Strong subadditivity of von Neumann entropy#14
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Mark `tensor_toLinearMap`, `tensor_tmul`, `tensor_one`, `tensor_mul`, `tensor_add_left`, `tensor_add_right` as `private` and drop the now-unused `Topology.Algebra.Module.FiniteDimension` and `RingTheory.TensorProduct.Finite` imports. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
Add `Complex.zero_le_ofReal`, `zero_le_one_sub_ofReal`, and `zero_le_natCast_inv` together with the `Mathlib.Analysis.Complex.Order` import. Used by spectral / density-matrix calculations that need `(0 : ℂ) ≤ (ε : ℂ)` for a real witness. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
…ty preservation Add `Matrix.reindexStarAlgEquiv` upgrading `Matrix.reindexAlgEquiv` to a `StarAlgEquiv` (with `conjTranspose_submatrix` for `map_star'`), trace-preservation lemmas (`trace_submatrix_eq`, `trace_reindexStarAlgEquiv`, `trace_conjStarAlgAut`), and Hermitian / `PosSemidef` / `PosDef` preservation under a `StarAlgEquiv` between complex matrix algebras (`map_starAlgEquiv` family) with reindex specialisations (`mapEquiv` family). Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
…lity Add `matrixLog_spectral_eq` (spectral expansion via the chosen Hermitian proof) and `matrixLog_map_starAlgEquiv` (PosDef + StarAlgEquiv → matrixLog commutes, proven via `StarAlgHomClass.map_cfc` after restricting scalars to ℝ). Wrap two existing chunks in `section JensenConvexity` / `section SpectralIdentities` and import the new `ForMathlib/Matrix/StarAlgEquiv` module. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
…onal matrices Add `Matrix.cfc_log_diagonal_pos` (`cfc Real.log (diagonal d) = diagonal (Real.log ∘ d)` for strictly positive real `d`) and `cfc_log_unitary_conj_diagonal` (commutation with unitary conjugation). Proven via the Pi C*-algebra structure on `m → ℂ` plus `StarAlgHomClass.map_cfc` for the diagonal-embedding star-algebra homomorphism. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
…regularization Add `DensityMatrix.map` (transport along a trace-preserving StarAlgEquiv) and `DensityMatrix.mapEquiv` (reindex specialisation via `Matrix.reindexStarAlgEquiv`), the maximally-mixed state `π = I/d` with `maximallyMixed_posDef`, and the standard regularization `regularize ρ ε := (1-ε) ρ + ε π` together with `regularize_posDef`, the cfc identity `regularize_eq_cfc`, and reindex compatibility `regularize_mapEquiv`. Flatten the redundant outer `namespace Matrix` (using `open Matrix`) and qualify the scoped notation namespace as `Matrix.QuantumInfo`. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
Add `LocalNet` (sites + per-site finite index types — data only, the Haag–Kastler axioms isotony / locality / covariance are not formalised in this commit; the `Haag–Kastler` mention in the docstring is a literature reference, not a proof). Provide `regionIdx`, `localAlgebra`, `densityMatrix`, the combiner equivalence `combineIdx` (`regionIdx Λ_total ≃ regionIdx Λ × regionIdx (Λ_total \ Λ)`), the algebra inclusion `includeAlgebra`, and the region-index split equivalences `regionIdxInsertEquiv` / `regionIdxPairEquiv` / `regionIdxTripleEquiv`. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
… ρ ↾ Λ notation Add the Schrödinger-picture restriction for a `LocalNet`: `Matrix.restrict` (linear), `Matrix.restrictKraus`, the bundled `Matrix.QuantumChannel.restrict` with CP / TP / channel structure, the iterated identity `restrict_restrict`, and the Heisenberg-picture trace identity `trace_mul_includeAlgebra`. Lift to density matrices via `DensityMatrix.restrict` and expose the paper-style notation `ρ ↾ Λ` / `ρ ↾[h]` (scoped under `LocalNet.QuantumInfo`). Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
…nd regularization continuity Bridge to Mathlib's CFC via `entropyFun_eq_negMulLog_of_nonneg`, `vonNeumannEntropy_eq_negMulLog_sum`, and `vonNeumannEntropy_eq_matrixFunction_re`. Add isomorphism invariance for PosDef states (`vonNeumannEntropy_map_starAlgEquiv_posDef` and the reindex specialisation `vonNeumannEntropy_mapEquiv_posDef`). Add the regularization stack: `vonNeumannEntropy_regularize_eq_negMulLog_sum`, `continuous_negMulLog_regularize_sum`, `tendsto_negMulLog_regularize_sum_zero`, and the PSD lift `vonNeumannEntropy_mapEquiv` via the ε → 0+ limit. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
…ving StarAlgEquiv Add `relativeEntropy_map_starAlgEquiv_posDef` (PosDef invariance under a trace-preserving StarAlgEquiv between complex matrix algebras — the isometric form of Lindblad–Uhlmann monotonicity restricted to invertible CPTP maps) and the reindex specialisation `relativeEntropy_mapEquiv_posDef`. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
…sor identity Add `DensityMatrix.kronecker` (independent-systems product state), preservation of Hermitian / unitary structure under `⊗ₖ`, the explicit Kronecker spectral decomposition, the **log-tensor identity** `matrixLog (A ⊗ₖ B) = matrixLog A ⊗ₖ 1 + 1 ⊗ₖ matrixLog B` for PosDef `A`, `B`, the equivalence-indexed partial trace `partialTrace`, and the Heisenberg duality at product type. The proof routes via `matrixFunction` and `StarAlgHomClass.map_cfc` to dodge the `CStarMatrix` instance diamond. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
…n on a LocalNet Add `Matrix.restrict_maximallyMixed` (the marginal of the maximally-mixed state is again maximally mixed after dimension cancellation) and the compatibility `restrict h (regularize ρ ε) = regularize (restrict h ρ) ε`. This is the key fact behind extending PosDef-only entropy theorems to PSD via the ε → 0+ limit. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
… form
Prove the bipartite mutual-information identity
`D(ρ_AB ‖ ρ_A ⊗ ρ_B) = -S(ρ_AB) + S(ρ_A) + S(ρ_B)` (PosDef) and use it together
with the data-processing inequality to prove **strong subadditivity** in the
AQFT-natural common-region-explicit form
S(ρ ↾ ΛAB) + S(ρ ↾ ΛBC) ≥ S(ρ) + S(ρ ↾ ΛB)
given `ΛAB ⊆ ΛABC`, `ΛBC ⊆ ΛABC`, `ΛB ⊆ ΛABC`, `ΛA ⊆ ΛAB`,
`ΛAB \ ΛA = ΛB`, `ΛABC \ ΛA = ΛBC`. The familiar three-site form on
`{a, b, c}` is a corollary.
Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
Drop the `ArakiRelativeEntropy`, `SegalEntropy`, `Channel/TraceClass`, and `TraceClassRelativeEntropy` import lines that became stale after `86ea6c2 chore: remove for now`, and add the modules introduced on this branch (`Algebra.LocalNet`, `Analysis.CFC.Diagonal`, `Analysis.Entropy.KroneckerProduct`, `Analysis.Entropy.Regularize`, `Analysis.Entropy.SSA`, `Analysis.Matrix.PartialTrace`, `ForMathlib.LinearAlgebra.Matrix.StarAlgEquiv`). Refresh the paper-notation table in `QuantumSystem/Notation.lean` for the `Matrix.QuantumInfo` namespace move and the new `ρ ↾ Λ` row. Co-Authored-By: Claude Opus 4.7 (1M context) <[email protected]>
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fix #6