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The maximally mixed state maximizes finite quantum entropy

Statement

Let $H$ be a nonzero finite-dimensional complex inner-product space of dimension $d$ , and let $ρ$ be a density operator on $H$ . Then

$S (ρ) \leq \ln d,$

with equality if and only if $ρ = I_{H} / d$ .

Facts & Assumptions

Given: The finite-dimensional state $ρ$ and nonzero space $H$ in the Statement.

[F1]

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

[F2]

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

[F3]

Nonnegativity and equality for finite quantum relative entropy: for finite-dimensional density operators $ρ, σ$ , $D (ρ ∥ σ) \geq 0$ , with equality exactly when $ρ = σ$ .

[F4]

Density operator and Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely: states are nonnegative trace-one operators and have finite orthonormal eigenbases.

[F5]

The basis-independent trace of an endomorphism of a finite-dimensional vector space: the trace of $I_{H}$ is $d$ and the trace is linear.

[F6]

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

Proof

1.1F2F4F5givenalgebra

The operator $τ : = I_{H} / d$ is a density operator: it is nonnegative and has trace $d / d = 1$ by [F4]–[F5]. Its support is all of $H$ , so $supp (ρ) \subseteq supp (τ)$ and the finite formula in [F2] applies.

2.1F1F2F3F4F6step 1.1algebra

Spectral calculus gives $\ln τ = \ln (1 / d) I_{H} = - (\ln d) I_{H}$ by [F4], [F6]. Hence [F1]–[F2] yield $D (ρ ∥ τ) = tr (ρ \ln ρ) - tr (ρ \ln τ) = - S (ρ) + \ln d$ , using $tr (ρ) = 1$ by [F4]. By [F3], this is nonnegative, so $S (ρ) \leq \ln d$ .

3.1F1F3step 2.1algebra∎

Equality $S (ρ) = \ln d$ is equivalent by step 2.1 to $D (ρ ∥ τ) = 0$ . The equality case of [F3] then gives $ρ = τ = I_{H} / d$ . Conversely, substituting $ρ = I_{H} / d$ into [F1] gives $S (ρ) = \ln d$ .

Depends on

Used by

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Sources

John Watrous, The Theory of Quantum Information, Proposition 5.22 and §5.2.1 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, second-edition prepublication v8, Theorem 11.8.2 (standard reference, not scraped)