Four-qubit open chain vs. closed ring

01What is being compared

Two branches, both starting at t=0 from |0000>, both evolved under their own fixed Hamiltonian for the whole observation window:

There is no bond switching, no quench, and no time-dependent term in either branch. Site labels 0,1,2,3 mean the same physical sites in both branches; the ring differs only by the extra periodic bond (3,0).

This is a topology comparison. It is a separate experiment from the pre-existing local-quench study with t_switch=1.688. Anything drawn from that older study appears only in the clearly marked supplementary section (§12) and uses the post-switch elapsed time tau, which must never be identified with the time origin t used everywhere else in this document.

02Hamiltonian and fixed conditions

• N = 4
• Omega = 1.0
• J = 0.7
• initial state |0000>
• time window 0 <= t <= 3.688
• primary step dt = 0.002; audit step dt = 0.001
• symmetric second-order Trotter propagation; exact diagonalization used for validation and for the modal analysis

03Numerical method and definitions

State distance. The global-phase-invariant distance sqrt(2 - 2 |<psi_open|psi_closed>|) is used throughout.

Local response. The primary site-resolved observable is

O_i = n_i = |1_i><1_i| = (I - Z_i)/2
R_i(t) = <psi(t)| n_i |psi(t)> = (1 - <Z_i>)/2

R_i is the probability that site i is in state 1, so R_i lies in [0,1] and is global-phase invariant. This is a definition introduced here as a common quantity for both the fixed and the quench analyses; it is not a claim that a site-resolved response existed in the older artifacts.

For nonzero total response, p_i = R_i / sum_j R_j. Raw L1/L2 and normalized L1/L2 distances are all stored; the primary distribution-shape distance quoted here is the normalized L1, and total variation is one half of it. Raw and normalized distances are reported separately so that a difference in distribution shape is never confused with a difference in total response. At t=0 all responses vanish, so p_i is stored as zero and PR/IPR are recorded as NaN.

One-body and two-body Pauli quantities. For every i<j the package stores the one-body expectations <X_i>, <Y_i>, <Z_i> and the raw two-body expectations <X_iX_j>, <Y_iY_j>, <Z_iZ_j>. The connected correlations are defined as

C^AA_ij = <A_i A_j> - <A_i><A_j>,   A = X, Y, Z

Raw and connected quantities are stored separately and must not be conflated. In the displayed 4x4 connected-correlation maps the diagonal is set to zero by convention; the diagonal is not an on-site variance and is not interpreted.

Correlation distances. For each channel, upper_L1 and upper_L2 are taken over the six i<j entries of the open-minus-closed difference. For a symmetric zero-diagonal matrix the Frobenius distance is sqrt(2)*upper_L2. The combined correlation distance is

D_corr = sqrt(D_XX_upper_L2^2 + D_YY_upper_L2^2 + D_ZZ_upper_L2^2)

Feature distances. Ordinary unnormalized Euclidean distances over, respectively: the 12 one-body values, the six ZZ-connected values, all 18 connected values, and the concatenation of 12 one-body plus 18 connected values. These are finite-observable feature distances. They are not reconstructions of the full state distance. Connected correlations here are correlation functions of specified Pauli observables and are not used as entanglement measures.

04State distance — what it shows

The two branches separate progressively from t=0 without any switching event, reach maximum separation at t=3.462, and remain near that level to the end of the window.

05Local response distribution — what it shows

Site-resolved responses at the time of maximum state distance, t = 3.462:

Closed is uniform across all four sites. Open is higher than closed at every site, with the end sites above the interior sites.

Key times along the window

At t=3.462 the normalized L1 distance is 0.040354 (total variation 0.020177) while the state distance is 0.546935. The shape of the normalized occupation distribution is comparatively close there, even though the raw L1 difference 0.474209 shows a substantial difference in total response.

Maximum-position behaviour. Fixed closed keeps all four sites equal at every saved time (fourfold tie throughout, zero position changes). Fixed open has R_0=R_3 and R_1=R_2 by reflection symmetry; after leaving the initial near-zero fourfold tie at t=0.008, the end pair 0|3 is maximal for the rest of the window, with no migration, reflection, or round trip. Ties are resolved with tolerance max(1e-10, 1e-8*|R_max|); the persistent ties here are symmetry-induced, not numerical noise.

Correspondence with the state distance: Pearson(D_state, normalized L1) = 0.688010, Spearman = 0.439491. Correlation coefficients here describe co-variation of time series only and carry no causal content.

06One-body and two-body Pauli correlations — what they show

Channel maxima over the window:

D_corr first exceeds 1e-8 at t = 0.012.

At the time of maximum state distance, t = 3.462, the combined correlation distance is D_corr = 0.974637690628, while the normalized local occupation L1 distance is only 0.040354. Connected correlations at that time:

The single largest connected-correlation difference at t=3.462 is the XX channel, pair (0,3), at 0.472156 — that is, the pair joined by the ring's extra bond.

Averaged over the interval where the state distance exceeds 0.5, the mean channel distances are XX = 0.564273, YY = 0.40003, ZZ = 0.63102. ZZ is the largest on average, but XX is of comparable size, so the separation in this window is not described by any single channel alone.

07Correspondence between state distance, local response, and correlation distances

The three maxima do not coincide: D_corr peaks at t=3.200, the state distance at t=3.462, and the normalized local occupation difference at t=2.154 (offset of 1.046 from the D_corr peak). These quantities track each other closely in rank order but are distinct functions of time with distinct extrema.

All coefficients above are descriptive measures of co-variation between time series. They do not license any causal statement, in either direction, about the ring bond and the observed differences.

08Symmetries of the two branches

The closed branch retains the rotation symmetry of the ring together with the rotation-invariant initial state |0000>; the open branch retains reflection symmetry. Both are verified numerically as residuals:

All symmetry residuals are at or below roughly 1e-14. In the closed branch the four nearest-neighbour pairs agree with each other and the two opposite pairs agree with each other. In the open branch reflection-related pairs agree, but pairs containing an end site differ from interior and separated pairs, and that difference is what appears in the correlation maps.

09Reconstruction from degenerate eigenspace pairs

Degenerate energies are grouped into eigenspaces and the analysis uses eigenspace projectors P_E, with contributions A_EF,i = Tr(P_E rho_0 P_F O_i). For E != F the real contribution of an unordered pair is 2 Re[A_EF,i exp(-i(E-F)t)]; the E=F diagonal term is counted once. Because degenerate levels are grouped before ranking, the resulting weights do not depend on the arbitrary choice of eigenvectors inside a degenerate space.

Connected correlations are not linear in the state, so they cannot be decomposed additively over eigenspace pairs. The procedure is: decompose the raw two-body terms and the one-body terms linearly over eigenspace pairs, reconstruct each, and only then form the connected quantity as raw minus the product of the reconstructed one-body terms. Because that product is nonlinear, the connected ranking score is a leave-one-eigenspace-pair-out RMS influence and is not claimed to be a unique additive decomposition.

Eigenspace pairs required to reach a given fraction of explained energy:

At the 99% level the open branch requires 23 to 37 pairs and the closed branch 11 to 14. The open branch in particular is more safely described as multi-mode competition than as a few-mode structure.

For the local response alone the counts are smaller: open reaches 95% of the total response with 6 eigenspace pairs and 99% with 7, while the spatially non-uniform component reaches 95% with 4 pairs but needs 15 for 99%. The closed response is spatially uniform, so it has no pairs carrying a maximum-position shift.

10Numerical audit

All audits below are reported as observed residuals; none of them is an interpretation.

Correlation analysis audit

State and local-response audit

Exact-diagonalization agreement, step-size convergence, norm conservation, Hermiticity, reality of expectation values, Pauli expectation bounds, global-phase invariance, and mode-sum completeness all pass. Only global-phase-invariant quantities — expectation values, connected correlations, and the distances built from them — are used for physical comparison.

11How to reproduce and audit

Tested with Python 3.14.5, NumPy 2.5.0, and Rust/Cargo 1.97.0. NumPy is pinned in requirements.txt; Cargo crates are included in vendor/, so the Rust stages do not require network access.

python -m venv .venv
.venv\Scripts\python -m pip install -r requirements.txt
.venv\Scripts\python verify_reproduction.py

On macOS/Linux, use .venv/bin/python instead. The verifier creates a clean temporary working directory, executes formatting, all 10 Rust tests, both Trotter runs, and all three Python analyses. It then compares all 40 regenerated CSV files against the packaged references with absolute tolerance 2e-12 and relative tolerance 1e-10, and writes reproduction_audit.json with the environment, command results, comparison maxima, and final verdict.

The Rust binary produces the time series and state distances at both step sizes; the Python scripts produce the modal, local-response, and correlation analyses. The modal audit uses exact diagonalization with degenerate eigenspaces grouped before ranking. Cargo/test PASS results are recorded by the verifier and are not hard-coded into the numerical CSV audit.

Everything needed to re-run is in the archive:

• four_qubit_direct_open_closed_two_body_correlation_reproducible_2026-07-20.zip

12Key output files

13What can and cannot be claimed

Supported by the numbers in this package

• Two fixed topologies starting from the same state separate without any switching event, reaching state distance 0.546935 at t=3.462.
• At t=3.462 the normalized local occupation distributions are comparatively close (normalized L1 0.040354) while the combined connected-correlation distance is 0.974638, and the largest single connected difference is XX pair (0,3) at 0.472156.
• D_corr and the state distance co-vary strongly (Pearson 0.953794, Spearman 0.990560) but peak at different times (3.200 vs 3.462), and neither coincides with the local-occupation peak at 2.154.
• Ring rotation symmetry and chain reflection symmetry hold to residuals of order 1e-14 or below.
• 99% reconstruction requires 23–37 eigenspace pairs on the open side and 11–14 on the closed side.
• All listed exactness, convergence, norm, Hermiticity, reality, bound, phase-invariance, and mode-sum audits pass.

Explicitly not claimed

• That the ring bond causes any transfer of correlation. Correlation coefficients here are descriptive only.
• That two-body correlations account for the full state difference. The feature distances are distances between finitely many observables.
• That the ZZ correlation difference is the cause of the state distance. On average ZZ is the largest channel above state distance 0.5, but XX is comparable.
• That entanglement has been detected. Connected correlations are correlation functions of chosen Pauli observables, not entanglement measures.
• Any general boundary mechanism, transport law, phase transition, or thermodynamic-limit statement.
• Any identity of mechanism with the N=7 Lindblad backflow study.

14Subsequent weak-link and symmetry extensions (2026-08-17)

The main extension uses

H(lambda) = Omega/2 sum_i X_i
          + J/2 [Z0Z1 + Z1Z2 + Z2Z3 + lambda Z3Z0]

with the same N=4, Omega=1, J=0.7, time window, and site ordering. The sampled grid is lambda=0,0.05,...,1; lambda=0 is the open-chain reference and lambda=1 is the closed-ring endpoint. This is a closing-bond-strength sweep, not a claim of continuous topology.

For connected correlations, the signed branch difference is

Delta C_ij^AA(t,lambda) = C_ij^AA(t,lambda) - C_ij^AA(t,0).

The pointwise squared-difference share of pair ij is

w_ij = sum_A (Delta C_ij^AA)^2
       / sum_{k<l,A} (Delta C_kl^AA)^2,

and Q_ij denotes the corresponding time-integrated numerator divided by the time-integrated denominator. Pointwise ratios are undefined where the all-channel difference norm is at or below 1e-8. Squaring discards sign, so every concentration result must be read together with the signed-channel results.

14.1 Weak-link sweep

The endpoint checks against the sealed open/closed baseline all pass. At lambda=1, the maximum state distance is 0.546934958021 at t=3.462, and its time RMS is 0.393869262797.

At lambda=1, the time-integrated seam shares for pair (0,3) are:

At t=3.462, the all-channel pointwise seam share is 0.514835. The signed seam differences are -0.472156818 in XX, +0.268163791 in YY, and +0.440689566 in ZZ. The seam is therefore prominent in squared-difference space, but the three channels do not move in one common signed direction.

Using the predeclared rule Q03 > 1/3 and requiring it to hold for at least 75% of nonzero lambda values and at least 50% of the valid nonzero-lambda time grid, the observed fractions are 0.550 and 0.761. The mechanical classification is therefore limited, not broadly robust.

14.2 Pair-space redistribution

The integrated top pair changes from the reflection-tied pair set 02|13 for lambda=0.05–0.15 to the seam pair 03 for lambda=0.20–1.00. Linear interpolation between sampled values places the seam-versus-02/13 crossings near lambda=0.166. The separate fixed-threshold event Q03=1/3 occurs later, near lambda=0.452.

At lambda=1, the integrated six-pair decomposition is:

When w03<1/3, the energy-weighted non-seam remainder is split 0.170 to the two seam-adjacent pairs 01+23 and 0.830 to 12+02+13. Thus the data do not show a simple seam-to-adjacent-to-remaining sequence. These are concurrent partitions of a squared observable-difference denominator, not evidence of propagation.

Reflection-related equalities are numerically retained: 01=23 and 02=13, with integrated-share residuals at approximately 1e-15.

15Signed versus squared connected-correlation differences

The signed audit separates the integral

I = integral Delta C dt

from the absolute integral

M = integral |Delta C| dt

and reports the signed balance P=I/M where defined. At lambda=1, the seam results are:

The large seam squared contribution is therefore compatible with persistent negative XX and predominantly positive YY/ZZ. Q measures magnitude concentration; P measures signed consistency and cancellation.

Sign-canceling components do occur. The strongest fractional cancellation is at lambda=0.30, pair 23, channel YY, with M=0.0496291 and P=-0.00503727, but its magnitude ranks only 309th of 360 components. Among components with channel share at least 0.1, the smallest observed |P| is about 0.662. The data therefore do not support the stronger statement that the dominant squared contributions are strongly canceled.

16End–interior occupation contrast versus seam concentration

For the original |0000> initial state, define the signed one-body contrast

E(t,lambda) = [R0 + R3 - R1 - R2] / 2.

The time RMS of E is nonincreasing across the sampled lambda grid, falling from 0.0848382 at lambda=0 to a numerical residual 6.922e-17 at lambda=1. In contrast, integrated Q03 is nondecreasing over its 20 defined nonzero-lambda values, from 0.235105 at lambda=0.05 to 0.451198 at lambda=1.

Across the 20 summaries, Spearman(E_RMS,Q03)=-1.000000 and Pearson=-0.980965. Across all valid (lambda,t) points, Spearman(|E|,w03)=-0.840902 and Pearson=-0.814780. These are descriptive relationships on a finite, serially related grid; they are not independent-sample inference or causal evidence.

The predeclared quartile check—|E| at or below its 25th percentile and w03 at or above its 75th percentile—selects 5,387 points, or 14.6657% of valid points, spanning lambda=0.20–1.00 and t=0.012–1.276. Dividing |E| by total occupation preserves the suppression trend where defined, so the raw trend is not explained only by a uniform change in total occupation.

The |E|, w03, and state-distance peaks generally occur at different times. None of these three quantities is used as a proxy for either of the others.

17Initial-state sensitivity

Only two additional initial states were introduced, with all Hamiltonian, grid, operator, floor, and per-state lambda=0 reference definitions held fixed.

For |1000>, E_RMS rises slightly rather than being suppressed, while Q03 is nonmonotonic and remains below the |0000> curve at every defined lambda. For |0100>, E_RMS decreases from a larger open-endpoint value. Both asymmetric single-site states reach the same three endpoint values at lambda=1, but equality of these selected metrics is not equality of the full states.

The full |1000> and |0100> curves agree to 4.382e-15 for Q03 and 1.221e-15 for time-RMS state distance. Their E_RMS curves differ for lambda<1. Thus the suppression of E together with growth of Q03, seen for |0000>, is initial-state sensitive rather than a universal Hamiltonian-only pattern.

17.1 Exact symmetry explanation

The Q03 and state-distance equalities for |1000> and |0100> are enforced by an antiunitary symmetry.

Let R implement the reflection r=(0 3)(1 2), and define

Gamma = Z0 Z1 Z2 Z3 X0 X2.

For every sampled lambda—indeed for the complete Hamiltonian family—

[R,H(lambda)] = 0,
Gamma H(lambda) Gamma^-1 = -H(lambda),
R Gamma |1000> = -|0100>.

Because H(lambda) is real in the computational basis, complex conjugation K gives the antiunitary map Theta=R Gamma K, which commutes with the same-time evolution:

Theta exp[-i H(lambda)t] = exp[-i H(lambda)t] Theta.

The same Theta maps both the lambda=0 and lambda>0 evolutions, so it preserves the overlap magnitude used in the state distance. For connected correlations it only permutes pairs and changes channel/site signs; squaring removes those signs, the all-pair denominator is invariant, and the seam pair 03 is fixed by reflection. Therefore pointwise w03 and integrated Q03 are symmetry invariants.

The occupation contrast is not invariant because some Z_i map to -Z_j, which sends R_i=(1-Z_i)/2 to 1-R_j. At lambda=1, the saved arrays satisfy E_0100(t)=-E_1000(t) to 8.33e-16, explaining the equal endpoint RMS values without implying pointwise equality of the occupations.

The operator identities have zero matrix residual. Stored-array residuals are 4.21e-15 for the all-channel difference norm, 1.00e-15 for reconstructed seam squared energy, and 4.40e-15 for integrated Q03. A larger pointwise ratio discrepancy near the 1e-8 floor is roundoff amplification, not symmetry breaking.

18Extension artifacts, audit status, and updated boundary

All extension stages report PASS for their numerical audits and independent regeneration checks. The sealed baseline and previously completed stage artifacts were hash-checked before and after the relevant work and were not modified.

Primary local reports:

• Weak-link sweep
• Pair-space redistribution
• Signed versus squared differences
• End–interior contrast versus seam concentration
• |1000> comparison
• |0100> three-state comparison
• Antiunitary symmetry explanation