Frozen-angle randomization and preparation convexification

Frozen-angle randomization and preparation convexification

A common least-favourable comparison for every horizon

For general positive gain floors1, efficiency floor 0 and 0∆4, let

cos ∆ sin ∆) sin ∆ cos ∆) sin 2∆

The admitted hard comparison fixes X at its worst positive offset and Y at its worst negative offset, with gain floors and efficiency. A mathematical comparison prior may correlate its fixed nuisance with the fixed source axis; it is a lower-bound device for the supremum over source/nuisance pairs, not a new physical calibration law. Put

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p =
b2 −c
a2 + b2 −2c,
z = pu + (1 −p)v,
m = ∥z∥=
ab cos 2∆
√
a2 + b2 + 2ab sin 2∆.

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Then z = mnθ∗, where
tan θ∗= a cos ∆+ b sin ∆
b cos ∆+ a sin ∆.

The directionlies in the common-positive interval

Let) be the error of majority with a fair tie after independent signs of positive bias including2. Define the full deadline risk, with all erasures counted, by

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B
∑︂
(︄
B
n
)︄
ηn(1 −η)B−nqn(β).
QB(η, β) =
n=0

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For k ≥0,
q′
2k+1(β) = −(2k + 1)
(︁2k
k
)︁
22k+1
(1 −β2)k,
q2k = q2k−1 (k ≥1).

Consequently) is decreasing and convex on 1]. Its second derivative is nonnegative term by term; no numerical curvature test is used to establish the statement.

Theorem 8.1 (All-horizon random-frozen-angle optimum and spread penalty). Before a block, choose one random nominal angle Θ, independently of the hidden source and nuisance. Retain the seed, repeat that angle for all trials, and allow any randomized terminal decoder using the full retained record. The exact robust minimax error in this class is, attained by the deterministic angle and fair-tie majority. Every such randomized reader has robust worst error at least

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QB(η0, m) + m[−∂βQB(η0, m)]E[1 −cos(Θ −θ∗)]
≥QB(η0, m) + 2m[−∂βQB(η0, m)]
π2
E[(Θ −θ∗)2].

This is not an optimum over varying-angle schedules, adaptive control, angles outside, or collective quantum access.

Proof. Choose the fair sign and axis probabilities of the fixed hard comparison above. Conditional on a seed angle, its two positive biases are cos+∆) and sin∆), both nonnegative. On any retained word, each axis component’s positive/negative sign likelihood difference has the sign of the nonnull signed count. Therefore their positive mixture has the same sign. Majority with a fair tie is Bayes-optimal even for the axis mixture and even if every ordered sign/null record is accessible. Its conditional Bayes risk is

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f(θ) = pQB(η0, βX) + (1 −p)QB(η0, βY ).

Any other decoder has at least this comparison-prior average error. Convexity in the bias gives

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f(θ) ≥QB(η0, pβX + (1 −p)βY ) = QB(η0, m cos(θ −θ∗)).

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Because QB is decreasing and convex, its tangent at m lower-bounds the last expression by QB(η0, m)+
m[−Q′
B(η0, m)](1 −cos(θ −θ∗)). Average over the retained seed. The robust supremum is at least
this fixed comparison average. Finally 1 −cos x ≥2x2/π2 for |x| ≤π, by concavity of sine on [0, π/2].
The interval I ensures this range. The preceding frozen reader attains QB(η0, m) uniformly over the
original uncertainty set, proving the equality as well as the quantitative penalties.

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For the original example,

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p = 654464
1478945,
m2 = 129574081296
369773223625.

Exact quadratic-field arithmetic gives

Trials Robust random-frozen optimum One-percent status

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15
.0124288300109 . . .
fails
16
.0102368487113 . . .
fails
17
.00844219845891 . . .
passes

Hence seventeen is the exact minimum in the enlarged random-frozen class. A phase lottery is not a route to improving that threshold. Any improvement must use another interface, such as varying settings within a block, feedback, or additional information. This does not prove that a varying nonadaptive schedule needs seventeen, nor that adaptation is advantageous at one percent.

When convexification makes the frozen reader globally optimal

Theorem 8.2 (An all-horizon convex-source benchmark). Let be a nonempty compact subset of the equatorial Bloch disk for the fixed antipodal source family. Suppose its closest convex-hull point is itself physically admissible:. With independent copies conditional on fixed , independent sign-blind erasures and a fixed deadline, both full continuous local adaptive and randomized nonadaptive minimax errors equal for every. One frozen direction attains it when; the value is

Proof. The supporting-plane inequality for every makes the frozen direction’s sign bias at least. Majority risk is decreasing in positive bias and efficiency, giving the upper bound. For the lower bound, the actual pair is admissible. Those states commute and are diagonal in the proposed measurement basis. Their independent tensor powers, including orthogonal erasure flags at, are a classical ternary likelihood problem. The likelihood ratio is monotone in the signed count, so even an arbitrary joint auxiliary-state measurement cannot beat fair-tie majority. This gives the same lower bound for every local protocol and its randomization. The zero vector has no sign information and gives 1

The assumption is essential to this proof. In the original asymmetric reader, the positive-target set consists of two separated near-axis regions and does not contain the diagonal point. Replacing it by its physical convex hull would admit a new independent-copy mixed source. That is a different, explicitly enlarged preparation class, not a permitted replacement of the original fixed hidden axis. It provides an exact benchmark and identifies the preparation distinction that a more advanced reader would have to exploit.

A retained-record witness of the preparation distinction

For the original hard support, let. The actual fixed-axis mixture and independent copies of that averaged one-copy state differ at two copies:

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Ω(2)
s
= pρ⊗2
s,u + (1 −p)ρ⊗2
s,v,
Ω(2)
s
−ρ⊗2
s,z = p(1 −p)
4
[(u −v) · σ] ⊗[(u −v) · σ].

Put. The trace distance is exactly2. Since) = 0, measuring each copy alonggives sign-product expectation in the fixed-axis mixture and zero in the independent averaged state. Coding a retained null as zero changes the difference to under the same independent sign-blind erasure law. No click postselection is used. For the original parameters,

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V = 134898283296
369773223625 ≃.364813552408,
η2
0V ≃.233480673541.

This is a standard covariance identity given an explicit operational readout, not a new physical law or a solution of the sign target itself. It makes the distinction between the two preparation laws experimentally meaningful under trusted preparation and the specified access contract. The comparison-prior stock is not asserted to be the actual unknown source in every deployment.