POVM steering threshold of higher-dimensional Werner states

Problem

What is the exact steering threshold for arbitrary POVMs on a higher-dimensional Werner state? For \(d\geq3\), let \(F\) be the swap operator on \(\mathbb C^d\otimes\mathbb C^d\) and define

\begin{equation} \rho_f^{(d)} :=\frac{(d-f)I+(df-1)F}{d(d^2-1)}, \qquad -1\leq f\leq1. \tag{1} \end{equation}

If Alice applies a POVM \(\{M_{a\mid x}\}_a\) to the state in Eq. (1), Bob’s subnormalized conditional states are

\begin{equation} \sigma_{a\mid x} :=\operatorname{Tr}_A\!\left[ (M_{a\mid x}\otimes I)\rho_f^{(d)} \right]. \tag{2} \end{equation}

The assemblage in Eq. (2) is unsteerable when it admits a local-hidden-state decomposition

\begin{equation} \sigma_{a\mid x} =\int_\Lambda\mu(d\lambda)\, p(a\mid x,\lambda)\tau_\lambda, \tag{3} \end{equation}

where \(\mu\) is a probability measure, \(p(a\mid x,\lambda)\) are response functions, and \(\tau_\lambda\) are density operators. Determine the critical value \(f_{\mathrm{POVM}}(d)\) characterized by

\begin{equation} \rho_f^{(d)}\ \text{is unsteerable from Alice to Bob for every POVM} \quad\Longleftrightarrow\quad f\geq f_{\mathrm{POVM}}(d). \tag{4} \end{equation}

In particular, decide whether the threshold in Eq. (4) equals the exact projective-measurement threshold

\begin{equation} f_{\mathrm{PVM}}(d)=-1+\frac1d+\frac1{d^2} \tag{5} \end{equation}

for every \(d\geq3\). Equation (3) fixes the notion of unsteerability used in both threshold statements.

Source

After resolving the qubit case, Zhang–Chitambar and Renner explicitly leave the arbitrary-POVM steering threshold of higher-dimensional Werner states open [ZC24], [Ren24].

Progress

The projective-measurement boundary is exactly Eq. (5) [Wer89], [JWD07].
Explicit local-hidden-state constructions give nontrivial arbitrary-POVM unsteerable intervals in every dimension, but for \(d\geq3\) their bounds do not reach Eq. (5) [NG20].
For \(d=2\), projective measurements and arbitrary POVMs have the same exact threshold. In the visibility parametrization \(\rho_W(r)=r\lvert\Psi^-\rangle\!\langle\Psi^-\rvert+(1-r)I/4\), the state is unsteerable for every POVM exactly when \(r\leq1/2\) [ZC24], [Ren24]. These qubit constructions have not determined Eq. (4) for \(d\geq3\).
Higher-dimensional steering inequalities detect steering for selected measurement families, but do not determine the all-POVM boundary [YQ26].

Comment

The exact arbitrary-POVM boundary is known in dimension two but not for any general \(d\geq3\). The remaining question is whether genuinely nonprojective POVMs lower the unsteerable Werner interval below the projective threshold.

References

[Wer89]: R. F. Werner, “Quantum States with Einstein–Podolsky–Rosen Correlations Admitting a Hidden-Variable Model,” Physical Review A 40, 4277–4281 (1989).DOI
[JWD07]: S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, EPR-Correlations, Bell-Nonlocality, and Steering,” Physical Review A 76, 052116 (2007).DOI arXiv
[NG20]: H. C. Nguyen and O. Gühne, “Some Quantum Measurements with Three Outcomes Can Reveal Nonclassicality Where All Two-Outcome Measurements Fail to Do So,” Physical Review Letters 125, 230402 (2020).DOI arXiv
[ZC24]: Y. Zhang and E. Chitambar, “Exact Steering Bound for Two-Qubit Werner States,” Physical Review Letters 132, 250201 (2024).DOI arXiv
[Ren24]: M. J. Renner, “Compatibility of Generalized Noisy Qubit Measurements,” Physical Review Letters 132, 250202 (2024).DOI arXiv
[YQ26]: M.-C. Yang and C.-F. Qiao, “Witness High-Dimensional Quantum Steering via Majorization Lattice,” npj Quantum Information 12, 55 (2026).DOI

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