Oai265-B Reference-Preserving Quantum-Memory Certification from Measured Correlations

We give a finite-sample, calibration-aware protocol for certifying a specified deterministic quantum-memory decoder. A logical Bell input and the three correlations $X\otimes X$, $Y\otimes Y$, and $Z\otimes Z$ produce confidence bounds $F_L\le F_e\le F_U$ on the normalized logical Choi overlap. Under the stated contract, with coverage at least $1-\alpha$, an identified target qubit channel has half-diamond error in

where $b$ is a separately justified half-diamond transfer allowance. For direct certification of the implemented laboratory channel, $b=0$. The upper bound controls every input and untouched reference; it does not establish decoder optimality.

We prove bounded-score Hoeffding and exact binomial procedures, calibration-bias conversions, complete failure accounting, and channel-confidence-region refinements. A supplied execution report for the supplementary version-1 script records coverage of all 140 simulated datasets: 20 repetitions for each of seven channels, with 40 shots per setting. It reports zero invalid acceptances at error threshold $0.1$. We show, however, that this sample count cannot accept that threshold even on perfect data; the absence of invalid acceptances therefore does not exercise an informative high-accuracy acceptance regime. The reported identity-channel upper endpoint is approximately $0.8226864813$, correcting a numerical interpretation in the report. Exact rational example endpoints are archived in the manuscript. The execution is supplied evidence, not an independent rerun. No laboratory calibration, laboratory data, device-independent guarantee, or gravitational identification is claimed.

. Quantum memory; reference preservation; Bell correlations; normalized Choi state; diamond distance; finite confidence; calibration; deterministic decoding; simulation audit.

Oai265-A separated selected output entropy from reference-preserving recovery Oai265A. The present question is operational:

Given a specified memory, a fixed implemented decoder, complete attempted-run records, and calibrated preparation and measurement, what finite-confidence statement can be made about arbitrary-input quantum recovery?

Direct fidelity estimation, quantum process certification, channel distances, and their statistical foundations are established subjects KlieschRoth,FlammiaLiu,DaSilvaEtAl,Watrous. Our contribution is a self-contained protocol and error budget assembled for this particular qubit-memory task. The Choi conversion and diamond-norm interpretation are not claimed as new.

The supplied Cheung manuscripts provide documentary interfaces: En07-C distinguishes selected diagnostics from complete channels; Hawk04-B gives dimension-explicit decoder certificates; En02-B/C retain all decoder branches; and En01-C separates complete-record statistics from calibration coverage CheungEn07C,CheungHawk04B,CheungEn02B,CheungEn02C,CheungEn01C. Their model-specific results are comparisons, not unprovided premises of the proofs below.

1. Pure-target trace distance is an inequality. For a generally mixed state $\rho$ and rank-one projector $P$,

   $$1-\mathrm{Tr}(P\rho )\le d_{\mathrm{tr}}(\rho ,P)\le \sqrt{1-\mathrm{Tr}(P\rho )}.$$

   The upper inequality is not generally an equality. It is an equality when both states are pure.

2. Replacement benchmarks depend on the decoder task. A qubit replacement channel has optimal recovery error $3/4$. A specified decoder returning a fixed pure state can have error $1$.

3. Confidence is a repeated-experiment guarantee. The theorem bounds the probability of an invalid certificate. It does not assign a posterior probability to a channel after observing a particular dataset.

4. The reference and tested channel must be identified. An uncalibrated reference operation can compensate a memory error. Preparation, reference isolation, measurement, stationarity, and failure accounting are part of the contract.

The user-supplied report (30).txt is incorporated as a reported execution of the version-1 supplementary script ExecutionReport. This revision:

1. records the parameters, reported coverage counts, and exact first-repetition certificate endpoints;

2. corrects the identity upper-endpoint approximation from “approximately $0.8225$” to approximately $0.8226864813$;

3. proves that no dataset at the reported sample count can satisfy the $0.1$ acceptance threshold;

4. distinguishes allowed statistical noncoverage from numerical bracketing errors and from failure of a coverage theorem;

5. supplies a diagnostic reporter and optional threshold and arithmetic tests, without asserting execution of those additions.

All spaces in the primary protocol are finite dimensional. For a Hilbert space $H$, $\mathcal{B}(H)$ denotes its matrix algebra. A channel is completely positive and trace preserving, abbreviated CPTP.

Let $C\cong {\mathbb{C}}^2$ be the logical input, $R$ the accessible memory output, and $C^{\prime }\cong C$ the returned logical output. The identification $C^{\prime }\cong C$, including its Pauli frame, is fixed before validation.

Let

include the declared encoding, storage, correction, and accessible collection operations. Let

be the implemented deterministic decoder. The tested logical channel is

The channel ends before the destructive certification analyzer. The analyzer is not part of the returned quantum-memory task.

We use

For any finite untouched reference $A$, input state ${\omega }_{AC}$, and joint output measurement, $e(\mathcal{E})$ bounds the trace distance, and hence the total variation distance of outcome laws, between

Before validation data are collected, specify:

1. input and output dimensions, basis identifications, Pauli frame, storage duration, and calendar-time window;

2. operations included in ${\mathcal{N}}_t$, the implemented decoder, all branches, and the output on every failure branch;

3. a fixed ancillary preparation and reset procedure sufficient to identify a logical CPTP channel, rather than a state-dependent system–environment assignment;

4. the reference $Q\cong C$, its isolation or calibrated disturbance, and any joint reference–device cross-talk allowance;

5. preparation and measurement calibration allowances and their simultaneous coverage probability;

6. the repetition model, sample counts, setting selection, optional-stopping rule if any, and permitted drift;

7. the complete outcome alphabet, including loss, leakage, unavailable readout, and heralded failure;

8. the error threshold $\epsilon$, total risk $\alpha$, and exact acceptance rule.

A decoder learned from training data is frozen before primary validation. Conditional on the training and calibration record, the validation contract must hold uniformly.

A separately identified target channel ${\mathcal{E}}_{\star }$ may obey

Both channels must have the same identified logical input and output. Direct laboratory certification sets $b=0$.

The allowance $b$ is already in half-diamond units. It is added outside the square root and is not multiplied by the input dimension, as in the specified-decoder convention of Hawk04-B CheungHawk04B (:certificate).

If a modeled decoder differs from the implemented decoder, that difference requires a converted channel allowance. Errors already inside the tested channel must not be charged again as calibration errors.

Suppose the device produces an instrument $\{{\mathcal{M}}_z{\}}_z$, where each ${\mathcal{M}}_z$ is completely positive and ${\sum }_z{\mathcal{M}}_z$ is trace preserving. Let $D_z$ be a declared CPTP output operation on branch $z$. Then

is the deterministic logical channel.

Equation (eq:branchsum) is CPTP. For device Kraus path maps $K_{zh}$, completeness is

A successful branch alone need not be trace preserving.

Proof. Compositions and sums preserve complete positivity. Trace preservation follows from trace preservation of every $D_z$ and of ${\sum }_z{\mathcal{M}}_z$. The Kraus identity is the trace-preservation condition for the complete instrument. Selecting only some branches generally removes terms from this identity.

This is the branch discipline used in Landmark I and En02-B/C CheungI,CheungEn02B,CheungEn02C. Destructive readout is an analyzer or instrument component; naming it “Enter” does not make it a decoder.

Prepare

Only $C$ enters the memory and decoder. Ideally measure the same Pauli operator on $Q$ and $C^{\prime }$.

Set

For local outcomes $u,v\in \{-1,1\}$, let $Z_j=uv$, and define the oriented pass indicator

For bounded-score certification, every other complete outcome receives a predeclared score in $[-1,1]$. Scoring unavailable readout by zero requires a proved bias allowance. No attempted trial is deleted.

Archive the attempt identifier, time, storage duration, decoder version, setting, local labels, flags, fallback action, and score. Settings must not change the pre-analyzer channel. Choosing the analyzer setting after the output has returned is one safeguard against setting-dependent device behavior.

The required normalization-sensitive facts are proved locally. They are established tools KlieschRoth,FuchsVanDeGraaf,Watrous.

For input dimension $d$, define

For a channel, $J_{\Phi }$ has trace one.

For channels with common input dimension $d$ and identified outputs,

Proof. The Bell input proves the lower bound. Put $\Delta =\Phi -\Psi$. For unit vectors $u,v$ in an arbitrary reference space tensored with the input, write

Hence ${\Vert A\Vert }_{\infty },{\Vert B\Vert }_{\infty }\le \sqrt{d}$, and

A singular-value decomposition extends this bound to arbitrary trace-norm-one input operators and all reference dimensions. Divide by two and use the channel-distance cap one.

This is also the normalization used in En07-C and Em05-B CheungEn07C,CheungEm05B. An unnormalized Choi matrix requires different factors.

For a state $\rho$, unit vector $\varphi$, and $P=|\varphi ⟩⟨\varphi |$, let $F=\mathrm{Tr}(P\rho )$. Then

Proof. The effect $P$ gives the lower bound. Write $\rho ={\sum }_k{p}_k|{\psi }_k⟩⟨{\psi }_k|$. Diagonalization on the span of $\varphi ,{\psi }_k$ gives

Convexity of trace norm and concavity of square root imply

For example, $\rho =I_4/4$ and a Bell projector $P$ give distance $3/4$, not $\sqrt{3/4}$.

For a $d$-dimensional logical channel, put $F_e=\mathrm{Tr}(P_d{J}_{\mathcal{E}})$. Then

Under (eq:transfer),

Proof. Combine Lemmas lem:choi and lem:pure. The reverse triangle inequality gives $\left|e({\mathcal{E}}_{\star })-e(\mathcal{E})\right|\le b$.

For a qubit channel,

Proof. Expand the Bell projector in the Pauli basis and trace against $J_{\mathcal{E}}$.

Define

A fixed implemented decoder proves only

For a qubit, the decoder that ignores its input and prepares $I_2/2$ has error $3/4$. Consequently an implemented-channel upper certificate $U$ gives

The $3/4$ cap is not a cap on the original implemented decoder.

Calibration assumptions must be in units that enter the overlap or channel bounds. Scalar entropy, energy, capacity, geometric, or gate-resource estimates do not supply those units without a conversion theorem.

Let $\mathcal{Z}$ be the training and calibration record. Suppose a calibration event $G$ satisfies

On $G$, the announced bounds below hold simultaneously. They may depend on $\mathcal{Z}$, but validation concentration must be valid conditional on that record.

Assume a fixed ancillary state independent of the logical-reference input, and let the actual preparation obey

Let the reference-storage channel satisfy

For the actual joint pre-analyzer map $\mathcal{G}$, assume

The last bound can charge cross-talk, but is not implied by local reference measurements.

Set

Contraction and the triangle inequality give

For each setting, let $\{M_{j,o}{\}}_o$ be the actual complete POVM and $z_j(o)\in [-1,1]$ its predeclared score. The score observable

has operator norm at most one. Assume

Let

Then

Proof. Measurement error changes the correlation combination by at most ${\sum }_j{\mu }_j/4$. The ideal combination for $\sigma$ equals $\mathrm{Tr}(P_2\sigma )$. Because $P_2$ is an effect, its expectation differs from $\mathrm{Tr}(P_2{J}_{\mathcal{E}})$ by at most $\delta$.

It is also sufficient to prove

Then

For separate state bounds $d_{\mathrm{tr}}({\sigma }_j,J_{\mathcal{E}})\le {\delta }_j$ and measurement bounds ${\mu }_j$, one may take ${\gamma }_j={\mu }_j+2{\delta }_j$. The factor two converts trace distance to trace norm.

The paper proves the conversion to $b_{\mathrm{cal}}$, not an apparatus calibration establishing these allowances. A reduced preparation bound alone does not identify a logical channel when the device environment is correlated with the input. Preparation independence or an appropriate joint bound is required. The simulation choice $b_{\mathrm{cal}}=0$ is a model assumption, not evidence of perfect laboratory calibration.

The trusted logical Pauli frame is part of the contract. Gate-set tomography has model assumptions and gauge freedom that must be addressed before reading a gauge-fixed estimate as an absolute distance to the identified identity channel GST.

This protocol is calibrated, not device-independent. A separate Bell/CHSH specimen, such as that in En06-B, does not certify an untraversed memory wire CheungEn06B (:Bellinstrument).

Conditional on $\mathcal{Z}$, each setting has a predeclared positive sample count $n_j$. Its scores $Z_{j1},\dots ,Z_{j{n}_j}$ are independent, in $[-1,1]$, and have common mean $m_j$. The same logical channel is tested in all settings. The calibration bias and transfer bounds hold on $G$.

Independence between settings is not required for the union bound. No fixed-sample theorem is used after an unaccounted adaptive stopping rule.

Choose ${\alpha }_{\mathrm{s}}>0$, $\beta \ge 0$, and ${\alpha }_{\mathrm{s}}+\beta \le \alpha <1$. Define

Write $[u{]}_{[0,1]}=\max \{0,\min \{1,u\}\}$.

Under the contract and Assumption ass:statistics, suppose $\left|F_{\mathrm{obs}}-F_e\right|\le b_{\mathrm{cal}}$ on $G$. Set

Uniformly over the channels and nuisance parameters satisfying the contract,

For direct certification $b=0$, the same event gives

If “certified” is declared only when $U\le \epsilon$, then

Proof. Hoeffding's inequality Hoeffding gives, conditional on $\mathcal{Z}$,

A union bound gives simultaneous coverage at least $1-{\alpha }_{\mathrm{s}}$. On this event,

Intersecting with $G$ gives $|\hat{F}-F_e|\le R_{\mathrm{s}}+b_{\mathrm{cal}}$. Clipping preserves coverage because $F_e\in [0,1]$. Proposition prop:bellbounds gives the error interval. Integrate the conditional statistical failure bound and add $\mathrm{Pr}(G^c)\le \beta$. An invalid acceptance is contained in the upper-certificate failure event.

Equation (eq:falseaccept) does not assert $\mathrm{Pr}\{e\le \epsilon \mid \text{certified}\}\ge 1-\alpha$. That interpretation requires a different statistical construction.

For $b<\epsilon <1$, acceptance is equivalent to

For $b=0,\epsilon =0.1$, this requires $F_L\ge 0.9975$.

For equal sample counts $n$, an all-ideal-score dataset is certified by the Hoeffding procedure whenever

Indeed $R_{\mathrm{s}}\le q$ under this allowance. If $q\le 0$, the announced calibration budget prevents this particular certificate from succeeding even on ideal scores.

These are sufficient allowances, not optimal sample lower bounds. Failure to certify does not show that the actual device error is large.

Binary pass records allow exact binomial inversion ClopperPearson.

For $0<\zeta <1$, $n\ge 1$, define

For $k>0$, $\ell (n,k;\zeta )$ solves

For $k<n$, $u(n,k;\zeta )$ solves

The relevant tails are strictly monotone on $0<p<1$, so these solutions are unique.

If $K\sim \mathrm{Bin}(n,p)$, then

Proof. For fixed $p$, the counts satisfying $p<\ell (n,K;\zeta )$ form an upper-tail event whose probability is at most $\zeta$, by (eq:cpL) and monotonicity. The corresponding lower-tail argument proves the upper-endpoint statement. The endpoint definitions cover $k=0,n$.

Conditional on the training and calibration record, assume iid binary indicators within each setting, with pass probability

Set $\zeta ={\alpha }_{\mathrm{s}}/6$, and

Define

The recovery interval and coverage conclusions of Theorem thm:main remain valid with these endpoints.

Proof. The six one-sided binomial failure probabilities sum to at most ${\alpha }_{\mathrm{s}}$, and

Combine the pass-probability intervals, calibration event, and channel conversion.

With $n$ passes out of $n$ in every setting,

For $q$ in (eq:samples), $0<q<3/2$, acceptance of the exact binomial endpoints on all-pass data requires

The integer sample allowance is the ceiling. For small $q$, this is of order $q^{-1}$, compared with the Hoeffding allowance of order $q^{-2}$. This comparison concerns favorable all-pass datasets.

Alternatively choose settings independently with probability $1/3$, after the device output has returned. Under a common stationary channel and measurement model, all pass indicators are iid Bernoulli with

A single binomial interval with $\zeta ={\alpha }_{\mathrm{s}}/2$ gives

The same coverage argument applies.

A zero score for unavailable readout is not a binary pass indicator. It must not be inserted into a binomial formula as a retained success.

A bounded score $z\in [-1,1]$ can be converted to a binary record using an independent trusted coin with pass probability $(1+s_{j}z)/2$. That randomization and its bias model must be declared. The bounded-score Hoeffding procedure needs no such conversion.

If the accessible output has a leakage or erasure sector, the primary qubit protocol certifies a specified CPTP completion

For example, retain the logical qubit on success and prepare $I_2/2$ on an erasure flag. The completion belongs to the implemented decoder and is included on every attempt.

The completed qubit channel and a larger flag-retaining channel are different tasks. Their ideal outputs and errors cannot be silently identified.

Suppose an ideal parity ${Z_j}^{\mathrm{ideal}}\in [-1,1]$ is defined on every attempt and agrees with the observed parity whenever readout is available. Score missing records by zero. If their unconditional probability is at most $q_j$, then

Proof. The pointwise score change is at most the missing-record indicator. Take expectations.

No independence between parity and missingness is required for this worst-case bound. The $q_j$'s need valid upper bounds and their own risk accounting. They contribute ${\sum }_j{q}_j/4$ to a componentwise fidelity-bias allowance.

Analyzer-record loss is not memory loss. The latter needs a declared output channel.

For

conditioning on success yields identity, but the deterministic channel is

Its error is positive for $p<1$.

With input-dependent success probability, normalization of a success branch need not even define a linear channel on all inputs. Failure branches remain in deterministic certification, as in the flagged interface of Em05-B CheungEm05B.

For independent nonidentically distributed scores, Hoeffding controls the average expectation, not an unidentified stationary channel.

If run $i$ in setting $j$ has

the correlation changes by at most $2{\xi }_{ji}$. A valid additional fidelity allowance is

Use $b_{\mathrm{cal}}+b_{\mathrm{drift}}$. The iid binomial model must not automatically be applied to this drifting model.

An independently justified bound

extends a point certificate by

The constant $L$ is a separate input, not derived from the three correlations CheungHawk04B (:interval).

If the $x,y,z$ blocks test ${\mathrm{Ad}}_X,{\mathrm{Ad}}_Y,{\mathrm{Ad}}_Z$, respectively, all oriented parities pass with certainty, so naive pooling gives $\hat{F}=1$. Each channel nevertheless has error one.

Proof. Each Pauli unitary preserves the correlation in its own axis. The observed signs are $1,-1,1$. Every nonidentity Pauli unitary sends a Bell state to an orthogonal Bell state. There is no common tested channel across the blocks.

If both the memory and reference apply $Z$,

The Bell test appears perfect although the memory-only channel has error one. The uncharged reference operation violates the contract.

Ordinary fixed-sample intervals cannot be reused after repeatedly looking for a favorable stopping point without additional control.

For each setting and every prefix $n\ge 1$, allocate

Their sum over all prefixes and three settings is ${\alpha }_{\mathrm{s}}$. The radii

therefore cover all prefixes simultaneously under the independent stream model. This permits stopping at a data-dependent prefix; it is not the fixed-sample formula (eq:statistics).

For input dimension $d$, every replacement channel has optimal recovery error $a_d=1-d^{-2}$. If $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{\omega })\le \eta$, every deterministic decoder has error at least $\max \{0,a_d-\eta \}$.

A qubit pure-output replacement followed by the identity decoder has implemented error one, although its optimal recovery error is $3/4$.

Proof. Every decoded replacement is a replacement. Its Bell output is $I_d/d\otimes \tau$, with Bell overlap $d^{-2}$, giving the lower bound $a_d$.

Preparing $I_d/d$ attains it: the corresponding replacement is the uniform average of $d^2$ Weyl unitary channels. Separating identity gives unhalved diamond norm at most $2a_d$; the Bell witness gives equality. Contraction and the reverse triangle inequality give the approximate lower bound. For a fixed pure output $|0⟩$, input $|1⟩$ gives orthogonal actual and ideal outputs.

These are the benchmarks in Oai265-A and En07-B/C Oai265A,CheungEn07B,CheungEn07C. Accessible-channel replacement closeness is a recovery lower bound, not a high-accuracy upper certificate.

If $\mathcal{N}$ is entanglement breaking, every deterministic decoder has error at least $1-d^{-1}$, hence at least $1/2$ for a qubit.

Proof. The decoded Bell output is separable. Every product state has Bell overlap at most $1/d$; convexity extends this to separable mixtures. The Bell-projector test proves the bound.

This is the relevant obstruction for the specified commuting control of Hawk04-B CheungHawk04B (:QND), not a theorem about every channel commuting with an arbitrary operator.

In the following table, the decoder is identity on the returned qubit. Let

Amplitude damping has Kraus operators

The table is exact.

Proof. For Kraus operators $K_a$,

This gives the overlaps.

For Pauli channels with probabilities $a_P,b_P$, the triangle inequality gives $d_{\diamond }(\Phi ,\Psi )\le \frac{1}{2}{\sum }_P|a_P-b_P|$. Their Bell outputs are mixtures of orthogonal Bell projectors, giving equality. This proves the dephasing and completed-loss errors; the replacement errors were proved above.

For a unitary rotation, a pure referenced input has output distance

The convex hull of the eigenvalues has minimum modulus $\left|\cos (\theta /2)\right|$, attained by equal weights. Convexity handles mixed inputs. Appendix app:AD proves the amplitude-damping distance.

The Pauli identity is also used in Em03-E CheungEm03E (:recovery). It does not justify replacing the general square-root bound by $1-F_e$ for arbitrary channels.

1. Identity and ${\mathcal{T}}_{{\rho }_0}$ agree on input ${\rho }_0$. One-state agreement is not channel certification.

2. For the isometry

   $$V|0⟩=|{\Phi }^{+}{⟩}_{RB},V|1⟩=|{\Phi }^{-}{⟩}_{RB},$$

   both computational-basis preparations have $B$ marginal $I_2/2$, but input $|+⟩$ maps to $|00⟩$. Selected residual data do not prove a constant complete complement. Every preparation purifier must be retained for a complete-complement claim.

3. Reference compensation and setting-dependent drift are the out-of-contract examples in Section sec:failure.

4. Magic resources, an area-law record, and a PEPS bond dimension are not measurements of $F_e$ or converted channel allowances CheungEn04A,Oai265A.

5. Classical capacity is a different task. No capacity theorem is used as a diamond-error premise.

The undated user-supplied file (30).txt states that the version-1 Appendix B script was extracted and executed ExecutionReport. We incorporate its recorded results as supplied execution evidence. We do not assert an independent rerun, authenticate an execution environment, or assign a run date not present in the report.

There are $120$ simulated binary records per dataset and $16,800$ across the reported suite. The records are classical Bernoulli passes generated from the benchmark correlations. They are not laboratory outcomes or a quantum-circuit simulation.

Every setting has $n=40$. The tuple $(k_x,k_y,k_z)$ below contains the reported first repetition's oriented pass counts, not raw same-sign counts for $Y$. “Covered” is the report's joint check of both $F_L\le F_e\le F_U$ and $L\le e\le U$.

These are reported outcomes, not independently reproduced results ExecutionReport. The execution report supplies one detailed repetition per case and aggregate coverage counts; it does not supply every dataset's full record.

The following fractions are transcribed from the supplied report. Here $L$ and $U_{\mathrm{rat}}$ are the returned rational error endpoints. The square-root bracketing makes $U_{\mathrm{rat}}$ at least the ideal mathematical upper expression, so it is conservative.

{}1.65

The exact fractions, rather than rounded decimal displays, define the recorded numerical certificates ExecutionReport. Their illustrative decimal values are:

These rounded numbers are not substituted into the acceptance test.

For all three identity settings, the report gives the outward Clopper–Pearson lower endpoint

Therefore

The rational upper endpoint is

This corrects the report's coarser interpretation “approximately $0.8225$” without changing its exact fraction. Moreover,

The exact acceptance test therefore fails.

For an arithmetic consistency check, the reported fractions satisfy

The spacing $2^{-39}$ is the spacing of twice a 40-step dyadic square-root bracket. This is a check of the recorded endpoint's algebraic role, not an independent execution of the script.

For $n_j=40$, ${\alpha }_{\mathrm{s}}=1/20$, and $b_{\mathrm{cal}}=b=0$, no dataset can be accepted at $\epsilon =0.1$ by the three-setting binomial procedure. This remains true with outward conservative numerical brackets.

Proof. The binomial lower endpoint is nondecreasing in the pass count: the upper-tail probability decreases with $k$, so its inverse endpoint increases with $k$. Thus all-pass counts maximize $F_L$.

For exact all-pass endpoints, let $p_0=120^{-1/40}$. The maximal lower fidelity is

Acceptance would require $p_0\ge 599/600$. But Bernoulli's inequality gives

hence $p_0<599/600$, excluding acceptance. Outward lower brackets cannot increase $F_L$, and outward upper square-root brackets cannot decrease the error upper endpoint.

Consequently the total number of acceptances is zero, not just the number of invalid acceptances. This is a deterministic deduction from the protocol and sample count. The report's zero-invalid-acceptance result is consistent with this deduction, but does not test whether the implementation correctly accepts a genuinely informative high-accuracy dataset.

For the same risk and allowances, exact binomial endpoints accept all-pass data at $\epsilon =0.1$ precisely when

shots per setting.

Proof. Use (eq:cpSamples) with $q=1/400$. The integer boundary can be checked without logarithm rounding by

This is $8613$ attempted shots across the three settings on an all-pass dataset. It is not a claim of laboratory execution, and noisy datasets can require more shots or fail this certificate. An actual outward-bracket implementation should check acceptance on its returned $F_L$, rather than assume acceptance solely from the analytic integer threshold.

The reported empirical coverage is $140/140=1$. This does not prove uniform coverage over all channels, establish the calibration assumptions, or replace Theorem thm:binomial.

For perspective, if a single case's repeated datasets were independent ideal draws with coverage probability $p_{\mathrm{cov}}$, observing 20 covered datasets gives the one-sided 95% exact lower confidence endpoint

This is an illustration of Monte Carlo uncertainty, not a new laboratory confidence statement. Seven separate cases require an additional multiplicity allowance for simultaneous Monte Carlo intervals. The heterogeneous suite cannot simply be treated as 140 iid observations of one channel's coverage probability.

The report's statement that any coverage miss would be a “failure of the statistical procedure itself” needs qualification. Even a correct procedure can miss on an allowed tail event. Under the model, its miscoverage probability is bounded, not zero. Outward arithmetic prevents inward numerical approximation from weakening the intended intervals; it does not eliminate statistical tail events, implementation bugs, or violations of assumptions.

The supplied report includes parameters, first-repetition details, and aggregate counts. It does not include a source hash, interpreter version, complete 140-dataset archive, or independently authenticated execution trace.

The version-1 listing printed aggregate results, whereas the supplied report includes richer example details. Those details are treated as reported diagnostic logging associated with the extraction, not as output that the unchanged version-1 print statements alone produced. Appendix app:code now supplies a reporter capable of emitting these details without changing the Bernoulli draw order.

The original suite models ideal correlations for completed loss, but does not simulate loss flags, readout missingness, drift, reference disturbance, or calibration estimation. The pure and mixed replacement cases have identical pass probabilities; their differing simulated counts arise from different random draws, not discrimination of their implemented errors by these three populations.

Three correlations bound fidelity, not the entire channel. Pure and mixed replacements have the same Bell overlap but different implemented errors.

Let $\{P_k{\}}_{k=1}^{15}$ be the nonidentity two-qubit Pauli basis, with $\mathrm{Tr}(P_k{P}_{\ell })=4{\delta }_{k\ell }$. Informationally complete local Pauli measurements can produce simultaneous calibrated intervals

Their statistical and calibration risks must be accounted for.

Define

This is a confidence region of CPTP Choi states, not a best-fit matrix. If empty, report incompatibility or an inconclusive result; do not silently enlarge error bars.

On $J_{\mathcal{E}}\in \mathcal{C}$, choose any feasible $\hat{J}\in \mathcal{C}$, and let