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Joint convexity of finite quantum relative entropy

Statement

Let $H$ be a finite-dimensional complex inner-product space, let $n \geq 1$ , let $A_{i}, B_{i} : H \to H$ be non-negative for $1 \leq i \leq n$ , and let $λ_{i} \geq 0$ with $\sum_{i} λ_{i} = 1$ . Then, in the extended-real order,

$D ⁣ (\sum_{i} λ_{i} A_{i} ∥ \sum_{i} λ_{i} B_{i}) \leq \sum_{i} λ_{i} D (A_{i} ∥ B_{i}),$

where zero-weight terms are omitted from the right side, and the order is that on the extended real line.

Facts & Assumptions

Given: $H$ , the finite families $(A_{i}), (B_{i})$ and the probability vector $(λ_{i})$ in the Statement.

[F1]

Finite-dimensional quantum relative entropy: for non-negative operators, $D$ is finite under support inclusion and uses natural spectral logarithms; otherwise it is $+ \infty$ .

[F2]

The compression Jensen inequality for the negative logarithm: $- \ln (V^{*} T V) ⪯ V^{*} (- \ln T) V$ for positive-definite $T$ and an isometry $V$ .

[F3]

Relative modular representation of finite quantum relative entropy: for positive-definite $A, B$ , $D (A ∥ B) = ⟨ (- \ln Δ_{B, A}) (A^{1 / 2}), A^{1 / 2} ⟩_{H S}$ with $Δ_{B, A} (X) = B X A^{- 1}$ .

[F4]

Support-sensitive regularization limit for quantum relative entropy: $D (A + ε I ∥ B + ε I) \to D (A ∥ B)$ as $ε ↓ 0$ , including the infinite support-failure branch.

[F5]

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.

[F6]

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

[F7]

The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ : $U_{i}^{*}$ is characterized by $⟨ U_{i} X, Y ⟩ = ⟨ X, U_{i}^{*} Y ⟩$ .

[F8]

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.

Proof

1.1givenF8F1F5F6F7algebracasesconstruct

All adjoints of the finite-dimensional linear maps used below exist uniquely by [F8]. Discard all indices with $λ_{i} = 0$ from both sides; at least one index remains because the weights sum to one. The remaining weights are strictly positive and still sum to one. If $H = {0}$ , all operators and relative entropies are zero by [F1], so the result holds. Now assume $H \neq {0}$ and first suppose every $A_{i}, B_{i}$ is positive definite. Set $A = \sum_{i} λ_{i} A_{i}$ and $B = \sum_{i} λ_{i} B_{i}$ . Positive-definite square roots and inverse square roots exist by [F6]. On the Hilbert-Schmidt spaces, with direct-sum inner product, define $U_{i} (X) : = \sqrt{λ_{i}} X A^{- 1 / 2} A_{i}^{1 / 2}$ and $U (X) : = (U_{i} (X))_{i}$ . Trace cyclicity [F5] and the adjoint definition [F7] in $⟨ U_{i} (X), Y ⟩_{H S}$ give $U_{i}^{*} (Y) = \sqrt{λ_{i}} Y A_{i}^{1 / 2} A^{- 1 / 2}$ . Therefore $U^{*} U (X) = X A^{- 1 / 2} (\sum_{i} λ_{i} A_{i}) A^{- 1 / 2} = X$ , so $U$ is an isometry.

2.1givenF2F3F6step 1.1algebra

Put $T : = ⨁_{i} Δ_{B_{i}, A_{i}}$ , positive definite by [F3]. For each $i$ , multiplication gives $U_{i}^{*} Δ_{B_{i}, A_{i}} U_{i} (X) = λ_{i} B_{i} X A^{- 1}$ , hence $U^{*} T U (X) = B X A^{- 1} = Δ_{B, A} (X)$ ; also $U (A^{1 / 2}) = (\sqrt{λ_{i}} A_{i}^{1 / 2})_{i}$ by [F6]. Apply [F2] to $U, T$ and evaluate the order inequality on $A^{1 / 2}$ in the Hilbert-Schmidt inner product. Using [F3] on both sides yields $D (A ∥ B) \leq \sum_{i} λ_{i} D (A_{i} ∥ B_{i})$ .

3.1givenF1F4step 2.1algebracases∎

For general non-negative $A_{i}, B_{i}$ , if some $D (A_{i} ∥ B_{i}) = + \infty$ , the right side is $+ \infty$ and the claim holds. Otherwise each $supp (A_{i}) \subseteq supp (B_{i})$ . If $v \in \ker B$ , where $B = \sum_{i} λ_{i} B_{i}$ , then $0 = ⟨ B v, v ⟩ = \sum_{i} λ_{i} ⟨ B_{i} v, v ⟩$ forces $B_{i} v = 0$ for every $i$ ; support inclusion then gives $A_{i} v = 0$ , so $A v = 0$ . Thus $supp (A) \subseteq supp (B)$ . Apply the positive-definite case to $A_{i} + ε I, B_{i} + ε I$ ; their weighted sums are $A + ε I, B + ε I$ . Let $ε ↓ 0$ and use [F4] for every term to obtain the desired inequality.

Depends on

Used by

Partial-trace data processing for finite quantum relative entropy Theorem

Dependency tree · two levels

31 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, §11.9 (standard reference, not scraped)