Technical Companion: Approximation and Accounting for Quantum Sources
Exact finite-record accuracy and a memoryless comparator
Exact finite-record accuracy and a memoryless comparator
An exact resource minimum does not alone establish that the resource is needed at a finite experimental accuracy. This section evaluates that distinction for the same full trine law, without changing its parameters or introducing an alternative Born rule. It is an analytical addition to the completion theorem’s rank-based certificates. It is not a claimed optimum over all approximate hidden-source realizations. Let
be the
-symbol stationary word law in the main resource manuscript, and let 
The latter is generated by the one-dimensional quantum instrument Φ
3, or a one-state classical source with fresh uniform trit emissions. Both have the same specified output alphabet and the same record length. Fresh randomness or its dilation still costs resources and is not contained in the one-dimensional memory count.
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Theorem 2.1 (Exact binomial reduction of the finite-record discrepancy). For N ≥2 set k = N −2
and R = Pk
t=1 1{Yt = Yt+2}. Then
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TV(Pλ,N, UN) = TV Bin(k, λ/3), Bin(k, 1/3) .
The count
is a sufficient statistic for this binary experiment. Thus even retaining every symbol gives no stronger test for these two models than retaining that count. This is an assertion about the declared classical word record, not about their microscopic output environments.
Proof. For each word with
repeats,
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k−r . Pλ,N(y) UN(y) = λr 3 −λ 2
Conditional on every positive-probability preceding history, the next repeat indicator has probability 
3. Iterating conditional probabilities proves that these
indicators are iid Bernoulli, despite the overlapping symbol comparisons. Under
their parameter is 1
3. The likelihood ratio is constant on each fibre of
; summing absolute signed masses fibre by fibre therefore gives the equality in TV, with no data-processing slack.
For 0
1 put
2 and
log
log
log
. Since the likelihood ratio decreases with
, the exact error is
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Fk,λ/3(r∗) −Fk,1/3(r∗),
where
is the binomial CDF. Equality at an integer crossing contributes zero signed mass, so either threshold convention is valid. At
= 0 the result is 1
, and at
= 1 it is zero. For
both word laws agree.
Two closed-form bounds are also available:
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s 1 + (1 −λ)2 k −1 1, 1 TV(Pλ,N, UN) ≤min , 2 2 "√ v u u t1 − #2k . λ + 2 p (3 −λ)/2 3 TV(Pλ,N, UN) ≤
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χ2 and the binomial likelihood second moment; the second follows by Cauchy–Schwarz from the product Bhattacharyya affinity. In particular, the one-state comparator meets TV tolerance ε whenever The first follows from TV ≤1
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2 p
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k ≤ log(1 + 4ε2) log(1 + (1 −λ)2/2).
This is a sufficient finite-horizon resource statement, not an exact minimum-memory frontier. The exact CDF expression gives the sharper decision for any specified (
What this changes. The completion theorem established that nine classical hidden states remain necessary for every exact non-iid trine law. The present result shows quantitatively when a one-state source already approximates its complete finite passive record. Thus an exact quantum/classical dimension separation may coexist with a memoryless approximation at the nominated tolerance. Resolving the exact qutrit question does not remove this operational qualification. A common fixed downstream reader, including a reader with an explicitly retained erasure symbol, contracts this discrepancy. For a source-independent erasure probability 1
on each emitted label, retaining the original un-erased word as well makes the joint discrepancy exactly the original TV: the channel
erased record) preserves it. Keeping only the erased report can reduce it. These statements compare the same admitted classical records on both sides; neither grants access to a microscopic bath or to a quantum source’s hidden memory.
Verification. The compact release supplies supplement/memory/checks/verify_consolidated_c onstants.py, which independently compares ordered-word enumeration and the binomial TV formula in twenty exact rational cases. The analytical proof above supplies the all-horizon statement. The historical diagnostic check_finite.py, preserved in the separate evidence archive, enumerated all words of lengths two through eight at six rational noise parameters (42 cases), and evaluated additional binomial CDF and retained-erasure examples. Those archived diagnostics are not advertised as extra checks run by the compact release’s reproduction command. These are mathematical calculations, not empirical data.