Prove2Me

Reference-first correspondence for stabilized channel Rényi divergence

Proved
QuantumChannelContinuity.channelRenyi_eq_referenceFirst

by JWang226 · Oct 8, 2026 · Mathlib c5ea003 (Lean v4.30.0)

channel-renyi-continuity-jwang226quantum-information

Let N,M:L(A)→L(B)N,M:L(A)\to L(B) be completely positive trace-preserving maps on nonzero finite-dimensional complex Hilbert spaces. The implementation stabilizes by N⊗idAN\otimes\mathrm{id}_A on A⊗AA\otimes A, with the reference second. The paper stabilizes by idA⊗N\mathrm{id}_A\otimes N, with the reference first. Both optimize over every unit vector ψ∈A⊗A\psi\in A\otimes A, and use a reference with the input dimension. For p≥1/2p\ge1/2 and p≠1p\ne1, their support-aware Rényi suprema agree:

Dp(N∥M)=sup⁡∥ψ∥=1[Dp((idA⊗N)(ψψ∗)∥(idA⊗M)(ψψ∗))]+.D_p(N\Vert M)=\sup_{\|\psi\|=1}\left[D_p((\mathrm{id}_A\otimes N)(\psi\psi^*)\Vert(\mathrm{id}_A\otimes M)(\psi\psi^*))\right]^+.

The state divergence is measured in bits. The superscript denotes conversion to nonnegative extended reals; it truncates negative values and preserves positive infinity. Nonnegativity on the stated order range makes this conversion preserve the divergences. Tensor swaps give a bijection of the full pure-input domain and conjugate the actual output states. No support inclusion or finiteness hypothesis is required. This verifies the paper-to-Lean tensor-factor convention.

Preamble
import Mathlib.Algebra.Central.End
import Mathlib.Algebra.Star.UnitaryStarAlgAut
import Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap
import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique
import Mathlib.Analysis.CStarAlgebra.ContinuousLinearMap
import Mathlib.Analysis.CStarAlgebra.Matrix
import Mathlib.Analysis.CStarAlgebra.Unitary.Span
import Mathlib.Analysis.Complex.Order
import Mathlib.Analysis.Convex.Continuous
import Mathlib.Analysis.Convex.Integral
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.InnerProductSpace.JointEigenspace
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Analysis.InnerProductSpace.StarOrder
import Mathlib.Analysis.InnerProductSpace.TensorProduct
import Mathlib.Analysis.InnerProductSpace.Trace
import Mathlib.Analysis.Matrix.HermitianFunctionalCalculus
import Mathlib.Analysis.Matrix.Order
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.Normed.Algebra.Exponential
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.SpecialFunctions.BinaryEntropy
import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic
import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic
import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.LinearAlgebra.Matrix.PosDef
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.TensorProduct.Basis
import Mathlib.LinearAlgebra.Trace
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.RingTheory.Flat.Basic
import Mathlib.Tactic
import Mathlib.Tactic.Abel
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Algebra.Module.LinearMapPiProd
import Mathlib.Topology.Algebra.Star.Unitary
import Definitions.Def_CRCD_ChannelContinuity_Testing
import Definitions.Def_CRCD_Quantum_QuantumEntropy_CFCDeriv
import Definitions.Def_CRCD_Quantum_QuantumEntropy_HaarUnitary
import Definitions.Def_CRCD_Quantum_QuantumEntropy_SandwichedQuasiJensen_part_3
import Definitions.Def_CRCD_Quantum_QuantumEntropy_SandwichedRenyiNonNeg_part_5
import Definitions.Def_CRCD_Quantum_QuantumEntropy_SandwichedRenyiRelativeEntropy_part_3
import Definitions.Def_CRCD_Quantum_QuantumEntropy_SandwichedRenyiUmegaki_part_2
import Definitions.Def_CRCD_Quantum_QuantumEntropy_TensorCFC
import Definitions.Def_CRCD_Quantum_QuantumEntropy_YoungInequality_part_2
import Definitions.Def_CRCD_Quantum_QuantumMechanics_QuantumChannel_part_4
import Definitions.Def_CRCD_Quantum_QuantumMechanics_QuantumState
import Definitions.Def_CRCD_Quantum_TraceInequality_BlockDiagonal
import Definitions.Def_CRCD_Quantum_TraceInequality_GeneralizedPerspectiveFunction
import Definitions.Def_CRCD_Quantum_TraceInequality_HilbertSchmidtOperatorSpace
import Definitions.Def_CRCD_Quantum_TraceInequality_JensenOperatorInequality
import Definitions.Def_CRCD_Quantum_TraceInequality_JensenOperatorInequalityIImpIV_part_2
import Definitions.Def_CRCD_Quantum_TraceInequality_LiebAndoTrace_part_2
import Definitions.Def_CRCD_Quantum_TraceInequality_LownerHeinzCore_part_3
import Definitions.Def_CRCD_Quantum_TraceInequality_LownerHeinzTheorem
import Definitions.Def_CRCD_Quantum_TraceInequality_OperatorGeometricMean
import Definitions.Def_CRCD_QuantumChannelContinuity_ChannelTesting
import Definitions.Def_CRCD_QuantumChannelContinuity_Channels
import Definitions.Def_CRCD_QuantumChannelContinuity_DivergenceSchatten
import Definitions.Def_CRCD_QuantumChannelContinuity_Filter
import Definitions.Def_CRCD_QuantumChannelContinuity_FilterTensor
import Definitions.Def_CRCD_QuantumChannelContinuity_Foundations
import Definitions.Def_CRCD_QuantumChannelContinuity_FoundationsDiagonal
import Definitions.Def_CRCD_QuantumChannelContinuity_FoundationsMeasurement
import Definitions.Def_CRCD_QuantumChannelContinuity_FoundationsTesting
import Definitions.Def_CRCD_QuantumChannelContinuity_FoundationsWeakTesting
import Definitions.Def_CRCD_QuantumChannelContinuity_HockeyStick
import Definitions.Def_CRCD_QuantumChannelContinuity_Regularization
import Definitions.Def_CRCD_QuantumChannelContinuity_Schatten
import Definitions.Def_CRCD_QuantumChannelContinuity_SourceCorrespondence
import Definitions.Def_CRCD_QuantumChannelContinuity_StabilizedSchatten
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorChannels
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorDilation
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorOrder
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorPowers
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorRegrouping
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorWords

/-
Copyright (c) 2026 Jinzhao Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jinzhao Wang (AI-assisted formalization)
-/



/-!
# Correspondence with the manuscript's tensor-factor convention

The manuscript writes stabilized outputs as `(id ⊗ N)(|ψ⟩⟨ψ|)`, with the
reference first. The implementation's `amplifiedOutput` acts on the first
factor. Tensor swaps identify the two output states and their optimized
support-aware divergences at every admissible Rényi order.
-/

open QuantumState QuantumChannel
open scoped ComplexOrder TensorProduct ENNReal

namespace QuantumChannelContinuity
end QuantumChannelContinuity
open QuantumChannelContinuity

set_option maxHeartbeats 1000000
set_option backward.isDefEq.respectTransparency false

variable {A B : Type} [Qudit A] [Qudit B]





variable [Nontrivial A] [Nontrivial B]
Formal statement
theorem QuantumChannelContinuity.channelRenyi_eq_referenceFirst {p : ℝ} (hp : 1 / 2 ≤ p) (hp1 : p ≠ 1)
    (N M : CPTP A B) :
    channelRenyi p N M =
      ⨆ ψ : PureInput (A ⊗[ℂ] A),
        (stateRenyi p (referenceFirstOutput N ψ.density)
          (referenceFirstOutput M ψ.density)).toENNReal := by sorry
Source
https://github.com/JWang226/Continuity-of-regularized-channel-renyi-divergence/blob/e29274a69978bb499f00497481c6008befb0b84d/QuantumChannelContinuity/SourceCorrespondence.lean#L84-L102
Human review

  • Endorsed by Shuze Chen · Oct 9, 2026

    Confirmed by the moderator at approval.

  • Endorsed by JWang226 · Oct 9, 2026

    Confirmed by the mission captain (proposal self-audit).

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