HS° Resource Consumption (v2.0)
Entropy-Production and Tick-Canonical Costs for Phase-Window Temporal Alignment.
Abstract
Quantum HS◦ models 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. A key point is that applying a deterministic group shift ρ ↦ → 𝑈(𝛼)ρ𝑈(𝛼)† does not consume HS◦; consumption arises from the estimation–correction loop and the residual uncertainty it leaves behind. When the measurement record is ignored, residual uncertainty induces an effective phase-diffusion channel (a mixture of shifts). We prove a general consumption identity for phase diffusion: Δ𝐴HS = 𝑆(ρout) − 𝑆(ρin), i.e. HS◦ consumption equals entropy production. To connect directly with phase-window stacks, we introduce a discrete “ticks” residual model (Z𝑅 mixture of shifts): for the maximally asymmetric input, the output eigenvalues equal the residual tick distribution 𝑝, hence Δ𝐴HS = 𝐻(𝑝) and Tr(ρ²out) = Σ𝑘 𝑝²𝑘 . This yields protocol-facing success-probability links and experimentally accessible proxies/bounds via coherence harmonics and purity, enabling reproducible reporting of HS◦ expenditure under stated assumptions.
1. Scope and Interoperability
This note is QIT-first: it quantifies HS◦ consumption under 𝑈(1)-covariant phase diffusion. For interoperability with phase-window stacks, residual uncertainty may be reported in a tick-canonical form consistent with Conventions and Q-Address.
2. Quantum HS° Recap
The HS◦ asymmetry monotone is defined as 𝐴HS(𝜌) = 𝑆(T(𝜌)) − 𝑆(𝜌). A deterministic group shift does not consume HS◦; consumption arises from estimation–correction with discarded outcomes, residual mixing, noise, or discarding subsystems.
3. Alignment as Phase Diffusion
Phase-diffusion channel: Let 𝑝(˜𝛼) be a normalized density on the circle. Define N(𝜌) = ∫d˜𝛼 𝑝(˜𝛼)𝑈(˜𝛼)𝜌𝑈(˜𝛼)†.
4. General Entropy-Production Identity
Twirl Invariance and Entropy Production: For the phase-diffusion channel, T(N(𝜌)) = T(𝜌). Consequently, Δ𝐴HS := 𝐴HS(𝜌) − 𝐴HS(N(𝜌)) = 𝑆(N(𝜌)) − 𝑆(𝜌) ≥ 0. If the input state 𝜌 is pure, then Δ𝐴HS = 𝑆(N(𝜌)).
5. Discrete Ticks Model and Closed-Form Cost
Tick Residual Model on Z𝑅: Fix a tick resolution R. A discrete residual model is N(𝜌) = Σ𝑝𝑘 𝑈𝑘 𝜌𝑈†𝑘.
Closed-Form Cost: Assuming R = M and a maximally asymmetric input, the output 𝜌out has eigenvalues equal to the residual tick distribution 𝑝. Therefore, the cost is Δ𝐴HS = 𝐻(𝑝), and the purity is Tr(𝜌²out) = Σ𝑝²𝑘.
6. Protocol-Facing Link to Phase Windows
Let a phase-window instruction accept a set of ticks W. The success probability is 𝑝succ := Σ𝑘∈W 𝑝𝑘. For any distribution 𝑝, the Shannon entropy satisfies 𝐻(𝑝) ≥ ℎ₂(𝑝succ), where ℎ₂ is the binary entropy. This is useful when a system reports only window success rates but not the full residual distribution.
7. Measurement Routes and Reporting
A reproducible report should specify: the log base, clock dimension, measurement conditions, measured proxies (like visibility or purity), and the residual model used.