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Lean statement

Additivity and capacity for generalized amplitude damping

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.GAD.main

Lean statement file: AmplitudeDamping.leanShow on the mapFamily 276. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theoremWho ran the proof-verification programReport 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 (γ ν : ℝ) (hγ : γ ∈ Set.Icc (0 : ℝ) 1)    (hν : ν ∈ Set.Icc (0 : ℝ) 1) :    (∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) * holevo γ ν 1) ∧    (∃ p : ℝ, Maximizes γ ν p ∧      ∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) / Real.log 2 * objective γ ν p) ∧    capacity γ ν = holevo γ ν 1 ∧    (∀ p : ℝ, Maximizes γ ν p → ∀ n : ℕ, 1 ≤ n →      (∀ s : Basis n, IsState (phaseState p s)) ∧      holevo γ ν n = phaseValue γ ν p 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.GAD.Basis line 13

    abbrev Basis (n : ℕ) := Fin n → Fin 2

  • abbreviation OAI.GAD.QMatrix line 14

    abbrev QMatrix (n : ℕ) := Matrix (Basis n) (Basis n) ℂ

  • definition OAI.GAD.entropy line 17

    def entropy {ι : Type u} [Fintype ι] [DecidableEq ι] (P : Matrix ι ι ℂ) : ℝ :=  (Matrix.trace (cfc Real.negMulLog P)).re

  • definition OAI.GAD.IsState line 20

    def IsState {ι : Type u} [Fintype ι] [DecidableEq ι] (P : Matrix ι ι ℂ) : Prop :=  P.PosSemidef ∧ Matrix.trace P = 1

  • definition OAI.GAD.kraus line 24

    def kraus (γ ν : ℝ) (r : Fin 4) : Matrix (Fin 2) (Fin 2) ℂ :=  if r = 0 then    !![(Real.sqrt (1 - ν) : ℂ), 0;       0, (Real.sqrt (1 - ν) * Real.sqrt (1 - γ) : ℂ)]  else if r = 1 then    !![0, (Real.sqrt (γ * (1 - ν)) : ℂ); 0, 0]  else if r = 2 then    !![(Real.sqrt ν * Real.sqrt (1 - γ) : ℂ), 0; 0, (Real.sqrt ν : ℂ)]  else    !![0, 0; (Real.sqrt (γ * ν) : ℂ), 0]

  • definition OAI.GAD.tensorKraus line 36

    def tensorKraus (γ ν : ℝ) (n : ℕ) (r : Fin n → Fin 4) : QMatrix n :=  fun i j ↦ ∏ k, kraus γ ν (r k) (i k) (j k)

  • definition OAI.GAD.channel line 40

    def channel (γ ν : ℝ) (n : ℕ) (P : QMatrix n) : QMatrix n :=  ∑ r : Fin n → Fin 4,    tensorKraus γ ν n r * P * (tensorKraus γ ν n r).conjTranspose

  • structure OAI.GAD.Ensemble line 45

    structure Ensemble (n : ℕ) where  size : ℕ  weight : Fin size → ℝ  weight_nonneg : ∀ a, 0 ≤ weight a  weight_sum : ∑ a, weight a = 1  signal : Fin size → QMatrix n  signal_state : ∀ a, IsState (signal a)

  • definition OAI.GAD.ensembleValue line 53

    def ensembleValue (γ ν : ℝ) {n : ℕ} (E : Ensemble n) : ℝ :=  (entropy (∑ a, (E.weight a : ℂ) • channel γ ν n (E.signal a)) -    ∑ a, E.weight a * entropy (channel γ ν n (E.signal a))) / Real.log 2

  • definition OAI.GAD.holevo line 58

    def holevo (γ ν : ℝ) (n : ℕ) : ℝ :=  sSup (Set.range (ensembleValue γ ν : Ensemble n → ℝ))

  • structure OAI.GAD.Code line 62

    structure Code (n : ℕ) where  messages : ℕ  messages_pos : 0 < messages  encoding : Fin messages → QMatrix n  encoding_state : ∀ a, IsState (encoding a)  decoding : Fin messages → QMatrix n  decoding_pos : ∀ a, (decoding a).PosSemidef  decoding_sum : ∑ a, decoding a = 1

  • definition OAI.GAD.averageError line 71

    def averageError (γ ν : ℝ) {n : ℕ} (C : Code n) : ℝ :=  1 - (∑ a, (Matrix.trace (C.decoding a * channel γ ν n (C.encoding a))).re) /    (C.messages : ℝ)

  • definition OAI.GAD.Achievable line 77

    def Achievable (γ ν R : ℝ) : Prop :=  ∃ C : (n : ℕ) → Code n,    Tendsto (fun n ↦ averageError γ ν (C n)) atTop (nhds 0) ∧    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop,      R - ε ≤ Real.log ((C n).messages : ℝ) / Real.log 2 / (n : ℝ)

  • definition OAI.GAD.capacity line 84

    def capacity (γ ν : ℝ) : ℝ := sSup {R : ℝ | Achievable γ ν R}

  • definition OAI.GAD.g line 86

    def g (u : ℝ) : ℝ := Real.binEntropy ((1 + Real.sqrt (1 - 4 * u)) / 2)

  • definition OAI.GAD.v line 88

    def v (γ ν p : ℝ) : ℝ := γ * ν * (1 - ν) + γ * (1 - γ) * (p - ν) ^ 2

  • definition OAI.GAD.objective line 90

    def objective (γ ν p : ℝ) : ℝ :=  Real.binEntropy ((1 - γ) * p + γ * ν) - g (v γ ν p)

  • definition OAI.GAD.Maximizes line 93

    def Maximizes (γ ν p : ℝ) : Prop :=  p ∈ Set.Icc (0 : ℝ) 1 ∧ ∀ q ∈ Set.Icc (0 : ℝ) 1, objective γ ν q ≤ objective γ ν p

  • definition OAI.GAD.phaseSignal line 97

    def phaseSignal (p : ℝ) (s b : Fin 2) : ℂ :=  if b = 0 then (Real.sqrt (1 - p) : ℂ)  else if s = 0 then (Real.sqrt p : ℂ) else -(Real.sqrt p : ℂ)

  • definition OAI.GAD.phaseProduct line 101

    def phaseProduct (p : ℝ) {n : ℕ} (s : Basis n) (i : Basis n) : ℂ :=  ∏ k, phaseSignal p (s k) (i k)

  • definition OAI.GAD.phaseState line 104

    def phaseState (p : ℝ) {n : ℕ} (s : Basis n) : QMatrix n :=  fun i j ↦ phaseProduct p s i * star (phaseProduct p s j)

  • definition OAI.GAD.phaseValue line 108

    def phaseValue (γ ν p : ℝ) (n : ℕ) : ℝ :=  (entropy (∑ s : Basis n, (((2 : ℝ) ^ n)⁻¹ : ℂ) • channel γ ν n (phaseState p s)) -    ∑ s : Basis n, ((2 : ℝ) ^ n)⁻¹ * entropy (channel γ ν n (phaseState p s))) /    Real.log 2

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: equiprobable, maximizes, signals, eq, products, sqrt, holevo and integer. Found in the paper’s title, which its main theorem states: generalized, entropy, capacity, amplitude and damping. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: block, information and attain.

Theorem 1.1 of Classical capacity and entropy inequalities for generalized amplitude damping

Theorem

For all \gamma,\nu\in[0,1] and every integer n\ge1,

\chi(\A_{\gamma,\nu}^{\otimes n}) =n\chi(\A_{\gamma,\nu}) =\frac n{\ln2}\max_{0\le p\le1} \{h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))\},

where

v_{\gamma,\nu}(p) =\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2.

In particular, C(\A_{\gamma,\nu})=\chi(\A_{\gamma,\nu}). If p maximizes (eq:main), independent products of the two equiprobable signals

\phi_\pm=\sqrt{1-p}\,\ket0\ \pm\sqrt p\,\ket1

attain the Holevo information at every block length.

Section 1, Introduction Read it in the paper TeX source, line 208
Show the TeX
For all $\gamma,\nu\in[0,1]$ and every integer $n\ge1$,
\begin{equation}
 \chi(\A_{\gamma,\nu}^{\otimes n})
 =n\chi(\A_{\gamma,\nu})
 =\frac n{\ln2}\max_{0\le p\le1}
 \{h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))\},
\end{equation}
where
\begin{equation}
 v_{\gamma,\nu}(p)
 =\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2.
\end{equation}
In particular, $C(\A_{\gamma,\nu})=\chi(\A_{\gamma,\nu})$.
If $p$ maximizes \eqref{eq:main}, independent products of the two
equiprobable signals
\begin{equation}
 \phi_\pm=\sqrt{1-p}\,\ket0\ \pm\sqrt p\,\ket1
\end{equation}
attain the Holevo information at every block length.

Other possible paper matches, closest wording first

  • Corollary 1.2 of Classical capacity and entropy inequalities for generalized amplitude damping. Words of the Lean statement, its names split into words, that also appear in the theorem: regularization, trace, unassisted, eq, finite, capacity and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: additivity and partner.
  • Corollary 7.1 of Classical capacity and entropy inequalities for generalized amplitude damping. Words of the Lean statement, its names split into words, that also appear in the theorem: kraus, operators, channel, damping, amplitude, sqrt and one more.
  • Corollary 7.2 of Classical capacity and entropy inequalities for generalized amplitude damping. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, products, sqrt, one and holevo. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: pair, attains and information.
  • Corollary 4.4 of Classical capacity and entropy inequalities for generalized amplitude damping. Words of the Lean statement, its names split into words, that also appear in the theorem: matrix, site and integer.
  • Theorem 3.1 of Classical capacity and entropy inequalities for generalized amplitude damping. Words of the Lean statement, its names split into words, that also appear in the theorem: state, tr, complex and one.
  • Proposition 4.3 of Classical capacity and entropy inequalities for generalized amplitude damping. Words of the Lean statement, its names split into words, that also appear in the theorem: basis, site and integer.

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 612 s; peak memory 6.5 GiB).

    Reported 11 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 276’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: Stated: GAD.main: for all γ, ν ∈ [0,1], holevo γ ν n = n * holevo γ ν 1 (additivity across n uses of the same channel), ∃ p, Maximizes γ ν p ∧ holevo n = n / log 2 * objective γ ν p (explicit one-variable optimization), capacity γ ν = holevo γ ν 1 (operational capacity as sup of achievable rates with unrestricted collective decoding) and that every maximizing phase-pair product ensemble attains the block value. Not stated: the summary's additivity of Holevo capacity, minimum output entropy and regularized capacity when tensored with ANY finite-dimensional channel (docs: 'Additivity with an arbitrary different partner channel ... not included').

    Definitions the group asks readers to compare with the paper: Hand-written information theory: entropy P := (trace (cfc Real.negMulLog P)).re, holevo γ ν n := sSup (range ensembleValue) over arbitrary finite ensembles, GAD Kraus operators kraus γ ν r, Achievable/capacity := sSup {R | Achievable γ ν R} with a Tendsto error and liminf-rate condition; standard definitions but not Mathlib's.

  • 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: Additivity with an arbitrary finite-dimensional partner channel (Holevo capacity, minimum output entropy, regularized capacity) is not stated.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result determines the unassisted classical capacity of every generalized amplitude-damping channel with parameters in $[0,1]^2$. Its unrestricted Holevo information is additive across every number of repeated uses, equals an attained scalar maximum, and agrees with the operational capacity per use. Every maximizing equiprobable phase-pair product ensemble attains the corresponding block value. Additivity with an arbitrary different partner channel and decoding by separate measurements are not included.”

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

127 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: 57 to its own declarations, 58 to Mathlib’s documentation. 23 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.

import Mathlibnamespace OAIuniverse unoncomputable sectionopen scoped BigOperators ComplexOrderopen Matrix Filternamespace GADabbrev Basis (n : ℕ) := Fin n → Fin 2abbrev QMatrix (n : ℕ) := Matrix (Basis n) (Basis n) ℂ/-- The spectral definition of -Tr(P log P), with 0 log 0 = 0. -/def entropy {ι : Type u} [Fintype ι] [DecidableEq ι] (P : Matrix ι ι ℂ) : ℝ :=  (Matrix.trace (cfc Real.negMulLog P)).redef IsState {ι : Type u} [Fintype ι] [DecidableEq ι] (P : Matrix ι ι ℂ) : Prop :=  P.PosSemidef ∧ Matrix.trace P = 1def kraus (γ ν : ℝ) (r : Fin 4) : Matrix (Fin 2) (Fin 2) ℂ :=  if r = 0 then    !![(Real.sqrt (1 - ν) : ℂ), 0;       0, (Real.sqrt (1 - ν) * Real.sqrt (1 - γ) : ℂ)]  else if r = 1 then    !![0, (Real.sqrt (γ * (1 - ν)) : ℂ); 0, 0]  else if r = 2 then    !![(Real.sqrt ν * Real.sqrt (1 - γ) : ℂ), 0; 0, (Real.sqrt ν : ℂ)]  else    !![0, 0; (Real.sqrt (γ * ν) : ℂ), 0]/-- Tensor products of the single-site Kraus operators, with the full Kraus index. -/def tensorKraus (γ ν : ℝ) (n : ℕ) (r : Fin n → Fin 4) : QMatrix n :=  fun i j ↦ ∏ k, kraus γ ν (r k) (i k) (j k)/-- The memoryless n-fold generalized amplitude-damping channel. -/def channel (γ ν : ℝ) (n : ℕ) (P : QMatrix n) : QMatrix n :=  ∑ r : Fin n → Fin 4,    tensorKraus γ ν n r * P * (tensorKraus γ ν n r).conjTranspose/-- All finite ensembles, without any product or purity restriction. -/structure Ensemble (n : ℕ) where  size : ℕ  weight : Fin size → ℝ  weight_nonneg : ∀ a, 0 ≤ weight a  weight_sum : ∑ a, weight a = 1  signal : Fin size → QMatrix n  signal_state : ∀ a, IsState (signal a)def ensembleValue (γ ν : ℝ) {n : ℕ} (E : Ensemble n) : ℝ :=  (entropy (∑ a, (E.weight a : ℂ) • channel γ ν n (E.signal a)) -    ∑ a, E.weight a * entropy (channel γ ν n (E.signal a))) / Real.log 2def holevo (γ ν : ℝ) (n : ℕ) : ℝ :=  sSup (Set.range (ensembleValue γ ν : Ensemble n → ℝ))/-- An unrestricted n-use quantum code with one collective output POVM. -/structure Code (n : ℕ) where  messages : ℕ  messages_pos : 0 < messages  encoding : Fin messages → QMatrix n  encoding_state : ∀ a, IsState (encoding a)  decoding : Fin messages → QMatrix n  decoding_pos : ∀ a, (decoding a).PosSemidef  decoding_sum : ∑ a, decoding a = 1def averageError (γ ν : ℝ) {n : ℕ} (C : Code n) : ℝ :=  1 - (∑ a, (Matrix.trace (C.decoding a * channel γ ν n (C.encoding a))).re) /    (C.messages : ℝ)/-- The liminf rate condition written as its eventual lower-bound characterization.There is no preshared entanglement, feedback, or restriction on encodings/decoding. -/def Achievable (γ ν R : ℝ) : Prop :=  ∃ C : (n : ℕ) → Code n,    Tendsto (fun n ↦ averageError γ ν (C n)) atTop (nhds 0) ∧    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop,      R - ε ≤ Real.log ((C n).messages : ℝ) / Real.log 2 / (n : ℝ)/-- Operational unassisted classical capacity, not a definition by regularization. -/def capacity (γ ν : ℝ) : ℝ := sSup {R : ℝ | Achievable γ ν R}def g (u : ℝ) : ℝ := Real.binEntropy ((1 + Real.sqrt (1 - 4 * u)) / 2)def v (γ ν p : ℝ) : ℝ := γ * ν * (1 - ν) + γ * (1 - γ) * (p - ν) ^ 2def objective (γ ν p : ℝ) : ℝ :=  Real.binEntropy ((1 - γ) * p + γ * ν) - g (v γ ν p)def Maximizes (γ ν p : ℝ) : Prop :=  p ∈ Set.Icc (0 : ℝ) 1 ∧ ∀ q ∈ Set.Icc (0 : ℝ) 1, objective γ ν q ≤ objective γ ν pdef phaseSignal (p : ℝ) (s b : Fin 2) : ℂ :=  if b = 0 then (Real.sqrt (1 - p) : ℂ)  else if s = 0 then (Real.sqrt p : ℂ) else -(Real.sqrt p : ℂ)def phaseProduct (p : ℝ) {n : ℕ} (s : Basis n) (i : Basis n) : ℂ :=  ∏ k, phaseSignal p (s k) (i k)def phaseState (p : ℝ) {n : ℕ} (s : Basis n) : QMatrix n :=  fun i j ↦ phaseProduct p s i * star (phaseProduct p s j)/-- Holevo value of the explicit 2^n equiprobable product signals. -/def phaseValue (γ ν p : ℝ) (n : ℕ) : ℝ :=  (entropy (∑ s : Basis n, (((2 : ℝ) ^ n)⁻¹ : ℂ) • channel γ ν n (phaseState p s)) -    ∑ s : Basis n, ((2 : ℝ) ^ n)⁻¹ * entropy (channel γ ν n (phaseState p s))) /    Real.log 2theorem main (γ ν : ℝ) (hγ : γ ∈ Set.Icc (0 : ℝ) 1)    (hν : ν ∈ Set.Icc (0 : ℝ) 1) :    (∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) * holevo γ ν 1) ∧    (∃ p : ℝ, Maximizes γ ν p ∧      ∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) / Real.log 2 * objective γ ν p) ∧    capacity γ ν = holevo γ ν 1 ∧    (∀ p : ℝ, Maximizes γ ν p → ∀ n : ℕ, 1 ≤ n →      (∀ s : Basis n, IsState (phaseState p s)) ∧      holevo γ ν n = phaseValue γ ν p n) := by  sorryend GADendend 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.

Classical capacity and entropy inequalities for generalized amplitude damping

Theorem

For all \gamma,\nu\in[0,1] and every integer n\ge1,

\chi(\A_{\gamma,\nu}^{\otimes n}) =n\chi(\A_{\gamma,\nu}) =\frac n{\ln2}\max_{0\le p\le1} \{h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))\},

where

v_{\gamma,\nu}(p) =\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2.

In particular, C(\A_{\gamma,\nu})=\chi(\A_{\gamma,\nu}). If p maximizes (eq:main), independent products of the two equiprobable signals

\phi_\pm=\sqrt{1-p}\,\ket0\ \pm\sqrt p\,\ket1

attain the Holevo information at every block length.

Section 1, Introduction Read it in the paper TeX source, line 208
Show the TeX
For all $\gamma,\nu\in[0,1]$ and every integer $n\ge1$,
\begin{equation}
 \chi(\A_{\gamma,\nu}^{\otimes n})
 =n\chi(\A_{\gamma,\nu})
 =\frac n{\ln2}\max_{0\le p\le1}
 \{h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))\},
\end{equation}
where
\begin{equation}
 v_{\gamma,\nu}(p)
 =\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2.
\end{equation}
In particular, $C(\A_{\gamma,\nu})=\chi(\A_{\gamma,\nu})$.
If $p$ maximizes \eqref{eq:main}, independent products of the two
equiprobable signals
\begin{equation}
 \phi_\pm=\sqrt{1-p}\,\ket0\ \pm\sqrt p\,\ket1
\end{equation}
attain the Holevo information at every block length.
Corollary (Additivity with a finite-dimensional partner)

Let \gamma,\nu\in[0,1], put \A=\A_{\gamma,\nu}, and let \mathcal N:\mathcal B(\mathcal H_{\mathrm{in}})\to \mathcal B(\mathcal H_{\mathrm{out}}) be completely positive and trace preserving, where \mathcal H_{\mathrm{in}} and \mathcal H_{\mathrm{out}} are nonzero finite-dimensional complex Hilbert spaces. Then

\begin{aligned}\chi(\A\otimes\mathcal N) &=\chi(\A)+\chi(\mathcal N),\\ S_{\min}(\A\otimes\mathcal N) &=S_{\min}(\A)+S_{\min}(\mathcal N),\\ C(\A\otimes\mathcal N) &=C(\A)+C(\mathcal N).\end{aligned}

Here \chi is the unconstrained one-shot Holevo capacity in (eq:holevo), and C is the regularized unassisted classical capacity in (eq:regularization).

Section 1, Introduction Read it in the paper TeX source, line 238
Show the TeX
Let $\gamma,\nu\in[0,1]$, put $\A=\A_{\gamma,\nu}$, and let
$\mathcal N:\mathcal B(\mathcal H_{\mathrm{in}})\to
\mathcal B(\mathcal H_{\mathrm{out}})$ be completely positive and
trace preserving, where $\mathcal H_{\mathrm{in}}$ and
$\mathcal H_{\mathrm{out}}$ are nonzero finite-dimensional complex
Hilbert spaces. Then
\begin{align}
 \chi(\A\otimes\mathcal N)
 &=\chi(\A)+\chi(\mathcal N),\\
 S_{\min}(\A\otimes\mathcal N)
 &=S_{\min}(\A)+S_{\min}(\mathcal N),\\
 C(\A\otimes\mathcal N)
 &=C(\A)+C(\mathcal N).
\end{align}
Here $\chi$ is the unconstrained one-shot Holevo capacity in
\eqref{eq:holevo}, and $C$ is the regularized unassisted classical
capacity in \eqref{eq:regularization}.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI lists this as a main result in its Lean catalogue.
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
22 local vocabulary declarations
Lines of Lean, without comments or blank lines
77
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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

Plectis Atlas

Additivity and capacity for generalized amplitude damping

Family 276: Classical capacity of generalized amplitude damping

Candidate pairing

Theorem

For all \gamma,\nu\in[0,1] and every integer n\ge1,

\chi(\A_{\gamma,\nu}^{\otimes n}) =n\chi(\A_{\gamma,\nu}) =\frac n{\ln2}\max_{0\le p\le1} \{h((1-\gamma)p+\gamma\nu)-g(v_{\gamma,\nu}(p))\},

where

v_{\gamma,\nu}(p) =\gamma\nu(1-\nu)+\gamma(1-\gamma)(p-\nu)^2.

In particular, C(\A_{\gamma,\nu})=\chi(\A_{\gamma,\nu}). If p maximizes (eq:main), independent products of the two equiprobable signals

\phi_\pm=\sqrt{1-p}\,\ket0\ \pm\sqrt p\,\ket1

attain the Holevo information at every block length.

Classical capacity and entropy inequalities for generalized amplitude damping

TeX label thm:main, build/paper.tex, line 208

The Lean statement says

theorem main (γ ν : ℝ) (hγ : γ ∈ Set.Icc (0 : ℝ) 1)    (hν : ν ∈ Set.Icc (0 : ℝ) 1) :    (∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) * holevo γ ν 1) ∧    (∃ p : ℝ, Maximizes γ ν p ∧      ∀ n : ℕ, 1 ≤ n → holevo γ ν n = (n : ℝ) / Real.log 2 * objective γ ν p) ∧    capacity γ ν = holevo γ ν 1 ∧    (∀ p : ℝ, Maximizes γ ν p → ∀ n : ℕ, 1 ≤ n →      (∀ s : Basis n, IsState (phaseState p s)) ∧      holevo γ ν n = phaseValue γ ν p n) := by  sorry

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Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.three independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

22 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/AmplitudeDamping.html

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