PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Finite relative entropy equals modular energy minus entropy

Statement

Let $H$ be a nonzero finite-dimensional complex inner-product space. Let $ρ, σ : H \to H$ be self-adjoint linear maps with nonnegative eigenvalues and trace one, and suppose every eigenvalue of $σ$ is strictly positive. Define $K_{σ} : = - \log σ$ by spectral calculus, and for any self-adjoint $τ : H \to H$ with nonnegative eigenvalues and trace one define $s (τ) : = - tr (τ \log τ) = - \sum_{λ_{j} (τ) > 0} λ_{j} (τ) \log λ_{j} (τ)$ , with the zero-eigenvalue term $0 \log 0 : = 0$ and natural logarithms, so the entropy is in nats. Define the modular-energy difference relative to $σ$ by $Δ_{K} (ρ, σ) : = tr ((ρ - σ) K_{σ})$ . Then

$D (ρ ∥ σ) = Δ_{K} (ρ, σ) - (s (ρ) - s (σ)) .$

For any nonzero finite-dimensional complex inner-product spaces $H_{A}, H_{B}$ with $H = H_{A} \otimes H_{B}$ , partial tracing both states over $H_{B}$ cannot increase relative entropy:

$D ({Tr}_{B} ρ ∥ {Tr}_{B} σ) \leq D (ρ ∥ σ) .$

The identity and inequality are finite-dimensional mathematical statements; no Hamiltonian, physical energy, physical units, continuum limit, or gravitational entropy is assumed.

Statement references

Finite-dimensional quantum relative entropy

Finite-dimensional von Neumann entropy

The basis-independent trace of an endomorphism of a finite-dimensional vector space

Partial trace over a finite tensor factor

Partial-trace data processing for finite quantum relative entropy

John Watrous, The Theory of Quantum Information, §§5.2.1–5.2.3

Aron C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, §2.5

Mark M. Wilde, From Classical to Quantum Shannon Theory, §§11.1 and 11.8

Facts & Assumptions

Given: A nonzero finite-dimensional complex inner-product space $H$ and self-adjoint $ρ, σ : H \to H$ with nonnegative eigenvalues and trace one, with $σ$ having strictly positive eigenvalues.

[F1]

Finite-dimensional quantum relative entropy: for nonnegative endomorphisms $A, B$ , if $supp (A) \subseteq supp (B)$ then $D (A ∥ B) = tr (A \log A) - tr (A \log B)$ using spectral logarithms on positive supports; otherwise $D (A ∥ B) = + \infty$ .

[F2]

Finite-dimensional von Neumann entropy defines the von Neumann entropy $S (τ) = - tr (τ \log τ)$ for a finite-dimensional density endomorphism using the natural logarithm. The function $s$ in the Statement is the same finite spectral sum, written with lowercase $s$ and the convention $0 \log 0 : = 0$ .

[F3]

The basis-independent trace of an endomorphism of a finite-dimensional vector space: $tr (T)$ is the basis-independent trace of a finite-dimensional endomorphism, given by the sum of diagonal matrix entries in any basis; it is linear by this finite-sum definition.

[F4]

Partial trace over a finite tensor factor: for nonnegative $R$ on $H_{A} \otimes H_{B}$ , the partial trace ${Tr}_{B} R$ is the endomorphism of $H_{A}$ whose product-basis entries are $({Tr}_{B} R)_{i i^{'}} = \sum_{j} R_{(i, j), (i^{'}, j)}$ .

[F5]

Partial-trace data processing for finite quantum relative entropy: for nonnegative $A, B$ on a finite tensor product $H_{A} \otimes H_{B}$ , $D ({Tr}_{B} A ∥ {Tr}_{B} B) \leq D (A ∥ B)$ .

Proof

1.1givenF1F2algebra

Since $σ$ has only positive eigenvalues, $supp (σ) = H$ and hence $supp (ρ) \subseteq supp (σ)$ . By [F1], $D (ρ ∥ σ)$ is finite and equals $tr (ρ \log ρ) - tr (ρ \log σ) = - s (ρ) + tr (ρ K_{σ})$ .

2.1F2F3step 1.1algebra

By [F2] and $K_{σ} = - \log σ$ , $s (σ) = tr (σ K_{σ})$ . Linearity of trace [F3] then gives $Δ_{K} (ρ, σ) - (s (ρ) - s (σ)) = tr (ρ K_{σ}) - tr (σ K_{σ}) - s (ρ) + s (σ) = tr (ρ K_{σ}) - s (ρ) = D (ρ ∥ σ)$ by step 1.1. This proves the finite modular-energy identity.

3.1givenF4F5step 2.1algebra∎

With the finite identity established in step 2.1, apply [F5] for the separate partial-trace claim to the nonnegative endomorphisms $ρ, σ$ on $H_{A} \otimes H_{B}$ , with the partial trace defined in [F4], to obtain the displayed inequality.

Finite relative entropy equals modular energy minus entropy

Statement

Statement references

Facts & Assumptions

Proof

Depends on

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Sources