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A Bell pair has negative quantum conditional entropy

Example

Let $H_{A} = H_{B} = C^{2}$ with computational bases $e_{0}, e_{1}$ , let $ϕ = (e_{0} \otimes e_{0} + e_{1} \otimes e_{1}) / \sqrt{2}$ , and define the pure state $ρ_{A B} x : = ⟨ x, ϕ ⟩ ϕ$ . Then the conditional entropy is $S (A ∣ B)_{ρ} = - \ln 2$ nats.

Facts & Assumptions

Given: The two-qubit vector $ϕ$ and the rank-one state $ρ_{A B}$ in the Example.

[F1]

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

[F2]

Finite-dimensional von Neumann entropy: $S (ρ) = - \sum_{λ > 0} λ \ln λ$ .

[F3]

Density operator: a state is non-negative with trace one.

[F4]

Partial trace over a finite tensor factor: ${Tr}_{B}$ sums the matching second-factor basis indices.

[F5]

Partial trace over the first tensor factor: ${Tr}_{A}$ sums the matching first-factor basis indices.

[F6]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\ln$ is strictly increasing, $\ln 1 = 0$ , and $\ln (1 / 2) = \ln 1 - \ln 2 = - \ln 2$ .

Verification

1.1givenF2F3construct

The vector $ϕ$ is a unit vector by orthonormality of the product basis, so the rank-one projection $ρ_{A B}$ has eigenvalues $1, 0, 0, 0$ and is a state by [F3]. Hence $S (ρ_{A B}) = 0$ by [F2].

2.1givenF4F5step 1.1algebra

Expanding $ρ_{A B}$ gives diagonal terms $\frac{1}{2} P_{e_{0} \otimes e_{0}} + \frac{1}{2} P_{e_{1} \otimes e_{1}}$ and two off-diagonal cross terms. The partial traces [F4]–[F5] remove those cross terms because their traced basis indices differ, and leave $ρ_{A} = ρ_{B} = \frac{1}{2} (P_{e_{0}} + P_{e_{1}}) = I_{2} / 2$ .

3.1givenF1F2F6step 1.1step 2.1algebra∎

The eigenvalues of $I_{2} / 2$ are $1 / 2, 1 / 2$ , so [F2] and [F6] give $S (ρ_{B}) = - 2 (\frac{1}{2} \ln \frac{1}{2}) = \ln 2$ . Since $2 > 1$ , strict increase in [F6] gives $\ln 2 > \ln 1 = 0$ . Substitution into [F1] and step 1.1 yields $S (A ∣ B)_{ρ} = 0 - \ln 2 = - \ln 2 < 0$ .

Depends on

Used by

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Sources

John Watrous, The Theory of Quantum Information, §5.2.2 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, arXiv:1106.1445v8, §11.7 (standard reference, not scraped)