Code, preparation, and coherent loading

Let

Fix an isometry

whose range lies in the subspace with at most $K$ particles, where $1\le K\le n$. For a physical parity-respecting implementation, require $\mathrm{Ran}W$ to have one fixed fermion parity.

An especially transparent choice is

with $1\le k<n$. This code has a common initial source energy $k{\omega }_0$.

The source and reservoir are initially in vacuum before loading. A separate carrier with one-particle space ${\mathbb{C}}^n$ holds $W\rho W^{†}$. Let $S$ be the one-particle unitary that swaps the carrier modes with the source modes. The fermionic second quantization $\Gamma (S)$ is a number-conserving, parity-even unitary.