bibkey: leebaekparkkimnha2022tensorizedresources
authors: Jaehak Lee; Kyunghyun Baek; Jiyong Park; Jaewan Kim; Hyunchul Nha
year: 2022
title: "Fundamental limits on concentrating and preserving tensorized quantum resources"
doi: 10.1103/PhysRevResearch.4.043070
url: https://arxiv.org/abs/2104.12307v2
claim: "With the maximal coherence η(ρ) = max_{i≠j} |ρ_ij|/√(ρ_ii ρ_jj) of Eq. (15) and the output maximal coherence η̃(Φ) = max_ρ η(Φ(ρ)) of Eq. (28), the authors conjecture the tensorization property η̃(Φ₁ ⊗ Φ₂) = max{η̃(Φ₁), η̃(Φ₂)} (Conjecture 1, Eq. (29))."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/OutputMaximalCoherenceTensorizationRefutation
license: citation-only
triage: anchor

Lee, Baek, Park, Kim and Nha, limits on concentrating and preserving tensorized quantum resources

J. Lee, K. Baek, J. Park, J. Kim and H. Nha, Fundamental limits on concentrating and preserving tensorized quantum resources, Phys. Rev. Research 4, 043070 (2022), DOI 10.1103/PhysRevResearch.4.043070; arXiv:2104.12307 [quant-ph] (v1 26 April 2021; v2 2 January 2024).

Verified locator

DOI: 10.1103/PhysRevResearch.4.043070 (Crossref: Physical Review Research 4, article 043070, published 31 October 2022). Primary version read: https://arxiv.org/abs/2104.12307v2; the text was read from the PDF.

• pages 5–6, §III.D: Eq. (15), the maximal coherence, with Theorem 4 on SIO distillation;
• page 6, §IV: Eq. (19), the output resource of a channel;
• page 7, §IV.B: Eqs. (28)–(29) and Conjecture 1;
• page 8: the numerical evidence of Fig. 3 and Conjecture 2.

The journal typesetting was not compared line by line.

Source statements

§III.D: “Lami et al. introduced a measure called the maximal coherence [23], defined as η(ρ) = max_{i≠j} |ρ_ij| / √(ρ_ii ρ_jj), (15) which satisfies the monotonicity (C2c) under SIO. Also it obeys the tensorization property (C4)”.

§IV.B: “We define the output maximal coherence of channels as η̃(Φ) = max_ρ {η(Φ(ρ))}. (28) We here make a conjecture on the tensorization property of η̃. Conjecture 1. The output maximal coherence of channels τ̃_m satisfies the tensorization property, that is, η̃(Φ₁ ⊗ Φ₂) = max{η̃(Φ₁), η̃(Φ₂)}. (29)”

Page 8: “Whether this conjecture holds true is not straightforward to see due to the optimization over all input states. However, we find a strong evidence to support this conjecture by conducting numerical calculations. For this purpose, we have generated pairs of random dynamical matrix (Choi matrix) … with the dimension of Hilbert space set as {d_A, d_B} = (a){2, 2}, (b){2, 3}”.

Reading used by the repository

The symbol τ̃_m in Conjecture 1 is a typo for η̃, as Eq. (29) shows. Channels are finite-dimensional completely positive trace-preserving maps, given by finite Kraus families, and Φ₁ ⊗ Φ₂ has the Kraus operators K ⊗ L. The maximum in Eq. (28) ranges over all density matrices, including entangled inputs of the product channel, and is read as a supremum. η is computed in the fixed computational basis over pairs i ≠ j with positive diagonal product, and is 0 when no such pair exists.