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Pauli-complementary qubit recovery fidelity inequality

PDF p. 9: Optimality argument after Eq. (S1.9)

Statement

Let $H = C^{2}$ with standard ordered basis, and let $R : L (H) \to L (H)$ be a quantum channel, meaning a complex-linear, completely positive, trace-preserving map. Let $U$ be a unitary matrix, let $Z = diag (1, - 1)$ , and write ${Ad}_{V} (X) = V X V^{†}$ . For entanglement fidelity $F_{e}$ at the maximally mixed input $I / 2$ ,

$F_{e} (R \circ {Ad}_{U}) + F_{e} (R \circ {Ad}_{Z U}) \leq 1.$

Here $V^{†}$ is the conjugate transpose, and composition applies the right-hand channel first.

Statement references

Finite operator and ket conventions

Entanglement fidelity on a finite logical space

Finite-dimensional quantum channel

Facts & Assumptions

Given: The qubit space, channel $R$ , unitary $U$ , and matrix $Z$ in the Statement.

Goal: Prove that the displayed sum of entanglement fidelities is at most one.

[F1]

The finite operator conventions use the standard basis and conjugate-transpose adjoints. Finite operator and ket conventions

[F2]

A quantum channel is complex-linear, completely positive and trace-preserving; $F_{e}$ is the maximally entangled-state overlap for the specified maximally mixed input. Finite-dimensional quantum channel Entanglement fidelity on a finite logical space

[F3]

For a qubit channel $R$ , unitary $U$ , and $Z = diag (1, - 1)$ , the sum $F_{e} (R \circ {Ad}_{U}) + F_{e} (R \circ {Ad}_{Z U})$ is at most one, with fidelities evaluated at $I / 2$ . Two orthogonal Pauli alternatives bound recovery fidelity

Proof

1.1givenF1F2F3

The given $H$ , $R$ , $U$ , and $Z$ satisfy the hypotheses of [F3]: the space is a qubit, $R$ is a quantum channel, $U$ is unitary, and $Z$ has the required diagonal entries. The adjoint and fidelity conventions are those in [F1] and [F2].

2.1step 1.1F3∎

Applying [F3] to these data gives the displayed inequality, with both fidelities evaluated at $I / 2$ , as required.

Depends on

Used by · 1

Optimal phase-corrected four-level recovery

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 9, Comparison of Recoveries and Fidelity Measures, paragraph after Eq. (S1.9)