Jump-time dephasing vanishes with the checking interval

PDF p. 16: Final paragraph of S4

Adopt the finite lifted model, its physical and spectral hypotheses, and the Axiom of Choice, Countable Choice and Dependent Choice. Fix its finite parameters, including $\gamma >0$. Write $V_K$ for its spin-$K$ block and $H_R={⨁}_{K=0}^N{V}_K$ for the full finite model space. Choose either the specified Æ code in the $J=5$ block or the specified signed Aydin–Barg code in the $J=4$ block, with its defining logical isometry $V:H_L\to V_J\subseteq H_R$, where $H_L={\mathbb{C}}^2$. For the complete selected coherent order-one tensor error span, choose a set $S\subseteq \{0,\dots ,N\}$ containing $J$ and $J-1$ with $\mathrm{\mid }K-J\mathrm{\mid }\le 1$ for each $K\in S$. The span is the complex linear span of the identity and all embedded coherent rank-zero and rank-one spherical-tensor components from $V_J$ into these selected blocks, in the stated error-span contract on ${⨁}_{K\in S}{V}_K$. Tensor rank refers to the transformation law, not matrix rank or individually resolved magnetic transitions. A density operator is a positive semidefinite operator of trace one. Let $\rho$ be a logical density operator, start from $V\rho V^{†}$, and perform the ideal block check after an interval of length $\tau >0$. For the single-decay history, the elapsed time between the jump into the $J-1$ block and the check is at most $\tau$.

A decoded recovery channel is a complex-linear, completely positive, trace-preserving map $\mathcal{R}:\mathcal{L}(H_R)\to \mathcal{L}(H_L)$, where $\mathcal{L}(K)$ is the space of linear endomorphisms of $K$. Exact decoded recovery of a noise channel $\mathcal{N}$ means $\mathcal{R}\circ \mathcal{N}\circ {\mathrm{Ad}}_V={\mathrm{id}}_{\mathcal{L}(H_L)}$, with ${\mathrm{Ad}}_V(X)=VX{V}^{†}$. The trace norm $\mathrm{\parallel }X{\mathrm{\parallel }}_1$ is the sum of the singular values of $X$.

For sufficiently small $\tau$, a recovery channel correcting the complete selected coherent order-one tensor error span decodes the exact normalized detected-decay branch to within $O(\tau )$ of $\rho$ in trace norm, uniformly over logical density operators. The detected-branch probability is $\gamma \tau +O({\tau }^2)$, and the corresponding unnormalized decoded error is $O({\tau }^2)$. A known diagonal phase correction for an elapsed time at most $\tau$ preserves these limits. For the special lift $\mathrm{\Delta }(J,m)={\mathrm{\Delta }}_{0}Jm$, a calibrated inverse of the known null-branch rotation returns a null-branch state to its state at the start of the interval, provided the unitary control is available. Calibration means that the relevant lift values and elapsed interval are known to the recovery rule; it does not reveal the jump time.

More specifically, for the separate phase-averaging assertion let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space and $s:\mathrm{\Omega }\to \mathbb{R}$ a measurable real random variable whose common input-independent law is supported on $I=[a,a+\tau ]$. Let $\eta$ be a density operator on ${\mathbb{C}}^r$ for an integer $r\ge 1$, and suppose a normalized conditional branch is represented in its fixed finite basis by $\sigma =\mathbb{E}[D(s)\eta D(s{)}^{†}]$, where $D(s)$ has diagonal entries $e^{i{d}_{j}s}$ for fixed real $d_j$. Here $\mathbb{E}$ denotes integration against $\mathbb{P}$, entrywise for matrices. Put $m=\mathbb{E}s$ and $M={\max }_{j,k}\mathrm{\mid }{d}_j-d_k\mathrm{\mid }$; assume that $m$ and the $d_j$ are supplied to the mean-phase correction. After the known mean-phase correction $\sigma \mapsto D(m{)}^{†}\sigma D(m)$, each relative-coherence multiplier differs from $1$ by at most $M^2{\tau }^2\mathrm{/}2$. Thus this phase-averaging residual vanishes as $\tau \downarrow 0$ for fixed $M$. This conditional diagonal representation and its common waiting law are additional hypotheses; they are not asserted for an arbitrary decoded branch by the entrywise estimate.

The single-branch bounds do not alone establish a repeated-check channel limit or a multiple-decay bound.

J equals five absorption-emission code

J equals four Aydin–Barg encoding

Order-one absorption–emission tensor error span

Finite-dimensional quantum channel

Trace Norm

Exact recovery of an encoded quantum code

Probability measures and probability spaces

Expectation of a nonnegative or integrable random variable

Random elements and real random variables