Secret key from every entangled state

Solved ID op_b952ec2dd8246a10 Last edited 7 October 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

Reports do not certify correctness or automatically change the problem's status. Progress policy.

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 identified the existence of an entangled state with zero distillable key as an open question in their 2026 paper [HSDW26].
OpenAI reports an explicit entangled state on $\mathbb C^{10}\otimes\mathbb C^{10}$ with $K_D=0$ (Theorem 1.1, p. 2) [OAI26]. The local-instrument protocols in Definition 4.1 include ordinary finite two-way protocols and the specified countable protocols that complete almost surely, with joint local processing of all copies and authenticated public communication. The input is the only shared private resource, and the eavesdropper holds its purification and the full public transcript.

Comment

The catalog designates this question Solved on the basis of the reported counterexample in [OAI26], a preprint. It gives a negative answer to Eq. (2) under the protocol convention stated in Progress. Measuring the key registers of asymptotically distilled private states would yield secret key, so the obstruction also excludes the private-state formulation above. No minimal local dimension is claimed.

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
[OAI26]: OpenAI, “Entanglement with zero distillable secret key in local dimension ten,” OpenAI Math Release preprint, 27 September 2026. manuscript (release version).link

Secret key from every entangled state

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