Raw-Fragment Benchmarking for Redundancy Tests in Quantum Darwinism and Spectrum Broadcast Structure

Quantum Darwinism and spectrum broadcast structure analyses address the emergence of effectively classical objectivity by asking how information about a preferred system variable is distributed into many environmental fragments [citation]. In the Darwinism language, redundancy is commonly probed by the rise of mutual information between a system and fragments of its environment. In the spectrum-broadcast language, objectivity is tied to a stronger state-structure condition in which different fragments carry distinguishable records of the same pointer value while remaining appropriately factorized within pointer sectors [citation]. Strong quantum Darwinism clarifies the information-theoretic part of this relation, while equivalence with spectrum broadcast structure requires the stated independence or factorization hypotheses [citation]. Recent analyses also separate redundancy, interpreted as repeated writing of information into the environment, from consensus, interpreted as the number of observers able to extract the information under their access rules [citation]. Photonic and solid-state implementations have probed Darwinism-style signatures, making raw fragment access an experimental design requirement rather than an after-the-fact summary [citation].

The empirical information required by these criteria is richer than an aggregate redundancy curve. A mean mutual-information plateau can be compatible with very different fragment-level situations: all subsets may pass, only selected subsets may pass, fragments may agree because of independent copies of a pointer value, or fragments may agree because a hidden common-mode error has been replicated into many channels. A plot of $I(S;F_m)$TeXI(S;F_m) versus fragment size can indicate where redundancy should be checked, but it does not by itself provide the raw per-shot fragment table required to evaluate all subsets, disagreement, common-mode controls, or negative controls that break fragment-system coupling.

The finite-window separation used here preserves the useful distinctions between three layers of a measurement chain. A measurement-map criterion $\Cmap$TeX\Cmap specifies the input-output probability model, for example a POVM or instrument [citation]. A readout criterion $\Crec$TeX\Crec specifies how activations become persistent, discriminable records under stated dark-count, reset, dead-time, and persistence conventions; detector platforms such as SPADs, TESs, and SNSPDs require such readout-level characterizations in addition to outcome probabilities [citation]. A fragment-access objectivity criterion $\Cobj$TeX\Cobj then concerns whether a specified pointer value can be recovered redundantly from disjoint fragments under the stated fragment readouts. These layers are not interchangeable: a POVM does not imply durable readout, and a durable local record does not imply fragment-access consensus.

The numerical study is confined to the fragment-access layer. A fixed binary pointer-copy sampler generates shot-indexed records for a system variable and eight fragments. The analysis procedure uses these records rather than aggregate curves, computes subset information and disagreement diagnostics, reproduces analytic threshold maps for independent binary channels, and requires controls that destroy fragment-system coupling to remain below threshold. The same analysis exposes the main fragility axis for future experiments: common-mode correlated errors can cap the information accessible from arbitrarily many fragments. Detector-platform or environmental-fragment objectivity remains an experimental inference rather than a consequence of the synthetic benchmark unless same-instance raw fragment-access data and common-mode controls are supplied.

The study combines a heterogeneous binary raw-shot generator with exhaustive subset mutual informations, pairwise disagreement statistics, analytic product-channel checks, independent-channel threshold maps, three coupling-breaking controls, a common-mode stress test, and finite-shot bootstrap checks. Section reference summarizes which conclusions follow from the synthetic analysis and which experimental inferences require physical raw fragment-access data.

Fragment-access data structure

Let $S\in\{0,1\}$TeXS\in\{0,1\} denote the specified pointer variable and let $E_i\in\{0,1\}$TeXE_i\in\{0,1\} denote the readout of fragment $i$TeXi in a family $\Ffam=\{\Frag_i\}_{i=1}^{M}$TeX\Ffam=\{\Frag_i\}_{i=1}^{M} of disjoint fragments. A raw fragment-access dataset consists of shot-indexed records

with a fixed subset-access rule. For a subset $A\subseteq\{1,\ldots,M\}$TeXA\subseteq\{1,\ldots,M\}, write $E_A=(E_i)_{i\in A}$TeXE_A=(E_i)_{i\in A}. The empirical distribution $\hat p_N(s,e_A)$TeX\hat p_N(s,e_A) gives

with the usual convention that zero-probability terms contribute zero [citation]. The normalized information fraction is

All estimates in Eqs. reference--reference are classical mutual informations of specified readout variables. If a quantum fragment ensemble $\{p_z,\rho_F^{(z)}\}$TeX\{p_z,\rho_F^{(z)}\} is used instead, the Holevo quantity upper-bounds accessible information under optimized fragment measurements; it is not interchangeable with the realized readout mutual information unless an attaining measurement or a sufficient condition such as mutual commutativity of the conditional fragment states is specified [citation].

The threshold used in the benchmark is $\etaobj=0.9$TeX\etaobj=0.9. Two finite-family crossings are reported:

The first quantity is a redundancy-curve crossing averaged over subsets. The second is a stricter all-subsets crossing. Their separation is important whenever fragment quality is heterogeneous.

definition: Synthetic raw-fragment benchmark. A synthetic raw-fragment benchmark at threshold $\etaobj$TeX\etaobj consists of the generator, the shot table $\mathcal D_N$TeX\mathcal D_N, the subset-access rule, the estimator in Eq. reference, the finite-family crossings in Eqs. reference--reference, pairwise disagreement diagnostics, negative controls that break fragment-system coupling, and a common-mode sensitivity test. The conclusion is model-level: it evaluates the analysis procedure on the fixed distribution and does not constitute an experimental objectivity result.

Specified pointer-copy sampler

The baseline model is a computational-basis pointer-copy sampler with a balanced binary pointer variable, $N=100000$TeXN=100000 synthetic shots, $M=8$TeXM=8 fragments, and a fixed reproducibility seed. The system readout error probability included in the raw table is $0.01$TeX0.01. Fragment $i$TeXi first receives a noisy copy of the pointer state with copy-error probability $\alpha_i$TeX\alpha_i, then undergoes fragment-readout error with probability $\beta_i$TeX\beta_i. The resulting effective pointer-error probability for an independent fragment is

The specified values are listed in Table reference. The synthetic table contains the pointer state, a noisy system readout, the eight fragment readouts, and the sampled copy/readout error indicators. The results below use the specified pointer variable $S$TeXS for the synthetic benchmark metric. No experimental data are included.

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Subset information and disagreement

The fragment-access diagnostics are computed for every subset $A$TeXA of the eight fragments, including all singletons and all $\binom{8}{m}$TeX\binom{8}{m} subsets at each size $m$TeXm. Pairwise disagreement is evaluated as

Disagreement is not a replacement for mutual information: high agreement among fragments can arise from a common-mode error, while low pairwise disagreement need not prove independent access to the pointer. It is therefore reported alongside the subset information fractions and the negative controls.

Independent binary-channel threshold map

For a balanced pointer variable and $m$TeXm conditionally independent fragments passing through identical binary symmetric channels with effective bit-flip probability $p$TeXp, let $K=\sum_{j=1}^{m}E_j$TeXK=\sum_{j=1}^{m}E_j be the number of observed ones in a fragment vector. The posterior probability is

With balanced prior, the count probability is

Therefore

and, for balanced $S$TeXS and logarithm base two,

These equations give the analytic IID threshold map over fragment count and noise.

For the heterogeneous baseline model, the analytic comparison sums exactly over the $2^{|A|}$TeX2^{|A|} fragment patterns for each subset $A$TeXA using the independent effective errors $q_i$TeXq_i in Eq. reference. Agreement between this analytic product-channel value and the empirical raw-shot estimate is used as a model-consistency check rather than as an experimental uncertainty statement.

Negative controls

Three controls are applied to distinguish pointer-coupled redundancy from analysis-induced threshold crossings. In the column-shuffled control, each fragment column is independently permuted against the pointer variable. In the row-shuffled control, full fragment vectors are permuted against the pointer variable, preserving within-row fragment dependence while destroying fragment-system coupling. In the independent-random control, fragment records are replaced by independent fair bits. A raw-fragment analysis that reports threshold-level redundancy under these controls fails the negative-control criterion.

Common-mode correlated noise

Common-mode errors are modeled by a hidden shared flip $C\sim\mathrm{Bernoulli}(c)$TeXC\sim\mathrm{Bernoulli}(c) applied to every fragment, together with independent idiosyncratic fragment noise. In the idealized limit of infinitely many otherwise noiseless fragments, the fragment majority reveals $S\oplus C$TeXS\oplus C rather than $S$TeXS. The limiting information fraction is therefore bounded by

where $\Htwo(c)=-c\log_2 c-(1-c)\log_2(1-c)$TeX\Htwo(c)=-c\log_2 c-(1-c)\log_2(1-c). Equation reference is a fragility result: adding more fragments cannot remove a shared error unless the shared error is independently measured, randomized, bounded, or otherwise controlled.

proposition: Common-mode ceiling. In the binary balanced model with fragment records $E_i=S\oplus C\oplus B_i$TeXE_i=S\oplus C\oplus B_i, where $C\sim\mathrm{Bernoulli}(c)$TeXC\sim\mathrm{Bernoulli}(c) is shared across all fragments and the $B_i$TeXB_i are conditionally independent idiosyncratic flips with a common error probability $b<1/2$TeXb<1/2---or, more generally, with a law under which the fragment ensemble recovers $S\oplus C$TeXS\oplus C with probability tending to one---the information accessible from an unbounded fragment family converges to the information in $Y=S\oplus C$TeXY=S\oplus C, namely the fraction $1-\Htwo(c)$TeX1-\Htwo(c).

proof. As $m\rightarrow\infty$TeXm\rightarrow\infty, the independent idiosyncratic flips are averaged out by the fragment ensemble under the stated recovery condition, so the asymptotically recoverable variable is $Y=S\oplus C$TeXY=S\oplus C. Since $S$TeXS is balanced and independent of $C$TeXC, $Y$TeXY is balanced and $H(S)=H(Y)=1$TeXH(S)=H(Y)=1 bit. The conditional entropy $H(S\mid Y)$TeXH(S\mid Y) equals $H(C)=\Htwo(c)$TeXH(C)=\Htwo(c) because $S=Y\oplus C$TeXS=Y\oplus C. Hence $I(S;Y)=H(S)-H(S\mid Y)=1-\Htwo(c)$TeXI(S;Y)=H(S)-H(S\mid Y)=1-\Htwo(c), giving Eq. reference.

Finite-shot stability

Bootstrap resampling is used only as an estimator-stability diagnostic for the synthetic dataset [citation]. Sample sizes $10000$TeX10000, $30000$TeX30000, and $100000$TeX100000 are evaluated with 24 bootstrap replicates per size. The reported quantiles therefore describe the stability of the finite-sample estimator under the fixed model and implementation; they do not represent experimental uncertainty.

Baseline raw-fragment redundancy and disagreement

The empirical pointer entropy is

The baseline redundancy curve is shown in Fig. reference. The mean subset information fraction first exceeds the $0.9$TeX0.9 threshold at $m=3$TeXm=3, where the mean is $0.935445$TeX0.935445. The all-subsets criterion first exceeds the threshold at $m=4$TeXm=4, where the minimum over all four-fragment subsets is $0.952245$TeX0.952245. At $m=3$TeXm=3, the minimum subset value is $0.898775$TeX0.898775, slightly below threshold, while $55$TeX55 of $56$TeX56 three-fragment subsets pass. The distinction between mean and all-subsets crossings is therefore resolved by the raw subset table rather than hidden in an aggregate curve.

The mean pairwise fragment disagreement is $0.122481428571$TeX0.122481428571. The observed pairwise range is $0.082430000000$TeX0.082430000000 to $0.159920000000$TeX0.159920000000. These rates are compatible with the specified heterogeneous copy/readout probabilities, but they are not used alone as an objectivity criterion.

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IID robustness threshold map

The analytic IID map gives the maximum effective bit-flip noise compatible with the $0.9$TeX0.9 threshold at each fragment count. Selected values are center

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center For selected noise levels, the minimum passing fragment counts are 22 at $p=0.30$TeXp=0.30 and 40 at $p=0.35$TeXp=0.35; at $p=0.40$TeXp=0.40, the threshold is not reached within 64 fragments. The full analytic threshold-map grid is reproduced by the benchmark calculation with maximum absolute discrepancy $4.999\times10^{-13}$TeX4.999\times10^{-13}.

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Heterogeneous analytic comparison, controls, and bootstrap stability

For the specified heterogeneous copy/readout model, exact product-channel calculations agree with the empirical raw-shot subset estimates to mean absolute discrepancy $0.000814$TeX0.000814 and maximum absolute discrepancy $0.004253$TeX0.004253. These values are consistent with finite sampling and the specified independent noise model.

In the rejection test, column-shuffled fragments, row-shuffled fragment vectors, and independent-random fragments all remain below the $0.9$TeX0.9 redundancy threshold at every subset size. No negative-control curve crosses the threshold, and the maximum information fractions in the three controls are below $0.002$TeX0.002.

The finite-shot bootstrap stability check preserves the relevant crossings at the full sample size. For $100000$TeX100000 shots, the $m=3$TeXm=3 mean information fraction has p05/p50/p95 values

and the $m=4$TeXm=4 minimum information fraction has p05/p50/p95 values

Thus the mean $m=3$TeXm=3 crossing and the all-subsets $m=4$TeXm=4 crossing are stable under the specified bootstrap procedure.

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Common-mode fragility

The common-mode model changes the interpretation of many agreeing fragments. Even with zero idiosyncratic fragment noise, the asymptotic ceiling in Eq. reference falls below the $0.9$TeX0.9 threshold near

At $m=4$TeXm=4, the selected thresholds give maximum idiosyncratic noise $0.112816531591$TeX0.112816531591 when $c=0$TeXc=0, $0.088481076570$TeX0.088481076570 when $c=0.005$TeXc=0.005, $0.055410571766$TeX0.055410571766 when $c=0.010$TeXc=0.010, $0.032557116142$TeX0.032557116142 when $c=0.012$TeXc=0.012, and no passing idiosyncratic-noise value at $c=0.015$TeXc=0.015. The common-mode axis is therefore a first-order diagnostic for fragment-access objectivity inferences.

The scope of the main conclusions is summarized in Table reference. The table separates conclusions established inside the synthetic benchmark from experimental inferences that require raw fragment-access records.

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A physical QD/SBS-style objectivity inference would require, at minimum, a same-instance table of pointer/reference records and disjoint fragment readouts; a definition of the fragment family and fragment-access masks; time stamps or window identifiers sufficient to test same-instance pairing; calibration and threshold metadata; readout-chain state variables relevant to drift, dark counts, recovery, latching, and dead time; and controls able to distinguish independent fragment copies from common-mode readout errors. For detector platforms such as SPADs, TESs, and SNSPDs, a readout criterion should be stated separately from the fragment-access objectivity criterion, because threshold crossing, latching, reset, and persistent discriminability are device-level properties rather than consequences of a POVM alone.

The negative-control and common-mode checks above are prerequisites for an experimental objectivity inference. A candidate experimental dataset should be processed with the same subset enumeration, disagreement statistics, and threshold convention used for the synthetic benchmark. It should also pass negative controls that break pointer-fragment coupling, preserve the reported crossing under finite-shot stability checks, and include independent common-mode controls for shared calibration shifts, timing offsets, threshold drift, electronic cross talk, source-state labeling errors, and common reconstruction choices. A failed control is not only a reduction in the numerical redundancy; it invalidates the inference from agreeing fragments to independent fragment-access objectivity.

The dataset is synthetic. The model is binary, pointer-copy oriented, and finite. It evaluates an analysis procedure and robustness benchmark for raw fragment-access data, not the physics of a detector, environment, or apparatus. The baseline model assumes independent copy/readout errors except where the common-mode stress test explicitly introduces a shared hidden flip. It leaves the need for physical shot-level fragment records unchanged.

The synthetic thresholds are also convention-dependent. The $0.9$TeX0.9 information-fraction threshold is a specified benchmark threshold, not a universal boundary. Changing the pointer prior, fragment-access rule, subset criterion, stationarity window, or readout convention changes the resulting criterion. The bootstrap quantiles describe estimator stability for resampled synthetic records and should not be interpreted as experimental uncertainties.

Aggregate QD-style curves may be useful summaries, but they do not establish raw fragment-access objectivity. In particular, they do not determine which subsets pass, how fragments disagree, whether within-row correlations persist after row shuffling, or whether hidden common-mode errors have been independently bounded. Experimental same-instance redundant-objectivity inference in a detector or environmental-fragment platform is therefore not warranted in the absence of raw fragment-access data and common-mode controls.

The accompanying data archive Raw-Fragment Benchmarking for Redundancy Tests in Quantum Darwinism and Spectrum Broadcast Structure - ancillary benchmark.zip contains raw synthetic shot data in NPZ and CSV.GZ formats; primary redundancy, fragment-access, independent-channel, and disagreement diagnostics; analytic IID threshold maps; heterogeneous analytic-vs-empirical comparisons; negative-control outputs; common-mode correlated-noise sensitivity maps; finite-shot bootstrap stability tables; prior-bias sensitivity outputs; and scripts that regenerate the synthetic shots and benchmark analyses. The central files are:

- raw_data/qdsbs_synthetic_shots_100000.npz; - raw_data/qdsbs_synthetic_shots_100000.csv.gz; - primary_results/redundancy_curve.csv; - primary_results/fragment_access_by_subset.csv; - primary_results/disagreement_pair_stats.csv; - validation_results/qdsbs_noise_fragment_threshold_map.csv; - validation_results/qdsbs_negative_controls_summary.csv; - validation_results/qdsbs_correlated_noise_selected_thresholds.csv; - validation_results/qdsbs_finite_shot_bootstrap_summary.csv; - scripts/generate_qdsbs_synthetic_shots.py; - scripts/sweep_qdsbs_threshold_map.py; - scripts/qdsbs_validation_suite.py.

These files constitute the reproducibility archive.

A raw-fragment benchmark for QD/SBS-style redundancy tests has been defined and evaluated on a fixed binary pointer-copy model. The baseline model shows a mean-over-subsets threshold crossing at $m=3$TeXm=3 and an all-subsets threshold crossing at $m=4$TeXm=4, while all negative controls remain below threshold. Analytic independent-channel thresholds are reproduced to numerical precision, and heterogeneous product-channel predictions agree with empirical raw-shot estimates inside finite-shot deviations. The main fragility axis is a hidden common-mode flip, which imposes the ceiling $1-\Htwo(c)$TeX1-\Htwo(c) and can block the $0.9$TeX0.9 threshold even as the number of fragments grows.

The resulting analysis is model-level rather than experimental. Raw per-shot fragment access, subset enumeration, disagreement diagnostics, negative controls, finite-shot stability, and common-mode bounds are required before experimental objectivity can be assessed. In their absence, aggregate redundancy curves and local detector records remain insufficient to establish fragment-access consensus.