Operational checking time

PDF p. 2: Checking-time definition after Eq. (5)

Let $H_L={\mathbb{C}}^d$ for an integer $d\ge 1$, and let $H_P,H_O$ be nonzero finite-dimensional complex inner-product spaces. Write $\mathcal{L}(H)$ for the linear operators on $H$. A code is specified by a linear isometry $V:H_L\to H_P$, with code projector $P=V{V}^{\ast }$. For each elapsed time $t\ge 0$, let the accumulated noise ${\mathcal{N}}_t:\mathcal{L}(H_P)\to \mathcal{L}(H_O)$ be a quantum channel, meaning a linear, completely positive, trace-preserving map. Let ${\mathcal{T}}_t:\mathcal{L}(H_O)\to \mathcal{L}(H_P)$ be a recovery channel supported on the code, meaning $P{\mathcal{T}}_t(Y)P={\mathcal{T}}_t(Y)$ for every $Y$. A fixed recovery is included by taking ${\mathcal{T}}_t=\mathcal{T}$ for all $t$.

Define the recovered logical channel by ${\mathcal{E}}_t(X)=V^{\ast }{\mathcal{T}}_t({\mathcal{N}}_t(VX{V}^{\ast }))V$. With a reference copy of $H_L$, set $\mathrm{\mid }{\mathrm{\Phi }}_d⟩=d^{-1\mathrm{/}2}{\sum }_{j=0}^{d-1}\mathrm{\mid }j⟩\otimes \mathrm{\mid }j⟩$, where the vectors $\mathrm{\mid }j⟩$ form the standard orthonormal basis. The recovered entanglement fidelity for the maximally mixed logical input is $F_{\mathcal{T}}(t)=⟨{\mathrm{\Phi }}_d\mathrm{\mid }(\mathrm{i}\mathrm{d}\otimes {\mathcal{E}}_t)(\mathrm{\mid }{\mathrm{\Phi }}_d⟩⟨{\mathrm{\Phi }}_d\mathrm{\mid })\mathrm{\mid }{\mathrm{\Phi }}_d⟩$.

For a target $F_0\in (0,1]$, the checking time is $\tau (F_0)=\inf \{t\ge 0:F_{\mathcal{T}}(t)<F_0\}$, with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. The article takes the recovery to be the transpose, or Petz, recovery; for a time-dependent noise family this may be constructed separately from each ${\mathcal{N}}_t$ and the same code projector.

Code restriction and decoded channel types

Entanglement fidelity on a finite logical space

Transpose (Petz) recovery on a code

Greatest lower bound (infimum)

Every nonempty set bounded below has an infimum

The code-supported recovery induces a decoder, and composing it with the noise and encoding gives the logical channel ${\mathcal{E}}_t$, by Code restriction and decoded channel types. The displayed overlap is therefore well-defined by Entanglement fidelity on a finite logical space. The crossing set is a subset of $[0,\mathrm{\infty })$; if nonempty, it is bounded below by zero and has an infimum by math:thm-infimum-property. The empty-set convention supplies the remaining case. The finite-dimensional transpose construction supplies a code-supported recovery for each noise channel as stated in Transpose (Petz) recovery on a code.

The article interprets this threshold as the time scale for checking often enough to maintain the target for the specified recovery. The definition alone does not establish a repeated-cycle performance bound. The infimum need not itself belong to the strict crossing set, and no monotonicity or continuity of fidelity is assumed. In particular, $\tau =+\mathrm{\infty }$ means that no time in the specified family has fidelity strictly below the target; a finite $\tau$ is the boundary of the first such times, not necessarily a time at which the strict inequality holds.