Detectable operators and distance of a finite-register quantum code

Let $V:{\mathcal{H}}_L\to {\mathcal{H}}_P$ be an isometric encoding into a fixed ordered $n$-factor register with $n\ge 1$, $K:=\dim {\mathcal{H}}_L\ge 1$, and code projector $P=V{V}^{\ast }$. An operator $E\in \mathrm{End}({\mathcal{H}}_P)$ is detectable by the code if there is a scalar ${\lambda }_E\in \mathbb{C}$ such that

$PEP={\lambda }_{E}P\mathrm{.}$

When $K\ge 2$, define the quantum distance by

$d:=\min \{\mathrm{wt}(E):E\in \mathrm{End}({\mathcal{H}}_P),PEP\mathrm{\notin }\mathbb{C}P\},$

where $\mathbb{C}P:=\{\lambda P:\lambda \in \mathbb{C}\}$.

When $K=1$, set $d:=n+1$ by convention; the non-detectable set is empty in that case, as proved in Detectable operators form a subspace and quantum distance is well defined ↗. The same theorem proves that the minimum for $K\ge 2$ exists, the condition is linear in $E$, and changing $V$ to $VU$ for a logical unitary $U$ leaves $d$ unchanged.

This definition distinguishes detection from recovery: no recovery map or correction criterion is asserted here. In particular the distance theorem on this page only proves detection below $d$; correction maps are deferred to QC19.

The convention $n+1$ for a one-dimensional logical space is a sentinel larger than every possible operator weight. It avoids treating a minimum of an empty set as an ordinary integer minimum.