Unitary return protocols: coherent histories, exact amplitudes and measurement timing

The measurement schedule is part of the model

A forced circuit applies a prescribed W to the mobile carrier and then tests return to the root singlet. A coherent return experiment instead lets a Hamiltonian generate waiting, backtracking and circulating amplitudes before one final test. An intermediate-measurement experiment records additional outcomes along the way. These procedures use related ingredients but have different probability laws.

For W=e^iθI, the first probability equals one for every θ. The second can resolve the phase because it compares amplitudes with different winding histories before the projection. That distinction is proved for the supplied Hamiltonian in Temporal completion, v20, Theorem 20.4 and equation (80), pp.61–62.

The root dual is an explicit comparison resource throughout the coherent walk. Removing it, measuring which path occurred or repeatedly resetting the carrier changes the experiment. Those changes cannot be represented merely by relabeling the final detector result.

Keep the preparation and failure penalty aligned explicitly

The auxiliary space is the direct sum of four mobile positions, each with a carrier and root dual. Its dimension is 4m². The root singlet is s=m^−1/2Σ_a|0,a,ā⟩, P=|s⟩⟨s|, and

All nonroot positions and root nonsinglets incur Δ. The same P defines the final success event. Identifying that event with the unpenalized state is a supplied microscopic choice. It yields linked predictions for static response and completion probabilities, but neither a detector label nor gauge invariance alone selects Δ or t. V20, §2.1, p.8, and §20.4, p.61.

Gauge three links to identity and put the forward holonomy on the final link. The mobile transfer operator K has eigenvalues λ_ak=2tcos[(θ_a+2πk)/4]. Each matched carrier–dual mode carries singlet weight 1/(4m). This spectral organization permits a much smaller exact calculation of the return than a generic 4m²-dimensional diagonalization.

Reduce the exact amplitude without discarding a contributing mode

List the 4m mobile eigenvalues, including repetitions, as λ_j. In the matched singlet-accessible representation let v have every component 1/√(4m). Then

The omitted dual directions have zero overlap with the prepared singlet and remain dynamically orthogonal to its cyclic subspace. This is an exact reduction for this preparation and return observable, not a general compression of every state or observable. The source calculates the corresponding cyclic and dark sectors in v20, Proposition 20.2, p.60.

If r identical free poles coincide, replace those entries by one pole with total weight r/(4m). The corresponding component of v is the square root of that weight. At a central holonomy only four distinct mobile eigenvalues are needed, subject to further special degeneracies. Dark states still matter when calculating the full auxiliary excitation gap; an amplitude-only reduced matrix should not silently replace the full gap calculation.

Finite-time return needs every contributing secular root

With distinct free poles λ_j and positive combined weights w_j, put R(z)=Σw_j/(z−λ_j). The rank-one resolvent in the singlet state is R(z)/(1−ΔR(z)). Its poles satisfy ΔR(z_ν)=1. There is one root between successive distinct free poles and one above the largest pole. The largest root gives the ground energy; the remaining roots supply excited singlet-accessible energies.

The residue formula follows by differentiating 1−ΔR at a simple root. It extends the source's ground-branch resolvent calculation to the full cyclic spectral measure. V20, Theorem 2.1 and proof, pp.9–10. Numerical diagonalization of H_bright is an alternative way to obtain the same energies and weights.

Keeping only the lowest energy would predict a unit-modulus amplitude and erase the return dynamics. Finite-time interference depends on energy differences among populated modes. At t=0 the preparation is an exact zero-energy eigenstate and p=1, providing a useful limiting check.

Derive the first holonomy-dependent return term

Let μ_k=⟨s|H^k|s⟩. The specified model has μ₁=0, μ₂=2t² and μ₄=2Δ²t²+t⁴[6+2ReTrW/m]. The six backtracking contributions here belong to the ordinary H⁴ moment; they are not the Q-filtered paths of a reduced static Feshbach coefficient.

Expanding the survival probability gives p_W(u)=1−μ₂u²+(μ₄/12+μ₂²/4)u⁴+O(u⁶). Subtract the identity-loop result. All common terms cancel:

The bound is uniform in the unitary holonomy and comes from the symmetric energy-difference distribution. V20, Theorem 20.4 and proof, pp.61–62. It is a short-time probability expansion, not a weak-hopping energy expansion. Its coefficient is dimensionless after multiplying by u⁴, and must not be substituted for the energy-valued holonomy Hessian.

A central-loop example shows both agreement and its limit

Choose m=3, Δ=1, t=0.2 and W=e^2πi/3I. The forced single-circuit return is exactly one. The leading coherent difference is t⁴u⁴/4 because 1−ReTrW/3=3/2. Independent full-Hamiltonian diagonalization gives:

At u=0.25 the leading difference is 0.0000015625. At u=4 it predicts 0.1024 and even misses the sign. The exact static inequality E(W)≥E(I) does not order survival probabilities at arbitrary times. Their populated-mode interference can reverse the difference.

These checkpoints are calculations from the supplied Hamiltonian, not measured data or claims made for all times by its short-time theorem. Use the exact matrix when the stated remainder no longer resolves the effect of interest.

Intermediate projections produce a different limit

For a time step τ/N, the two probe operators are M₁=Pe^{−iτH/(Nℏ)} and M₀=Qe^{−iτH/(Nℏ)}. A recorded sequence has probability ‖M_rN⋯M_r1s‖². Ignoring intermediate outcomes after they were physically recorded does not restore the coherent cross terms of uninterrupted evolution. V21, §8, p.17.

For the branch where every rank-one return succeeds, each success restores s up to a scalar. Its probability is [p_W(u/N)]^N. At fixed total u, the leading loss is 2t²u²/N, so this all-success probability approaches one as N→∞. The leading identity-versus-loop difference scales as t⁴u⁴[1−ReTrW/m]/(6N³). The frequent projections suppress the very histories that carry the loop signal.

This calculation explains the manuscript's warning about a possible Zeno limit. V20, §20.4, p.62. A fresh reset after every outcome is another resource-dependent protocol. Neither repeated measurement nor reset can be inserted into the static or coherent-return theorem without recomputing the dynamics.

Use the shared Hamiltonian to make distinct predictions

The return experiment offers checks unavailable from a static energy alone. The initial quadratic loss estimates t²; the fourth-order holonomy difference checks the trace response; the full time curve checks the contributing energy differences and weights. The same calibrated Δ and t should predict all three when the preparation, links and measurement policy remain fixed.

A mismatch can therefore be localized. An incorrect initial loss may indicate hopping or preparation error. Correct early-time coefficients with an incorrect later curve can expose missing modes, dephasing or timing assumptions. A different response after intermediate measurements is expected and tests a different protocol. Those diagnoses require independently measured controls rather than unrestricted parameter refitting.

The source proposes a single projector as both completion event and energetic binding criterion, while keeping that identification and its calibration explicit. V20, §20.4–20.5, pp.61–62. Verifying the associated finite predictions would support that implementation. It would still leave the source-selection mechanism, physical units, autonomous gauge dynamics and continuum interpretation as separate research questions.

Continue reading

• The specified singlet Hamiltonian
• The static physical root and auxiliary gap
• What different loop probes determine
• Use held-out tests of the finite model