Lean statement

Dimension-ten PPT pair with non-entanglement-breaking composition

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.DimensionTen.main_pair

Lean statement file: DimensionTenPair.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 272 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main_pair :    ∃ Φ₁ Φ₂ : Mat 10 → Mat 10,      Φ₁ = phiOne ∧ Φ₂ = phiTwo ∧ PPT Φ₁ ∧ PPT Φ₂ ∧      (let Z := choi (Φ₂ ∘ Φ₁)       Z ≠ 0 ∧       (∀ u v : Fin 10 → ℂ, ∀ w : Fin 10 × Fin 10 → ℂ,         Z *ᵥ w = productVector u v → u = 0 ∨ v = 0) ∧       ¬ EntanglementBreaking (Φ₂ ∘ Φ₁)) := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

definition OAI.DimensionTen.blocksZ line 10

def blocksZ : Fin 4 → Matrix (Fin 6) (Fin 4) ℤ :=  ![ ![ ![6,0,0,0], ![0,6,0,0], ![0,0,6,0], ![0,0,0,6], ![0,0,0,0], ![0,0,0,0] ],     ![ ![-12,0,0,0], ![0,-6,0,0], ![0,0,6,0], ![0,0,0,12], ![-6,0,6,6], ![-6,-6,6,0] ],     ![ ![0,6,-2,0], ![6,6,0,2], ![-2,0,10,0], ![0,2,0,24], ![0,0,0,6], ![0,-6,6,-6] ],     ![ ![0,0,-4,-3], ![0,-2,-3,-4], ![-4,-3,11,6], ![-3,-4,6,25], ![6,6,0,0], ![0,-6,6,6] ] ]

definition OAI.DimensionTen.blocks line 16

def blocks (i : Fin 4) : Matrix (Fin 6) (Fin 4) ℂ :=  (blocksZ i).map (Int.castRingHom ℂ)

definition OAI.DimensionTen.pencilMap line 19

def pencilMap (A : Matrix (Fin 4) (Fin 4) ℂ) : Matrix (Fin 4) (Fin 4) ℂ :=  ∑ i : Fin 4, ∑ j : Fin 4, A i j • ((blocks i)ᵀ * blocks j)

abbreviation OAI.DimensionTen.Mat line 22

abbrev Mat (n : ℕ) := Matrix (Fin n) (Fin n) ℂ

definition OAI.DimensionTen.IsComplexLinear line 24

def IsComplexLinear {a b : ℕ} (F : Mat a → Mat b) : Prop :=  (∀ A B, F (A + B) = F A + F B) ∧  (∀ (c : ℂ) A, F (c • A) = c • F A)

definition OAI.DimensionTen.amplify line 28

def amplify {a b : ℕ} (F : Mat a → Mat b) (k : ℕ)    (X : Matrix (Fin k × Fin a) (Fin k × Fin a) ℂ) :    Matrix (Fin k × Fin b) (Fin k × Fin b) ℂ :=  fun u v => F (fun i j => X (u.1, i) (v.1, j)) u.2 v.2

definition OAI.DimensionTen.CompletelyPositive line 33

def CompletelyPositive {a b : ℕ} (F : Mat a → Mat b) : Prop :=  ∀ k : ℕ, 0 < k → ∀ X : Matrix (Fin k × Fin a) (Fin k × Fin a) ℂ,    X.PosSemidef → (amplify F k X).PosSemidef

definition OAI.DimensionTen.PPT line 37

def PPT {a b : ℕ} (F : Mat a → Mat b) : Prop :=  IsComplexLinear F ∧ CompletelyPositive F ∧    CompletelyPositive (fun X => (F X)ᵀ)

definition OAI.DimensionTen.tensorMap line 55

def tensorMap {a b c d : ℕ} (F : Mat a → Mat b) (G : Mat c → Mat d)    (X : Matrix (Fin a × Fin c) (Fin a × Fin c) ℂ) :    Matrix (Fin b × Fin d) (Fin b × Fin d) ℂ :=  ∑ i : Fin c, ∑ j : Fin c,    Matrix.kronecker (F (fun a b => X (a, i) (b, j))) (G (Matrix.single i j 1))

definition OAI.DimensionTen.hsAdjoint line 61

def hsAdjoint {a b : ℕ} (F : Mat a → Mat b) (Y : Mat b) : Mat a :=  fun i j => ∑ u : Fin b, ∑ v : Fin b, star (F (Matrix.single i j 1) u v) * Y u v

definition OAI.DimensionTen.symmetricPairs line 64

def symmetricPairs : Fin 10 → Fin 4 × Fin 4 :=  ![(0,0), (0,1), (0,2), (0,3), (1,1), (1,2), (1,3), (2,2), (2,3), (3,3)]

definition OAI.DimensionTen.exteriorPairs line 67

def exteriorPairs : Fin 6 → Fin 4 × Fin 4 :=  ![(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)]

definition OAI.DimensionTen.symmetricIsometry line 70

def symmetricIsometry : Matrix (Fin 4 × Fin 4) (Fin 10) ℂ :=  fun ij a =>    let p := (symmetricPairs a).1    let q := (symmetricPairs a).2    if p = q then (if ij = (p,p) then 1 else 0)    else ((if ij = (p,q) then 1 else 0) + (if ij = (q,p) then 1 else 0)) /      (Real.sqrt 2 : ℂ)

definition OAI.DimensionTen.exteriorIsometry line 78

def exteriorIsometry : Matrix (Fin 4 × Fin 4) (Fin 6) ℂ :=  fun ij a =>    let p := (exteriorPairs a).1    let q := (exteriorPairs a).2    ((if ij = (p,q) then 1 else 0) - (if ij = (q,p) then 1 else 0)) /      (Real.sqrt 2 : ℂ)

definition OAI.DimensionTen.hodgeComplement line 85

def hodgeComplement : Mat 6 :=  ![ ![0,0,0,0,0,1], ![0,0,0,0,-1,0], ![0,0,0,1,0,0],     ![0,0,1,0,0,0], ![0,-1,0,0,0,0], ![1,0,0,0,0,0] ]

definition OAI.DimensionTen.firstSix line 89

def firstSix : Matrix (Fin 10) (Fin 6) ℂ :=  fun i j => if i.val = j.val then 1 else 0

definition OAI.DimensionTen.exteriorMap line 92

def exteriorMap (A : Mat 10) : Mat 6 :=  exteriorIsometryᴴ *    tensorMap pencilMap pencilMap (symmetricIsometry * A * symmetricIsometryᴴ) *      exteriorIsometry

definition OAI.DimensionTen.complementaryMap line 97

def complementaryMap (A : Mat 10) : Mat 6 :=  hodgeComplement * (exteriorMap A)ᵀ * hodgeComplementᴴ

definition OAI.DimensionTen.phiOne line 100

def phiOne (A : Mat 10) : Mat 10 :=  firstSix * exteriorMap Aᵀ * firstSixᴴ

definition OAI.DimensionTen.phiTwo line 103

def phiTwo (B : Mat 10) : Mat 10 :=  hsAdjoint complementaryMap (firstSixᴴ * B * firstSix)

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: breaking, composition, entanglement, explicitly, maps, nonzero, product, range and two more.

Theorem 1.2 of Entanglement with zero distillable secret key in local dimension ten

Theorem

There are explicitly specified PPT maps \Phi_1,\Phi_2:\Mat{10}\to\Mat{10} for which

Z=J(\Phi_2\circ\Phi_1)

is nonzero and its range contains no nonzero product vector in \C^{10}\otimes\C^{10}. In particular, the composition \Phi_2\circ\Phi_1 is not entanglement breaking.

Section 1, Introduction Read it in the paper TeX source, line 80

Show the TeX

There are explicitly specified PPT maps
$\Phi_1,\Phi_2:\Mat{10}\to\Mat{10}$ for which
\[
 Z=J(\Phi_2\circ\Phi_1)
\]
is nonzero and its range contains no nonzero product vector in
$\C^{10}\otimes\C^{10}$.  In particular, the composition
$\Phi_2\circ\Phi_1$ is not entanglement breaking.

Other possible paper matches, closest wording first

Proposition 9.1 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: adjoint, ij, linear, integers, complex, matrix and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: vectors.
Theorem 10.5 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: pair, first, sqrt, matrix, positive, isometry and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: choi, consequences, composition, vectors, breaking and entanglement.
Corollary 9.2 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric, sqrt, complex, one, phi, real and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: vector.
Proposition 6.1 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: tensor, positive, phi, maps, ppt and mat. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonzero, explicitly, breaking and entanglement.
Corollary 9.3 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: first, isometry, real, mat and ppt. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: composition, breaking and entanglement.
Proposition 8.1 of Entanglement with zero distillable secret key in local dimension ten. Words of the Lean statement, its names split into words, that also appear in the theorem: isometry, real, maps, ppt and mat. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: explicitly, breaking and entanglement.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 272’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Stated: DimensionTenChannel.exists_channel_fin21 : ∃ Θ : Map (Fin 21) (Fin 21), PPT Θ ∧ TracePreserving Θ ∧ ¬ EntanglementBreaking (Θ.comp Θ) (the PPT-square counterexample with PPT F := CP F ∧ CP (T ∘ F) and EntanglementBreaking via separability of every amplification) and DimensionTenPair.main_pair (two explicit PPT maps on 10x10 matrices whose composition is not entanglement breaking and whose Choi matrix has no nonzero product vector in its range). Not stated: the summary's first claim, an entangled state on ℂ¹⁰⊗ℂ¹⁰ with zero distillable secret key under the specified local-instrument/authenticated-communication protocols (docs: 'outside these two selected Comparator statements').
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The zero-distillable-secret-key state (the headline's lead claim).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The linked formalization covers these consequences: two explicitly specified PPT maps on $10\times10$ complex matrices have a composition that is not entanglement breaking, and the nonzero Choi matrix of that composition has no nonzero product vector in its range.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

125 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 74 to its own declarations, 103 to Mathlib’s documentation. 26 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main_pair exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 24 of the file’s 25 other declarations; with the theorem itself they span 84 lines. 7 of them are named in the statement itself.

Names in the namespace OAI.DimensionTen are listed without it.

def blocksZline 10library
def blocksline 16library
def pencilMapline 19library
abbrev Matline 22in the statementlibrary
def IsComplexLinearline 24library
def amplifyline 28library
def CompletelyPositiveline 33library
def PPTline 37in the statementlibrary
def separableline 41library
def EntanglementBreakingline 46in the statementlibrary
def choiline 51in the statementlibrary
def tensorMapline 55library
def hsAdjointline 61library
def symmetricPairsline 64library
def exteriorPairsline 67library
def symmetricIsometryline 70library
def exteriorIsometryline 78library
def hodgeComplementline 85library
def firstSixline 89library
def exteriorMapline 92library
def complementaryMapline 97library
def phiOneline 100in the statementlibrary
def phiTwoline 103in the statementlibrary
def productVectorline 109in the statementlibrary
theorem main_pairline 112selected theoremlibrary

1 declaration the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

def compositeChoi line 106

import Mathlibnamespace OAInoncomputable sectionopen scoped Matrix ComplexOrderopen Matrix Complexnamespace DimensionTendef blocksZ : Fin 4 → Matrix (Fin 6) (Fin 4) ℤ :=  ![ ![ ![6,0,0,0], ![0,6,0,0], ![0,0,6,0], ![0,0,0,6], ![0,0,0,0], ![0,0,0,0] ],     ![ ![-12,0,0,0], ![0,-6,0,0], ![0,0,6,0], ![0,0,0,12], ![-6,0,6,6], ![-6,-6,6,0] ],     ![ ![0,6,-2,0], ![6,6,0,2], ![-2,0,10,0], ![0,2,0,24], ![0,0,0,6], ![0,-6,6,-6] ],     ![ ![0,0,-4,-3], ![0,-2,-3,-4], ![-4,-3,11,6], ![-3,-4,6,25], ![6,6,0,0], ![0,-6,6,6] ] ]def blocks (i : Fin 4) : Matrix (Fin 6) (Fin 4) ℂ :=  (blocksZ i).map (Int.castRingHom ℂ)def pencilMap (A : Matrix (Fin 4) (Fin 4) ℂ) : Matrix (Fin 4) (Fin 4) ℂ :=  ∑ i : Fin 4, ∑ j : Fin 4, A i j • ((blocks i)ᵀ * blocks j)abbrev Mat (n : ℕ) := Matrix (Fin n) (Fin n) ℂdef IsComplexLinear {a b : ℕ} (F : Mat a → Mat b) : Prop :=  (∀ A B, F (A + B) = F A + F B) ∧  (∀ (c : ℂ) A, F (c • A) = c • F A)def amplify {a b : ℕ} (F : Mat a → Mat b) (k : ℕ)    (X : Matrix (Fin k × Fin a) (Fin k × Fin a) ℂ) :    Matrix (Fin k × Fin b) (Fin k × Fin b) ℂ :=  fun u v => F (fun i j => X (u.1, i) (v.1, j)) u.2 v.2def CompletelyPositive {a b : ℕ} (F : Mat a → Mat b) : Prop :=  ∀ k : ℕ, 0 < k → ∀ X : Matrix (Fin k × Fin a) (Fin k × Fin a) ℂ,    X.PosSemidef → (amplify F k X).PosSemidefdef PPT {a b : ℕ} (F : Mat a → Mat b) : Prop :=  IsComplexLinear F ∧ CompletelyPositive F ∧    CompletelyPositive (fun X => (F X)ᵀ)def separable {a b : ℕ} (X : Matrix (Fin a × Fin b) (Fin a × Fin b) ℂ) : Prop :=  ∃ r : ℕ, ∃ A : Fin r → Mat a, ∃ B : Fin r → Mat b,    (∀ i, (A i).PosSemidef ∧ (B i).PosSemidef) ∧    X = ∑ i, Matrix.kronecker (A i) (B i)def EntanglementBreaking {a b : ℕ} (F : Mat a → Mat b) : Prop :=  CompletelyPositive F ∧  ∀ k : ℕ, 0 < k → ∀ X : Matrix (Fin k × Fin a) (Fin k × Fin a) ℂ,    X.PosSemidef → separable (amplify F k X)def choi {a b : ℕ} (F : Mat a → Mat b) :    Matrix (Fin a × Fin b) (Fin a × Fin b) ℂ :=  fun u v => F (Matrix.single u.1 v.1 1) u.2 v.2def tensorMap {a b c d : ℕ} (F : Mat a → Mat b) (G : Mat c → Mat d)    (X : Matrix (Fin a × Fin c) (Fin a × Fin c) ℂ) :    Matrix (Fin b × Fin d) (Fin b × Fin d) ℂ :=  ∑ i : Fin c, ∑ j : Fin c,    Matrix.kronecker (F (fun a b => X (a, i) (b, j))) (G (Matrix.single i j 1))def hsAdjoint {a b : ℕ} (F : Mat a → Mat b) (Y : Mat b) : Mat a :=  fun i j => ∑ u : Fin b, ∑ v : Fin b, star (F (Matrix.single i j 1) u v) * Y u vdef symmetricPairs : Fin 10 → Fin 4 × Fin 4 :=  ![(0,0), (0,1), (0,2), (0,3), (1,1), (1,2), (1,3), (2,2), (2,3), (3,3)]def exteriorPairs : Fin 6 → Fin 4 × Fin 4 :=  ![(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)]def symmetricIsometry : Matrix (Fin 4 × Fin 4) (Fin 10) ℂ :=  fun ij a =>    let p := (symmetricPairs a).1    let q := (symmetricPairs a).2    if p = q then (if ij = (p,p) then 1 else 0)    else ((if ij = (p,q) then 1 else 0) + (if ij = (q,p) then 1 else 0)) /      (Real.sqrt 2 : ℂ)def exteriorIsometry : Matrix (Fin 4 × Fin 4) (Fin 6) ℂ :=  fun ij a =>    let p := (exteriorPairs a).1    let q := (exteriorPairs a).2    ((if ij = (p,q) then 1 else 0) - (if ij = (q,p) then 1 else 0)) /      (Real.sqrt 2 : ℂ)def hodgeComplement : Mat 6 :=  ![ ![0,0,0,0,0,1], ![0,0,0,0,-1,0], ![0,0,0,1,0,0],     ![0,0,1,0,0,0], ![0,-1,0,0,0,0], ![1,0,0,0,0,0] ]def firstSix : Matrix (Fin 10) (Fin 6) ℂ :=  fun i j => if i.val = j.val then 1 else 0def exteriorMap (A : Mat 10) : Mat 6 :=  exteriorIsometryᴴ *    tensorMap pencilMap pencilMap (symmetricIsometry * A * symmetricIsometryᴴ) *      exteriorIsometrydef complementaryMap (A : Mat 10) : Mat 6 :=  hodgeComplement * (exteriorMap A)ᵀ * hodgeComplementᴴdef phiOne (A : Mat 10) : Mat 10 :=  firstSix * exteriorMap Aᵀ * firstSixᴴdef phiTwo (B : Mat 10) : Mat 10 :=  hsAdjoint complementaryMap (firstSixᴴ * B * firstSix)def compositeChoi : Matrix (Fin 10 × Fin 10) (Fin 10 × Fin 10) ℂ :=  choi (phiTwo ∘ phiOne)def productVector {a b : ℕ} (u : Fin a → ℂ) (v : Fin b → ℂ) : Fin a × Fin b → ℂ :=  fun ij => u ij.1 * v ij.2theorem main_pair :    ∃ Φ₁ Φ₂ : Mat 10 → Mat 10,      Φ₁ = phiOne ∧ Φ₂ = phiTwo ∧ PPT Φ₁ ∧ PPT Φ₂ ∧      (let Z := choi (Φ₂ ∘ Φ₁)       Z ≠ 0 ∧       (∀ u v : Fin 10 → ℂ, ∀ w : Fin 10 × Fin 10 → ℂ,         Z *ᵥ w = productVector u v → u = 0 ∨ v = 0) ∧       ¬ EntanglementBreaking (Φ₂ ∘ Φ₁)) := by  sorryend DimensionTenendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Entanglement with zero distillable secret key in local dimension ten

Theorem

There is an explicitly specified entangled density operator \rho on \C^{10}\otimes\C^{10} whose range contains no nonzero product vector and for which K_D(\rho)=0. More precisely, let n\geq1 and let \tau_{K_AK_BE'} be the normalized output of any admitted completed bit-output protocol of Definition def:protocol on \rho^{\otimes n}, where E' retains an initial purification and the complete public transcript. For every state \sigma_{E'},

\left\lVert \tau_{K_AK_BE'}- \frac12\sum_{i=0}^1|i,i\rangle\langle i,i|\otimes\sigma_{E'} \right\rVert_1\geq\frac15.

Thus the corresponding trace distance, with its conventional factor 1/2, is at least 1/10.

Section 1, Introduction Read it in the paper TeX source, line 45

Show the TeX

There is an explicitly specified entangled density operator $\rho$ on
$\C^{10}\otimes\C^{10}$ whose range contains no nonzero product vector and
for which $K_D(\rho)=0$.  More precisely, let $n\geq1$ and let
$\tau_{K_AK_BE'}$ be the normalized output of any admitted completed bit-output protocol of
Definition~\ref{def:protocol} on $\rho^{\otimes n}$, where $E'$ retains an
initial purification and the complete public transcript.  For every state
$\sigma_{E'}$,
\begin{equation}
 \left\lVert \tau_{K_AK_BE'}-
 \frac12\sum_{i=0}^1|i,i\rangle\langle i,i|\otimes\sigma_{E'}
 \right\rVert_1\geq\frac15.
\end{equation}
Thus the corresponding trace distance, with its conventional factor
$1/2$, is at least $1/10$.

Theorem

There are explicitly specified PPT maps \Phi_1,\Phi_2:\Mat{10}\to\Mat{10} for which

Z=J(\Phi_2\circ\Phi_1)

is nonzero and its range contains no nonzero product vector in \C^{10}\otimes\C^{10}. In particular, the composition \Phi_2\circ\Phi_1 is not entanglement breaking.

Section 1, Introduction Read it in the paper TeX source, line 80

Show the TeX

There are explicitly specified PPT maps
$\Phi_1,\Phi_2:\Mat{10}\to\Mat{10}$ for which
\[
 Z=J(\Phi_2\circ\Phi_1)
\]
is nonzero and its range contains no nonzero product vector in
$\C^{10}\otimes\C^{10}$.  In particular, the composition
$\Phi_2\circ\Phi_1$ is not entanglement breaking.

Theorem

There is an explicitly specified trace-preserving PPT channel \Theta:\Mat{21}\to\Mat{21} whose square \Theta\circ\Theta is not entanglement breaking.

Section 1, Introduction Read it in the paper TeX source, line 101

Show the TeX

There is an explicitly specified trace-preserving PPT channel
$\Theta:\Mat{21}\to\Mat{21}$ whose square $\Theta\circ\Theta$ is not
entanglement breaking.

Configuration and file facts

File: lean/ComparatorChallenges/DimensionTenPair.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 272 link it.
Solution module: OAI.Analysis.Quantum.DimensionTen.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 25 local vocabulary declarations
Lines of Lean, without comments or blank lines: 86
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas