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Matrix elements of the fast detected-decay state

PDF p. 15: Eqs. (S4.7)–(S4.9)

Statement

Adopt the finite lifted model with integer blocks $J = 0, \dots, N$ , $N \geq 1$ , its stated physical and spectral hypotheses, and the Axiom of Choice, Countable Choice and Dependent Choice. In particular $γ > 0$ . Let $J_{0} \in {1, \dots, N}$ and let the density operator $ρ_{0}$ be supported in the $J_{0}$ block. A density operator is positive semidefinite with trace one. For $m, m^{'} = - J_{0}, \dots, J_{0}$ write $ρ_{m m^{'}} = ⟨ J_{0}, m ∣ ρ_{0} ∣ J_{0}, m^{'} ⟩$ . For $δ \in {- 1, 0, 1}$ let $C_{m, δ} = C_{1, δ; J_{0}, m}^{J_{0} - 1, m + δ}$ be the ordered-coupling coefficient, with out-of-range terms zero.

For a checking interval $τ \geq 0$ , let $K_{δ} = \sqrt{τ} L_{δ}$ be the first-order jump operators and perform the ideal two-projector check of the $J_{0}$ block. Its approximate unnormalized decay branch is

$B_{d e c} = γ τ \frac{2 J_{0} + 1}{2 J_{0} - 1} \sum_{m, m^{'}, δ} C_{m, δ} C_{m^{'}, δ}^{*} ρ_{m m^{'}} ∣ J_{0} - 1, m + δ ⟩ ⟨ J_{0} - 1, m^{'} + δ ∣ .$

Here and below the sums run over the displayed magnetic labels and $δ \in {- 1, 0, 1}$ ; a term with an out-of-range output label is zero. Its trace is

$Tr B_{d e c} = γ τ \frac{2 J_{0} + 1}{2 J_{0} - 1} \sum_{m} ρ_{m m} \sum_{δ} ∣ C_{m, δ} ∣^{2} = γ τ .$

When $τ > 0$ , the normalized approximate decay state is the displayed expression for $B_{d e c}$ with the factor $γ τ$ removed. At $τ = 0$ this branch has zero weight and no conditional state. These are first-order branch formulas; the exact checked channel has an $O (τ^{2})$ unnormalized output error at fixed finite model parameters.

Statement references

Finite operator and ket conventions

The trace tr(A) as the sum of the diagonal entries

General bounded diagonal lift for fast-checking analysis

First-order fast-check channel

Projective code-manifold syndrome check

Angular-momentum basis and tensor labels

Clebsch–Gordan total-rate identity

Facts & Assumptions

[F1]

The finite lifted model postulates $L_{δ} = \sqrt{γ} \sum_{J = 1}^{N} \sum_{m} \sqrt{(2 J + 1) / (2 J - 1)} C_{1, δ; J, m}^{J - 1, m + δ} ∣ J - 1, m + δ ⟩ ⟨ J, m ∣$ , with out-of-range terms zero, and $γ > 0$ . General bounded diagonal lift for fast-checking analysis

[F2]

In the first-order description $K_{δ} = \sqrt{τ} L_{δ}$ ; at fixed finite model parameters the exact channel differs from the first-order Kraus sum by $O (τ^{2})$ in induced matrix spectral norm. First-order fast-check channel

[F3]

For an input supported in the $J_{0}$ block, the ideal two-projector check selects $\sum_{δ} K_{δ} ρ_{0} K_{δ}^{†}$ as the approximate unnormalized decay branch. Its conditional state is obtained by dividing by its trace only when that trace is positive. Projective code-manifold syndrome check

[F4]

The magnetic basis is orthonormal, and the specified ordered-coupling coefficient is zero outside the target magnetic-label range. Angular-momentum basis and tensor labels

[F5]

For every admissible $m$ and $J_{0} \geq 1$ , $\sum_{δ} ∣ C_{m, δ} ∣^{2} = (2 J_{0} - 1) / (2 J_{0} + 1)$ . Clebsch–Gordan total-rate identity

[F6]

Bras, matrix elements, outer products and adjoints use the finite operator conventions: $⟨ v ∣ x ⟩ = v^{†} x$ , $∣ u ⟩ ⟨ v ∣ x = u (v^{†} x)$ , and $A^{†}$ is the conjugate transpose in the specified orthonormal bases. Finite operator and ket conventions

[F7]

In the specified orthonormal coordinates, the matrix spectral norm is the operator norm $∥ A ∥ = \max_{∥ v ∥_{2} = 1} ∥ A v ∥_{2}$ , where $∥ v ∥_{2}^{2} = \sum_{J, m} ∣ v_{J, m} ∣^{2}$ . For compatible finite matrices, $∥ A B ∥ \leq ∥ A ∥ ∥ B ∥$ . The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖ The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||²

[F8]

For a finite square matrix $A = (a_{i j})$ over $C$ , its trace is $Tr A = \sum_{i} a_{i i}$ in the specified coordinates. The trace tr(A) as the sum of the diagonal entries

[A1]

Assume the Axiom of Choice, Countable Choice and Dependent Choice, together with the physical and spectral assumptions inherited from the finite lifted model. The Axiom of Choice The Axiom of Countable Choice (AC_ω) The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain General bounded diagonal lift for fast-checking analysis

Proof

Given: The finite lifted model, supported density operator, interval and ideal check in the Statement.

Goal: Calculate the approximate decay branch, its trace and its normalized state, and state the scope of the approximation.

1.1givenF1F2F3F4F6algebra

Expand $ρ_{0} = \sum_{m, m^{'}} ρ_{m m^{'}} ∣ J_{0}, m ⟩ ⟨ J_{0}, m^{'} ∣$ in the orthonormal basis. On this block, [F1] gives $L_{δ} ∣ J_{0}, m ⟩ = \sqrt{γ (2 J_{0} + 1) / (2 J_{0} - 1)} C_{m, δ} ∣ J_{0} - 1, m + δ ⟩$ , with zero for an out-of-range label. Substituting this and $K_{δ} = \sqrt{τ} L_{δ}$ into the branch in [F3] gives the displayed double magnetic sum.

2.1givenF3F4F5F6F8step 1.1algebra

Using the diagonal-sum trace definition in [F8], for fixed $δ$ orthonormality makes the trace of $∣ J_{0} - 1, m + δ ⟩ ⟨ J_{0} - 1, m^{'} + δ ∣$ vanish unless $m = m^{'}$ . Thus the trace is the first displayed trace sum. Applying [F5] cancels the dimension ratio, and $\sum_{m} ρ_{m m} = Tr ρ_{0} = 1$ , giving $γ τ$ . For $τ > 0$ this is positive, so [F3] gives the normalized expression; for $τ = 0$ it gives no conditional state.

3.1F2F3F4F6F7step 1.1algebra∎

Write $P = Π_{J_{0}}$ and $Q = I - P$ for the check projectors. In the orthonormal block coordinates of [F4] and [F6], $Q$ deletes the $J_{0}$ coordinates, so every vector satisfies $∥ Q v ∥_{2}^{2} = \sum_{J \neq J_{0}, m} ∣ v_{J, m} ∣^{2} \leq \sum_{J, m} ∣ v_{J, m} ∣^{2} = ∥ v ∥_{2}^{2}$ . The norm definition in [F7] gives $∥ Q ∥ \leq 1$ , and its submultiplicativity gives $∥ Q X Q ∥ \leq ∥ Q ∥^{2} ∥ X ∥ \leq ∥ X ∥$ for every matrix $X$ . Let $E_{τ} = N_{τ} (ρ_{0}) - {\tilde{N}}_{τ} (ρ_{0})$ , where the two maps are the exact channel and first-order Kraus sum of [F2]. By [F3] and step 1.1, the exact decay branch minus $B_{d e c}$ is $Q E_{τ} Q$ . Thus $∥ Q E_{τ} Q ∥ \leq ∥ E_{τ} ∥ \leq ∥ N_{τ} - {\tilde{N}}_{τ} ∥ ∥ ρ_{0} ∥ = O (τ^{2})$ by the induced-norm bound in [F2], for this fixed input and finite model. The statement does not claim the normalized expression is exact at finite time.

Depends on

Used by · 2

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF page 15, Eqs. (S4.7)–(S4.9)