ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Entanglement-assisted capacity of a noiseless qubit

Example

For the identity channel on a qubit, the entanglement-assisted classical capacity is $2 \ln 2$ nats per use, or two classical bits per transmitted qubit. A shared Bell pair and the four-message dense-coding protocol attain the value.

Facts & Assumptions

Given: The noiseless identity channel ${id}_{C^{2}}$ and prior shared entanglement.

[F1]

Additive entanglement-assisted classical capacity formula: $C_{E} (N) = \max_{ρ} [S (ρ) + I_{c} (ρ, N)]$ .

[F2]

Complementary channels and coherent information: $I_{c} (ρ, N) = S (B) - S (R B)$ for a purification.

[F3]

The maximally mixed state maximizes finite quantum entropy: for a qubit, $S (ρ) \leq \ln 2$ , with equality at $I / 2$ .

[F4]

The Bell vectors: the four Bell vectors form an orthonormal basis of $C^{2} \otimes C^{2}$ .

[F5]

Dense-coding correctness and resources: the four Pauli encodings on one half of a shared Bell pair produce the four orthogonal Bell states, which Bob distinguishes with zero error using one transmitted qubit.

Verification

1.1F1F2F3givenalgebra

For any qubit input $ρ$ , the identity channel leaves its purification $ψ_{R A}$ pure. Hence $S (R B) = 0$ and $S (B) = S (ρ)$ , so [F2] gives $I_{c} (ρ, id) = S (ρ)$ . By [F1] and [F3], $C_{E} (id) = \max_{ρ} 2 S (ρ) = 2 \ln 2$ .

2.1F4F5step 1.1algebra∎

For a maximally entangled Bell pair, Alice's four Pauli choices produce the four orthogonal Bell states by [F4]–[F5]. Bob's Bell-basis measurement identifies the message without error, so four messages are sent per channel use and the rate is $\ln 4 = 2 \ln 2$ . This attains the capacity computed in step 1.1.

Depends on

Used by

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Dependency tree · two levels

55 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, §6.3.1 and Theorem 8.41 (standard reference, not scraped)
Bennett, Shor, Smolin, and Thapliyal, Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem, §II (standard reference, not scraped)