Secret key from every entangled state

Unsolved ID op_b952ec2dd8246a10 Last edited 4 September 2026

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Fields: Quantum Cryptography
Quantum Resource Theory
Topics: Secret-key distillation
Bound entanglement
Local operations and classical communication

Problem

Does every finite-dimensional entangled bipartite state have positive asymptotic distillable secret key? For a state $\rho_{AB}$, with an adversary holding a purification, define its distillable-key rate by

\begin{equation} K_D(\rho_{AB}) :=\sup\left\{R\geq0: \begin{array}{l} \text{there are LOPC protocols $\Lambda_n$ and private states $\gamma^{(n)}_{K_n}$, with $K_n=2^{\lfloor nR\rfloor}$, such that}\\[-1mm] \displaystyle \lim_{n\to\infty} \left\|\Lambda_n(\rho_{AB}^{\otimes n}) -\gamma^{(n)}_{K_n}\right\|_1=0 \end{array} \right\}, \tag{1} \end{equation}

where each target $\gamma^{(n)}_{K_n}$ may have its own shield system and has key registers of dimension $K_n$. With the operational convention in Eq. (1), the question is whether the implication

\begin{equation} \rho_{AB}\ \text{entangled} \quad\Longrightarrow\quad K_D(\rho_{AB})>0 \tag{2} \end{equation}

holds for every finite-dimensional $\rho_{AB}$.

Source

Horodecki, Sikorski, Das, and Wilde explicitly identify the question whether every entangled state has positive distillable key [HSDW26].

Progress

Horodecki, Horodecki, Horodecki, and Oppenheim constructed bound-entangled states with positive distillable key. Thus zero distillable entanglement does not provide a counterexample to Eq. (2) [HHHO05].
Horodecki, Sikorski, Das, and Wilde explicitly state that the existence of an entangled state with zero distillable key remains open. Their 2026 result therefore confirms the unresolved scope of Eq. (2) close to the present cutoff [HSDW26].

Comment

The unresolved alternative is either to prove Eq. (2) or to construct an entangled state with $K_D=0$. The question concerns the asymptotic key rate, not one-shot key extraction or distillable entanglement.

References

[HHHO05]: K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, “Secure Key from Bound Entanglement,” Physical Review Letters 94, 160502 (2005).DOI arXiv
[HSDW26]: K. Horodecki, L. Sikorski, S. Das, and M. M. Wilde, “Cost of Quantum Secret Key,” Quantum 10, 2098 (2026).DOI arXiv

Secret key from every entangled state

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