Subadditivity of the von Neumann entropy #
For a bipartite density matrix ρ on nA ⊗ nB with reduced density matrices
ρ_A = Tr_B ρ and ρ_B = Tr_A ρ, the von Neumann entropy is subadditive:
S(ρ) ≤ S(ρ_A) + S(ρ_B).
The proof is the elementary route through the scalar Klein / Peierls inequality
(klein_scalar, Carlen, Trace Inequalities and Quantum Entropy, Thm 2.11; Nielsen–Chuang
§11.3) — no Lieb concavity and no matrix logarithm / continuous functional calculus.
Writing M = G diag(p) Gᴴ, ρ_A = U diag(λ) Uᴴ, ρ_B = V diag(μ) Vᴴ from the spectral
theorem and W = U ⊗ V, the doubly stochastic matrix D k m = |‖(Gᴴ W)ₖₘ‖² together with the
product eigenvalue vector s (i,j) = λ i · μ j feeds Klein's inequality. The key linear-algebra
input is the conjugation identity Tr_B(Wᴴ M W) = Uᴴ (Tr_B M) U (and its left analogue), which
lets the marginals of D be read off as λ and μ.
The reduced density matrix on the left (A) factor: Tr_B ρ.
Equations
- ρ.partialTraceRight = { val := Oseledets.OperatorEntropy.partialTraceRight ρ.val, posSemidef := ⋯, trace_one := ⋯ }
Instances For
The reduced density matrix on the right (B) factor: Tr_A ρ.
Equations
- ρ.partialTraceLeft = { val := Oseledets.OperatorEntropy.partialTraceLeft ρ.val, posSemidef := ⋯, trace_one := ⋯ }
Instances For
Subadditivity of the von Neumann entropy.
S(ρ) ≤ S(Tr_B ρ) + S(Tr_A ρ) for a bipartite density matrix ρ.