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bibkey: johnstonrussosikora2025antidistinguishability
authors: Nathaniel Johnston, Vincent Russo, Jamie Sikora
year: 2025
title: Tight bounds for antidistinguishability and circulant sets of pure quantum states
doi: 10.22331/q-2025-02-04-1622
url: https://arxiv.org/abs/2311.17047v2
claim: Theorem 5.1 gives a spectral sufficient condition for antidistinguishability; Corollary 5.4 gives an overlap bound whose tightness for n ≥ 5 is questioned in Section 6.
strata_touched:
  - D5/S3/Quantum/Measurement/AntidistinguishabilityGram
  - D5/S3/Quantum/Measurement/IncoherentCone
  - D5/S3/Quantum/Measurement/LocallyPositiveSpectrum
  - D5/S3/Quantum/Measurement/AntidistinguishabilitySpectrum
  - D5/S3/Quantum/Measurement/OverlapLowRankDistance
  - D5/S3/Quantum/Measurement/AntidistinguishabilityOverlapNonTight
license: citation-only
triage: anchor

Verified locator

DOI: 10.22331/q-2025-02-04-1622

Source: https://arxiv.org/abs/2311.17047v2

The arXiv v2 record identifies this work as Quantum 9, 1622 (2025) and supplies the DOI above.

Statements

Section 2.1, page 3: “The set of states is antidistinguishable if there exists a POVM such that ⟨ψi|Mi|ψi⟩ = 0 for all 0 ≤ i ≤ n − 1.” A POVM consists of positive semidefinite effects whose sum is the identity.

Theorem 5.1, page 12: “Let n ≥ 2 be an integer and let G ∈ Pos(ℂⁿ) be the Gram matrix of a set S of n pure states, and let λ₀ ≥ λ₁ ≥ ⋯ ≥ λₙ₋₁ be the eigenvalues of G.” The displayed condition is

$$\sqrt{{\lambda }_0}\le \sum _{j=1}^{n-1}\sqrt{{\lambda }_j}.$$

It implies that S is antidistinguishable, and is necessary for circulant G.

Corollary 5.4, page 15: “Let n ≥ 2 be an integer and let S = {|ψ₀⟩, |ψ₁⟩, . . . , |ψₙ₋₁⟩}. If”

$$|⟨{\psi }_i|{\psi }_j⟩|\le \frac{1}{\sqrt{2}}\sqrt{\frac{n-2}{n-1}}\text{for all }0\le i\ne j\le n-1,$$

“then S is antidistinguishable.”

Section 6, page 17: “We still do not know whether or not Corollary 5.4 is tight when n ≥ 5.”

Encoding and boundary

The formal states have type Fin n → EuclideanSpace ℂ (Fin d). A finite family of positive semidefinite matrices summing to the identity supplies the POVM. The Gram matrix is the existing Matrix.gram ℂ construction. The squared overlap cap is (n - 2) / (2 * (n - 1)), with subtraction in ℝ.

The repository-derived improvement uses δₙ = (n − 9)/(16n(n − 1)(n − 3)) for every n ≥ 10. It does not decide the source’s cases n = 5,…,9 or alter the source’s circulant characterization and sharp Frobenius bound.