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An orthogonal ensemble attains its Holevo information

Example

Let $X$ have a finite probability vector $(p_{x})$ and let $H_{B} = C^{∣ X ∣}$ have orthonormal basis $(e_{x})_{x \in X}$ . Set $ρ_{x} : = P_{x}$ , where $P_{x} v : = ⟨ v, e_{x} ⟩ e_{x}$ , and measure with the projective POVM $(P_{y})_{y \in X}$ . Define the joint law $p_{x y} : = p_{x} tr (ρ_{x} P_{y})$ and its output marginal $p_{Y} (y) : = \sum_{x} p_{x y}$ . With $H (p) : = - \sum_{x : p_{x} > 0} p_{x} \ln p_{x}$ and $I (X : Y) : = \sum_{x, y : p_{x y} > 0} p_{x y} \ln (p_{x y} / (p_{x} p_{Y} (y)))$ , the Holevo information and the measurement mutual information both equal $H (p)$ .

Facts & Assumptions

Given: The probability vector, orthonormal basis, ensemble, and POVM in the Example.

[F1]

Finite classical-quantum states and Holevo information: $χ = S (\overset{ˉ}{ρ}) - \sum_{p_{x} > 0} p_{x} S (ρ_{x})$ .

[F2]

Finite-dimensional von Neumann entropy: a state with eigenvalue list $(λ_{j})$ has entropy $- \sum_{λ_{j} > 0} λ_{j} \ln λ_{j}$ .

[F3]

Finite POVMs and their outcome law and POVM outcome weights are probabilities: the outcome weight of $P_{y}$ in state $ρ_{x}$ is $tr (ρ_{x} P_{y})$ , and the weights form a probability vector.

[F4]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: for $p_{x} > 0$ , $\ln (1 / p_{x}) = - \ln p_{x}$ .

Verification

1.1givenF1F2algebra

The average state $\overset{ˉ}{ρ} = \sum_{x} p_{x} P_{x}$ is diagonal in $(e_{x})$ with eigenvalues $p_{x}$ , so $S (\overset{ˉ}{ρ}) = H (p)$ by [F2]. Each $P_{x}$ has eigenvalues $1$ and zeros, hence $S (P_{x}) = 0$ ; [F1] gives $χ = H (p)$ .

2.1givenF3F4step 1.1algebra

By [F3], $p_{x y} = p_{x} tr (P_{x} P_{y}) = p_{x} δ_{x y}$ . Thus $p_{Y} (y) = p_{y}$ , and in the sum defining $I (X : Y)$ only $x = y$ with $p_{x} > 0$ contributes, giving $I (X : Y) = \sum_{x : p_{x} > 0} p_{x} \ln (1 / p_{x}) = H (p)$ by [F4].

3.1step 1.1step 2.1algebra∎

Combining steps 1.1 and 2.1 gives $I (X : Y) = χ$ for the displayed orthogonal ensemble. This computation does not assert equality for a non-orthogonal ensemble.

Depends on

Used by

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Sources

John Watrous, The Theory of Quantum Information, §5.3.3 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, arXiv:1106.1445v8, §11.1 (standard reference, not scraped)