Quantum code as an isometric encoding and its code projector

Let ${\mathcal{H}}_L$ be a finite-dimensional complex inner-product space and let ${\mathcal{H}}_P={⨂}_{j<n}{\mathcal{H}}_j$ be a fixed ordered finite register tensor product with $n\ge 1$, supplied by the QC register interface. A quantum code is an isometric encoding $V:{\mathcal{H}}_L\to {\mathcal{H}}_P$. Its code subspace is $\mathcal{C}:=\mathrm{im}V$, its logical dimension is $K:=\dim {\mathcal{H}}_L$, and its physical block length is the number $n$ of tensor factors. When each physical factor is a qubit, ${\mathcal{H}}_P\cong ({\mathbb{C}}^2{)}^{\otimes n}$; a logical $k$-qubit register has $K=2^k$. Require $K\ge 1$.

Using the root library's adjoint convention, the code projector is $P:=V{V}^{\ast }\in \mathrm{End}({\mathcal{H}}_P)$. The theorem An isometric quantum encoding gives an orthogonal code projector ↗ proves $V^{\ast }V=I_{{\mathcal{H}}_L}$, $P^{\ast }=P=P^2$, and $\mathrm{im}P=\mathcal{C}$, including one-dimensional logical spaces and the full-space code.

The QC02 suppliers set $Q_d={\mathbb{C}}^d$ for $d\ge 1$, order qubit strings lexicographically with the first wire most significant, and distinguish the empty register $\mathbb{C}$ from ${\mathbb{C}}^0$ (Finite registers and their ordered computational bases). Ordered tensor registers use the factorwise product pairing, linear in the first argument, and declared wire permutations have inverse permutations (Ordered tensor registers, their inner product, and wire permutations). These are the conventions used here.