bibkey: wangzhouchenfei2026conical authors: H.-F. Wang; W. Zhou; L. Chen; S.-M. Fei year: 2026 title: Estimating the concurrence for quantum states via symmetric measurements doi: null url: https://arxiv.org/abs/2606.31010v2 claim: “The lower bounds of concurrence induced by arbitrary two distinct conical 2-designs are comparable.” strata_touched:

• D5/S3/Quantum/Entanglement/ConicalDesignConcurrenceComparability license: citation-only triage: anchor

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Source: https://arxiv.org/abs/2606.31010v2

Section IV, Theorem 2 and the conjecture immediately following it, PDF p. 8. The two-constant definition of the bound is the source’s reference [LBGEM], Siudzińska, J. Phys. A 58 (2025) 375302, Section 7, Theorem 6. The Crossref title query found no exact journal record for this title; no journal DOI is asserted.

Comparability of conical-design concurrence bounds

The source conjecture reads:

The lower bounds of concurrence induced by arbitrary two distinct conical 2-designs are comparable.

A finite positive semidefinite family on the d-dimensional complex Hilbert space satisfies ${\sum }_i{E}_i\otimes E_i=\alpha I+\beta F$ with $\alpha \ge \beta >0$, where $F$ swaps the tensor factors. Its measurement correlation matrix has entries $P_E(\rho {)}_{ij}=\mathrm{tr}[\rho (E_i\otimes E_j)]$. Reference [LBGEM] gives the bound $B_E(\rho )=\sqrt{2/[d(d-1)]}(\Vert P_E(\rho ){\Vert }_1-\alpha -\beta )/\beta$. The trace norm is the sum of singular values, equivalently $\mathrm{tr}\sqrt{P_E(\rho {)}^{\ast }{P}_E(\rho )}$.

Comparability means one uniform pointwise ordering over all positive semidefinite trace-one bipartite states. The same family is used on both subsystems. The claim includes equal designs and equal ratios; the source’s distinct-design assertion is contained in that statement. This comparison does not assert that the bound is nonnegative, equals concurrence, detects all PPT entanglement, or has a maximizing finite design.