TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Tomography correctness and finite-sample stability

Statement

Let $d, m, n \geq 1$ , let $(F_{a})_{a = 1}^{m}$ be a finite POVM on $C^{d}$ whose real span is $H_{d}$ , and let $ρ \in H_{d}$ satisfy $ρ \geq 0$ and $Tr ρ = 1$ . Put $p_{a} = Tr (ρ F_{a})$ , form the canonical dual $D_{a}$ from the Gram pseudoinverse, and write $D : R^{m} \to H_{d}$ for $D c = \sum_{a} c_{a} D_{a}$ .

Define the frequency simplex $Δ_{m} = {f \in R^{m} : f_{a} \geq 0 for all a, \sum_{a} f_{a} = 1}$ and the real-linear map $X : R^{m} \to H_{d}$ by $X (f) = \sum_{a} f_{a} D_{a}$ . On $Δ_{m}$ , this is the output of the previously defined linear-inversion procedure; on all of $R^{m}$ , $X$ denotes its linear extension. Use the induced operator norm $∥ D ∥_{2 \to H S} = \sup_{∥ c ∥_{2} = 1} ∥ D c ∥_{H S}$ .

Then exact probabilities reconstruct the state: $X (p) = ρ$ . For every real frequency vector $f \in R^{m}$ ,

∥ X (f) - ρ ∥_{H S} \leq ∥ D ∥_{2 \to H S} ∥ f - p ∥_{2} .

Let $(Ω, F, P)$ be a probability space and let $Y_{1}, \dots, Y_{n} : Ω \to {1, \dots, m}$ be measurable with the independent product law $P (Y_{1} = a_{1}, \dots, Y_{n} = a_{n}) = \prod_{k} p_{a_{k}}$ for every outcome tuple. Define $f_{a} = n^{- 1} \sum_{k = 1}^{n} 1_{{Y_{k} = a}}$ . Then for every $0 < δ < 1$ ,

P ⁣ (\max_{1 \leq a \leq m} ∣ f_{a} - p_{a} ∣ \leq \sqrt{\frac{\log (2 m / δ)}{2 n}}) \geq 1 - δ .

Consequently, with probability at least $1 - δ$ ,

∥ X (f) - ρ ∥_{H S} \leq ∥ D ∥_{2 \to H S} \sqrt{m} \sqrt{\frac{\log (2 m / δ)}{2 n}} .

These are conditional mathematical statements under the specified product sampling law; no empirical data or calibrated apparatus is asserted.

Statement references

Finite Hermitian frames and their Gram pseudoinverse

Informationally complete linear tomography

Finite effects and instruments: normalization and non-determination of the update

Two-sided Hoeffding bound for Bernoulli frequencies

Facts & Assumptions

Given: The finite POVM and spanning, density-matrix, and independent product-sampling hypotheses in the Statement.

[A1]

The sample outcomes have the independent product law with one-trial probabilities $(p_{a})_{a = 1}^{m}$ stated in the Statement.

[F1]

The Gram pseudoinverse reconstructs a spanning Hermitian frame: for every $H \in H_{d}$ , $\sum_{a} Tr (H F_{a}) D_{a} = H$ .

[F2]

Finite effects and instruments: normalization and non-determination of the update: for a density operator $ρ$ and finite POVM $(F_{a})$ , $p_{a} = Tr (ρ F_{a}) \geq 0$ and $\sum_{a} p_{a} = 1$ .

[F3]

Informationally complete linear tomography: on an admissible frequency vector $f \in Δ_{m}$ , the procedure returns $X (f) = \sum_{a} f_{a} D_{a}$ ; the Statement defines the same formula as a linear map on all of $R^{m}$ .

[F4]

Two-sided Hoeffding bound for Bernoulli frequencies: for $n$ independent Bernoulli $(p)$ variables, their frequency $f$ satisfies $P (∣ f - p ∣ \geq t) \leq 2 e^{- 2 n t^{2}}$ for every $t > 0$ , without a choice assumption.

[F5]

Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors: the Euclidean and Hilbert–Schmidt norms satisfy Cauchy–Schwarz.

[F6]

The Cauchy-sequence reals have the least-upper-bound property: every nonempty bounded-above set of real numbers has a supremum.

[F7]

The real trace pairing on finite Hermitian matrices is an inner product: the Hilbert–Schmidt trace pairing on $H_{d}$ is a real inner product and its induced norm is $∥ A ∥_{H S} = \sqrt{Tr (A^{2})}$ .

Proof

1.1givenF1F2F3algebra

Apply [F1] with $H = ρ$ . Then $\sum_{a} p_{a} D_{a} = ρ$ , so [F3] gives $X (p) = ρ$ . The probabilities are in $[0, 1]$ and sum to one by [F2].

1.2givenA1F2F3F4algebra

For each sample $k$ , $\sum_{a = 1}^{m} 1_{{Y_{k} = a}} = 1$ , so $f_{a} \geq 0$ and $\sum_{a} f_{a} = 1$ , as required by [F3]. Fix an outcome $a$ and set $Z_{k}^{a} = 1_{{Y_{k} = a}}$ . For every $z \in {0, 1}^{n}$ , summing [A1] over the finite outcome tuples with $1_{{Y_{k} = a}} = z_{k}$ gives $P (Z_{1}^{a} = z_{1}, \dots, Z_{n}^{a} = z_{n}) = \prod_{k = 1}^{n} \sum_{b : 1_{{b = a}} = z_{k}} p_{b} = \prod_{k = 1}^{n} b_{p_{a}} (z_{k}),$ since [F2] gives $\sum_{b} p_{b} = 1$ . Thus these indicators have the Bernoulli product law, and [F4] yields $P (∣ f_{a} - p_{a} ∣ \geq t) \leq 2 e^{- 2 n t^{2}}$ . No independence between indicators for different outcomes is used.

2.1givenF5F6F7step 1.1algebra

By linearity, $X (f) - ρ = D (f - p)$ . Set $C = (\sum_{a} ∥ D_{a} ∥_{H S}^{2})^{1 / 2}$ . By [F7], this is the norm from a real inner product, so its triangle inequality follows from [F5] by expanding the square of a sum. Thus $∥ D c ∥_{H S} \leq \sum_{a} ∣ c_{a} ∣ ∥ D_{a} ∥_{H S} \leq C ∥ c ∥_{2}$ . Since $m \geq 1$ , the unit sphere in $R^{m}$ contains $e_{1}$ , so the set ${∥ D c ∥_{H S} : ∥ c ∥_{2} = 1}$ is nonempty; the bound makes it bounded above. Its supremum exists by [F6] and is the finite induced operator norm in the Statement. Scaling $c$ (and treating $c = 0$ directly) gives $∥ D c ∥_{H S} \leq ∥ D ∥_{2 \to H S} ∥ c ∥_{2}$ , which proves the deterministic estimate.

3.1givenA1step 2.1step 1.2algebra∎

A finite union bound gives $P (\exists a : ∣ f_{a} - p_{a} ∣ \geq t) \leq 2 m e^{- 2 n t^{2}}$ . Set $t = \sqrt{\log (2 m / δ) / (2 n)}$ , which is positive since $2 m / δ > 1$ ; then the right side is $δ$ . On the complementary event, $∥ f - p ∥_{2} \leq \sqrt{m} t$ , and step 2.1 gives the asserted state-error bound.

Tomography correctness and finite-sample stability

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources