Tensor additivity of extended-real quantum relative entropy

Proved

QuantumChannelContinuity.stateRelative_tensor

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

channel-renyi-continuity-jwang226quantum-information

Let $\rho,\sigma$ be density operators on a nonzero finite-dimensional complex Hilbert space $A$ and let $\tau,\omega$ be density operators on another such space $B$ . Then

D(\rho\otimes\tau\Vert\sigma\otimes\omega)=D(\rho\Vert\sigma)+D(\tau\Vert\omega).

The entropies are measured in bits and equality is in the extended real line. Support mismatches giving infinity are included. This supplies the relative-entropy block superadditivity used by regularization.

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 Mathlib.Topology.UniformSpace.HeineCantor
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_ChannelProducts
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_FoundationsInfiniteLimit
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_StabilizedSchatten
import Definitions.Def_CRCD_QuantumChannelContinuity_StateSupportLimits
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorChannels
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorDilation
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorOrder
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorRegrouping
import Definitions.Def_CRCD_QuantumChannelContinuity_TensorStates
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)
-/




/-! # Relative-entropy additivity from the proved singular-state order-one limit -/

open QuantumState QuantumChannel
open scoped ComplexOrder TensorProduct ENNReal Topology

namespace QuantumChannelContinuity
end QuantumChannelContinuity
open QuantumChannelContinuity

set_option maxHeartbeats 1000000
set_option maxRecDepth 8192
set_option backward.isDefEq.respectTransparency false

variable {A B C D : Type} [Qudit A] [Qudit B] [Qudit C] [Qudit D]
  [Nontrivial A] [Nontrivial B] [Nontrivial C] [Nontrivial D]

Formal statement

theorem QuantumChannelContinuity.stateRelative_tensor (ρ σ : DensityState A) (τ ω : DensityState B) :
    stateRelative (ρ.tensor τ) (σ.tensor ω) = stateRelative ρ σ + stateRelative τ ω := by sorry

Source

https://github.com/JWang226/Continuity-of-regularized-channel-renyi-divergence/blob/e29274a69978bb499f00497481c6008befb0b84d/QuantumChannelContinuity/TensorRelative.lean#L24-L46

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