Batch episode PSI01–05 (outside OpenAI’s math catalogue), October 2026

Five Open Quantum Problems

What Physical Superintelligence’s five AI-assisted quantum information results claim, and what they don’t

8:57. English captions available. The video loads from YouTube when you press play.

Chapters

The release, and how AI was used

PSI’s page says only that its researchers worked with its AI research system; it does not name that system. The repository names models. For each result it says the mathematics was developed by PSI researchers working with AI models, from Anthropic’s Claude family and OpenAI’s GPT family, in automated research workflows, and that AI agents wrote the Lean, under PSI’s direction. Each paper says the authors used frontier AI models to develop, derive and compose it. The named authors, Ian MacCormack, David Turturean and Ning Bao, are responsible for correctness.

PSI’s release page (October 23, 2026) does not name its AI system. The repository’s result guides name the models for each result. Last checked October 27, 2026.

PSI01 · Problem 2: Undistillability implies PPT?

Three copies of the Werner state: the sharp curve in dimensions three and four and a bound in every dimension, Ian MacCormack, Ning Bao; 14 pages; 2026-10-23.

The claim. For local dimensions 3 and 4 and 0 ≤ α ≤ 1/2, the minimum of the three-copy form on unit matrices of rank at most two is (1 − 2α)(1 − α)², so three copies of the Werner state are distillable exactly when α > 1/2 (proved by hand from published d = 3 and d = 4 certificates by others); in every finite local dimension three copies are not distillable for α ≤ 37/80 (a new computer-verified certificate), so NPT Werner states that survive three copies exist for every d ≥ 3. A partial result on Problem 2.

Not claimed.

Undistillability from any number of copies (the paper: ’untouched’; PSI’s page: ’remains open’)
For d ≥ 5, the range 37/80 < α ≤ 1/2 stays open
37/80 is not claimed optimal

Preprint: PSI GitHub repository, released October 23, 2026
Named expert review: None independent. Co-author Ning Bao reviewed and edited as a paid contractor to PSI; the paper thanks an unnamed referee for a report on an earlier version, which the first author obtained.
Peer reviewed: No journal in any source we read (as of October 27, 2026)
Lean by PSI: Qutrit case only: Theorems 1.1 and 1.2 at d = 3 (endpoint from PSI’s own exact certificate), Proposition 4.2(a), Corollary 5.3 for k = 3; 48 Comparator statements. Theorem 1.3 (37/80) is not formalized. Not run by us.

In the video. PSI01 is problem 2: undistillability implies PPT? A state of two systems is distillable if local operations and classical communication can extract near-perfect entangled pairs from many copies. States with a positive partial transpose, PPT, are never distillable; the problem asks for the converse. The list notes it is enough to study Werner states, a family with one parameter, α. In the paper’s convention they fail to be PPT exactly when α exceeds 1/d, the local dimension, and one copy is distillable exactly when α exceeds 1/2. The paper, by MacCormack and Bao, starts from two computer certificates it cites, published in October: in dimensions three and four, three copies are not enough at α = 1/2. From them it proves by hand the exact three-copy curve, (1 − 2α)(1 − α)²; so in those dimensions three copies are distillable exactly when α exceeds 1/2. A new certificate of its own, checked by computer, shows that in every dimension three copies are not enough for α up to 37/80. Not claimed: more copies. The paper calls the many-copy question untouched; and for dimension five and up, α between 37/80 and 1/2 stays open.

PSI02 · Problem 12: Bell inequalities for long range vacuum correlations

The vacuum Bell correlation of the massive free scalar field attains the classical bound for bounded regions beyond a finite separation, David Turturean; 17 (letter); 59 (full manuscript) pages; 2026-10-22 (letter); 2026-10-21 (full manuscript).

The claim. For the real free scalar field of any mass m > 0 in 3+1 dimensions there is one finite C* > 2 such that two equal-time double cones of radius R with centres at distance L ≥ C*R have vacuum Bell correlation exactly 1 (no CHSH violation); any two bounded regions moved far enough along a fixed spacelike direction likewise. The vacuum stays entangled. ’This answers Problem 12 in the negative for fixed bounded regions.’

Not claimed.

A value of C* (finite, not explicit)
The smallest such separation
Regions that grow with the separation; wedges at a positive gap (open)

Preprint: PSI GitHub repository, released October 23, 2026
Named expert review: None independent. Ning Bao (not an author) wrote a referee report on an earlier draft as a paid contractor to PSI.
Peer reviewed: No journal in any source we read (as of October 27, 2026)
Lean by PSI: Theorem 1.1 and Corollary 1.2 of the full manuscript as stated, the CHSH form, a direction-uniform version, vacuum existence and noncommutativity; 6 Comparator statements. Not run by us.

In the video. PSI02 is problem 12. Vacuum fluctuations of a quantum field maximally violate the CHSH Bell inequality for suitable spacelike separated observables, and the violation goes to zero as the regions move apart. The list asks whether some small violation is possible for regions arbitrarily far apart, for a massive free scalar field. The letter, by David Turturean, proves: in 3+1 dimensions, two equal-time double cones of radius R show no CHSH violation once their centres are at least C* times R apart, with one finite constant C* for every mass and radius. For any two bounded regions, moving one far enough in a fixed spacelike direction does the same. The Bell correlation is then one, its classical value, though the regions stay entangled. The letter says this answers problem 12 in the negative for fixed bounded regions. Not claimed: a value for C*, which is not explicit, or the smallest separation; for regions that grow with the separation, and for wedges at a positive gap, the question remains open.

PSI03 · Problem 27: The power of CGLMP inequalities

The optimal CGLMP measurements on maximally entangled states, David Turturean, Ning Bao; 15 pages; 2026-10-21 (draft).

The claim. For two settings and d outcomes, projective measurements on a maximally entangled state have CGLMP value at most the Fourier-phase value I_F(d) (2√2 at d = 2, 2.8729… at d = 3, → 32G/π² ≈ 2.9698), and every maximizer is the Fourier-phase family up to one common unitary (on Φ_n, equality needs d | n). ’This resolves the first assertion of Problem 27.B in the projective class.’ The paper says Mishra and Senthilkumar prove the same projective value, rigidity and divisibility statements.

Not claimed.

General (POVM) measurements
Other (non-maximally-entangled) states
Part A
Noise-resistance optimality beyond the fixed CGLMP inequality’s own threshold (where the maximizers do minimize it), and Kullback–Leibler optimality

Preprint: PSI GitHub repository, released October 23, 2026
Named expert review: None independent. The paper thanks Ian MacCormack (PSI) for reviewing earlier drafts and Jim Halverson for coordinating the review (reviewers not named); co-author Ning Bao commented as a paid contractor to PSI.
Peer reviewed: No journal in any source we read (as of October 27, 2026)
Lean by PSI: Projective bound and equality case (Theorems 1 and 3), attainment, witness tolerance; also statements not claimed by the current paper (tensor-CHSH noise and statistical-strength comparisons at d = 2^m, m ≥ 955 and m ≥ 41; Part A facet statements), from an earlier manuscript; 16 Comparator statements. Not run by us.

In the video. PSI03 is problem 27, on CGLMP Bell inequalities. Part B says that numerically, the measurements that violate them most on a maximally entangled state have a specific form, built from the discrete Fourier transform and diagonal phases, and asks to show this is necessarily so; and also that they resist noise best, and discriminate best in the Kullback–Leibler sense. The paper, by Turturean and Bao, proves the first part for projective measurements: none beats the Fourier-phase value, and every maximizer is the Fourier-phase family up to one common change of basis. That value is 2√2 at d = 2, about 2.873 at d = 3, and it tends to about 2.97. Our script rebuilt them and got the same values. The paper says this resolves the first assertion of problem 27.B in the projective class. Not claimed: general measurements, other states, part A, or noise and Kullback–Leibler optimality beyond this one inequality’s noise threshold. It credits Mishra and Senthilkumar with the same value and rigidity statements, by another route.

PSI04 · Problem 38: The PPT squared conjecture

Entanglement Breaking after 2(d−1) PPT Maps, David Turturean, Ning Bao; 4 pages; 2026-10-23 (repository snapshot; no date on the paper).

The claim. For every d ≥ 2, any 2(d − 1) PPT maps on d × d complex matrices (which may differ and need not preserve trace) compose to an entanglement-breaking map; so the entanglement-breaking index of every PPT map is at most 2(d − 1). A bound, not the PPT-squared conjecture, which OpenAI’s preprint of September 27, 2026 (openai/math result 272) reported false in dimension 10; PSI cites it.

Not claimed.

That 2(d − 1) is tight: the paper says it is suboptimal in dimension 3, where two maps suffice
Optimality for large d (left for future work)
The PPT-squared conjecture itself

Preprint: PSI GitHub repository, released October 23, 2026
Named expert review: None independent. The paper thanks Adam Levine for reviewing an earlier draft of the proof and Jim Halverson for coordinating the expert review (reviewers not named); co-author Ning Bao verified the result as a paid contractor to PSI.
Peer reviewed: No journal in any source we read (as of October 27, 2026)
Lean by PSI: Theorem 1 (both sentences), N ≥ 2(d − 1) maps, and an interleaved version; 13 Comparator statements. Not run by us.

In the video. PSI04 is problem 38, the PPT squared conjecture of Matthias Christandl. In the form the paper uses: compose any two PPT maps and the result is entanglement breaking, destroying entanglement with any reference system. OpenAI’s preprint dated September 27, result 272 in its catalogue, reported a counterexample in dimension ten; PSI’s page and paper cite it, and our episode 272 covers it. The 4-page note by Turturean and Bao proves a bound instead: on a d-level system, any 2(d − 1) PPT maps, which may differ and need not preserve trace, compose to an entanglement-breaking map. The key step, following a suggestion of Christandl, Müller-Hermes and Wolf, shows that each pair of PPT maps breaks entanglement with a qubit reference; each pair then lowers the Schmidt number by one, from at most d down to one. Is the bound tight? The paper says it is not in dimension three, where two maps already suffice and the bound gives four; whether it is optimal for large d, they leave for future work.

PSI05 · Problem 43: Are all extensibly causal processes purifiable?

An extensibly causal quantum process with no purification, David Turturean, Ning Bao; 8 pages; 2026-10-23 (repository snapshot; no date on the paper).

The claim. An explicit four-qubit process S = (1/80)(20 IIII + 10 ZIII + 10 IIZI + 5 ZIZI + 4 IZZI + 4 ZIXZ) is a valid process of full rank, extensibly causal and not causally separable, hence (Refs. [5, 6]) not purifiable. ’This answers Problem 43 in the negative, in its finite-dimensional bipartite form and with the notion of purity of’ Araújo, Feix, Navascués and Brukner.

Not claimed.

That the process is unphysical in every sense
Multipartite or infinite-dimensional forms of the question

Preprint: PSI GitHub repository, released October 23, 2026
Named expert review: None independent. Co-author Ning Bao wrote a referee report (including improvements suggested by an AI review) as a paid contractor to PSI; the paper thanks Jim Halverson for coordinating the expert review (reviewers not named).
Peer reviewed: No journal in any source we read (as of October 27, 2026)
Lean by PSI: The counterexample, its Pauli form, the dimension dichotomy, purifiable ⇒ causally separable, ordered ⇒ purifiable and more; 11 Comparator statements. Not run by us.

In the video. PSI05 is problem 43. In the process matrix framework, two laboratories are linked without a fixed causal order. A process is extensibly causal if it never violates a causal inequality, even with entangled ancillas shared by the labs. It is purifiable if it comes from a larger pure process that turns reversible local operations into a reversible global evolution. The list asks whether every extensibly causal process is purifiable. The 8-page letter by Turturean and Bao says no. It writes down an explicit process on four qubits, a short sum of Pauli terms, and proves it is a valid process of full rank, extensibly causal, but not causally separable, which for two parties, by earlier theorems, means not purifiable. A causal witness gives −1/5; our script reproduces that number and the smallest eigenvalue, about 0.0019. The key tool is a sum of squares identity covering every ancilla at once. The paper says it answers problem 43 in the negative, in its finite-dimensional bipartite form, with the purity notion of Araújo and coauthors; it does not claim the process is unphysical in every sense.

Where they stand

Last checked October 27, 2026. Where they stand, as of October 27. All five are preprints in PSI’s repository, and none says it has been peer reviewed. No source names an independent expert review. Ning Bao, co-author of four and a paid contractor to PSI, reviewed, refereed, verified or commented on each; PSI01 thanks an unnamed referee; PSI03 thanks Ian MacCormack, and PSI04 Adam Levine, for reviewing earlier drafts; and three thank Jim Halverson for coordinating a review, naming no reviewers. PSI says Lean’s kernel checked each formalization. Scope varies: PSI01 covers dimension three only, and PSI03’s Lean also states results its current paper does not claim. Our scan found sorry only in the challenge files. We did not build or run any of it.

The numbers in the video

Ours: the three-copy curve (1 − 2α)(1 − α)² is attained by 2^(−1/2) P_E ⊗ |u⟩⟨u| ⊗ |v⟩⟨v| at d = 3 for α = 0, 0.1, 0.25, 0.4, 0.5, and the paper’s continuation identity checks symbolically.
Ours: CHSH in the letter’s normalization has local bound 1 and quantum value √2 ≈ 1.414.
Ours: the Fourier-phase CGLMP value I_F(d) is 2.8284, 2.8729, 2.8962, 2.9105 for d = 2 to 5, rebuilt from the paper’s measurement formula; the limit 32G/π² ≈ 2.9698.
Ours: the four-qubit process S has trace 4, smallest eigenvalue (10 − √97)/80 ≈ 0.0019, causal witness value Tr(WS) = −1/5, and the sum-of-squares identity (23) holds on random non-commuting operators.
Ours: Lean files in each lean/ folder: 3,619; 1,017; 471; 43; 80 (counted, not run). Sorry appears only in Challenge.lean.

Transcript

Read the full script

0:00The list and the release

This is a batch episode, outside OpenAI’s math catalogue. Open Quantum Problems is a numbered list of open questions in quantum information, which began as Reinhard Werner’s list and is now maintained by IQOQI Vienna. On October 23, 2026, the company Physical Superintelligence, or PSI, released results on five of them: problems 2, 12, 27, 38 and 43. each with a paper and Lean, in one GitHub repository. We call them PSI01 to PSI05, in the list’s order. PSI’s page says only that its researchers worked with its AI research system; it does not name that system. The repository names models. For each result it says the mathematics was developed by PSI researchers working with AI models, from Anthropic’s Claude family and OpenAI’s GPT family, in automated research workflows, and that AI agents wrote the Lean, under PSI’s direction. Each paper says the authors used frontier AI models to develop, derive and compose it. The named authors, Ian MacCormack, David Turturean and Ning Bao, are responsible for correctness.

1:16PSI01 · Problem 2: three copies of a Werner state

PSI01 is problem 2: undistillability implies PPT? A state of two systems is distillable if local operations and classical communication can extract near-perfect entangled pairs from many copies. States with a positive partial transpose, PPT, are never distillable; the problem asks for the converse. The list notes it is enough to study Werner states, a family with one parameter, α. In the paper’s convention they fail to be PPT exactly when α exceeds 1/d, the local dimension, and one copy is distillable exactly when α exceeds 1/2. The paper, by MacCormack and Bao, starts from two computer certificates it cites, published in October: in dimensions three and four, three copies are not enough at α = 1/2. From them it proves by hand the exact three-copy curve, (1 − 2α)(1 − α)²; so in those dimensions three copies are distillable exactly when α exceeds 1/2. A new certificate of its own, checked by computer, shows that in every dimension three copies are not enough for α up to 37/80. Not claimed: more copies. The paper calls the many-copy question untouched; and for dimension five and up, α between 37/80 and 1/2 stays open.

2:40PSI02 · Problem 12: Bell tests on the vacuum

PSI02 is problem 12. Vacuum fluctuations of a quantum field maximally violate the CHSH Bell inequality for suitable spacelike separated observables, and the violation goes to zero as the regions move apart. The list asks whether some small violation is possible for regions arbitrarily far apart, for a massive free scalar field. The letter, by David Turturean, proves: in 3+1 dimensions, two equal-time double cones of radius R show no CHSH violation once their centres are at least C* times R apart, with one finite constant C* for every mass and radius. For any two bounded regions, moving one far enough in a fixed spacelike direction does the same. The Bell correlation is then one, its classical value, though the regions stay entangled. The letter says this answers problem 12 in the negative for fixed bounded regions. Not claimed: a value for C*, which is not explicit, or the smallest separation; for regions that grow with the separation, and for wedges at a positive gap, the question remains open.

3:51PSI03 · Problem 27: optimal CGLMP measurements

PSI03 is problem 27, on CGLMP Bell inequalities. Part B says that numerically, the measurements that violate them most on a maximally entangled state have a specific form, built from the discrete Fourier transform and diagonal phases, and asks to show this is necessarily so; and also that they resist noise best, and discriminate best in the Kullback–Leibler sense. The paper, by Turturean and Bao, proves the first part for projective measurements: none beats the Fourier-phase value, and every maximizer is the Fourier-phase family up to one common change of basis. That value is 2√2 at d = 2, about 2.873 at d = 3, and it tends to about 2.97. Our script rebuilt them and got the same values. The paper says this resolves the first assertion of problem 27.B in the projective class. Not claimed: general measurements, other states, part A, or noise and Kullback–Leibler optimality beyond this one inequality’s noise threshold. It credits Mishra and Senthilkumar with the same value and rigidity statements, by another route.

5:04PSI04 · Problem 38: PPT maps

PSI04 is problem 38, the PPT squared conjecture of Matthias Christandl. In the form the paper uses: compose any two PPT maps and the result is entanglement breaking, destroying entanglement with any reference system. OpenAI’s preprint dated September 27, result 272 in its catalogue, reported a counterexample in dimension ten; PSI’s page and paper cite it, and our episode 272 covers it. The 4-page note by Turturean and Bao proves a bound instead: on a d-level system, any 2(d − 1) PPT maps, which may differ and need not preserve trace, compose to an entanglement-breaking map. The key step, following a suggestion of Christandl, Müller-Hermes and Wolf, shows that each pair of PPT maps breaks entanglement with a qubit reference; each pair then lowers the Schmidt number by one, from at most d down to one. Is the bound tight? The paper says it is not in dimension three, where two maps already suffice and the bound gives four; whether it is optimal for large d, they leave for future work.

6:16PSI05 · Problem 43: causality and purification

PSI05 is problem 43. In the process matrix framework, two laboratories are linked without a fixed causal order. A process is extensibly causal if it never violates a causal inequality, even with entangled ancillas shared by the labs. It is purifiable if it comes from a larger pure process that turns reversible local operations into a reversible global evolution. The list asks whether every extensibly causal process is purifiable. The 8-page letter by Turturean and Bao says no. It writes down an explicit process on four qubits, a short sum of Pauli terms, and proves it is a valid process of full rank, extensibly causal, but not causally separable, which for two parties, by earlier theorems, means not purifiable. A causal witness gives −1/5; our script reproduces that number and the smallest eigenvalue, about 0.0019. The key tool is a sum of squares identity covering every ancilla at once. The paper says it answers problem 43 in the negative, in its finite-dimensional bipartite form, with the purity notion of Araújo and coauthors; it does not claim the process is unphysical in every sense.

7:36Where they stand

Where they stand, as of October 27. All five are preprints in PSI’s repository, and none says it has been peer reviewed. No source names an independent expert review. Ning Bao, co-author of four and a paid contractor to PSI, reviewed, refereed, verified or commented on each; PSI01 thanks an unnamed referee; PSI03 thanks Ian MacCormack, and PSI04 Adam Levine, for reviewing earlier drafts; and three thank Jim Halverson for coordinating a review, naming no reviewers. PSI says Lean’s kernel checked each formalization. Scope varies: PSI01 covers dimension three only, and PSI03’s Lean also states results its current paper does not claim. Our scan found sorry only in the challenge files. We did not build or run any of it. If they hold: some NPT Werner states survive three copies in every dimension from three up; distant vacuum regions of fixed size never violate CHSH; Fourier-phase measurements are the best projective CGLMP measurements on maximally entangled states; 2(d − 1) PPT maps always break entanglement; and extensible causality does not imply purifiability. Many copies stay open.

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date. AI Proved What? is not affiliated with PSI.