LemmaStatement: AI-generatedProof: AI-generated

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Kraus trace formula for maximally mixed entanglement fidelity

Statement

Let $H = C^{d}$ , $d \geq 1$ , and let $E (X) = \sum_{a} K_{a} X K_{a}^{†}$ be a channel on $H$ . Then $F_{e} (E) = d^{- 2} \sum_{a} ∣ tr K_{a} ∣^{2}$ .

Statement references

Finite-dimensional quantum channel

Facts & Assumptions

Given: The spaces, maps and operators in the Statement.

Goal: Prove the asserted identities and conclusions.

[F1]

Entanglement fidelity is the overlap with the normalized vector $Φ_{d} = d^{- 1 / 2} \sum_{j} ∣ j ⟩ \otimes ∣ j ⟩$ . Entanglement fidelity on a finite logical space

[F2]

A Kraus representation expresses the channel as a finite sum of conjugations. Finite Kraus representation

[F3]

The trace of an endomorphism is the trace of its matrix in any ordered basis, hence the sum of its diagonal entries. The basis-independent trace of an endomorphism of a finite-dimensional vector space

[F4]

The tensor map $I \otimes K$ is the unique linear map with $(I \otimes K) (u \otimes v) = u \otimes K v$ . Tensor products of finitely many linear maps

Proof

technique · direct

1.1givenF1F3F4

For any operator $K$ on $H$ , expansion in the standard basis gives $⟨ Φ_{d} ∣ (I \otimes K) ∣ Φ_{d} ⟩ = d^{- 1} \sum_{j} ⟨ j ∣ K ∣ j ⟩ = d^{- 1} tr K$ .

2.1F1F2F4step 1.1∎

Apply the Kraus expansion to the second tensor factor of $∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣$ . Each summand contributes the squared modulus of the overlap in step 1.1. Summing proves the stated trace formula.

Depends on

Used by · 1

Feasibility and range of transpose entanglement fidelity

Dependency tree · 2 levels

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