1 · The essentials
Entanglement is a kind of shared quantum connection, but this manuscript shows that it need not provide a usable shared secret. It constructs an entangled state in which each of two participants has a quantum system of dimension ten. For the procedures specified in the manuscript, they cannot extract secret key from this state. They may process all their copies together on each side and exchange unlimited authenticated public messages in both directions. However, they have no other shared private resource, and an eavesdropper holds both the full public record and a quantum system completing the description of their input. The result applies to the specified procedures that finish with probability one, not automatically to every conceivable procedure. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine two people receiving linked puzzle pieces and trying to turn their connection into a password while an observer hears every message they exchange. The result says that, for this particular quantum resource and the allowed procedures, having a connection is not enough to obtain a secret key.
Why it matters
The construction separates the presence of entanglement from the ability to extract secret key under the stated operational rules. Allowing joint local processing of all copies and unlimited two-way public communication does not remove that separation. The source also reports counterexamples to two conjectures about composing quantum channels.
2 · Going further
The precise state-construction problem is to find an entangled state on with zero distillable secret key for the specified local-instrument protocols. These protocols permit joint local processing of all copies and unlimited authenticated two-way public communication, and must complete almost surely. The input is the only shared private resource; the eavesdropper holds a purification of it and the complete public record.
The abstract describes a construction rather than merely a restriction on possible states. It also states that this construction disproves the unrestricted two-map PPT-composition conjecture. Neither the state itself nor the intermediate argument establishing zero distillable key is supplied in the excerpt, so the mechanism cannot be reconstructed from this source alone.
A separate construction reported in the same abstract is a trace-preserving PPT channel on whose square is not entanglement breaking. The repository summary identifies this as a counterexample to Christandl's PPT-square conjecture. Only one manuscript abstract is supplied, so no relationship to companion manuscripts can be specified. The main result has a Lean formalization, but the supplied material does not delineate whether that formalization also covers these channel claims.
Useful notions
- Quantum states and entanglement
- Local quantum operations and public communication
- Secret-key extraction
- Purification of a quantum state
- Quantum channels and their composition
What was known before / what is new
Before
The source identifies Christandl's PPT-square conjecture and an unrestricted two-map PPT-composition conjecture as the conjectures addressed. It supplies no further historical context or account of earlier partial results.
New
The manuscript constructs an entangled state of local dimension ten with zero distillable secret key for its specified protocols. It also reports that the construction disproves the unrestricted two-map PPT-composition conjecture. Separately, it gives a trace-preserving PPT channel on whose square is not entanglement breaking.
Limits and assumptions
The zero-key conclusion is restricted to the specified local-instrument protocols that complete almost surely. Joint processing of copies and unlimited authenticated two-way public communication are allowed, but no additional shared private resource is available, and the eavesdropper holds the input purification and public record. The main result has a Lean formalization; the supplied excerpt does not specify its coverage of the separate channel claims. It provides neither construction details nor a list of remaining open problems, and it does not justify extending the conclusion beyond the stated protocol class.
Mini glossary
- Local dimension ten —
- Each participant's quantum system is described on the space .
- Zero distillable secret key —
- The state provides no extractable secret key under the protocol class specified in the manuscript.
- Almost-sure completion —
- A procedure finishes with probability one.
- Public record —
- The complete record of public communication, which is available to the eavesdropper.
- Square of a channel —
- The composition obtained by applying the same channel twice.
Keywords : Entanglement · Distillable secret key · PPT-square conjecture
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs an entangled state on with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Entanglement with zero distillable secret key in local dimension ten
27/09/2026We construct an entangled state on with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A separate trace-preserving PPT channel on has a square that is not entanglement breaking.