Experimentally Measurable Quantification of Quantum HS° Resource Consumption

Release label: v2.0 (entropy-production identity + tick-canonical residual model aligned with phase-window stacks). This note provides a measurement-facing way to quantify temporal/phase alignment consumption in the Quantum HS° framework, which models alignment without an external synchronized clock as a U(1)-asymmetry resource theory. Free states are time-translation invariant, and free operations are time-translation covariant channels. A central clarification is that applying a deterministic group shift \( \rho \mapsto U(\alpha)\rho U(\alpha)^\dagger \) 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 (unconditioned output), residual uncertainty induces an effective phase-diffusion channel (a mixture of shifts). The key v2.0 contribution is a general identity for this setting: Entropy-production identity: for phase diffusion, HS° consumption equals entropy production, \[ \Delta A_{\mathrm{HS}} = A_{\mathrm{HS}}(\rho_{\mathrm{in}})-A_{\mathrm{HS}}(\rho_{\mathrm{out}}) = S(\rho_{\mathrm{out}})-S(\rho_{\mathrm{in}}), \] and for pure inputs this reduces to \( \Delta A_{\mathrm{HS}} = S(\rho_{\mathrm{out}}) \). To connect directly with implementation-facing phase-window systems, v2.0 introduces a discrete “ticks” residual model (a \(\mathbb{Z}_R\) mixture of shifts) aligned with tick-canonical phase conventions used in practical coordination stacks. In this model, for a finite-dimensional maximally asymmetric clock input, the output eigenvalues equal the residual tick distribution \(p\), yielding closed forms: \( \Delta A_{\mathrm{HS}} = H(p) \) (Shannon entropy of the residual tick error), - \( \mathrm{Tr}(\rho_{\mathrm{out}}^2) = \sum_k p_k^2 \) (purity directly tied to the residual distribution). This enables protocol-facing reporting: for a phase-window acceptance set \(W\), the success probability is \(p_{\mathrm{succ}}=\sum_{k\in W}p_k\), giving conservative lower bounds on the consumed resource from window success rates alone. The note also provides experimentally accessible proxies and bounds via coherence harmonics/visibility and via purity (Rényi-2 lower bound), supporting reproducible reporting under explicitly stated assumptions. Interoperability bridge (related specs): - Phase-Coordination Series Conventions (tick-canonical windows): https://doi.org/10.5281/zenodo.18068999 - Q-Address (macro window + micro slot, tick-canonical): https://doi.org/10.5281/zenodo.18068997 DOI (this version): https://doi.org/10.5281/zenodo.18071181 Keywords: quantum resource theory, U(1) asymmetry, temporal alignment, phase diffusion, entropy production, ticks, phase windows, visibility, purity, quantum timing, distributed quantum systems.