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Independent transpose Gram fidelity formula

Statement

Let $H_{L} = C^{d}$ with $d \geq 1$ , and let $H_{P}, H_{O}$ be nonzero finite-dimensional complex coordinate spaces. Let $V : H_{L} \to H_{P}$ be an isometry, $P = V V^{†}$ , and let $N : L (H_{P}) \to L (H_{O})$ be a quantum channel with finite Kraus representation $N (X) = \sum_{l} E_{l} X E_{l}^{†}$ . Let $H_{E}$ have an orthonormal basis indexed by the Kraus operators, and define $A : H_{L} \otimes H_{E} \to H_{O}$ by $A ∣ μ, l ⟩ = E_{l} V ∣ μ ⟩$ . Put $M = A^{†} A$ and $Q = N (P)$ . Let $Q^{- 1 / 2}$ be the inverse square root on the support of $Q$ , extended by zero on its kernel, and let $T$ be the support-completed transpose recovery with uncompleted Kraus operators $P E_{a}^{†} Q^{- 1 / 2}$ . Decode its code-supported output by $V^{†}$ and call the resulting logical channel $E$ .

Then the entanglement fidelity of $E$ at the maximally mixed logical input is $F_{e} (E) = d^{- 2} ∥ {Tr}_{L} \sqrt{M} ∥_{F}^{2}$ . Here ${Tr}_{L}$ traces the first, logical factor, and $∥ B ∥_{F}^{2} = \sum_{a, l} ∣ B_{a, l} ∣^{2}$ in the specified error basis. The formula includes singular $M$ and $Q$ .

Statement references

Finite operator and ket conventions

Code restriction and decoded channel types

Entanglement fidelity on a finite logical space

Finite support square roots and generalized inverses

QEC matrix as an error-vector Gram operator

Support-normalized transpose recovery and completion

Finite Kraus representation

Partial trace over the first tensor factor

Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison

Facts & Assumptions

[F1]

The coordinate conventions use conjugate-transpose adjoints and ordered product bases. Finite operator and ket conventions

[F2]

The isometric encoding, code projector and decoding of a code-supported recovery give a logical channel. Code restriction and decoded channel types

[F3]

Entanglement fidelity is the overlap of the maximally entangled vector $∣ Φ_{d} ⟩ = d^{- 1 / 2} \sum_{μ} ∣ μ, μ ⟩$ with its image under the identity on a reference tensor the logical channel. Entanglement fidelity on a finite logical space

[F4]

A positive semidefinite finite operator has an orthonormal eigenbasis, its positive square root acts by the square roots of its eigenvalues, and its support inverse square root acts by reciprocal square roots on positive eigenvalues and zero on the kernel. Finite support square roots and generalized inverses

[F5]

The error-vector map has positive semidefinite Gram operator $M = A^{†} A$ , with entries $M_{[μ, a], [ν, l]} = ⟨ μ ∣ V^{†} E_{a}^{†} E_{l} V ∣ ν ⟩$ . QEC matrix as an error-vector Gram operator

[F6]

The completed transpose recovery is a channel with code-supported output, and its completion has no effect on noise outputs of code-supported operators. Support-normalized transpose recovery and completion

[F7]

A Kraus representation acts by summing conjugations by its operators. Finite Kraus representation

[F8]

First-factor partial trace has entries $({Tr}_{L} X)_{a, l} = \sum_{μ} X_{[μ, a], [μ, l]}$ . Partial trace over the first tensor factor

[F9]

The squared Frobenius norm of a finite matrix is the sum of the squared moduli of its entries. Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison

Proof

Given: The finite spaces, isometry, Kraus operators, Gram operator and support-completed recovery in the statement.

Goal: Compute the entanglement fidelity of the decoded recovered channel as the stated squared Frobenius norm.

1.1givenF1F4F5algebra

In the product basis, $A A^{†} = \sum_{l} E_{l} V V^{†} E_{l}^{†} = Q$ . Choose an orthonormal eigenbasis $(x_{j})$ of $M$ , with eigenvalues $λ_{j} \geq 0$ . If $λ_{j} = 0$ , then $∥ A x_{j} ∥^{2} = ⟨ M x_{j}, x_{j} ⟩ = 0$ , so $A x_{j} = 0$ . If $λ_{j} > 0$ , then $Q A x_{j} = A M x_{j} = λ_{j} A x_{j}$ , whence $Q^{- 1 / 2} A x_{j} = λ_{j}^{- 1 / 2} A x_{j}$ . Thus $A^{†} Q^{- 1 / 2} A x_{j} = \sqrt{λ_{j}} x_{j}$ for every $j$ , including zero eigenvalues. Consequently $A^{†} Q^{- 1 / 2} A = \sqrt{M}$ .

2.1givenF1F2F6F7F8step 1.1

By the support-completion result, the decoded channel agrees on every logical operator with the Kraus sum whose operators are $B_{a, l} = V^{†} E_{a}^{†} Q^{- 1 / 2} E_{l} V$ . The isometry identity $V^{†} P = V^{†}$ gives this expression after decoding. For each pair $a, l$ , step 1.1 and the definition of $A$ give $(\sqrt{M})_{[μ, a], [ν, l]} = ⟨ μ ∣ B_{a, l} ∣ ν ⟩$ . Therefore $tr B_{a, l} = \sum_{μ} (\sqrt{M})_{[μ, a], [μ, l]} = ({Tr}_{L} \sqrt{M})_{a, l}$ .

3.1F3F7F9step 1.1step 2.1algebra∎

Applying the Kraus sum to the second half of $∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣$ gives $F_{e} (E) = \sum_{a, l} ∣ ⟨ Φ_{d} ∣ (I \otimes B_{a, l}) ∣ Φ_{d} ⟩ ∣^{2}$ . Expanding the normalized matching-basis vector gives $⟨ Φ_{d} ∣ (I \otimes B_{a, l}) ∣ Φ_{d} ⟩ = d^{- 1} tr B_{a, l}$ . Substitution from step 2.1 and the Frobenius norm formula yield $F_{e} (E) = d^{- 2} \sum_{a, l} ∣ ({Tr}_{L} \sqrt{M})_{a, l} ∣^{2} = d^{- 2} ∥ {Tr}_{L} \sqrt{M} ∥_{F}^{2}$ . The zero-eigenvalue case was included in step 1.1, so no invertibility assumption is needed.

Depends on

Used by · 1

Transpose fidelity from the QEC matrix

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

The Near-optimal Performance of Quantum Error Correction Codes · Accepted manuscript, Theorem 1, Eq. (4), and discussion following it

The logical partial-trace expression gives a channel fidelity achievable by transpose recovery, with a QEC matrix built from encoded Kraus operators.

Hypotheses: A finite-dimensional encoded code and a quantum noise channel with a Kraus representation; channel fidelity is evaluated on a purification of the maximally mixed logical input.

Reading limits: The cited theorem and adjacent discussion were read in the accepted manuscript; the present finite-dimensional formulation, support completion and proof are supplied locally.

Sources

The Near-optimal Performance of Quantum Error Correction Codes · Accepted manuscript, Theorem 1, Eq. (4), and discussion following it