Durable Records and Redundant Objectivity in Quantum Measurement Chains

A laboratory measurement description can combine several physically distinct specifications: an input--output probability model, a detector response process, a durable outcome record, and an output-label rule. These specifications, however, are not logically equivalent. A POVM or instrument specifies the probabilities assigned to outcome labels. Detector tomography and calibration protocols can certify such a measurement map with stated uncertainty. Yet an event counter, a superconducting nanowire, a transition-edge sensor, or an avalanche photodiode is also characterized by ready-state preparation, signal response, latching or amplification, discriminable output production, recovery, and specified dark-count, afterpulse, and dead-time tolerances. Finally, a durable local record does not by itself establish classical objectivity in the sense studied in quantum Darwinism and spectrum broadcast structure: such objectivity involves independently accessible, redundant, predominantly classical information about a specified pointer variable, with the relevant coherence and consistency tolerances stated.

These distinctions are represented by finite-window criteria for three layers of a measurement chain. Each layer is specified by finite-window quantities, thresholds, tolerances, and operating windows appropriate to that layer. The criteria use standard POVM and instrument statistics while leaving the detector micro-Hamiltonian unspecified. Accordingly, the following implications do not hold without the corresponding readout or objectivity conditions:

- a measurement-map certificate, such as a POVM, implies a durable-record certificate; - a threshold crossing or activation event implies record-bearing readout; - a local durable record implies consensus-level objectivity.

Two considerations motivate the separation. On the objectivity side, quantum Darwinism, strong quantum Darwinism, and spectrum broadcast structure distinguish decoherence, redundancy, consensus, and state-structure objectivity through fragment-access and broadcast-structure conditions [citation]. Decoherence can suppress off-diagonal terms without making records independently accessible. Redundancy can be discussed through mutual-information plateaus, while the stronger criteria used below add, according to level, classical accessibility under specified fragment readouts, pointer-sector coherence control, bounded disagreement, and, for structural results, the required independence or factorization hypothesis. For strong-quantum-Darwinism/spectrum-broadcast-structure equivalence results, the relevant independence or factorization condition is part of the structural hypothesis [citation]. On the detector side, SPADs, TESs, and SNSPDs are not fully characterized by threshold events; they are nonequilibrium devices with dark-count envelopes, afterpulsing or relaxation channels, latching or reset constraints, and finite recovery times [citation].

The resulting finite-window description consists of a layer-indexed tuple; a ready--activated--latched readout model with reset; finite-family objectivity tests based on realized readout mutual information, or on stated Holevo-level bounds where appropriate; basis-fixed correlation-surplus and disagreement diagnostics; redundancy and consensus counts; and compatibility conditions for composing layer-specific certifications. All quantities are defined on finite operating windows using standard quantum-measurement, quantum-information, and detector-characterization notation.

Let $S$TeXS denote the source or measured system and let $Z=\sum_z z\Pi_z$TeXZ=\sum_z z\Pi_z be a candidate pointer observable when objectivity is discussed. Unless otherwise stated, logarithms in entropy expressions are natural logarithms; using another fixed base only rescales the entropy and information thresholds. A detector or apparatus can be described at several levels. At the measurement-map level, a measurement with outcomes $m\in\mathcal X$TeXm\in\mathcal X is represented by POVM effects $E_m\ge 0$TeXE_m\ge 0 with $\sum_m E_m=I$TeX\sum_m E_m=I, or by a quantum instrument $\{\mathcal I_m\}$TeX\{\mathcal I_m\} whose probabilities are

A measurement-map certificate $\Cmap$TeX\Cmap means that the effects, instrument, calibration map, or equivalent response model have been specified with a stated uncertainty or tolerance on a stated operating domain.

At the readout level, a detector is modeled as a device that turns activations into durable, discriminable output records. The minimal reduced model used below has states $R$TeXR (ready), $A$TeXA (activated), and $L$TeXL (record-bearing output class, including latched outputs when appropriate). Here $L$TeXL denotes the resolved output class accepted as an event under the specified readout convention, without assuming permanent material trapping.

At the objectivity level, a family $\Ffam=\{\Frag_a\}_{a=1}^{M_f}$TeX\Ffam=\{\Frag_a\}_{a=1}^{M_f} of disjoint fragments is fixed, together with a fragment readout rule for each fragment. The question is whether the specified fragment readouts recover the same pointer value with readout mutual information close to the pointer entropy, low basis-fixed correlation surplus, controlled pointer-sector coherence, and low inter-fragment disagreement.

All criteria below are indexed by a finite operating window, an output-label convention, tolerances, and, when objectivity is discussed, a fragment-access rule. Accordingly, redundancy, sector coherence, population drift when invoked, and consensus are treated as finite-window properties relative to stated thresholds and access rules.

The measurement-map layer specifies the model connecting input states to outcome probabilities. It is indispensable for assigning outcome statistics, but it does not address persistence, reset, or fragment-level accessibility.

definition: Measurement-map certificate. A measurement-map certificate $\Cmap$TeX\Cmap for an outcome set $\mathcal X$TeX\mathcal X consists of a POVM $\{E_m\}_{m\in\mathcal X}$TeX\{E_m\}_{m\in\mathcal X}, or a quantum instrument $\{\mathcal I_m\}_{m\in\mathcal X}$TeX\{\mathcal I_m\}_{m\in\mathcal X}, together with a calibration or tomography specification of the operating domain, uncertainty convention, and outcome-label convention under which Eq. reference is used.

Quantum instruments, POVM descriptions, and detector tomography supply possible implementations of $\Cmap$TeX\Cmap [citation]. The criteria do not depend on a particular implementation; each such description answers the measurement-map question of which probabilities are assigned to outcome labels under a stated convention.

proposition: A measurement-map certificate is not a durable-record certificate. A POVM or instrument certificate does not by itself establish that an accepted output event is durably recorded, nor that it satisfies the stated consensus criterion for objectivity.

proof. Two implementations can have the same POVM on $S$TeXS while differing on their record registers. Let one implementation write the outcome $m$TeXm into an ancillary memory $M$TeXM and leave the conditional memory states distinguishable throughout $[t_L,t_L+\taupers]$TeX[t_L,t_L+\taupers]. Let a second implementation apply the same instrument on $S$TeXS but erase or thermalize $M$TeXM before the persistence window, mapping the conditional memory states to a common blank state. Tracing over $M$TeXM gives the same effects $E_m$TeXE_m and hence the same probabilities $p(m|\rho)$TeXp(m|\rho) in both cases, but only the first implementation satisfies a persistence criterion. A third implementation can have the same local record while all specified fragments satisfy $\rho_{\Frag_a}^{(z)}=\rho_{\Frag_a}$TeX\rho_{\Frag_a}^{(z)}=\rho_{\Frag_a}, giving no redundant pointer information. Thus the measurement map does not entail either durable readout or objectivity diagnostics.

For outcome-statistics applications, $\Cmap$TeX\Cmap may be the only required certificate. Durable events, repeated counting, classical records, or objectivity require the corresponding readout or objectivity diagnostics rather than an implicit enlargement of the measurement-map certificate.

A record-bearing detector output is not specified by a threshold condition alone. The minimal class considered here is a metastable amplification device: it maintains a ready state, undergoes signal-assisted activation, moves probability into a record-bearing output class, and resets.

Minimal ready--activated--latched model

Let $p_R(t)$TeXp_R(t), $p_A(t)$TeXp_A(t), and $p_L(t)$TeXp_L(t) be the probabilities of ready, activated, and record-bearing output states on a specified readout window. In the minimal closed three-state reduction used here, relaxation from the activated sector returns the device to the ready sector without producing a record. The model is

Here $\Gdark$TeX\Gdark is the dark or spontaneous activation rate, $\Gsw$TeX\Gsw is the signal-assisted activation rate, $\Glat$TeX\Glat is the rate into the resolved output class, $\Grel$TeX\Grel is relaxation from the activated sector back to the ready sector without record formation, and $\Greset$TeX\Greset is the reset rate. With this convention, Eqs. reference--reference conserve $p_R+p_A+p_L$TeXp_R+p_A+p_L. Platform-dependent models may refine these rates, add non-Markovian kernels, or resolve many internal states. If relaxation from $A$TeXA represents probability loss rather than return to the ready sector, the closed reduction is replaced by a model with an explicit loss state. Equations reference--reference define only the reduced readout layer.

definition: Record flux and first-record hazard. The record-entry flux is

It is the unconditional probability current into the record-bearing output class in the reduced model, not the raw activation flux $[\Gsw(t)+\Gdark(t)]p_R(t)$TeX[\Gsw(t)+\Gdark(t)]p_R(t).

On a single-hit window with $p_L(0)=0$TeXp_L(0)=0 and with reset and retriggering neglected until the observation window closes, the durable-record probability is

If a renewal or first-arrival analysis is used instead, one may introduce a separate first-record hazard $h_{\rm rec}(t)$TeXh_{\rm rec}(t) and write $P_{\rm rec}^{(1)}[0,T]=1-\exp[-\int_0^T h_{\rm rec}(t)\,\dd t]$TeXP_{\rm rec}^{(1)}[0,T]=1-\exp[-\int_0^T h_{\rm rec}(t)\,\dd t]. In that case $h_{\rm rec}$TeXh_{\rm rec} is defined by the renewal model and is distinct from the unconditional flux $\Jrec$TeX\Jrec unless an additional approximation relates them. When afterpulsing, retriggering, or recovery effects matter, Eq. reference is not a complete counting theory; the envelope is included as a separate readout component.

If the activated state is quasi-steady on the relevant time scale, Eq. reference gives

where

is a record-formation efficiency. The distinction between activation and record formation is visible in $\etarec$TeX\etarec: two devices may have the same activation statistics but different durable-record statistics.

Persistence, event accounting, reset thermodynamics, and dead time

A durable-record condition requires that the output class remain discriminable on a persistence window. Let $d_L(t)$TeXd_L(t) be a platform-specific discriminability measure for the output class $L$TeXL under the output-discrimination convention. A minimal persistence condition is

where $t_L$TeXt_L is the record-entry time and $d_\ast$TeXd_\ast is the stated output-discrimination threshold. The choice of $d_L$TeXd_L is platform dependent. In an SNSPD it may be tied to pulse recovery and latching behavior; in a TES to electrothermal output discrimination; in a SPAD to quench/recharge and afterpulse control. The threshold and persistence window are therefore part of the readout convention.

A simple event-accounting mismatch can be quantified as

where $N_{\mathrm{trig}}$TeXN_{\mathrm{trig}} is the number of threshold or activation candidates under a stated trigger definition and $N_{\mathrm{rec}}$TeXN_{\mathrm{rec}} is the number of accepted durable records. Other conventions may be preferable in a given experiment; in each case the accounting convention is part of the record-level certificate. Without such an accounting convention, activation-level data do not determine record-level data.

For repeated operation, thermodynamic bookkeeping gives a consistency check rather than a detector-independent acceptance test. For an ideal quasistatic erasure or reinitialization of an $M$TeXM-state logical register with uniform prior uncertainty, so that the prior Shannon entropy is $\log M$TeX\log M, coupled to a reservoir at temperature $T_{\mathrm{res}}$TeXT_{\mathrm{res}}, Landauer's bound [citation] gives the heat scale

For reset to a fixed standard logical state with a general prior distribution $P$TeXP over the logical states, the corresponding entropy-change lower bound is $\kB T_{\mathrm{res}}H(P)$TeX\kB T_{\mathrm{res}}H(P) under the same logarithmic convention. For nonequilibrium detector cycles this is a lower thermodynamic scale, not a calorimetric model of the detector; practical amplification, readout, and reset channels can dissipate substantially more. Repeated operation is accompanied by a dead-time or recovery compatibility condition, for example

in a low-occupancy regime, or an explicit dead-time correction model outside that regime. Here $r_{\mathrm{use}}$TeXr_{\mathrm{use}} is the intended use rate and $\taudead$TeX\taudead is the platform-specific recovery time.

definition: Readout certificate. A readout certificate $\Crec$TeX\Crec for one outcome class consists of a record-flux convention $\Jrec$TeX\Jrec or an explicitly defined first-record hazard $h_{\rm rec}$TeXh_{\rm rec}, a record-formation efficiency or bound $\etarec$TeX\etarec, an event-accounting mismatch $\DeltaEA$TeX\DeltaEA or bound, a persistence condition on $\taupers$TeX\taupers, a specified readout envelope for dark counts, afterpulsing or retriggering, latching, reset, and dead time on the stated operating window, and, where thermodynamic or reset-energy bounds are included, a reset-thermodynamic consistency convention.

proposition: Threshold crossing is not sufficient for durable readout. A threshold crossing or activation event supports a durable-readout certificate only when the readout certificate is also satisfied.

proof. In the reduced model, choose $\Gsw+\Gdark>0$TeX\Gsw+\Gdark>0 on the observation window but set $\Glat=0$TeX\Glat=0. Activation events then occur, but $\Jrec(t)=\Glat p_A(t)=0$TeX\Jrec(t)=\Glat p_A(t)=0 and, for $p_L(0)=0$TeXp_L(0)=0, no probability enters the record-bearing sector. More generally, nonzero activation flux constrains $p_A$TeXp_A but does not by itself impose persistence, event-accounting accuracy, or reset/dead-time compatibility. A threshold crossing is therefore an activation-level datum unless the readout certificate is also satisfied.

A local durable record does not by itself establish consensus-level objectivity. On a finite observation window, consensus-level objectivity requires redundant and mutually consistent access to the recorded pointer value through the specified fragments, without supplementing the fragment-access measurements by a new measurement of the source system. The finite-family standard uses readout information, basis-fixed correlation surplus, pointer-sector coherence, and inter-fragment disagreement, with a population-drift bound included whenever stationarity of pointer-sector probabilities is part of the stated criterion; stronger state-structure criteria apply only under their stated assumptions.

Fragment information and basis-fixed correlation surplus

Fix a pointer observable $Z=\sum_z z\Pi_z$TeXZ=\sum_z z\Pi_z of the source system and a family $\Ffam=\{\Frag_a\}_{a=1}^{M_f}$TeX\Ffam=\{\Frag_a\}_{a=1}^{M_f} of disjoint fragments. Let

be the pointer distribution. For each fragment, define the conditional fragment state associated with the specified pointer conditioning by

whenever $p_z>0$TeXp_z>0; zero-probability terms are omitted from conditional-state expressions and entropy sums. The two-sided projection specifies the pointer-ensemble conditioning used by the fragment diagnostics; it is not an additional detector-dynamical assumption and does not require the full $S\Frag_a$TeXS\Frag_a state to be block diagonal in $Z$TeXZ. The fragment Holevo information about the fixed pointer ensemble is [citation]

where $\rho_{\Frag_a}=\Tr_S\rho_{S\Frag_a}$TeX\rho_{\Frag_a}=\Tr_S\rho_{S\Frag_a}; equivalently, because the partial trace over $S$TeXS removes cross-sector terms, $\rho_{\Frag_a}=\sum_{z:p_z>0} p_z\rho_{\Frag_a}^{(z)}$TeX\rho_{\Frag_a}=\sum_{z:p_z>0} p_z\rho_{\Frag_a}^{(z)} with zero-probability sectors omitted. A specified fragment readout is a POVM or response map $\mathsf R_a=\{R_{a,y}\}_y$TeX\mathsf R_a=\{R_{a,y}\}_y on $\Frag_a$TeX\Frag_a, giving

and the readout mutual information

For the ensemble $\{p_z,\rho_{\Frag_a}^{(z)}\}_z$TeX\{p_z,\rho_{\Frag_a}^{(z)}\}_z, the Holevo quantity upper-bounds the accessible mutual information $\sup_{\mathsf R} I(Z:Y_{\mathsf R})$TeX\sup_{\mathsf R} I(Z:Y_{\mathsf R}), where $Y_{\mathsf R}$TeXY_{\mathsf R} is the classical outcome generated by a candidate fragment measurement $\mathsf R$TeX\mathsf R. Replacing the realized $\Ird_a$TeX\Ird_a by $\Holevo(\Frag_a)$TeX\Holevo(\Frag_a) therefore yields a Holevo-level redundancy bound, rather than a realized-readout value, unless a readout rule attaining the stated value, or an attainability condition such as mutually commuting conditional fragment states under the specified pointer ensemble, is supplied. The corresponding basis-fixed correlation surplus is

where $I(S:\Frag_a)$TeXI(S:\Frag_a) is the quantum mutual information. The quantity $\SurpZ$TeX\SurpZ is not a basis-optimized quantum-discord measure in the sense of Refs. [citation]; it is defined relative to the specified pointer observable, fragment family, and access rule. Small $\SurpZ$TeX\SurpZ indicates that correlations available in the fragment are predominantly accounted for by the specified classical pointer ensemble.

To compare two fragment inferences, let

denote the posterior distributions produced by the specified fragment readouts for outcomes with nonzero marginal readout probability. For a pair of disjoint fragments define

whenever $p_z>0$TeXp_z>0; zero-probability terms are omitted from the conditional pair expressions. Under the same pointer-conditioning convention, use the induced joint readout distribution

This joint distribution is the law used for the disagreement diagnostic; it does not add a block-diagonality assumption on the unconditioned state. Define pairwise disagreement as the expected total-variation distance

where the expectation is evaluated only for pairs with $p(y_a,y_b)>0$TeXp(y_a,y_b)>0; on this support the marginal denominators in Eq. reference are positive. Sums are replaced by integrals for continuous readout variables. Finally, define a pointer-sector coherence functional on the observation window by the trace norm of the off-block-diagonal component in the specified pointer decomposition,

This functional controls coherence between pointer sectors rather than stationarity of the pointer-sector probabilities. If the finite-window criterion also requires stationarity of those probabilities over the observation window, a separate population-drift diagnostic may be used, for example

or the fragment diagnostics are indexed by the specified preparation and readout times. For a nondegenerate pointer basis Eq. reference reduces to an off-diagonal coherence diagnostic. For degenerate pointer sectors it is invariant under changes of basis within each sector. Upper bounds based on sums of off-block trace norms may be used in place of Eq. reference when the bound is part of the certificate. A calibrated witness can replace direct access to Eq. reference only if it supplies a bound or monotone relation for the same pointer variable and observation window.

Finite-family redundancy and consensus

For tolerances $0<\delta\ll1$TeX0<\delta\ll1 and $0<\epsilon\ll1$TeX0<\epsilon\ll1, fix an entropy floor $\Hmin>0$TeX\Hmin>0. The relative redundancy diagnostic is evaluated only when

If $H(Z)<\Hmin$TeXH(Z)<\Hmin, the entropy-floor condition fails and the normalized redundancy fraction is not used under the stated convention. When Eq. reference holds, a fragment is called $\delta$TeX\delta-redundant for $Z$TeXZ under the specified readout if

with $\Ird_a$TeX\Ird_a defined by Eq. reference. When a realized fragment readout is not part of the certificate and $\Holevo(\Frag_a)$TeX\Holevo(\Frag_a) is substituted for $\Ird_a$TeX\Ird_a, the resulting entry is a Holevo-level redundancy bound with the corresponding attainability convention, such as mutual commutativity of the conditional fragment states for the specified pointer ensemble. The redundancy count $\Red$TeX\Red is the number of disjoint fragments satisfying Eq. reference under the fixed fragment family and readout convention. A subset $\mathcal C\subset\Ffam$TeX\mathcal C\subset\Ffam satisfies the $(\delta,\epsilon)$TeX(\delta,\epsilon) consensus condition if every member is $\delta$TeX\delta-redundant and

The consensus count $\Con$TeX\Con is the size of the largest such subset under the fixed fragment family and joint readout convention, or a lower bound stated under that convention.

definition: Objectivity certificate levels. Fix $Z$TeXZ, $\Ffam$TeX\Ffam, $\{\mathsf R_a\}_{a=1}^{M_f}$TeX\{\mathsf R_a\}_{a=1}^{M_f}, $\tauobs$TeX\tauobs, $\delta$TeX\delta, $\epsilon$TeX\epsilon, an entropy floor $\Hmin>0$TeX\Hmin>0, thresholds $\Rmin,\Cmin\ge2$TeX\Rmin,\Cmin\ge2, and a pointer-sector coherence tolerance $\sigmax$TeX\sigmax. If stationarity of pointer-sector probabilities is part of the stated criterion, also fix a population-drift tolerance $\Delta_Z^{\mathrm{pop},\ast}$TeX\Delta_Z^{\mathrm{pop},\ast} for Eq. reference. The objectivity certificate levels are:

- $\Cobj^{(1)}$TeX\Cobj^{(1)}: redundant encoding, satisfied if $H(Z)\ge\Hmin$TeXH(Z)\ge\Hmin, $\Red\ge\Rmin$TeX\Red\ge\Rmin, $\SigZ\le\sigmax$TeX\SigZ\le\sigmax, and, when stationarity of pointer-sector probabilities is part of the stated criterion, $\Delta_Z^{\mathrm{pop}}\le\Delta_Z^{\mathrm{pop},\ast}$TeX\Delta_Z^{\mathrm{pop}}\le\Delta_Z^{\mathrm{pop},\ast}, where the counted fragments satisfy Eq. reference using specified readout information $\Ird_a$TeX\Ird_a, or using a Holevo-level bound together with an attainability condition such as mutual commutativity of the conditional fragment states for the specified pointer ensemble. - $\Cobj^{(2)}$TeX\Cobj^{(2)}: consensus-class objectivity, satisfied if $\Cobj^{(1)}$TeX\Cobj^{(1)} is satisfied and $\Con\ge\Cmin$TeX\Con\ge\Cmin under Eq. reference, using the specified disagreement convention from Eq. reference or its stated alternative. - $\Cobj^{(3)}$TeX\Cobj^{(3)}: structural objectivity, satisfied by an approximate spectrum-broadcast-structure or strong-quantum-Darwinism condition specified with (i) a distance or divergence, (ii) a tolerance, (iii) a fragment-access rule, and (iv) the independence or factorization condition required by the stated structural result. For equivalence results between strong quantum Darwinism and spectrum broadcast structure, the structural certificate uses the independence or factorization hypothesis of the selected theorem; a pairwise conditional-independence condition is used only when that theorem establishes sufficiency from the same pairwise condition [citation]. The structural condition is evaluated for the same $Z$TeXZ, $\Ffam$TeX\Ffam, $\tauobs$TeX\tauobs, and fragment-access convention as the lower-level diagnostics, with its approximation tolerance stated for the selected metric. If the structural metric does not itself bound pointer-sector coherence, the certificate includes the corresponding $\SigZ$TeX\SigZ tolerance; it also includes any population-drift bound required by the stated criterion.

The separation between $\Cobj^{(1)}$TeX\Cobj^{(1)} and $\Cobj^{(2)}$TeX\Cobj^{(2)} follows because redundant information in separate fragments need not imply compatible posterior distributions under the specified readouts. Consensus is an additional condition on fragment-level inferences.

proposition: Durable records are not sufficient for consensus-level objectivity. A durable local record is not sufficient for consensus-level objectivity. Consensus-level objectivity requires at least $\Cobj^{(2)}$TeX\Cobj^{(2)}, or the stronger $\Cobj^{(3)}$TeX\Cobj^{(3)}, on the same pointer variable, fragment family, observation window, and readout convention.

proof. Let a detector memory $L$TeXL store the pointer value $z$TeXz with perfect local persistence on $[t_L,t_L+\taupers]$TeX[t_L,t_L+\taupers], but let every specified fragment be independent of $z$TeXz, so that $\rho_{\Frag_a}^{(z)}=\rho_{\Frag_a}$TeX\rho_{\Frag_a}^{(z)}=\rho_{\Frag_a} for all $a$TeXa and $z$TeXz. Then $\Holevo(\Frag_a)=0$TeX\Holevo(\Frag_a)=0 for each fragment, hence no fragment is $\delta$TeX\delta-redundant when $H(Z)\ge\Hmin$TeXH(Z)\ge\Hmin, and $\Red=\Con=0$TeX\Red=\Con=0. The local readout certificate can therefore hold while the redundancy and consensus diagnostics fail. Local durability is necessary for a recorded outcome but does not entail consensus-level objectivity.

For a fixed outcome-label convention or coarse-grained pointer class, the layer quantities are collected in a measurement-chain tuple.

definition: Measurement-chain tuple. For a fixed outcome-label convention or coarse-grained pointer class, define

The tuple is accompanied by the outcome-label map, the pointer observable $Z$TeXZ or its coarse-grained surrogate, $\Ffam$TeX\Ffam, $\tauobs$TeX\tauobs, $\delta$TeX\delta, $\epsilon$TeX\epsilon, $\Hmin$TeX\Hmin, $\Rmin$TeX\Rmin, $\Cmin$TeX\Cmin, $\sigmax$TeX\sigmax, the readout and persistence windows, the trigger and discrimination-threshold conventions, the fragment-access rule, and the reset-thermodynamic convention when a thermodynamic or reset-energy bound is used. When stationarity of pointer-sector probabilities is part of the finite-window criterion, the pair $(\Delta_Z^{\mathrm{pop}},\Delta_Z^{\mathrm{pop},\ast})$TeX(\Delta_Z^{\mathrm{pop}},\Delta_Z^{\mathrm{pop},\ast}) is included as additional tuple data. Platform-specific envelopes such as inter-arrival distributions, dark-count and afterpulse envelopes, reset traces, and witness-calibration maps may be appended when they enter the operating convention.

proposition: Layered criterion for consensus-level objectivity. For the finite-window convention associated with the measurement-chain tuple, consensus-level objectivity for an outcome class requires

with all entries evaluated for the same outcome class and readout convention. Neither $\Cmap$TeX\Cmap alone, nor $\Cmap\wedge\Crec$TeX\Cmap\wedge\Crec, nor $\Cmap\wedge\Crec\wedge\Cobj^{(1)}$TeX\Cmap\wedge\Crec\wedge\Cobj^{(1)} is sufficient.

proof. The measurement-map certificate fixes the measurement map but does not certify durable records. The readout certificate fixes durable record formation but does not certify redundant and mutually consistent fragment access. The level $\Cobj^{(1)}$TeX\Cobj^{(1)} establishes redundancy but not necessarily inter-fragment agreement. Consensus-level objectivity begins at $\Cobj^{(2)}$TeX\Cobj^{(2)}, while $\Cobj^{(3)}$TeX\Cobj^{(3)} is a stronger structural criterion under a stated approximation metric. Thus the conjunction in Eq. reference is required for consensus-level objectivity.

corollary: Layer separation. A POVM does not by itself provide a durable-record certificate unless the readout certificate is also supplied. A detector click or threshold crossing does not by itself support consensus-level objectivity; the corresponding readout and objectivity certificates are also required. A decoherence witness by itself does not furnish an objectivity certificate; such a certificate includes bounds or calibrated witnesses for redundancy, consensus, basis-fixed correlation surplus, and pointer-sector coherence under the same pointer variable, fragment family, and observation window; when stationarity of pointer-sector probabilities is part of the criterion, the corresponding population-drift bound is also included.

Single-photon and superconducting detectors

In SPAD, TES, and SNSPD platforms, the measurement-map layer can be specified through a detector response model or POVM/tomography, whereas the readout layer is fixed by device-specific recovery and output-discrimination physics. A SPAD event convention depends on quench/recharge behavior, dead time, dark-count and afterpulse envelopes, and the pulse-acceptance rule. A TES event convention depends on electrothermal response and recovery. An SNSPD event convention depends on hotspot-triggered switching, latching or nonlatching bias regime, kinetic-inductance-limited recovery, and reset compatibility. These conditions do not replace the POVM specification; they determine whether an activation becomes an accepted record under the specified readout convention.

For detector-characterization applications, the measurement-chain tuple can be truncated at the measurement-map and readout entries. The objectivity entries are relevant only for analyses concerning an accessible pointer record across a specified fragment family, an environmental branching structure, redundant environmental records, or consensus among fragment readouts.

Quantum Darwinism and spectrum broadcast structure

Quantum Darwinism and spectrum broadcast structure provide structural benchmarks for the objectivity entries of the tuple. A mutual-information plateau is a redundancy diagnostic, but finite-window objectivity also requires a pointer variable, a fragment family, a fragment-access rule, and tolerances for readout information, basis-fixed correlation surplus, pointer-sector coherence, inter-fragment disagreement, and any population-drift bound required by a stationarity condition. This distinction is consistent with recent redundancy/consensus analyses [citation]. Strong quantum Darwinism and approximate spectrum broadcast structure supply $\Cobj^{(3)}$TeX\Cobj^{(3)}-type structural conditions only when an approximation metric, tolerance, fragment-access rule, and the relevant independence or factorization assumptions are fixed [citation].

Layer-wise and cross-platform composition

In practice, the three layers may be probed in different platform classes. A detector-tomography experiment may determine a measurement-map certificate, a time-resolved readout experiment may determine a readout certificate, and a photonic simulator or superconducting-circuit experiment may determine bounds entering an objectivity certificate. Such cross-platform composition defines compatibility among layer diagnostics rather than device-level realization of the full measurement-chain tuple. It is well defined only when all imported layers refer to the same pointer variable or to a stated coarse-grained surrogate, the same outcome-label convention, and compatible readout windows.

The following consistency checks state compatibility requirements for combining layer-specific evidence under the assumptions defining each diagnostic.

F0: Pointer and outcome mismatch..

If no map relates the measurement-map, readout, and objectivity layers to one pointer variable and one readout-compatible outcome class, the composed tuple does not define a single finite-window certificate.

F1: Uncalibrated witness substitution..

A witness or surrogate estimate for $\Ird_a$TeX\Ird_a, $\Holevo$TeX\Holevo, $\SurpZ$TeX\SurpZ, $\epsab$TeX\epsab, $\SigZ$TeX\SigZ, or, when used, $\Delta_Z^{\mathrm{pop}}$TeX\Delta_Z^{\mathrm{pop}} certifies the corresponding diagnostic only when it supplies a bound or monotone calibration on the same pointer variable, fragment family, and observation window.

F2: Measurement-map sufficiency..

A measurement-map certificate alone does not imply persistence, reset behavior, event accounting, or objectivity diagnostics; these require readout and objectivity information not contained in $\Cmap$TeX\Cmap.

F3: Threshold-only sufficiency..

A threshold-only model certifies durable readout only when it also reproduces the persistence, dark-count/afterpulse structure, and reset/dead-time behavior required by the readout convention.

F4: Redundancy without consensus..

If fragments satisfy the readout-information threshold in Eq. reference but fragment posterior distributions disagree beyond the stated tolerance, the evidence may support $\Cobj^{(1)}$TeX\Cobj^{(1)} but not $\Cobj^{(2)}$TeX\Cobj^{(2)}.

F5: Decoherence without accessibility..

If off-diagonal suppression is observed but no disjoint fragment family is shown to carry accessible pointer information, the evidence supports decoherence but does not by itself establish the finite-family objectivity criteria.

F6: Structural criterion without the required assumptions..

An approximate spectrum-broadcast-structure or strong-quantum-Darwinism structural condition supplies $\Cobj^{(3)}$TeX\Cobj^{(3)} only when its approximation metric, tolerance, fragment-access rule, and required independence or factorization condition are specified for the same pointer variable, fragment family, and observation window.

The finite-window criteria separate three components that are often combined under a single measurement label: an input--output probability component, a record-bearing readout component, and a redundant-objectivity component. The separation is at the level of finite-window certification and leaves the underlying quantum dynamics unchanged. Each layer can be assessed independently, provided that the operating window, tolerances, and conventions attached to that layer are stated.

At the detector level, the separation clarifies why detector tomography and physical readout characterization certify different diagnostics: the former fixes an input--output map, whereas the latter concerns latching, persistence, reset-energy consistency, and reset dynamics. At the objectivity level, mutual-information plateaus, decoherence witnesses, and persistent local memories support different diagnostics unless fragment access, posterior agreement, basis-fixed correlation surplus, and pointer-sector coherence are all controlled. The same separation permits modular comparison across platforms, but cross-platform composition requires explicit maps among pointer variables, outcome labels, and readout windows.

Quantum measurement chains separate into measurement-map, readout, and objectivity layers. A POVM or instrument fixes measurement-map statistics; it does not certify durable records. A threshold crossing supports record-bearing readout only together with the readout certificate. A durable local record satisfies the consensus criterion for objectivity only when the fragment-access, coherence, and agreement conditions are also satisfied. In the finite-window setting, that criterion requires redundant fragment access, specified fragment readout information, low inter-fragment disagreement, bounded basis-fixed correlation surplus, and control of pointer-sector coherence; a population-drift bound is additionally required whenever stationarity of pointer-sector probabilities is part of the stated criterion.

Thus measurement-map calibration, record-bearing readout, and redundant-objectivity criteria remain separate unless the corresponding finite-window conditions are supplied.

No new experimental or numerical datasets were generated or analyzed for this work.