bibkey: liu2025quantumtriangular authors: Yupan Liu year: 2025 title: “Quantum state testing beyond the polarizing regime and quantum triangular discrimination” doi: 10.48550/arXiv.2303.01952 url: https://arxiv.org/abs/2303.01952v5 claim: “We observe that Theorem 1.4(ii) is not tight. Numerical simulations indicate that the tight bound is QTD²(ρ₀, ρ₁) ≤ QJS₂(ρ₀, ρ₁) ≤ QTD(ρ₀, ρ₁) for any states ρ₀ and ρ₁.” strata_touched:

• D5/S3/Divergence/QuantumTriangularJensenShannonRefutation license: citation-only triage: anchor

Quantum state testing beyond the polarizing regime and quantum triangular discrimination

Yupan Liu, Quantum state testing beyond the polarizing regime and quantum triangular discrimination, arXiv:2303.01952v5; published in Computational Complexity 34 (2025), article 11.

§1.3, “Improved inequalities on the quantum triangular discrimination”, printed p. 7:

We observe that Theorem 1.4(ii) is not tight. Numerical simulations indicate that the tight bound is QTD²(ρ₀, ρ₁) ≤ QJS₂(ρ₀, ρ₁) ≤ QTD(ρ₀, ρ₁) for any states ρ₀ and ρ₁.

Immediately after Theorem 3.4:

Numerical simulations suggest that the tight bound is QTD²(ρ₀, ρ₁) ≤ QJS₂(ρ₀, ρ₁) ≤ QTD(ρ₀, ρ₁), while we only managed to prove a slightly weaker bound ½QTD²(ρ₀, ρ₁) ≤ QJS₂(ρ₀, ρ₁) ≤ QTD(ρ₀, ρ₁) in Theorem 3.4.

Definition 3.1 (p. 13) reads in part:

QTD(ρ₀, ρ₁) := ½ Tr[(ρ₀ − ρ₁)(ρ₀ + ρ₁)^{−1/2}(ρ₀ − ρ₁)(ρ₀ + ρ₁)^{−1/2}]

Furthermore, if ρ₀ + ρ₁ is not full-rank, then the inverse is defined only on its support.

Definition 2.10 (p. 10) reads in part:

QJS(ρ₀, ρ₁) := S((ρ₀+ρ₁)/2) − (S(ρ₀) + S(ρ₁))/2

Here, S(ρ) := −Tr(ρ ln ρ) denotes the von Neumann entropy of the quantum state ρ

QJS₂ uses the base-2 logarithm. The lower proposed inequality is false already for full-rank diagonal qubit states; the upper inequality is not addressed by this refutation.

Locator

Author text: https://arxiv.org/abs/2303.01952v5.

• arXiv v5, §1.3, “Improved inequalities on the quantum triangular discrimination”, printed p. 7.
• arXiv v5, paragraph immediately after Theorem 3.4.
• arXiv v5, Definition 3.1 (QTD and the inverse on its support).
• arXiv v5, Definition 2.10 (QJS and von Neumann entropy).
• DOI: https://doi.org/10.48550/arXiv.2303.01952