A physically admissible weighted completion-design frontier

A physically admissible weighted completion-design frontier

The deterministic collision test identifies whether a target is exactly fixed by a record. A physical quantum reader also needs a quantitative error analysis when different targets can produce the same finite record. Here rational record likelihoods and exact optimization close that gap for one finite, explicitly specified phase-source experiment. This is a new programme-specific construction, independent of the ETH theorem; it demonstrates a substantive consumer of the weighted-completion interface. It is not a faster implementation of the ETH algorithm.

Source, allowed interventions and retained outputs

Use a two-mode one-excitation register with computational basisand equal populations. The finite preparation promise is

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ρX± = (I ± X)/2,
ρY ± = (I ± Y )/2.

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Equivalently their phases are 0, π, π/2, 3π/2 in (|0⟩+ eiφ|1⟩)/
√

2. All four preparations have the same separate-mode intensity readout (12). The target is only the sign class: and. This target is an externally nominated function on the finite preparation set, not the eigenvalue of a quantum observable assigned to all states. A source label is chosen once and held fixed for an experiment. Conditional on, the source supplies independent copies , independent of the reader ready stock. This is a preparation promise, not a conclusion from matching one-copy marginals or from resetting a device. The source controller retaining is declared preparation stock and is unavailable to the decoder; no generator-memory separation is inferred from this experiment. The permitted experiment menu consists of a fixed, nonadaptive schedule: copies measured with the reader and with the reader. The two incompatible observables act on different copies. Neither is silently assumed jointly measurable on the same source. The intensity reader alone yields exactly uniform outcome sequences for all and cannot improve worst-case sign error below 1 A phase-controlled beam splitter implements the ideal reader: rotate by = exp(2) with 2 under a fixed phase convention, then apply a Hadamard beam splitter and detect the two terminal paths. The convention can equivalently be fixed by the effects below. A

controlled HamiltonianΩ2 realizes the phase rotation in durationΩ, with the sign of the drive declared. The beam splitter and detector have separately allotted durations; there is no instantaneous-pulse or speed optimum claim. The calibrated noisy reader has terminal effects

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E(a)
± = η
2(I ± vσa),
E(a)
0
= (1 −η)I,
0 ≤η, v ≤1,
(1)

where . A realization measures the ideal Pauli observable, flips its reported sign with independent probability (12, and independently replaces it with a null with probability 1 Fresh independent coin/environment stock per copy is admitted and counted as apparatus resources. The ordered record retains every setting, terminal sign, null and time. Environments may remain retained but unread; no assertion equates their complete quantum channels. This specifies effects and a conventional realization rather than importing a complete apparatus-response certificate from terminal counts. It is an ideal/specified-noise reader construction, not a solution of arbitrary detector mismatch. A separate phase-to-record model can supply perturbation budgets only when its preparation, access and output assumptions match. No delayed-trine resource bound is inferred from this copy-allocation task.

An exact ideal fixed-schedule optimum

Let ) be the minimum over all randomized terminal decoders of the maximum sign error over the four possible source labels. It includes the entire original population, with no click postselection. In this subsection

Theorem 1.1 (Target-relative phase-reader allocation). For

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e∗(nX, nY ) =
1
2nX + 2nY .
(2)

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If either count is zero, e∗= 1/2. At total copy budget B ≥2, the optimal permitted allocation is
(1, B −1) or its swap, giving e∗
B = 1/(2 + 2B−1). The minimum worst-case error for identifying the
full four-valued label under the same fixed schedule is 2/(2nX + 2nY ) when both counts are positive.

Proof. An+ source gives only plus signs at the reader and fair independent signs at the reader; exchange signs or axes for the other sources. Apart from the two opposing corners

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A = (X all +, Y all −),
B = (X all −, Y all +),

the record determines the target sign. Corner has probability under+ and under. If a decoder chooses + there with probability, its worst error is at least max whose minimum is). Choosing at, and the uniquely determined sign elsewhere attains this bound on all four sources. If one axis is absent, the two opposite signs of that axis induce identical records, which forces worst error at least 12; a fair guess attains it. For fixed and positive counts, maximize 2 . This convex function has its maximum on the integer interval 11 at an endpoint. For full-label discrimination, all four constant-sign corners have two possible labels, one from each axis. Assign the label with probability at each corner. Each source then incurs error 2). For a matching lower bound, use prior weights2 on each label and (12 on each label with). At each corner the two weighted likelihoods are equal, contributing)] to every decoder’s Bayes error. Summing four corners gives the claimed lower bound.

For eight source copies, the target-optimal schedule (1 7) has error 10076923, while the balanced (4 4) schedule has error 103125. The comparison charges the same eight copies, one readout per copy, both calibrated axes and the same retained archive. It is not comparison against an optimal arbitrary POVM or adaptive protocol. For a one-percent sign-error target the optimal menu schedule needs eight copies, while balanced or nearly balanced schedules need twelve. Full-label identification optimized over the same menu needs nine. Thus target choice and experiment allocation both change the accessible-resource requirement. The eight-copy one-percent guarantee does not require an exact biased decision coin: with (1 7), return the common sign if all seven results agree, and otherwise return the single result. Its worst error is 1100. This deterministic decoder gives a nearby constructive repair when ideal rational randomization is unavailable; the sharper 1 remains the randomized-class optimum. Extend the rule to every record by returning + whenever a null occurs. Those records have zero nominal probability but must have a declared decision before applying an actual-device perturbation bound. Every finite positive-count experiment still has a record compatible with opposite target values. On the admissible relation of source labels and possible outcome records, an exact support-only completion test correctly rejects zero-error sufficiency at every finite budget. A small weighted error is a different target and is not called an exact repair of that collision. This is why the likelihood/weight extension adds something the Boolean collision oracle alone cannot supply.

Exact rational noisy design with matching certificates

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From (1), let
p(a)
h (±) = η(1 ± vra(h))/2,
p(a)
h (0) = 1 −η,

where1 and the cross-axis values are zero. For rational these are rational probabilities. A setting’s histogram) has probability

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n!
k+!k−!k0!p(+)k+p(−)k−p(0)k0.

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The product of the two setting histograms is sufficient for this likelihood task: under the declared
conditional product law, all permutations with the same histogram have identical likelihood ratios.
The complete ordered physical archive is retained; the decoder’s sufficient statistic does not erase
or return it to the source. For fixed setting counts, there are only
nX+2
2

nY +2
2

 histograms. Their
rational probabilities can already be computed in polynomial time here; the ETH algorithm is not
needed or claimed to speed this simple special case.
Write the four likelihoods at record o as PX+(o), PX−(o), PY +(o), PY −(o). For any α ∈[0, 1], every
decoder has worst-case error at least

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L(α) = 1
X
o
min{αPX+(o) + (1 −α)PY +(o), αPX−(o) + (1 −α)PY −(o)}.
(3)
2

This is the Bayes risk for an explicit prior, not an assumption that the source actually uses that prior. The function is concave piecewise linear. Its maximum occurs at an endpoint or at a rational crossing of its two affine record weights.

Exact construction and certificate. Enumerate those breakpoints and select a maximizing On non-tied records use the weighted likelihood-ratio decision. On tied records construct two policies minimizing and maximizing the difference between the-axis and -axis average errors. Symmetrize under global sign reversal so the errors of the two signs within each axis agree. Every such policy attains) at its nominated prior. If 0 1, optimality of the concave maximum makes

the two attainable error-difference endpoints bracket zero; mix them to equalize the axis errors. At = 0, choose the endpoint whose error does not exceed its error; at= 1 choose the converse. One-sided maximality supplies the respective endpoint condition. Thus the constructed policy’s worst error equals the lower bound). This proves exact minimax optimality for each schedule. The implementation retains a simpler independently checkable certificate: all rational likelihoods, the prior, and a rational decision probability) for each histogram. Recomputing each of the four errors gives an upper bound. Evaluating (3) gives a lower bound. Equality is an exact zero primal/dual gap; it requires no floating optimizer or assumption of numerical convergence. For any histogram absent from the union of nominal supports, the exported decoder returns +; this completes it on actual perturbed records. With calibrated5 and10, enumerate every fixed schedule of at most copies at unit cost:

Budget Certified optimal ( Exact minimax error 2 4 6 8 10

This table minimizes over a single deterministic X/Y count allocation and its minimax decoder. It does not optimize mixtures over different allocations; the randomized-schedule comparator is a broader class. In particular, at budget eight this fixed-allocation optimum error is 008270910464, while the formerly ideal-optimal (1 7) schedule has error 7268219370011513134900. Noise changes the optimal design. Balanced allocation is not asserted optimal at all budgets: the exact ten-copy result is asymmetric. The data are mathematical model outputs, not experimental measurements.

Controlled deviations and fair physical accounting

Suppose deterministic bounds 1] control the total variation between actual and nominated conditional outcome laws at every matched past history. Sequential maximal coupling keeps the records matched at step, conditional on matching so far, with probability at least 1. Consequently the complete classical-record distance is at most

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B
Y
t=1
(1 −δt) ≤
X
κB = 1 −
t
δt.

Independence is not needed under this stronger uniform conditional contract; one-copy marginal bounds do not imply it. The product bound is sharp as a record-distance statement: compare a nominal all-zero word with independent per-position flips of probabilities. Every fixed classifier’s error increases by at most. This is not a claim of robust minimax optimality and covers classical outputs, not residual quantum baths. For example, the eight-copy ideal (1 7) decoder retains a one-percent worst-case error guarantee whenever the actual conditional laws obey4000 at every history: the exact rational bound is strictly below 1100. This makes the finite design’s calibration allowance quantitative without inferring future stability from past calibration alone. The deterministic decoder also passes: replace 1128 in that bound, obtaining approximately 0 without a randomized-decision port. For the specific reader family, changing () changes each one-copy outcome law by at most

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Trace-distance preparation error δS adds at most δS. These are explicit consumers for a certified
calibration/drift budget. An unconditional one-copy marginal estimate alone does not justify the
sequential conditional promise or independent fresh apparatus stock.
The resource optimized above is only source-copy/readout count within the declared menu. Each
trial consumes an independent preparation and reader port, a calibrated phase setting and clock interval.
The archived setting and three-outcome record remain append-only and unavailable to future source
operations. Histogram counters and the final decoder are additional classical resources. Randomized
decisions also consume independent ready stock. An exact rational coin can be admitted as a separately
calibrated qubit prepared by a finite rotation: Hc = ħΩcY/2 for duration 2 arcsin √q/Ωc, followed by a
terminal computational measurement. Its preparation and parameter accuracy are not free. With only
fair bits, exact rational rejection sampling has finite expected cost but unbounded stopping time; a
finite deadline requires an explicit residual error margin. No total energy, hardware or thermodynamic
optimum follows.
For example, a continuous equatorial bisector-basis measurement is outside this menu and can
outperform some small-budget axis schedules. There is also a useful unrestricted lower comparison for
every always-deciding measurement on B copies, collective or otherwise:

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eB ≥1 −
√
1 −2−B
2
> 0.

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Indeed restrict the four-label minimax problem to the opposite-target pair X+, Y −, whose B-copy
squared overlap is 2−B. Their trace distance is
√

, obtained in their two-dimensional span. Maximizing Tr) over 0 gives this trace distance, so the optimal pair-average error is the displayed value; four-label worst error cannot be smaller. This is the established two-state trace-distance bound, not a new global optimum. An unambiguous task with abstentions has a different objective. Collective measurements and adaptive basis choices are also excluded from the present optimum. These limitations are essential: arbitrary quantum-state discrimination is established theory, and the new result is a sharply typed target-relative design example with exact noise certificates. General discrimination and multiple-copy local/adaptive comparisons have established treatments[1, 3]. The result here is a target-relative weighted access-design example with exact noise certificates under the declared preparation and measurement restrictions. No worldwide novelty claim is made for BB84-family discrimination or the general idea of allocating measurements.