Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture

authored by: Henrik Wilming
Abstract: It is well known that if a (finite-dimensional) density matrix ρ has smaller entropy than ρ0, then the tensor product of sufficiently many copies of ρ majorizes a quantum state arbitrarily close to the tensor product of correspondingly many copies of ρ0. In this short note I show that if additionally rank(ρ) ≤ rank(ρ0), then n copies of ρ also majorize a state where all single-body marginals are exactly identical to ρ0 but arbitrary correlations are allowed (for some sufficiently large n). An immediate application of this is an affirmative solution of the exact catalytic entropy conjecture introduced by Boes et al. [PRL 122, 210402 (2019)]: If H(ρ) < H(ρ0) and rank(ρ) ≤ rank(ρ0) there exists a finite dimensional density matrix σ and a unitary U such that Uρ⊗ σU has marginals ρ0 and σ exactly. All the results transfer to the classical setting of probability distributions over finite alphabets with unitaries replaced by permutations.
Organisation(s): Institute of Theoretical Physics
CRC 1227 Designed Quantum States of Matter (DQ-mat)
Type: Article
Journal: Quantum
Volume: 6
No. of pages: 5
ISSN: 2521-327X
Publication date: 10.11.2022
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Atomic and Molecular Physics, and Optics, Physics and Astronomy (miscellaneous)
Electronic version(s): https://doi.org/10.48550/arXiv.2205.08915 (Access: Open)
https://doi.org/10.22331/Q-2022-11-10-858 (Access: Open)

BibTeX

@article{b96e75b0664048688a0587ae117599e7,
title = "Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture",
abstract = "It is well known that if a (finite-dimensional) density matrix ρ has smaller entropy than ρ0, then the tensor product of sufficiently many copies of ρ majorizes a quantum state arbitrarily close to the tensor product of correspondingly many copies of ρ0. In this short note I show that if additionally rank(ρ) ≤ rank(ρ0), then n copies of ρ also majorize a state where all single-body marginals are exactly identical to ρ0 but arbitrary correlations are allowed (for some sufficiently large n). An immediate application of this is an affirmative solution of the exact catalytic entropy conjecture introduced by Boes et al. [PRL 122, 210402 (2019)]: If H(ρ) < H(ρ0) and rank(ρ) ≤ rank(ρ0) there exists a finite dimensional density matrix σ and a unitary U such that Uρ⊗ σU has marginals ρ0 and σ exactly. All the results transfer to the classical setting of probability distributions over finite alphabets with unitaries replaced by permutations.",
author = "Henrik Wilming",
note = "Funding Information: I would like to thank Niklas Galke for rekindling my interest in this problem and pointing out a mistake in a previous version of the argument. I would also like to thank Paul Boes, Thomas Cope, Patryk Lipka-Bartosik, Nelly H.Y. Ng and Reinhard F. Werner for discussions as well as Roberto Rubboli and two anony- mous referees for useful feedback on a previous version of the paper. Support by the DFG through SFB 1227 (DQ-mat), Quantum Valley Lower Saxony, and funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germanys Excellence Strat- egy EXC-2123 QuantumFrontiers 390837967 is also ac- knowledged.",
year = "2022",
month = nov,
day = "10",
doi = "10.48550/arXiv.2205.08915",
language = "English",
volume = "6",
journal = "Quantum",
issn = "2521-327X",
publisher = "Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften",
}