Theorem: Finite Enter instrument

Freeze the interface of Section sec:interface, adopt Born semantics and Definition def:names, and impose Assumption ass:statistics for the estimation statements. Then the following hold.

1. Complete instrument identity. The instrument $\mathcal{I}$ is a complete CP instrument with classical output. For every finite untouched retained factor $T$, including an arbitrary reference,

   $${\mathcal{I}}^T={{\mathcal{J}}_{\mathcal{B}}}^T\circ (\Phi \otimes {\mathrm{id}}_T)$$

   on all input operators. Discarding its record leaves the unconditional $T$-marginal unchanged.

2. Consumption and residual scope. Both measured data names are consumed. The newly returned object is the classical word $r$; no quantum $N$- or $A$-output belongs to the instrument. A conditional spectator or inaccessible implementation complement is not a return of those names.

3. Declared likelihood. The one-shot probabilities are

   $$\begin{array}{rl}{p}_{00}(\theta )& =\theta (1-t)(1-ht)+(1-\theta )t(\gamma +ht),\\ p_{01}(\theta )& =ht(1-t),\\ p_{10}(\theta )& =\theta t(1-ht)+(1-\theta )(1-t)(\gamma +ht),\\ p_{11}(\theta )& =h[\theta t^2+(1-\theta )(1-t{)}^2].\end{array}$$

   In particular, $Z_i$ are independent Bernoulli variables with

   $$q_{\theta }=\underset{\theta }{\mathrm{Pr}}\{Z_i=1\}=a+b\theta .$$

4. Separate finite-shot inference. The estimator (eq:estimator) is the maximum-likelihood estimator for the coarsened Bernoulli data $Z_1,\dots ,Z_n$. It is not asserted to be the maximum-likelihood estimator for the complete four-outcome record.

   For every $\epsilon >0$,

   $$\underset{\theta }{\mathrm{Pr}}\{|\hat{\theta }-\theta |>\epsilon \}\le 2e^{-2n{b}^2{\epsilon }^2}.$$

   For $0<\alpha <1$, define

   $$R_n(\alpha )=\min \left\{1,\frac{1}{b}\sqrt{\frac{\log (2/\alpha )}{2n}}\right\}.$$

   Uniformly over $\theta \in [0,1]$,

   $$\underset{\theta }{\mathrm{Pr}}\{|\hat{\theta }-\theta |\le R_n(\alpha )\}\ge 1-\alpha .$$

   Thus a sufficient shot budget for a radius at most $0<\epsilon <1$ is

   $$n\ge \lceil \frac{\log (2/\alpha )}{2b^2{\epsilon }^2}\rceil .$$

5. Fisher-information ledger. For $0<\theta <1$, the achieved one-shot information in the coarsened data is

   $$F_A(\theta )=\frac{b^2}{q_{\theta }(1-q_{\theta })}.$$

   The complete word has information

   $$F_C(\theta )=d^2\left[\frac{1}{p_{00}(\theta )}+\frac{{\gamma }^2}{p_{10}(\theta )}+\frac{h^2}{p_{11}(\theta )}\right].$$

   They satisfy

   $$F_A(\theta )\le F_C(\theta )\le B_{\theta }:=\frac{1}{\theta (1-\theta )}.$$

   The last bound also applies to any POVM on the post-step data state $\Phi ({\rho }_{\theta })$. Independent repetition multiplies the respective Fisher informations by $n$.

The output interpretation is finite-model estimate under declared access. The theorem supplies neither a quantum-return decoder nor a physical identification with Galactic transport.

Proof. We prove each claim on the declared interface.

and consumption. Lemma lem:identity proves the complete identity and its reference extensions. Equivalently, the $r$-branch on a record-only interface has Kraus rows

Their completeness is

Lemma lem:consumption proves the name statement.

. The squared moduli of both $V_{\varphi }$ and ${V_{\varphi }}^{†}$ are the transition matrix

The diagonal gate $\mathrm{CZ}$ changes phases but not computational populations.

Conditional on source component $00$, the normal output has probabilities $1-t,t$. The antiqubit's probability of value $1$, after relaxation, is $ht$. Conditional on source component $11$, the normal output has probabilities $t,1-t$, and the antiqubit's probability of value $1$ is $h(1-t)$.

The diagonal output populations conditional on either source component factor accordingly. This remains true despite coherences in the quantum state: amplitude damping transfers diagonal populations as stated, and its off-diagonal matrix elements do not contribute to these computational probabilities. Mixing the two source components gives (eq:probabilities). Summing the outcomes with $y=0$ gives

The product preparation and product device assumptions give independence.

maximization. For a count $K=k$, the Bernoulli likelihood is proportional to

Its unconstrained maximizing value is $k/n$; restricting it to the interval gives the projection of $k/n$ onto $[a,a+b]$. Applying the affine inverse gives (eq:estimator), including endpoint counts.

. For completeness, let $Z$ be Bernoulli with mean $q$, and set

Then $g(0)=g^{\prime }(0)=0$, and $g^{\prime \prime }(u)$ is the variance of a Bernoulli variable under an exponentially tilted law. Hence $g^{\prime \prime }(u)\le 1/4$, and integration yields

Independence and exponential Markov bounds therefore give

This is the Bernoulli instance of Hoeffding's inequality Hoeffding1963.

Projection onto $[0,1]$ cannot increase distance from $\theta \in [0,1]$. Consequently,

Taking $v=b\epsilon$ proves (eq:tail). Taking $v=\sqrt{\log (2/\alpha )/(2n)}$, and using $|\hat{\theta }-\theta |\le 1$, proves (eq:radius)–(eq:coverage). Equation (eq:shots) follows directly.

Fisher information. For a Bernoulli law with derivative ${q^{\prime }}_{\theta }=b$,

Differentiating (eq:probabilities) gives

which proves (eq:FC). All four probabilities are strictly positive under the calibration assumptions.

comparisons. Let $J$ denote the hidden source component, with ${\mathrm{Pr}}_{\theta }\{J=0\}=\theta$. Its score is

Thus

For any outcome $R$ obtained through a $\theta$-independent conditional law given $J$, the outcome score equals

Conditional Jensen gives

This applies to the complete word $C$. It also applies to any POVM on $\Phi ({\rho }_{\theta })$: its probabilities are

which have exactly the required mixture form.

Finally, $Z$ is a deterministic coarsening of $C$. Its score is the conditional expectation of the complete-word score given $Z$. A second application of conditional Jensen gives $F_A\le F_C$.

For independent repetitions, scores add and have mean zero, so the squared-score expectations add. This proves the $n$-shot information statements and completes the proof.