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Entanglement with zero distillable secret key in local dimension ten
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Abstract
We construct an entangled state on $\mathbb C^{10}\otimes\mathbb C^{10}$ 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 $M_{21}(\mathbb C)$ has a square that is not entanglement breaking.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Entanglement without distillable secret key (Lean scope notes) (lean/docs/272.md, 2026). The release's Lean comparator statements cover the dimension-ten PPT pair whose composition is not entanglement breaking (Theorem 1.2) and the 21-dimensional trace-preserving PPT channel whose square is not entanglement breaking (Theorem 1.3); the scope notes explicitly exclude the zero-distillable-key statement (Theorem 1.1).
- M. Krenn (issue author), Formalize Open Quantum Problem #38: The PPT-squared conjecture (google-deepmind/formal-conjectures issue #3455, opened 2026-03-06). Open request to formalize the PPT-squared conjecture that Theorems 1.2–1.3 address; when checked it had no comments and no linked pull request, and its text describes the general case as open.
Related prior literature mentioned by others
- Ion Nechita and Sang-Jun Park, Random covariant quantum channels (arXiv:2403.03667, 6 March 2024). Cited in the QIQCOP Zoo discussion of the PPT-squared counterexample, this work establishes generic entanglement breaking for compositions of two random diagonal-orthogonal-covariant channels, providing context for the manuscript's unrestricted counterexample.
Possibly relevant literature (found by the agent, not checked)
- Karol Horodecki, Leonard Sikorski, Siddhartha Das, and Mark M. Wilde, Cost of quantum secret key (arXiv:2402.17007; Quantum 10, 2098, 2026). This resource-theoretic treatment discusses the possibility of entangled states with zero distillable key and states that its results remain valid if such states exist, relevant to Theorem 1.1.
- B. Collins, Z. Yin, P. Zhong, The PPT square conjecture holds generically for some classes of independent states (arXiv:1803.00143, J. Phys. A 51 (2018) 425301). Shows that for certain random PPT inputs, entanglement swapping generically gives separable output in the large-dimension limit; context for the explicit non-generic counterexample in Theorems 1.2–1.3.
- R. Jin, Investigation of the PPT Squared Conjecture for High Dimensions (arXiv:2010.15554, 2020). An earlier search for higher-dimensional counterexamples via decomposition and composition of PPT channels, which reports a potential counterexample; relevant to Theorems 1.2–1.3.
- W. A. Majewski, On PPT Square Conjecture (arXiv:2108.01588, 2021). Proposes a positive-maps and tensor-cone strategy for the PPT-squared conjecture, which Theorem 1.2 claims to refute in the unrestricted two-map form.
- A. Müller-Hermes, S. Singh, Bi-PPT channels are entanglement breaking (arXiv:2204.01685, 2022). Proves that a related class of PPT-type channels is entanglement breaking; relevant to the paper's survey of affirmative cases of PPT composition.
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 272
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026,
author = {{OpenAI}},
title = {{Entanglement with zero distillable secret key in local dimension ten}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf}{OAI:Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026}},
year = {2026}
}From OpenAI's README for this paper:
Verification
The optional supplementary checks use Python 3 and only its standard library, with exact integer arithmetic. Run with assertions enabled, without -O or -OO. The commands below use a temporary copy and compare the replayed outputs with the included recorded outputs:
set -eu
unset PYTHONOPTIMIZE
verification_run=$(mktemp -d "${TMPDIR:-/tmp}/zero-key-verification.XXXXXX")
cp -R verification "$verification_run/verification"
python3 "$verification_run/verification/combinatorics_exact_checks.py" > "$verification_run/combinatorics.stdout"
cmp "$verification_run/verification/combinatorics_exact_checks.json" verification/combinatorics_exact_checks.json
python3 "$verification_run/verification/gadget_exact_check.py" > "$verification_run/gadget.json"
cmp "$verification_run/gadget.json" verification/gadget_exact_check.json
python3 "$verification_run/verification/exact_certificate.py" > "$verification_run/certificate-output.txt"
cmp "$verification_run/certificate-output.txt" verification/certificate-output.txt
Success means all programs and comparisons exit zero. These finite checks cover only the arithmetic components described in the paper; they do not establish the full paper or instantiate the enormous global matrices. The PDF build does not run them.
Lean formalization
OpenAI provides Lean files for this family: lean/docs/272.md. The Lean statements have not been checked by people against the paper's claims.
Family 272: Entanglement without distillable secret key
Constructs an entangled state on $\mathbb C^{10}\otimes\mathbb C^{10}$ 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 $M_{21}(\mathbb C)$ whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.
This is the only paper in this family.