Clock basis and calibration

Choose an orthonormal clock basis

Choose an injective calibration

The stored object is the finite code $k$. The real number $t_k$ is its declared interpretation; no finite register is asserted to contain every real number exactly.

Let $B_C$ be the calibrated basis-change unitary satisfying

Reading the clock means applying $B_C$, then destructive computational-basis Enter to every name in $C$. If the clock basis is already the computational basis, $B_C=I_C$.

The logical clock Hamiltonian need not be diagonal in the computational presentation. Indeed, confusing an energy basis with a time-reading basis can destroy the desired dynamical interpretation; see Example ex:energy-basis.