Experimentally Measurable Quantification of Quantum HS° Resource Consumption (v1.0)

Quantum HS° formalizes temporal (phase) alignment without an external clock as a U(1)-asymmetry resource theory: free states are time-translation invariant, and free operations are time-translation covariant channels. This note provides a compact, measurement-facing way to quantify resource consumption during an alignment attempt. A key point is that a deterministic group shift (rho → U(alpha) rho U(alpha)†) does not consume HS° resource; consumption arises from the estimation–correction loop and the residual uncertainty it leaves behind. When the measurement record is ignored, the residual uncertainty induces an effective phase-diffusion channel, i.e., a mixture of shifts. For a finite-dimensional clock of dimension M prepared in a maximally asymmetric state, we derive: (i) the analytic baseline A_HS(rho_in) = log M; (ii) the identity Delta A_HS = S(rho_out) under phase diffusion for this input family; and (iii) experimentally accessible proxies and lower bounds based on adjacent-mode coherence visibility and output purity. These tools enable reproducible reporting of HS° expenditure under clearly stated noise assumptions.