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Coherent-information concavity and subadditivity for degradable channels

Statement

Let $N_{i} : A_{i} \to B_{i}$ for $i = 1, 2$ be finite-dimensional degradable channels with complements $N_{i}^{c} = D_{i} \circ N_{i}$ . Then $ρ \mapsto I_{c} (ρ, N_{i})$ is concave and nonnegative. For every joint input state $ρ_{A_{1} A_{2}}$ ,

$I_{c} (ρ_{A_{1} A_{2}}, N_{1} \otimes N_{2}) \leq I_{c} (ρ_{A_{1}}, N_{1}) + I_{c} (ρ_{A_{2}}, N_{2}) .$

Consequently the supremum coherent information of $N_{1} \otimes N_{2}$ is the sum of the two single-channel suprema.

Facts & Assumptions

Given: The finite-dimensional degradable channels and joint input state in the Statement.

[F1]

Degradable and antidegradable channels: $N_{i}^{c} = D_{i} \circ N_{i}$ for a finite-dimensional channel $D_{i}$ .

[F2]

Private information is an average coherent-information difference: $I_{p} (E, N) = I_{c} (\overset{ˉ}{ρ}, N) - \sum_{x} p_{x} I_{c} (ρ_{x}, N)$ .

[F3]

Finite quantum mutual information identities and the Holevo identity: for a finite classical-quantum state, its mutual information equals its Holevo information.

[F4]

Data processing for finite quantum mutual information: local channels on either register cannot increase finite quantum mutual information.

[F5]

Coherent information equals output entropy minus environment entropy: $I_{c} (ρ, N) = S (N (ρ)) - S (N^{c} (ρ))$ .

[F6]

Finite quantum mutual, conditional, and conditional mutual information: quantum mutual information is defined by $I (X : Y) = S (X) + S (Y) - S (X Y)$ .

[F7]

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely: every finite-dimensional density operator is a finite mixture of pure eigenstates.

[F8]

Product additivity of coherent and private information: coherent information is additive on product input states for product channels.

[F9]

Schmidt decomposition: the receiver and environment marginals of the pure Stinespring output of a pure input have the same positive eigenvalues.

[F10]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\ln (x y) = \ln x + \ln y$ for positive $x, y$ ; it supports the cq entropy identity stated in [F3].

Proof

1.1F1F2F3F4F10givenalgebra

Fix a finite input ensemble $E = {p_{x}, ρ_{x}}$ with average $\overset{ˉ}{ρ}$ . By [F2] and [F3], $I_{p} (E, N_{i}) = I (X; B_{i}) - I (X; E_{i})$ , where $B_{i}$ and $E_{i}$ denote the receiver and complementary output registers for the ensemble. The finite cq entropy calculation uses the block eigenvalues $p_{x} λ_{x j}$ and the logarithm product law [F10]. Since $E_{i}$ is obtained from $B_{i}$ by $D_{i}$ in [F1], [F4] gives $I (X; E_{i}) \leq I (X; B_{i})$ . Thus $I_{p} (E, N_{i}) \geq 0$ .

1.2F1F2F4F5F6algebra

For a joint state $ρ_{A_{1} A_{2}}$ , write $ρ_{i}$ for its marginals, $ω_{B_{1} B_{2}} = (N_{1} \otimes N_{2}) (ρ)$ , and $η_{E_{1} E_{2}} = (N_{1}^{c} \otimes N_{2}^{c}) (ρ)$ . Expanding the entropies in [F5] and using [F2], [F6] gives $I_{c} (ρ, N_{1} \otimes N_{2}) - I_{c} (ρ_{1}, N_{1}) - I_{c} (ρ_{2}, N_{2}) = - I (B_{1} : B_{2})_{ω} + I (E_{1} : E_{2})_{η}$ . The tensor product $D_{1} \otimes D_{2}$ maps $ω$ to $η$ , so [F4] gives $I (E_{1} : E_{2})_{η} \leq I (B_{1} : B_{2})_{ω}$ . The right side is therefore nonpositive, proving the claimed inequality.

2.1F2step 1.1algebra

Rewrite the inequality $I_{p} (E, N_{i}) \geq 0$ using [F2]. For every finite ensemble with average $\overset{ˉ}{ρ}$ , this yields $I_{c} (\overset{ˉ}{ρ}, N_{i}) \geq \sum_{x} p_{x} I_{c} (ρ_{x}, N_{i})$ , which is exactly concavity of coherent information on the finite-dimensional state space.

3.1F5F7F9step 2.1algebraconstruct

Let $ρ = \sum_{j} λ_{j} ∣ u_{j} ⟩ ⟨ u_{j} ∣$ be a spectral decomposition from [F7]. For each pure $∣ u_{j} ⟩$ , an isometric Stinespring dilation produces a pure receiver-environment vector, so the two marginal entropies agree by [F9] and [F5] gives $I_{c} (∣ u_{j} ⟩ ⟨ u_{j} ∣, N_{i}) = 0$ . Concavity from step 2.1 therefore implies $I_{c} (ρ, N_{i}) \geq \sum_{j} λ_{j} \cdot 0 = 0$ .

4.1F8step 1.2algebrachoose∎

The inequality in step 1.2 gives the upper bound on the optimized coherent information. Product inputs give equality by [F8]; choosing inputs within arbitrary $δ > 0$ of each finite supremum and letting $δ ↓ 0$ proves additivity of the supremum.

The proof uses the finite entropy-identity and strong-subadditivity interfaces listed above. Their source status does not alter the stated mathematical hypotheses.

Depends on

Used by

Single-letter quantum and private capacities of degradable channels Theorem

Dependency tree · two levels

60 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, Exercise 8.3 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, second-edition prepublication v8, §§13.5 and 24.8 (standard reference, not scraped)