Lean statement

Entropy photon-number inequality for finite-energy bosonic inputs

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Theorem selected for verification: OAI.EntropyPhotonNumber.entropy_photon_number_inequality

Lean statement file: EntropyPhotonNumber.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 273 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 entropy_photon_number_inequality    (n : ℕ) (hn : 1 ≤ n) (ρA ρB ρC : State n)    (hA : FiniteEnergy ρA) (hB : FiniteEnergy ρB)    (η : ℝ) (hη : η ∈ Set.Icc 0 1)    (hC : IsBeamSplitterOutput η ρA ρB ρC) :    η * gInv (entropy ρA / n) + (1 - η) * gInv (entropy ρB / n) ≤      gInv (entropy ρC / n) := 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.

abbreviation OAI.EntropyPhotonNumber.NumberIndex line 12
```
abbrev NumberIndex (n : ℕ) := Fin n → ℕ
```

abbreviation OAI.EntropyPhotonNumber.Fock line 15

abbrev Fock (n : ℕ) := lp (fun _ : NumberIndex n => ℂ) 2

definition OAI.EntropyPhotonNumber.numberKet line 18

def numberKet {n : ℕ} (k : NumberIndex n) : Fock n := lp.single 2 k 1

definition OAI.EntropyPhotonNumber.entry line 21

def entry {n : ℕ} (T : Fock n →L[ℂ] Fock n) (k l : NumberIndex n) : ℂ :=  inner ℂ (numberKet k) (T (numberKet l))

structure OAI.EntropyPhotonNumber.State line 27

structure State (n : ℕ) where  op : Fock n →L[ℂ] Fock n  positive : op.IsPositive  trace_one : HasSum (fun k => (entry op k k).re) 1

definition OAI.EntropyPhotonNumber.totalNumber line 33

def totalNumber {n : ℕ} (k : NumberIndex n) : ℕ := ∑ j, k j

definition OAI.EntropyPhotonNumber.FiniteEnergy line 36

def FiniteEnergy {n : ℕ} (ρ : State n) : Prop :=  Summable (fun k => (totalNumber k : ℝ) * (entry ρ.op k k).re)

definition OAI.EntropyPhotonNumber.entropy line 41

def entropy {n : ℕ} (ρ : State n) : ℝ :=  ∑' k, (entry (cfc (fun z : ℂ => (-(z.re * Real.log z.re) : ℂ)) ρ.op) k k).re

definition OAI.EntropyPhotonNumber.g line 45

def g (t : ℝ) : ℝ := (t + 1) * Real.log (t + 1) - t * Real.log t

definition OAI.EntropyPhotonNumber.gInv line 49

def gInv (s : ℝ) : ℝ := sInf {t : ℝ | 0 ≤ t ∧ s ≤ g t}

definition OAI.EntropyPhotonNumber.oneModeBeamCoefficient line 55

def oneModeBeamCoefficient (η : ℝ) (k e a b : ℕ) : ℝ :=  if k + e = a + b then    Real.sqrt (((k.factorial : ℝ) * e.factorial) / (a.factorial * b.factorial)) *      ∑ p ∈ Finset.range (a + 1), ∑ q ∈ Finset.range (b + 1),        if p + q = k then          (a.choose p : ℝ) * b.choose q *            Real.sqrt η ^ p * (-Real.sqrt (1 - η)) ^ (a - p) *            Real.sqrt (1 - η) ^ q * Real.sqrt η ^ (b - q)        else 0  else 0

definition OAI.EntropyPhotonNumber.beamCoefficient line 67

def beamCoefficient {n : ℕ} (η : ℝ) (k e a b : NumberIndex n) : ℝ :=  ∏ j, oneModeBeamCoefficient η (k j) (e j) (a j) (b j)

abbreviation OAI.EntropyPhotonNumber.SectorSplit line 71

abbrev SectorSplit {n : ℕ} (k e : NumberIndex n) := ∀ j, Fin (k j + e j + 1)

definition OAI.EntropyPhotonNumber.splitLeft line 73

def splitLeft {n : ℕ} {k e : NumberIndex n} (a : SectorSplit k e) : NumberIndex n :=  fun j => a j

definition OAI.EntropyPhotonNumber.splitRight line 76

def splitRight {n : ℕ} {k e : NumberIndex n} (a : SectorSplit k e) : NumberIndex n :=  fun j => k j + e j - a j

definition OAI.EntropyPhotonNumber.outputBlock line 82

def outputBlock {n : ℕ} (η : ℝ) (ρA ρB : State n)    (k l e : NumberIndex n) : ℂ :=  ∑ a : SectorSplit k e, ∑ b : SectorSplit l e,    (beamCoefficient η k e (splitLeft a) (splitRight a) : ℂ) *      entry ρA.op (splitLeft a) (splitLeft b) *      entry ρB.op (splitRight a) (splitRight b) *      (beamCoefficient η l e (splitLeft b) (splitRight b) : ℂ)

definition OAI.EntropyPhotonNumber.IsBeamSplitterOutput line 94

def IsBeamSplitterOutput {n : ℕ} (η : ℝ) (ρA ρB ρC : State n) : Prop :=  ∀ k l, HasSum (outputBlock η ρA ρB k l) (entry ρC.op k l)

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: inequality, photon, reduced, beam, passive, operators, splitter, bosonic and ten more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: eta and rho.

Theorem 1.1 of The entropy photon-number inequality

Theorem (Entropy photon-number inequality)

Let n\ge1 be finite, and let \rho_A,\rho_B be states on n bosonic modes, with annihilation operators a_1,\ldots,a_n and b_1,\ldots,b_n, respectively, satisfying

\Tr\rho_A\sum_{j=1}^n a_j^\dagger a_j<\infty, \qquad \Tr\rho_B\sum_{j=1}^n b_j^\dagger b_j<\infty.

The joint input is \rho_A\otimes\rho_B; no independence assumption is made among the modes within either state. For 0\le\eta\le1, let \rho_C be the reduced state of the modes

c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad 1\le j\le n,

after passive beam-splitter mixing. Then

g^{-1}\!\left(\frac{S(\rho_C)}n\right) \ge \eta\,g^{-1}\!\left(\frac{S(\rho_A)}n\right) +(1-\eta)\,g^{-1}\!\left(\frac{S(\rho_B)}n\right),

where S(\rho)=-\Tr\rho\log\rho.

Section 1, Introduction Read it in the paper TeX source, line 10

Show the TeX

Let \(n\ge1\) be finite, and let \(\rho_A,\rho_B\) be states on \(n\) bosonic modes, with annihilation operators \(a_1,\ldots,a_n\) and \(b_1,\ldots,b_n\), respectively, satisfying
\[
 \Tr\rho_A\sum_{j=1}^n a_j^\dagger a_j<\infty,
 \qquad
 \Tr\rho_B\sum_{j=1}^n b_j^\dagger b_j<\infty.
\]
The joint input is \(\rho_A\otimes\rho_B\); no independence assumption is made among the modes within either state. For \(0\le\eta\le1\), let \(\rho_C\) be the reduced state of the modes
\[
 c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad 1\le j\le n,
\]
after passive beam-splitter mixing. Then
\begin{equation}
 g^{-1}\!\left(\frac{S(\rho_C)}n\right)
 \ge \eta\,g^{-1}\!\left(\frac{S(\rho_A)}n\right)
 +(1-\eta)\,g^{-1}\!\left(\frac{S(\rho_B)}n\right),
\end{equation}
where \(S(\rho)=-\Tr\rho\log\rho\).

Other possible paper matches, closest wording first

Proposition 4.2 of The entropy photon-number inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: real, energies, functional, value, two, output and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: rho.
Proposition 2.3 of The entropy photon-number inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: mode, ports, thermal, mixing, sqrt, independent and two more.
Corollary 7.1 of The entropy photon-number inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: pairs, natural, per, nonnegative, two and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: eta.
Corollary 1.2 of The entropy photon-number inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: thermal, tr, entropy, finite, independent, modes and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: eta and rho.
Proposition 4.3 of The entropy photon-number inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: extension, spaces, output and bounded.
Proposition 5.1 of The entropy photon-number inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: positive, independent and states.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 381 s; peak memory 6.6 GiB).
Reported 10 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 273’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.

MathVet and cert-machine disagree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: entropy_photon_number_inequality (n) (hn : 1 ≤ n) (ρA ρB ρC : State n) (hA : FiniteEnergy ρA) (hB : FiniteEnergy ρB) (η ∈ [0,1]) (hC : IsBeamSplitterOutput η ρA ρB ρC) : η * gInv (S ρA / n) + (1 - η) * gInv (S ρB / n) ≤ gInv (S ρC / n) on the untruncated Fock space ℓ²(ℕⁿ) with arbitrary (entangled) inputs and gInv the inverse of the thermal entropy g t = (t+1) log(t+1) - t log t. The summary's remark that product thermal inputs attain equality is not separately stated. Caveat: the output state is specified only by the hypothesis IsBeamSplitterOutput (explicit matrix-element formula), so the theorem is vacuous if that formula were wrong or unsatisfiable.
Definitions the group asks readers to compare with the paper: Hand-written quantum optics: State n = positive operator with HasSum diagonal 1 on lp ℂ 2 over ℕⁿ, entropy ρ := ∑' k, (entry (cfc (-t log t) ρ.op) k k).re (real tsum, junk 0 if not summable; finite energy ensures it is), gInv s := sInf {t ≥ 0 | s ≤ g t}, and the beam-splitter output defined by explicit coefficients oneModeBeamCoefficient η k e a b (binomial/factorial formula) and IsBeamSplitterOutput: ∀ k l, HasSum (outputBlock ..) (entry ρC k l).
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: The equality (sharpness) clause, that product thermal inputs attain equality, is not stated. It is a direct computation rather than a hard theorem, but it is a headline claim.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the entropy photon-number inequality for two independent bosonic inputs with finite mean energy and any finite positive number $n$ of modes. Write $N(\rho)=g^{-1}(S(\rho)/n)$, where $S$ is von Neumann entropy and $g(x)=(x+1)\log(x+1)-x\log x$ is the thermal entropy per mode. For a beam splitter of transmissivity $0\le\eta\le1$, its output satisfies $N(\rho_C)\ge\eta N(\rho_A)+(1-\eta)N(\rho_B)$.”
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

112 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: 56 to its own declarations, 21 to Mathlib’s documentation. 18 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 entropy_photon_number_inequality 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 all 17 other declarations of the file; with the theorem itself they span 50 lines. 5 of them are named in the statement itself.

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

abbrev NumberIndexline 12library
abbrev Fockline 15library
def numberKetline 18library
def entryline 21library
structure Stateline 27in the statementlibrary
def totalNumberline 33library
def FiniteEnergyline 36in the statementlibrary
def entropyline 41in the statementlibrary
def gline 45library
def gInvline 49in the statementlibrary
def oneModeBeamCoefficientline 55library
def beamCoefficientline 67library
abbrev SectorSplitline 71library
def splitLeftline 73library
def splitRightline 76library
def outputBlockline 82library
def IsBeamSplitterOutputline 94in the statementlibrary
theorem entropy_photon_number_inequalityline 99selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperators ComplexConjugatenamespace EntropyPhotonNumber/-- The number-basis indices for `n` bosonic modes. -/abbrev NumberIndex (n : ℕ) := Fin n → ℕ/-- The full, untruncated bosonic Fock space `ℓ²(ℕⁿ, ℂ)`. -/abbrev Fock (n : ℕ) := lp (fun _ : NumberIndex n => ℂ) 2/-- A number-basis vector. -/def numberKet {n : ℕ} (k : NumberIndex n) : Fock n := lp.single 2 k 1/-- Matrix entries of an operator in the number basis. -/def entry {n : ℕ} (T : Fock n →L[ℂ] Fock n) (k l : NumberIndex n) : ℂ :=  inner ℂ (numberKet k) (T (numberKet l))/-- Positive trace-one operators. For a positive bounded operator on thisHilbert space, summability of its diagonal is the positive trace-classcriterion; `HasSum ... 1` includes summability as well as normalization. -/structure State (n : ℕ) where  op : Fock n →L[ℂ] Fock n  positive : op.IsPositive  trace_one : HasSum (fun k => (entry op k k).re) 1/-- Total number of quanta in a number-basis vector. -/def totalNumber {n : ℕ} (k : NumberIndex n) : ℕ := ∑ j, k j/-- Exactly finiteness of the expectation of the nonnegative number operator. -/def FiniteEnergy {n : ℕ} (ρ : State n) : Prop :=  Summable (fun k => (totalNumber k : ℝ) * (entry ρ.op k k).re)/-- `−Tr ρ log ρ`, using the continuous extension of `−t log t` at zeroand continuous functional calculus for the positive operator `ρ`. -/def entropy {n : ℕ} (ρ : State n) : ℝ :=  ∑' k, (entry (cfc (fun z : ℂ => (-(z.re * Real.log z.re) : ℂ)) ρ.op) k k).re/-- The thermal entropy per mode, with the convention `0 log 0 = 0`. -/def g (t : ℝ) : ℝ := (t + 1) * Real.log (t + 1) - t * Real.log t/-- The inverse of increasing `g : [0,∞) → [0,∞)`, expressed by its lowerlevel cut. Only its values on `[0,∞)` occur in the main theorem. -/def gInv (s : ℝ) : ℝ := sInf {t : ℝ | 0 ≤ t ∧ s ≤ g t}/-- One-mode passive beam-splitter coefficient, in number bases. It is thecoefficient of `(c†)^k (d†)^e` in`(√η c† − √(1−η) d†)^a (√(1−η) c† + √η d†)^b`,with the normalized number-vector factorials. -/def oneModeBeamCoefficient (η : ℝ) (k e a b : ℕ) : ℝ :=  if k + e = a + b then    Real.sqrt (((k.factorial : ℝ) * e.factorial) / (a.factorial * b.factorial)) *      ∑ p ∈ Finset.range (a + 1), ∑ q ∈ Finset.range (b + 1),        if p + q = k then          (a.choose p : ℝ) * b.choose q *            Real.sqrt η ^ p * (-Real.sqrt (1 - η)) ^ (a - p) *            Real.sqrt (1 - η) ^ q * Real.sqrt η ^ (b - q)        else 0  else 0/-- The same passive rotation is performed on all mode indices. -/def beamCoefficient {n : ℕ} (η : ℝ) (k e a b : NumberIndex n) : ℝ :=  ∏ j, oneModeBeamCoefficient η (k j) (e j) (a j) (b j)/-- Choices of the first input number in the total-number sector of `(k,e)`. -/abbrev SectorSplit {n : ℕ} (k e : NumberIndex n) := ∀ j, Fin (k j + e j + 1)def splitLeft {n : ℕ} {k e : NumberIndex n} (a : SectorSplit k e) : NumberIndex n :=  fun j => a jdef splitRight {n : ℕ} {k e : NumberIndex n} (a : SectorSplit k e) : NumberIndex n :=  fun j => k j + e j - a j/-- A single discarded-number diagonal block of `U(ρA ⊗ ρB)U†`.The sums are finite because passive mixing preserves number in each pairedmode. The two unrestricted density matrices allow all internal entanglement. -/def outputBlock {n : ℕ} (η : ℝ) (ρA ρB : State n)    (k l e : NumberIndex n) : ℂ :=  ∑ a : SectorSplit k e, ∑ b : SectorSplit l e,    (beamCoefficient η k e (splitLeft a) (splitRight a) : ℂ) *      entry ρA.op (splitLeft a) (splitLeft b) *      entry ρB.op (splitRight a) (splitRight b) *      (beamCoefficient η l e (splitLeft b) (splitRight b) : ℂ)/-- Reduced state after passive beam-splitter mixing of independent ports.The second row of the orthogonal rotation is `(-√(1−η),√η)`; changing itssign has no effect on this reduced state. `HasSum` records the actual partialtrace, not a default value for a possibly divergent series. -/def IsBeamSplitterOutput {n : ℕ} (η : ℝ) (ρA ρB ρC : State n) : Prop :=  ∀ k l, HasSum (outputBlock η ρA ρB k l) (entry ρC.op k l)/-- The entropy photon-number inequality for arbitrary finite-energy inputsand any positive finite number of modes. -/theorem entropy_photon_number_inequality    (n : ℕ) (hn : 1 ≤ n) (ρA ρB ρC : State n)    (hA : FiniteEnergy ρA) (hB : FiniteEnergy ρB)    (η : ℝ) (hη : η ∈ Set.Icc 0 1)    (hC : IsBeamSplitterOutput η ρA ρB ρC) :    η * gInv (entropy ρA / n) + (1 - η) * gInv (entropy ρB / n) ≤      gInv (entropy ρC / n) := by  sorryend EntropyPhotonNumberendend 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.

The entropy photon-number inequality

Theorem (Entropy photon-number inequality)

Let n\ge1 be finite, and let \rho_A,\rho_B be states on n bosonic modes, with annihilation operators a_1,\ldots,a_n and b_1,\ldots,b_n, respectively, satisfying

\Tr\rho_A\sum_{j=1}^n a_j^\dagger a_j<\infty, \qquad \Tr\rho_B\sum_{j=1}^n b_j^\dagger b_j<\infty.

The joint input is \rho_A\otimes\rho_B; no independence assumption is made among the modes within either state. For 0\le\eta\le1, let \rho_C be the reduced state of the modes

c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad 1\le j\le n,

after passive beam-splitter mixing. Then

g^{-1}\!\left(\frac{S(\rho_C)}n\right) \ge \eta\,g^{-1}\!\left(\frac{S(\rho_A)}n\right) +(1-\eta)\,g^{-1}\!\left(\frac{S(\rho_B)}n\right),

where S(\rho)=-\Tr\rho\log\rho.

Section 1, Introduction Read it in the paper TeX source, line 10

Show the TeX

Let \(n\ge1\) be finite, and let \(\rho_A,\rho_B\) be states on \(n\) bosonic modes, with annihilation operators \(a_1,\ldots,a_n\) and \(b_1,\ldots,b_n\), respectively, satisfying
\[
 \Tr\rho_A\sum_{j=1}^n a_j^\dagger a_j<\infty,
 \qquad
 \Tr\rho_B\sum_{j=1}^n b_j^\dagger b_j<\infty.
\]
The joint input is \(\rho_A\otimes\rho_B\); no independence assumption is made among the modes within either state. For \(0\le\eta\le1\), let \(\rho_C\) be the reduced state of the modes
\[
 c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad 1\le j\le n,
\]
after passive beam-splitter mixing. Then
\begin{equation}
 g^{-1}\!\left(\frac{S(\rho_C)}n\right)
 \ge \eta\,g^{-1}\!\left(\frac{S(\rho_A)}n\right)
 +(1-\eta)\,g^{-1}\!\left(\frac{S(\rho_B)}n\right),
\end{equation}
where \(S(\rho)=-\Tr\rho\log\rho\).

Corollary (Constrained entropy minimum for identical thermal attenuators)

Let n\ge1 be finite, let s,N_B\ge0 be finite, and let 0\le\eta\le1. Every state \rho on n modes with \Tr\rho\sum_{j=1}^n a_j^\dagger a_j<\infty satisfies

\frac1n S\!\left(\mathcal E_{\eta,N_B}^{\otimes n}(\rho)\right) \ge g\!\left(\eta g^{-1}\!\left(\frac{S(\rho)}n\right) +(1-\eta)N_B\right).

In particular, the exact constrained minimum is

\begin{gathered}\min_{\substack{\rho\text{ a state on }n\text{ modes}\\ \Tr\rho\sum_{j=1}^n a_j^\dagger a_j<\infty\\ S(\rho)=ns}} \frac1n S\!\left(\mathcal E_{\eta,N_B}^{\otimes n}(\rho)\right) =g\!\left(\eta g^{-1}(s)+(1-\eta)N_B\right).\end{gathered}

It is attained by the product thermal input \tau_{g^{-1}(s)}. No independence assumption is imposed among the modes of \rho; only the environment is required to be independent of that input.

Section 1, Introduction Read it in the paper TeX source, line 34

Show the TeX

Let \(n\ge1\) be finite, let \(s,N_B\ge0\) be finite, and let \(0\le\eta\le1\). Every state \(\rho\) on \(n\) modes with
\(\Tr\rho\sum_{j=1}^n a_j^\dagger a_j<\infty\) satisfies
\begin{equation}
 \frac1n S\!\left(\mathcal E_{\eta,N_B}^{\otimes n}(\rho)\right)
 \ge g\!\left(\eta g^{-1}\!\left(\frac{S(\rho)}n\right)
                   +(1-\eta)N_B\right).
\end{equation}
In particular, the exact constrained minimum is
\begin{equation}
 \min_{\substack{\rho\text{ a state on }n\text{ modes}\\
                  \Tr\rho\sum_{j=1}^n a_j^\dagger a_j<\infty\\
                  S(\rho)=ns}}
 \frac1n S\!\left(\mathcal E_{\eta,N_B}^{\otimes n}(\rho)\right)
 =g\!\left(\eta g^{-1}(s)+(1-\eta)N_B\right).
\end{equation}
It is attained by the product thermal input \(\tau_{g^{-1}(s)}\). No independence assumption is imposed among the modes of \(\rho\); only the environment is required to be independent of that input.

Configuration and file facts

File: lean/ComparatorChallenges/EntropyPhotonNumber.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 273 link it.
Solution module: OAI.InformationTheory.PhotonNumber.Inequality
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: 17 local vocabulary declarations
Lines of Lean, without comments or blank lines: 50
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.

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