Feedback and Robust Local Readout of a Fixed Four-State Quantum Source
Task, contributions and operational contract
Task, contributions and operational contract
This paper develops local quantum-reader decision problems within Shadow Theory’s pre-existing source–readout programme [9]. The results below quantify target access under the declared preparation, measurement and retained-record conditions. Let
label
2 and 
2. The target is the common sign. One label remains fixed for a complete experiment: the source provides
, rather than redrawing the axis on each specimen. The preparation controller is unavailable. A trial destructively measures one fresh copy and all commanded settings, outcomes, nulls and specified timings remain retained. Every fixed deadline requires a sign; no postselection or abstention is allowed. Randomized nonadaptive protocols may choose a joint distribution of entire schedules before outcomes and retain their seed. Adaptive protocols choose settings from past accessible records and ready coins. Neither class has coherent quantum memory returning between specimens. The effects specified below and independent ready stock define the statistical comparison; they do not identify unread detector baths. Equatorial phase control is realized by the declared two-mode phase rotation, mixing and detection interface, with phase, clock and reference costs separate from specimen count. The results give constructive source distinctions inaccessible to separate-mode intensities, together with restrictions under which feedback, randomization and calibration actually change attainable readout risk. Adaptive multiple-copy discrimination and minimax quantum decision theory have established treatments[5, 7]. The known-binary comparison in the former does not supply a bound for the present grouped target or for all arbitrary preselected measurement schedules. Here the hidden axis stays fixed across the complete conditional-copy stock. Standard quantum discrimination, minimax symmetrization and representation methods have antecedents cited throughout; the contributions are certified portions of the declared-task frontiers, rational constructions and independently checkable
continuous certificates. There is no worldwide novelty or empirical validation claim. Ordinary Born probabilities are assumptions.
Decision risks and the retained classical interface
Write
and
for the two policy classes above. A policy Π has an independent, retained seed
commands
and a terminal sign
may depend on (
its entire schedule depends on
alone. The terminal decoder may use all recorded outcomes in either class. Its label risk and minimax value are
Searchable transcription
rh(Π; η, v) =
Pr
h,Π;η,v{DB ̸= R(h)},
eC
B(η, v) = inf
Π∈C max
h
rh(Π; η, v).For a prior
, Bayes risk instead averages the label risks with weights
; a Bayes-to-minimax equality is used only when a stated symmetry or least-favourable comparison proves it. In the robust task, the maximum additionally ranges over one fixed nuisance tuple for the complete experiment, and one policy is chosen before that tuple is revealed. The calibrated shared-noise comparisons use the conditional trial kernels
Searchable transcription
Kθ,h(±) = η 2(1 ± v bh · nθ), Kθ,h(0) = 1 −η,
where
is the pure preparation’s equatorial Bloch vector and
cos
sin
). Conditional on the fixed label and admitted nuisance, the fresh specimen and ready apparatus stock produce this kernel at each history. Equal one-copy marginals alone would not establish that sequential law. The retained decision record contains commanded settings, ternary outcomes, ready seeds and declared timings. Timings included in this statistical interface are determined by those records or have a source-independent conditional law; additional source-dependent arrival-time information would require an enlarged kernel and a new comparison. The theorem does not identify unread baths, time-resolved absorbing instruments or residual quantum outputs merely from their terminal effects.
Organization and conventions. Sections 2 and 3 give the small-horizon continuous-menu optima and attainable four-copy policies. The nonadaptive certificates prove the cross-class separation. Sections 5 and 6 then treat the fixed asymmetric uncertainty; larger-horizon bounds and compact policies follow. The appendices explicitly restrict access to the X/Y menu. Angles are in radians, probabilities and Bloch gains are dimensionless, and physical pulse times and angular rates retain the units stated in their interaction models. All certificate paths below are relative to supplement/readers/; reproduction expands compressed inputs in a disposable tree.