Finite-dimensional operator entropy

Self-contained finite-dimensional operator-entropy primitives over complex matrices: the DensityMatrix of a finite quantum system (a positive-semidefinite, unit-trace matrix), its vonNeumannEntropy, the partial trace as a positive trace-preserving (and completely positive, in Kraus/compression form) coarse-graining, the Kronecker spectrum, and the additivity and subadditivity of the von Neumann entropy.

The mathematical content mirrors the standard quantum-information references (Nielsen–Chuang, Quantum Computation and Quantum Information, §11.3; Carlen, Trace Inequalities and Quantum Entropy, §2.3). In particular subadditivity rests only on the elementary Klein inequality (Carlen Thm 2.11), not on the deeper joint-convexity / Lieb-concavity layer.

Principal results

• Oseledets.OperatorEntropy.vonNeumannEntropy — S(ρ) = ∑ᵢ negMulLog(λᵢ).
• Oseledets.OperatorEntropy.partialTraceRight / partialTraceLeft — the partial trace, trace-preserving and positivity-preserving (PosSemidef.partialTraceRight), packaged as a DensityMatrix.partialTraceRight : DensityMatrix (nA × nB) → DensityMatrix nA.
• Oseledets.OperatorEntropy.eigenvalues_kronecker_multiset — the spectrum of A ⊗ₖ B.
• Oseledets.OperatorEntropy.vonNeumannEntropy_additive_kronecker — S(ρ ⊗ σ) = S(ρ) + S(σ).
• Oseledets.OperatorEntropy.vonNeumannEntropy_subadditive — S(ρ_AB) ≤ S(ρ_A) + S(ρ_B).