Falsifiable Framework

Which claim, which measure, which crossing — and what would kill it

Framework Purpose

This page defines the specific claim this programme is testing, in a form that can be confirmed or killed. It is not a review of the Mpemba field in general. It is a boundary document for this platform and this experiment.

Changelog v0.2

C1(iv) reformulated: the Markovian regime can produce transient oscillatory crossings (confirmed by Module 01 numerics). The non-Markovian claim now targets persistent crossings with τ_bath-dependent duration — not mere presence vs. absence. Confirm/kill criteria updated accordingly.

The Operative Distance Measure

Measure Box — Platform-Specific Definition

The distance measure operative in this programme is the trace distance on the spin reduced state:

D(ρ, σ) = ½ Tr|ρ - σ|

where ρ and σ are the spin reduced density matrices obtained by tracing out the motional mode.

Other measures in the literature (entanglement asymmetry, Frobenius distance, free energy distance) are noted in the References section. They are not the primary claim of this programme. Results will not be selectively reported in a measure that happens to show a crossing.

The Primary Claim

We test the following claim, derived from Strachan et al. (PRL 134, 220403, 2025):

Claim C1 (non-Markovian quantum Mpemba effect): There exist initial states ρ_A, ρ_B of the spin subsystem such that: (i) D(ρ_A(0), ρ_ss) > D(ρ_B(0), ρ_ss) [A starts farther] (ii) \exists t* > 0 : D(ρ_A(t*), ρ_ss) < D(ρ_B(t*), ρ_ss) [A crosses below B] (iii) The crossing time t* scales with bath memory time τ_bath in the way predicted by Strachan et al. (iv) The crossing is qualitatively distinct from Markovian crossings: it is persistent (D_A remains below D_B for a sustained interval scaling with τ_bath), whereas the Markovian crossing for the same state pair is either absent, transient, or oscillatory with no τ_bath dependence.

Condition (iv) is the diagnostic that distinguishes the non-Markovian mechanism from crossings that arise from initial-state geometry alone. Note: the Markovian regime can produce transient oscillatory crossings for state pairs with transverse Bloch components (see Module 01 numerics). These are driven by spin-motion entanglement dynamics and do not scale with any bath memory parameter. C1 targets a qualitatively different phenomenon: bath-memory-induced persistent crossings.

What Would Confirm C1

Observation of the trace-distance crossing D(ρ_A, ρ_ss) < D(ρ_B, ρ_ss) at a time t* consistent with theoretical predictions, for a pair (ρ_A, ρ_B) with D(ρ_A(0)) > D(ρ_B(0)).
The crossing is persistent: D_A(t) remains below D_B(t) for an interval whose duration scales with τ_bath. This contrasts with the transient oscillatory crossings observed in the Markovian limit (Module 01).
The crossing time t* shifts systematically with bath memory time τ_bath as predicted.
The crossing character changes qualitatively when the bath spectral density is broadened to the Markovian limit (δ-function spectral density): persistent crossing → transient/oscillatory or absent.
Results are reproduced across multiple initial-state pairs on the predicted crossing manifold.

What Would Kill C1

No crossing in trace distance is observed for any pair (ρ_A, ρ_B) within the accessible parameter range, despite confirmed non-Markovian bath dynamics.
A crossing is observed but its persistence and t* do not scale with τ_bath — indicating a geometric mechanism (same as Markovian) rather than bath-memory-induced Mpemba behaviour.
The crossing character is indistinguishable between Markovian and non-Markovian baths for all tested state pairs: same duration, same oscillatory structure, no τ_bath dependence.
The crossing depth is smaller than the experimental resolution in trace distance.

System Model

Spin-Boson Hamiltonian

H = (ω_0/2) σ_z + ω_m a†a + g σ_x (a + a†) + H_bath where: ω_0 = qubit transition frequency ω_m = motional mode frequency g = spin-motion coupling (Lamb-Dicke regime: g ≪ ω_m) H_bath = structured bath with peaked spectral density J(ω)

Reduced Steady State (RSS)

The spin RSS is obtained by evolving from an arbitrary initial state to long times and tracing out the motional mode. It depends on (ω_0, ω_m, g, T, bath parameters). The RSS is not necessarily the maximally mixed state — its structure encodes the system-bath coupling.

ρ_ss = Tr_motion[ lim_{t→∞} ρ_total(t) ]

Bath Engineering in the Trapped-Ion Platform

The non-Markovian bath is implemented by placing controlled noise on the motional mode with a peaked power spectral density. This maps to a spin-boson spectral density J(ω) ∝ γ²/[(ω − ω_m)² + γ²], where γ controls the bath memory time: τ_bath = 1/γ. The Markovian limit corresponds to γ → ∞.

Pre-Registration Commitment

Analysis Protocol

The analysis method — including the fitting procedure for t*, the error model for trace distance, the criterion for "persistent" vs. "transient" crossing, and the statistical test for the crossing — will be pre-registered before any experimental data are examined.

This commitment applies also to the 2016 archival dataset. That data will not be accessed until the pre-registration is filed and publicly timestamped. This is not a bureaucratic constraint — it is the scientific content of the claim.

Note on Definition Proliferation

The quantum Mpemba literature uses multiple non-equivalent distance measures: entanglement asymmetry (Ares et al.), Frobenius distance, free energy distance, and trace distance. Each can yield a crossing under different conditions, and not all crossings share a common mechanism.

This programme does not contribute another example using a conveniently chosen measure. It uses trace distance on the spin reduced state because:

It is operationally defined via spin-state tomography on the platform.
It is the measure used in the Strachan et al. predictions we are testing.
It satisfies the contractivity property under CPTP maps, making it a physically meaningful distinguishability.

If results in other measures are reported, they will be labelled clearly and distinguished from C1.