Technical Companion: Reader Methods and Resource Interfaces

A larger rational measurement menu: the copy gap closes but feedback still helps

Jeremy RodgersSection 3 of 8Original PDF ↗

The axis-restricted result proved a nineteen-versus-twenty-one copy threshold inside its X/Y menu. That statement does not imply a separation after additional phase settings become available. We make a strict, exactly computable expansion of the same physical reader and compare both protocol classes in that same expanded menu. The result distinguishes a surviving error advantage from a copy threshold that no longer separates the classes.

Declared settings, noise and symmetry

Admit exactly four equatorial axes, listed by their unit direction vectors:

M = {(1, 0), (0, 1), (3/5, 4/5), (4/5, 3/5)}. — Source page 6.

These are the original X/Y settings and two additional phase rotations $θ =$ arctan $(4/$ 3) and arctan $(3/4),$ implemented by the specified controlled two-mode phase rotation, beam splitter and terminal detector.

The effects are $E±|a = η(I ± v(caX + saY ))/$ 2 and $E0|a = (1 −η)I$ . We use the same known, fixed common parameters $η = 4/5, v = 9/$ 10 for every setting. No setting-dependent calibration or switching cost is silently inferred from that assumption. All four source preparations, unavailable source label, independent fresh copies, fixed deadline and retained signs/nulls/settings keep the specified contract. Different observables act on different copies. This is a finite physical menu, not a discretization claimed to equal the continuous equatorial class. The menu is closed under exchanging X and Y. Global source-sign reversal is implemented in a protocol by reversing its recorded signs and final decision. These two symmetries act transitively on the source labels. Consequently the uniform-prior Bayes optimum is a lower bound for minimax risk and averaging an optimal policy over the four symmetries attains it with equal error on each label. Two fair root bits suffice; our implementation resolves terminal posterior ties deterministically as plus before applying that root symmetry. This avoids charging an unneeded additional decision coin. The root seed must remain available to the decoder and is a counted controller resource.

Exact adaptive and randomized-nonadaptive certificates

For this menu all likelihoods have denominator $D$ = 125. For the X reader the plus numerators $on (X+, X−, Y +, Y −$ ) are (95 $, 5, 50,$ 50), and for the two new directions they are (77 $, 23, 86,$ 14) and $(86, 14, 77,$ 23). Minus swaps each sign pair and every null numerator is 25. Let $ka,o$ denote these four-vectors. The exact recurrence on integer prior weights is

V0(w) = min(w1 + w3, w2 + w4),
Vt(w) = min
a∈M
1
125
X
o∈{+,−,0}
Vt−1(w ⊙ka,o). — Source page 7.

$Homogeneity permits division by the integer gcd of a weight vector, giving an exact rational memoized computation. Every action is included at each node. Selecting a minimizing action and then a terminal Bayes decision attains the lower bound. The preceding symmetry promotes the root value VB(1, 1, 1, 1)/4 to minimax equality over all randomized, nonanticipating protocols using this menu. For nonadaptive protocols, conditional product likelihoods imply that an action sequence’s Bayes risk depends only on its four action counts. Enumerate all �B+3 3 count allocations and all ternary histograms within each setting. At a histogram the multiplicity-weighted contribution is the minimum of its two target likelihood sums, divided by four. This is an integer sum over 125B, with exact multinomial multiplicities. A mixture of schedules has average Bayes risk at least the smallest component risk. Root symmetrization of a minimizing schedule gives the matching minimax upper bound, so this comparator includes every randomized nonadaptive schedule and terminal decision rule in the same menu.$

Technical passage, source page 7.

Searchable transcription

Homogeneity permits division by the integer gcd of a weight vector, giving an exact rational memoized
computation. Every action is included at each node. Selecting a minimizing action and then a
terminal Bayes decision attains the lower bound. The preceding symmetry promotes the root value
VB(1, 1, 1, 1)/4 to minimax equality over all randomized, nonanticipating protocols using this menu.
For nonadaptive protocols, conditional product likelihoods imply that an action sequence’s Bayes
risk depends only on its four action counts. Enumerate all
B+3
3
 count allocations and all ternary
histograms within each setting. At a histogram the multiplicity-weighted contribution is the minimum of
its two target likelihood sums, divided by four. This is an integer sum over 125B, with exact multinomial
multiplicities. A mixture of schedules has average Bayes risk at least the smallest component risk. Root
symmetrization of a minimizing schedule gives the matching minimax upper bound, so this comparator
includes every randomized nonadaptive schedule and terminal decision rule in the same menu.

Theorem 2.1 (Finite expanded-menu frontier). For the specified four-setting menu and detector law, the exact minimax errors are:

B
Adaptive
Randomized nonadaptive
4
2062489/19531250
414539/3906250
8
177227695861393/4768371582031250
35860640527277/953674316406250 — Source page 7.

Both inequalities are strict, so feedback retains a genuine advantage within this larger menu. Nevertheless both classes require exactly fourteen copies for one-percent worst-case sign error.

Proof. The recurrence and full count-allocation enumeration above give exact lower and attaining upper certificates. The saved four/eight-copy Bellman DAGs contain both terminal losses and every action value, and the checker verifies all equalities with rational arithmetic. Direct evaluation of the selected policy’s four label errors supplies an additional upper check before group averaging. Nonadaptive histogram sums are independently compared with ordered records on small horizons.

For the copy threshold, at $B$ = 13 the errors are

20475493238962971614182459
1818989403545856475830078125,
20862994861778108626565401
1818989403545856475830078125, — Source page 8.

respectively, both greater than 1 $/100. At B$ = 14 they are

162424078504800687452994223
18189894035458564758300781250,
165670382020832988542373811
18189894035458564758300781250, — Source page 8.

both less than 1 $/$ 100. The exact computation enumerates all 560 and 680 nonadaptive count allocations at those horizons and covers all adaptive actions by the same Bellman recursion. Its code, inputs and exact values are delivered; large threshold DAGs can be regenerated instead of being duplicated as a large static archive. More copies cannot worsen either optimum because an extra record can be ignored. The adjacent exact comparisons therefore establish the minimum budgets.

At eight copies the errors are approximately 0 $.$ 03716734 and 0 $.$ 03760261, both substantially lower than the old X/Y values 0 $.$ 06714193 and 0 $.$ 07977783. At fourteen they are 0 $.$ 00892936 and 0 $.00910783.$ The best nonadaptive fourteen-copy allocation uses seven copies at each new oblique setting and none at X or Y. Thus a feasible broader nonadaptive reader already beats the nineteen-copy restricted adaptive budget. The original theorem remains correct in its original class, but its copy saving is not a measurement-menu-independent advantage. There is a constructive repair at a stricter declared target. The fourteen-copy gap gives an entire interval of tolerances for which the adaptive reader passes and every nonadaptive reader fails. For example, at error tolerance 9 $/$ 1000 the exact minimum budgets are fourteen adaptive versus fifteen randomized nonadaptive copies. The thirteen-copy adaptive lower bound already exceeds this tolerance, and the fourteen-copy nonadaptive lower bound does too. For attainment at fifteen use the explicit allocation (0 $, 0, 7,$ 8) and its root-symmetrized counterpart; its exact error is

206141219809325120449786527454
28421709430404007434844970703125 ≃0.00725294938 < 9/1000. — Source page 8.

Thus an expanded-menu copy advantage survives on a specified accuracy frontier, although the original one-percent separation closes. The 0 $.$ 9% illustration was selected within the proved risk-gap interval after calculation; it is not presented as a pre-registered experimental target or a universal advantage.

Implementation and limits

research9/menu/expanded_reader.py supplies the exact kernels, generic finite-menu Bellman calculation, direct policy risks, schedule optimization and certificate checker. Its public minimax interface requires exact axis-swap closure; it rejects an asymmetric menu rather than incorrectly reusing uniform-prior minimax equality. Nominally unreachable branches have declared defaults. research9/menu/verify_menu.py checks 48 normalized-posterior reference problems, 16 ordered-record/histogram comparisons, malformed menus, a forged certificate, the four/eight-copy certificates and the thirteen/fourteen-copy thresholds. The replay runtime is hardware dependent; no asymptotic speed claim follows from a particular execution time. The added settings, feedback controller, root coins, phase accuracy and switching times are additional apparatus resources. The current theorem optimizes copies and error within the declared menu, not total physical cost. Unequal setting efficiencies, common phase drift and uncertain controls are separate physical parents requiring new bounds. These exact rational results supply a useful restriction and benchmark for the continuous-menu investigation, not a global continuous optimum. The methods extend established Bayesian/minimax decision theory and the previous programme’s source–readout interface; no worldwide priority or new Born, trine or consciousness result is asserted.