Fig. 2
From: Achieving the Heisenberg limit in quantum metrology using quantum error correction
Schematic illustration of HNLS and code optimization. a \(G_ \bot \) is the projection of G onto \({\cal S}\) in the Hilbert space of Hermitian matrices equipped with the Hilbert-Schmidt norm \(\sqrt {{\mathrm{tr}}(O \cdot O)}\). \(G_ \bot \ne 0\) if and only if \(G \notin {\cal S}\), which is the HNLS condition. b \(\tilde G^\diamondsuit \) is the projection of G onto \({\cal S}\) in the linear space of Hermitian matrices equipped with the operator norm \(\left\| O \right\| = {\mathrm{max}}_{\left| \psi \right\rangle }{\kern 1pt} \left\langle \psi \right|O\left| \psi \right\rangle \). In general, the optimal QEC code can be contructed from \(\tilde G^\diamondsuit \) and \(\tilde G^\diamondsuit \) is not necessarily equal to \(G_ \bot \)
