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Quantum HS◦: A Resource Theory of Temporal Alignment (v1.1)

This document defines Quantum HS◦ (Harmony Segment Degree) as a minimal quantum resource theory for temporal (phase) alignment when no external synchronized clock is available.

Status: Official Spec (v1.1)
DOI: 10.5281/zenodo.18035886

Abstract

We define Quantum HS◦ (Harmony Segment Degree) as a minimal quantum resource theory for temporal (phase) alignment when no external synchronized clock is available. The physical restriction is captured by a U(1) time/phase-translation symmetry: free operations are time-translation covariant quantum channels, and free states are time-translation invariant states. As a canonical monotone, we use the relative-entropy of temporal asymmetry AHS(ρ) = S(T(ρ)) − S(ρ), where T is the group-twirling (symmetry-averaging) channel. We show (i) equivalence to the relative-entropy distance to the invariant set, (ii) monotonicity under covariant channels, and (iii) an operational interpretation: for a uniform unknown phase offset encoding ensemble, AHS(ρ) equals the Holevo information. This frames temporal asymmetry as an operational and consumable resource for alignment tasks without invoking time operators or thermodynamic assumptions.

1. Introduction

Quantum resource theories provide an operational language for quantifying what can or cannot be achieved under restricted sets of operations [1]. When no external time reference is available, physically implementable processing is naturally constrained by time-translation symmetry. The resulting framework is an instance of the resource theory of asymmetry / quantum reference frames [2, 3, 4].

This note presents a minimal, symmetry-based resource theory that treats time/phase-translation asymmetry as the consumable resource underlying temporal alignment. We use “time” in the cyclic/phase-reference sense (compact U(1) symmetry).

2. HS◦ symmetry model

Let H be a Hilbert space and consider a compact one-parameter time/phase-translation symmetry group U(1) represented unitarily by

U(α) = e-iHα, α ∈ [0, 2π), (1)

where H is a (dimensionless) generator. We assume the representation factors through U(1), i.e. U(α + 2π) = U(α). The group action on states is

Uα(ρ) = U(α)ρU(α)†. (2)

Remark 1 (Why U(1)?): The parameterization α ∈ [0, 2π) models a cyclic time/phase reference (time modulo a period). This compact choice ensures the existence of a normalized Haar measure used in symmetry averaging (twirling). Non-compact time translations (e.g. R) can be treated with additional technical conditions but are outside the scope of this minimal note [2, 4].

3. Free operations and free states

Definition 1 (HS◦-free operations: covariant channels):

A completely positive trace-preserving (CPTP) map E is HS◦-free if it is covariant under time translations:

E ◦ Uα = Uα ◦ E, ∀α ∈ [0, 2π). (3)

Definition 2 (Twirling channel):

Define the symmetry-averaging (twirling) channel

T(ρ) = (1/2π) ∫02π dα Uα(ρ) = (1/2π) ∫02π dα U(α)ρU(α)†. (4)

Definition 3 (HS◦-free states: invariant states):

A state σ is HS◦-free if it is invariant under time translations:

Uα(σ) = σ, ∀α ⇐⇒ σ = T(σ). (5)

Remark 2 (Structure of invariant states): Invariance under U(α) = e-iHα implies that σ is block-diagonal with respect to the eigenspace decomposition of H (coherences between distinct eigenvalues vanish; coherences within degenerate eigenspaces may remain). This is the standard “no preferred time origin” condition in symmetry-based resource theories [2, 4].

4. HS◦ monotone: relative-entropy temporal asymmetry

Let S(ρ) := − Tr(ρ log ρ) denote the von Neumann entropy. The base of the logarithm determines units (base 2 for bits, base e for nats).

We use the quantum relative entropy

S(ρ∥σ) := Tr(ρ log ρ) − Tr(ρ log σ),

defined when supp(ρ) ⊆ supp(σ) and +∞ otherwise.

Definition 4 (Relative-entropy temporal asymmetry):

Define the HS◦ resource of a state ρ as

AHS(ρ) := S(T(ρ)) − S(ρ). (6)

Proposition 1 (Distance to the invariant set)

Let Free denote the set of HS◦-free (invariant) states. Then

AHS(ρ) = minσ∈Free S(ρ∥σ), (7)

and the minimizer is σ⋆ = T(ρ).

Proposition 2 (Monotonicity under HS◦-free operations)

For any HS◦-free operation E and any state ρ,

AHS(E(ρ)) ≤ AHS(ρ). (8)

5. Operational content: unknown phase-offset encoding

A standard alignment model is: an unknown offset α is applied to a probe state ρ, producing the ensemble

Eρ := {p(α) = 1/2π, ρ_α = U_α(ρ)}. (9)

A receiver attempts to infer α from ρα via a measurement and classical post-processing. The accessible classical information is upper bounded by the Holevo information χ(Eρ) [6, 7].

Proposition 3 (Asymmetry equals Holevo information for uniform offsets)

For the uniform U(1) prior above,

χ(Eρ) = AHS(ρ). (10)

Remark 3 (Interpretation): AHS(ρ) quantifies how much classical information about an unknown time/phase offset can be encoded (in principle) into the ensemble {ρ_α}. In this precise sense, temporal asymmetry is an operational resource for alignment.

6. No free creation of temporal asymmetry

Theorem 1 (No free creation of temporal asymmetry)

If the input state is HS◦-free, then any HS◦-free operation outputs an HS◦-free state:

ρ ∈ Free ⇒ E(ρ) ∈ Free for all HS◦-free E. (11)

In particular, AHS(E(ρ)) = 0 whenever AHS(ρ) = 0.

Corollary 1 (Resource requirement for deterministic alignment): Any protocol that deterministically outputs a state with nonzero temporal asymmetry (usable as a time/phase reference) must have access to nonzero HS◦ resource in its inputs (states, ancillas, or via catalytic access).

7. Non-normative use case: phase-window coordination

This section illustrates one generic application of the U(1)-asymmetry framework without introducing any deployment-specific protocol.

Remark 4 (From U(1) phase to normalized phase on S1): Many coordination stacks represent a cyclic offset using a normalized phase ϕ ∈ [0, 1) ~= S1 rather than an angle α ∈ [0, 2π). These parameterizations are equivalent via α = 2πϕ.

Definition 5 (Phase-window triggering (generic)):

Fix a cycle duration Tcycle > 0 and a target phase ϕtarget ∈ [0, 1). Given a tolerance (half-width) ∆ϕ ∈ (0, 1/2), define the circular phase window

W(ϕtarget, ∆ϕ) = {ϕ ∈ [0, 1) : dS1(ϕ, ϕtarget) ≤ ∆ϕ},

where dS1(ϕ, ψ) = min{|ϕ − ψ|, 1 − |ϕ − ψ|}. The corresponding physical duration is ∆tw = 2∆ϕ Tcycle. A node triggers an action when its local phase estimate enters W.

Remark 5 (Resource interpretation): In phase-window coordination, the operational goal is to estimate and compensate unknown relative phase offsets between parties so that their local phase estimates enter the same window. In the symmetry-based model, unknown offsets correspond to a U(1) action ρα = U(α)ρU(α)†. The asymmetry monotone AHS(ρ) quantifies the amount of usable phase-reference resource in ρ and, for a uniform unknown offset, equals the Holevo information of the orbit ensemble.

8. Scope and extensions

Remark 6: This document defines a minimal symmetry-based resource theory. Extensions may include finite-dimensional clock models, non-uniform priors over offsets, multi-party alignment networks, noise models, or non-compact translation groups. These extensions do not change the core free operations (covariant channels) and free states (invariant states) introduced here [2, 4].

9. Conclusion

Quantum HS◦ formalizes temporal alignment as a consumable quantum resource given by time/phase-translation asymmetry. The canonical monotone AHS(ρ) = S(T(ρ)) − S(ρ) is both symmetry-theoretic (distance to invariants) and operational (Holevo information for uniform offsets), providing a minimal, QIT-compatible foundation for temporal coordination.

References