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Exact state evolution from a four-level codeword

PDF p. 8: Eq. (S1.1)

Statement

Assume the four-level model with real $ω_{0}, Δ$ , decay rate $γ \geq 0$ , and $t \geq 0$ . For a normalized initial code state $∣ ψ ⟩ = a ∣ 0 ⟩ + b ∣ 1 ⟩$ , its evolved density operator is

$ρ (t) = e^{- γ t} ∣ ψ ⟩ ⟨ ψ ∣ + (1 - e^{- γ t}) (∣ a ∣^{2} ∣ 2 ⟩ ⟨ 2 ∣ + ∣ b ∣^{2} ∣ 3 ⟩ ⟨ 3 ∣) + [a b^{*} c (t) ∣ 2 ⟩ ⟨ 3 ∣ + h . c .],$

where $c (t) = γ (e^{i Δ t} - e^{- γ t}) / (γ + i Δ)$ when $(γ, Δ) \neq (0, 0)$ and $c (t) = 0$ at $(γ, Δ) = (0, 0)$ . Here h.c. denotes the Hermitian conjugate of the preceding off-diagonal term. The value at the origin is the continuous extension.

Facts & Assumptions

[F1]

In the postulated model, $H = ω_{0} (∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣) + Δ ∣ 3 ⟩ ⟨ 3 ∣$ , $L = \sqrt{γ} (∣ 2 ⟩ ⟨ 0 ∣ + ∣ 3 ⟩ ⟨ 1 ∣)$ , and the state evolves by the corresponding finite GKSL channel. Four-level spontaneous-decay model

[F2]

The ordered basis is orthonormal; bras, adjoints and outer products act according to the finite operator conventions. Finite operator and ket conventions

[F3]

The finite GKSL interface defines $N_{t} = e^{t L}$ for the stated Hermitian Hamiltonian and jump. Finite GKSL evolution and the QEC channel interface

[F4]

For finite GKSL data, the channel is the sum of the no-jump term and ordered integrals of jump amplitudes, with $G = - i H - L^{†} L / 2$ and $E (u) = e^{u G}$ . Finite no-jump and ordered-jump expansion

[F5]

A density operator on a nonzero finite-dimensional complex inner-product space is a non-negative endomorphism with trace one. Density operator

[F6]

A pure state is a ray, and a unit representative $u$ has operator representative $∣ u ⟩ ⟨ u ∣$ . Use this operator representative for the initial code state. Pure state and ray

[F7]

On a nonzero finite-dimensional complex inner-product space with the root pairing linear in its first argument, the outer product $∣ u ⟩ ⟨ u ∣$ of a unit vector is non-negative and has trace one, hence is a density operator. A unit-vector outer product is a density operator

Proof

Given: The model, normalized $∣ ψ ⟩$ , and nonnegative time in the statement.

Goal: Calculate $N_{t} (∣ ψ ⟩ ⟨ ψ ∣)$ and establish the stated extension.

1.1givenF1F2F3F5F6F7algebra

By [F6], the initial pure code state has operator representative $ρ (0) = ∣ ψ ⟩ ⟨ ψ ∣$ . Its underlying space is $C^{4}$ with the root pairing linear in its first argument, and the normalized vector $∣ ψ ⟩$ is unit. The outer-product action in [F2] is $∣ ψ ⟩ ⟨ ψ ∣ x = ψ (ψ^{†} x) = ⟨ x, ψ ⟩_{r o o t} ψ$ , which agrees with the convention in [F7]. Thus [F7] and [F5] make $ρ (0)$ a density operator. By [F1] and [F3], its physical model evolution is $ρ (t) = N_{t} (ρ (0))$ .

2.1givenF1F2F4step 1.1algebra

By [F1] and [F2], $L^{†} L = γ (∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣)$ . Thus $E (u)$ acts by $e^{(- i ω_{0} - γ / 2) u}$ on both code basis vectors, by $1$ on $∣ 2 ⟩$ , and by $e^{- i Δ u}$ on $∣ 3 ⟩$ . In particular, the no-jump contribution in [F4] is $e^{- γ t} ∣ ψ ⟩ ⟨ ψ ∣$ .

3.1givenF1F2F4step 2.1algebra

For $0 \leq s \leq t$ , the single-jump amplitude on the initial state is $E (t - s) L E (s) ∣ ψ ⟩ = \sqrt{γ} e^{(- i ω_{0} - γ / 2) s} (a ∣ 2 ⟩ + b e^{- i Δ (t - s)} ∣ 3 ⟩)$ . Its image lies in the span of $∣ 2 ⟩, ∣ 3 ⟩$ , which is preserved by $E (u)$ and annihilated by $L$ . Hence every term with two or more jumps vanishes on the initial density operator. The single-jump diagonal coefficients are $∣ a ∣^{2} (1 - e^{- γ t})$ and $∣ b ∣^{2} (1 - e^{- γ t})$ .

4.1givenF3F4step 2.1step 3.1algebra∎

The single-jump $∣ 2 ⟩ ⟨ 3 ∣$ coefficient is $a b^{*} γ \int_{0}^{t} e^{- γ s} e^{i Δ (t - s)} d s$ . For $(γ, Δ) \neq (0, 0)$ , direct integration gives $a b^{*} γ (e^{i Δ t} - e^{- γ t}) / (γ + i Δ)$ ; the opposite coefficient is its complex conjugate. At $(0, 0)$ the integral is zero, and its absolute value is at most $γ t$ , so this value is continuous as $(γ, Δ) \to (0, 0)$ within $γ \geq 0$ . Combining the no-jump and single-jump terms gives the displayed density operator.

Depends on

Used by · 3

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 8, Eq. (S1.1) and opening model paragraph