Alan Kwan author 5fca0887-a1db-11eb-95d1-ca9d5e0453b30000-0002-4003-7567 Marcus Michelen author MaKw department Randomness and structure in combinatorics project Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state |ψ. Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform |ψ into another state |φ. Famously, Nielsen’s theorem [28] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the entanglement spectrum of |φ should majorise the entanglement spectrum of |ψ). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states |ψ, |φ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states |ψ, |φ are entangled in fundamentally different ways, that cannot be converted to each other via local operations and classical communication.ViaNielsen’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 × m random matrices whose entries are i.i.d. standard complex Gaussians; then Nielsen’s conjecture says that the probability that the spectrum of XX†/ tr(XX†) majorises the spectrum of YY†/ tr(YY†) tends to zero as both n and m grow large. We prove this conjecture, and we also confirm some related predictions of Cunden et al. [12]. https://research-explorer.ista.ac.at/download/22813/22888/2026_CommMathPhysics_Jain.pdf application/pdfno Springer2026 eng Communications in Mathematical Physics 0010-3616 1432-0916 2406.03335 4266866310.1007/s00220-026-05711-4 40710 V. Jain, M.A. Kwan, M. Michelen, Communications in Mathematical Physics 407 (2026). Jain, Vishesh, Matthew Alan Kwan, and Marcus Michelen. “Entangled States Are Typically Incomparable.” Communications in Mathematical Physics. Springer, 2026. <a href="https://doi.org/10.1007/s00220-026-05711-4">https://doi.org/10.1007/s00220-026-05711-4</a>. V. Jain, M. A. Kwan, and M. Michelen, “Entangled states are typically incomparable,” Communications in Mathematical Physics, vol. 407, no. 10. Springer, 2026. Jain V, Kwan MA, Michelen M. 2026. Entangled states are typically incomparable. Communications in Mathematical Physics. 407(10), 214. Jain, Vishesh, et al. “Entangled States Are Typically Incomparable.” Communications in Mathematical Physics, 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>. Jain, V., Kwan, M. A., & Michelen, M. (2026). Entangled states are typically incomparable. Communications in Mathematical Physics. Springer. <a href="https://doi.org/10.1007/s00220-026-05711-4">https://doi.org/10.1007/s00220-026-05711-4</a> Jain V, Kwan MA, Michelen M. Entangled states are typically incomparable. Communications in Mathematical Physics. 2026;407(10). doi:<a href="https://doi.org/10.1007/s00220-026-05711-4">10.1007/s00220-026-05711-4</a> 228132026-09-06T22:01:56Z2026-09-09T12:35:24Z