Two-copy distillability at the Werner-state threshold

Is the NPT Werner state at the one-copy threshold two-copy nondistillable? A reduction expresses the issue as nonnegativity of a rank-two partial-trace quadratic form, including the studied two-ququart case.

Why it matters and what counts as progress

Why it matters: The two-copy threshold tests a concrete boundary of entanglement distillation without settling arbitrary-copy behavior.

What counts as progress: A nonnegativity proof of the exact reduced form or a rank-two witness violating it; a two-copy theorem does not automatically extend to all copy counts.

How the problem developed

2022 · Open problem [1]
Ryszard Horodecki, Paweł Horodecki, Michał Horodecki and Karol Życzkowski Listed two-copy distillability of the two-ququart Werner state at the one-copy threshold as an open quantum-information problem.
26 Jul 2026 · Reported claim [2]
Zhiwei Song and Lin Chen Reported two-copy undistillability of NPT Werner states in every local dimension, which includes the two-ququart threshold case; this does not settle distillability for arbitrarily many copies.

AI contribution

A partial-trace matrix inequality and Werner-state distillability

Zhiwei Song, Lin Chen

GPT-5.5 Pro’s Proposition 2 expresses the rank-two partial-trace quadratic form q⁽²⁾₋₁/₂(C) as projection terms. Song and Chen then use a separate human-developed lemma to prove nonnegativity and two-copy nondistillability at the Werner threshold.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull