Finite observable accuracy and an explicit rank test

Finite observable accuracy and an explicit rank test

The exact source separations have an accessible finite-accuracy consequence against qubit memories and classical sources with at most eight causal states. This is a known spectral-rank technique newly specialized to the delayed law; it does not improve the difficult three-versus-four quantum boundary. It supplies a controlled, finite-data interface for the established quantum and classical memory claims.

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Proposition 3.1 (Observable approximation floor). Let H be a normalized joint distribution of a past
and a future block, viewed as a matrix. Let γc(H) = (P
i>c σi(H)2)1/2. For any complete competing
law P ′ factoring through a causal cut with matrix rank at most c,

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TV(P ′, P) ≥γc(H)/
√
2.

A classical-state source has this rank bound; a quantum memory of Hilbert dimension has rank at most. Every future-accessed, history-correlated system belongs to that cut. The statement allows nonstationary competitors and extra null symbols.

Proof. The classical joint table is a sum of at most past/future product matrices. Quantum subnormalized conditional states and future effects pair in the-dimensional real Hermitian space. Pad the target with zeros for extra outcomes, preserving its nonzero singular values. The matrix difference has zero total sum; its positive and negative entries each sum to TV). Consequently its squared Frobenius norm is at most 2. The least Frobenius error of any rank- approximant is ), proving the assertion. The latter identity follows by truncating the singular-value decomposition for attainment; for the lower bound, project onto the orthogonal complement of the approximant’s column space and minimize the discarded quadratic mass.

For the four-record trine cut, put. The singular values of9 are

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1/9,
r/18 (four times),
r2/36 (four times).

Every qubit-cut competitor therefore obeys

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TV(P ′, Pλ,4) ≥r
√
1 + r2
18
√
2
,

while any at-most-eight-state classical competitor obeys

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TV(P ′, Pλ,4) ≥
r2
36
√
2.

These are outer bounds, not attained optimal approximation frontiers. They vanish at the iid endpoint, as required. They do not exclude a qutrit, because its ordinary rank ceiling is already nine.

Independent four-record trials and finite confidence

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Assume N independently prepared trials from one fixed complete four-record law, with every trial and
null retained. The preparation/reset mechanism, clock and independence are physical conditions, not
inferred from a reset flag. Pool all nontarget past pairs into one extra row and all nontarget future
pairs into one extra column; this coarse-graining preserves the rank bound and gives a finite complete
table. Let bH be its empirical frequency matrix and

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eN(α) = 1 +
p
log(1/α)
√
N
.

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Then
Pr{∥bH −H∥F > eN(α)} ≤α.

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Indeed E∥bH −H∥2
F = (1 −∥H∥2
F)/N ≤1/N; replacing one trial changes the norm by at most
√

. Bounded differences gives Pr, and Jensen gives the mean bound. The concentration step can be derived by exposing one independent trial at a time and bounding each conditional centered moment-generating function by the exponential of one eighth its squared range; optimizing its exponential Markov bound yields the displayed inequality. Distance to the rank- matrix set is one-Lipschitz, so the test

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γc( bH) > eN(α)

has false-rejection probability at most for every admitted rank- source. Against the exact target its rejection probability is at least 1 whenever). This supplies a stated sufficient trial budget. It is conservative and is not claimed sample-optimal. It certifies an observable rank obstruction, not a particular microscopic channel or the full infinite word law. Spectral identification, singular-value-tail thresholding and finite confidence are established tech-niques. Du Roy de Chaumaray et al., JMLR 25(415), 1–37 (2024), Eqs. (5)–(6), Theorem 4, uses the same tail statistic for nonparametric hidden Markov order estimation under a different, mixing single-sequence sampling model. The present contribution is the exact trine specialization, joint classical/quantum cut interpretation and complete-null four-record interface. Original least-rank approximation is Eckart–Young (1936). Hsu–Kakade–Zhang (2012), Lemma 8, already gives the same empirical probability-vector/Frobenius concentration method for independently sampled observation triples. No first spectral memory test is claimed. Neither this test nor its target constitutes an experimental distinction between Shadow Theory and ordinary quantum theory. At the new noise endpoint, the calculated qubit-cut TV floor is approximately 000961570 and the eight-state classical floor is 000111370. For one-percent false-rejection risk and at least 99-percent power against the exact target, the sufficient independent four-record trial budgets are 214,080 and 15,958,893, respectively. Each trial costs four emitted records plus its independently prepared stock. These conservative sufficiency budgets are not practical hardware estimates or sample-optimal claims. The supplied synthetic million-trial table rejects the qubit rank cap but does not reject the eight-state cap at the stated threshold, as expected.