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Finite quantum mutual information identities and the Holevo identity

Statement

For a finite-dimensional bipartite state $ρ_{A B}$ ,

$I (A : B)_{ρ} = S (ρ_{A}) + S (ρ_{B}) - S (ρ_{A B}) = D (ρ_{A B} ∥ ρ_{A} \otimes ρ_{B}) .$

For a finite classical-quantum state $ρ_{X B} = \sum_{x} p_{x} Π_{x} \otimes ρ_{x}$ , its mutual information equals its Holevo information:

$I (X : B)_{ρ} = S ⁣ (\sum_{x} p_{x} ρ_{x}) - \sum_{x : p_{x} > 0} p_{x} S (ρ_{x}) = χ ({p_{x}, ρ_{x}}) .$

Facts & Assumptions

Given: The finite-dimensional states in the Statement, with the root linear-first convention.

[F1]

Finite quantum mutual, conditional, and conditional mutual information: $I (A : B)_{ρ} = S (ρ_{A}) + S (ρ_{B}) - S (ρ_{A B})$ and $I (X : B)_{ρ} = S (ρ_{X}) + S (ρ_{B}) - S (ρ_{X B})$ .

[F2]

Finite-dimensional quantum relative entropy: on support inclusion, $D (A ∥ B) = tr (A \ln A) - tr (A \ln B)$ with support-restricted spectral logarithms.

[F3]

Joint state support is contained in the tensor product of its marginal supports: $supp (ρ_{A B}) \subseteq supp (ρ_{A} \otimes ρ_{B})$ , so the QRE in the Statement is finite.

[F4]

Finite classical-quantum states and Holevo information: $ρ_{X B} = \sum_{x} p_{x} Π_{x} \otimes ρ_{x}$ , $ρ_{B} = \sum_{x} p_{x} ρ_{x}$ , and $χ = S (ρ_{B}) - \sum_{p_{x} > 0} p_{x} S (ρ_{x})$ .

[F5]

The partial trace is characterized by the trace pairing and is basis-independent: $tr (({Tr}_{B} R) T) = tr (R (T \otimes I_{B}))$ .

[F6]

Partial trace over the first tensor factor: ${Tr}_{A}$ is the first-factor partial trace and has entries $({Tr}_{A} R)_{j j^{'}} = \sum_{i} R_{(i, j), (i, j^{'})}$ .

[F7]

Partial trace over a finite tensor factor: the second-factor partial trace has entries $({Tr}_{B} R)_{i i^{'}} = \sum_{j} R_{(i, j), (i^{'}, j)}$ .

[F8]

Ordered tensor registers, their inner product, and wire permutations: tensor product bases retain the displayed factor order and use the product inner product.

[F9]

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: non-negative operators have orthonormal eigenbases with nonnegative eigenvalues.

[F10]

Finite-dimensional von Neumann entropy: entropy is $- tr (R \ln R)$ , with zero-eigenvalue terms omitted.

[F11]

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

Proof

1.1givenF2F3

By [F3], the support condition in [F2] holds for $ρ_{A B}$ and $ρ_{A} \otimes ρ_{B}$ , so the relative entropy is evaluated by its finite trace formula.

2.1givenF2F8F9F11step 1.1algebrachoose

Choose orthonormal eigenbases of the marginals using [F9], with positive eigenvalues $a_{i}$ for $ρ_{A}$ on its support and $b_{j}$ for $ρ_{B}$ on its support. Their product vectors form an orthonormal eigenbasis of $ρ_{A} \otimes ρ_{B}$ on its support by [F8], with eigenvalues $a_{i} b_{j} > 0$ . Since [F11] gives $\ln (a_{i} b_{j}) = \ln a_{i} + \ln b_{j}$ , spectral calculus gives $\ln (ρ_{A} \otimes ρ_{B}) = (\ln ρ_{A}) \otimes I_{B} + I_{A} \otimes (\ln ρ_{B})$ on that support.

3.1F1F2F3F5F6step 2.1algebra

Extend each marginal logarithm by zero on its kernel, which does not affect the support-restricted traces by [F2] and [F3]. Taking $T = \ln ρ_{A}$ in [F5] gives $tr (ρ_{A B} ((\ln ρ_{A}) \otimes I_{B})) = tr (ρ_{A} \ln ρ_{A})$ . For the second marginal, expanding the finite product-basis trace and using the entries in [F6] gives $tr (ρ_{A B} (I_{A} \otimes \ln ρ_{B})) = tr (ρ_{B} \ln ρ_{B})$ . Combining these identities with [F2] and step 2.1 yields $D (ρ_{A B} ∥ ρ_{A} \otimes ρ_{B}) = S (ρ_{A}) + S (ρ_{B}) - S (ρ_{A B}) = I (A : B)_{ρ}$ .

4.1givenF1F4F7F9F10F11step 2.1algebra∎

In the orthogonal blocks of $ρ_{X B}$ , the eigenvalues are $p_{x} λ_{x j}$ , where $λ_{x j}$ are the eigenvalues of $ρ_{x}$ ; zero-probability blocks contribute only zeros. By [F7] and $tr ρ_{x} = 1$ , $ρ_{X} = \sum_{x} p_{x} Π_{x}$ ; its eigenvalues are $(p_{x})$ , and the $B$ -marginal is ${\overset{ˉ}{ρ}}_{B} = \sum_{x} p_{x} ρ_{x}$ by [F4]. The block eigenvalues follow from [F9]. For $p_{x} > 0$ and $λ_{x j} > 0$ , applying [F11] and expanding $- (p_{x} λ_{x j}) \ln (p_{x} λ_{x j})$ and using $\sum_{j} λ_{x j} = 1$ gives $S (ρ_{X B}) = S (ρ_{X}) + \sum_{x : p_{x} > 0} p_{x} S (ρ_{x})$ by [F10]. Substitution into the definition [F1] cancels $S (ρ_{X})$ and proves $I (X : B)_{ρ} = χ ({p_{x}, ρ_{x}})$ .

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Sources

John Watrous, The Theory of Quantum Information, §§5.2.1–5.2.2 and §5.3.3 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, arXiv:1106.1445v8, §§11.1 and 11.8 (standard reference, not scraped)