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Relative modular representation of finite quantum relative entropy

Statement

Let $A, B$ be positive-definite operators on a nonzero finite-dimensional complex inner-product space $H$ . On $L (H)$ , with Hilbert-Schmidt inner product $⟨ X, Y ⟩_{H S} : = tr (X Y^{*})$ , define $Δ_{B, A} (X) : = B X A^{- 1}$ . Then $Δ_{B, A}$ is positive definite and

$D (A ∥ B) = ⟨ (- \ln Δ_{B, A}) (A^{1 / 2}), A^{1 / 2} ⟩_{H S} .$

The logarithm of $Δ_{B, A}$ is defined by its positive eigenvalues.

Facts & Assumptions

Given: $H, A, B$ as in the Statement.

[F1]

Finite-dimensional quantum relative entropy: for positive-definite $A, B$ , $D (A ∥ B) = tr (A \ln A) - tr (A \ln B)$ .

[F2]

A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S: positive-definite operators have orthonormal eigenbases with positive eigenvalues.

[F3]

A non-negative operator has a unique non-negative square root: $A$ has a positive-definite square root $A^{1 / 2}$ .

[F4]

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ : trace is cyclic for compatible finite-dimensional maps.

[F5]

Real and complex inner-product spaces and their induced length: the inner product is linear in its first argument and conjugate-linear in its second, so $E_{j i} x = ⟨ x, u_{i} ⟩ v_{j}$ is linear.

[F6]

Every linear map between finite-dimensional inner product spaces has a unique adjoint: every linear map between finite-dimensional inner-product spaces has a unique linear adjoint, including zero-dimensional spaces.

[F7]

The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ : the adjoint is characterized by $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ .

[F8]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: for positive real $x, y$ , $\ln (x / y) = \ln x - \ln y$ .

Proof

1.1givenF6F7F8F2F5algebrachoose

All adjoints of the finite-dimensional linear maps used below exist uniquely by [F6]. Choose orthonormal eigenbases $(u_{i})$ and $(v_{j})$ for $A$ and $B$ , with $A u_{i} = a_{i} u_{i}$ , $B v_{j} = b_{j} v_{j}$ and $a_{i}, b_{j} > 0$ , using [F2]. For each $i, j$ define $E_{j i} x : = ⟨ x, u_{i} ⟩ v_{j}$ . These linear matrix units form an orthonormal basis of $L (H)$ for $⟨ \cdot, \cdot ⟩_{H S}$ by the adjoint pairing [F7] and direct trace evaluation. They satisfy $B E_{j i} = b_{j} E_{j i}$ and $E_{j i} A^{- 1} = a_{i}^{- 1} E_{j i}$ , so $Δ_{B, A} E_{j i} = (b_{j} / a_{i}) E_{j i}$ . The quotient law [F8] gives $- \ln (b_{j} / a_{i}) = - \ln b_{j} + \ln a_{i}$ , so spectral calculus on this basis shows that $Δ_{B, A}$ is positive definite and gives $(- \ln Δ_{B, A}) (X) = - (\ln B) X + X (\ln A)$ .

2.1F1F3F4step 1.1algebra∎

Applying the formula in step 1.1 to $A^{1 / 2}$ gives $⟨ (- \ln Δ_{B, A}) (A^{1 / 2}), A^{1 / 2} ⟩_{H S} = tr (- A \ln B + A^{1 / 2} (\ln A) A^{1 / 2})$ . By trace cyclicity [F4], the second trace is $tr (A \ln A)$ . The result equals $D (A ∥ B)$ by [F1], proving the identity.

Depends on

Used by

Joint convexity of finite quantum relative entropy Theorem

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, The Theory of Quantum Information, §5.2.3 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, arXiv:1106.1445v8, §11.8 (standard reference, not scraped)