bibkey: champagnejohnstonmacdonaldpipes2022spectral
authors: Gabriel Champagne; Nathaniel Johnston; Mitchell MacDonald; Logan Pipes
year: 2022
title: "Spectral properties of symmetric quantum states and symmetric entanglement witnesses"
doi: 10.1016/j.laa.2022.05.004
url: https://arxiv.org/abs/2108.10405v1
claim: "A symmetric entanglement witness is a Hermitian W on Cd ∨ Cd with (v ⊗ v)*W(v ⊗ v) ≥ 0 for all v ∈ Cd (§2.3); positive semidefinite W are not excluded (footnote 1). For d = 2, Theorem 3 shows that real µ1 ≥ µ2 ≥ µ3 with µ2 ≥ 0 and µ3 ≥ −√(µ1µ2) are the eigenvalues of a (decomposable) symmetric witness, that every decomposable symmetric witness satisfies these two inequalities, and that every symmetric witness satisfies µ2 ≥ 0 and a weaker bound (c). The authors conjecture that µ2 ≥ 0 and µ3 ≥ −√(µ1µ2) characterize the eigenvalues of every symmetric witness in L(C2 ∨ C2) (after Theorem 3 and §6(c))."
strata_touched:
  - D5/S3/Quantum/Entanglement/AbsoluteSeparability/SymmetricQubitWitnessSpectrum
license: citation-only
triage: anchor

Champagne, Johnston, MacDonald and Pipes, spectral properties of symmetric quantum states and witnesses

G. Champagne, N. Johnston, M. MacDonald and L. Pipes, Spectral properties of symmetric quantum states and symmetric entanglement witnesses, Linear Algebra Appl. 649, 273–300 (2022); arXiv:2108.10405 [quant-ph] (v1 of 23 August 2021, the only version listed).

Verified locator

DOI: 10.1016/j.laa.2022.05.004

Source: https://arxiv.org/abs/2108.10405v1

The DOI and the journal reference are listed in the arXiv metadata, and Crossref lists the same title, authors, volume and pages. The text was read from the PDF https://arxiv.org/pdf/2108.10405v1:

• p. 3, footnote 1;
• p. 4, §2.3, the definition of a symmetric entanglement witness;
• p. 14, Theorem 3 and the sentence that follows it;
• p. 20, §6, open problem (c).

Source statements

§2.3 (p. 4): “A symmetric entanglement witness is a Hermitian operator W ∈ L(Cd ∨ Cd) with the property that (v ⊗ v)∗W(v ⊗ v) ≥ 0 for all v ∈ Cd, or equivalently it is a Hermitian matrix W = P∨WP∨ ∈ L(Cd ⊗ Cd) with the same property.”

Footnote 1 (p. 3): “Many sources also require that W not be positive semidefinite in order to qualify as an entanglement witness. We do not impose that restriction here, …”

Theorem 3 (p. 14): “Consider the following inequalities involving real numbers µ1 ≥ µ2 ≥ µ3: (a) µ2 ≥ 0, (b) µ3 ≥ −√(µ1µ2), and (c) µ3 ≥ −µ1/4 − µ2 if µ2 < µ1/4, −√(µ1µ2) if µ2 ≥ µ1/4. If inequalities (a) and (b) hold then there exists a (decomposable) symmetric entanglement witness in L(C2 ∨ C2) with eigenvalues µ1 ≥ µ2 ≥ µ3. Conversely, the eigenvalues of every decomposable symmetric entanglement witness in L(C2 ∨ C2) satisfy inequalities (a) and (b), and the eigenvalues of every symmetric entanglement witness in L(C2 ∨ C2) satisfy inequalities (a) and (c). We conjecture that the eigenvalues of every symmetric entanglement witness in L(C2 ∨ C2) must satisfy inequalities (a) and (b) (instead of the slightly weaker (a) and (c)), but have been unable to prove it.”

§6, item (c) (p. 20): “Can we strengthen Theorem 3 into an exact characterization of the eigenvalues of a symmetric entanglement witness in L(C2 ∨ C2)? We conjecture that inequalities (a) and (b) from that theorem provide the correct characterization.”

Reading used by the repository

C2 ∨ C2 is identified with C3 through the orthonormal Dicke basis e00, (e01 + e10)/√2, e11, in which v ⊗ v has coordinates (v0², √2 v0v1, v1²). A symmetric witness is a Hermitian 3 × 3 matrix W with nonnegative expectation on these vectors, and “eigenvalues µ1 ≥ µ2 ≥ µ3” means that the characteristic polynomial of W is (X − µ1)(X − µ2)(X − µ3). The conjecture of §6(c) is read as: for all real µ1 ≥ µ2 ≥ µ3, such a W exists if and only if µ2 ≥ 0 and µ3 ≥ −√(µ1µ2).