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The learning theorem
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Experimental learning of quantum states

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The theorem proved in (14) states:

Theorem 1. Let ρ be an n-qubit state, let D be a distribution over two-outcome measurements, and let ε = (E1, …, Em) consist of m measurements drawn independently from D. Suppose that we are given bits B = (b1, …, bm), where each bi is 1 with independent probability Tr(Eiρ) and 0 with probability 1 − Tr(Eiρ). Suppose also that we choose a hypothesis state σ to minimize the quadratic functional $f(σ)=∑i=1m(Tr(Eiσ)−bi)2$. Then, there exists a positive constant K such that$PrE∈D[|Tr(Eσ)−Tr(Eρ)|>γ]≤ε$with a probability of at least 1 − δ over E and B, provided that$m≥Kγ4ε2(nγ4ε2log21γε+log1δ)$

Here, rather than working with single-measurement outcomes bi, we are concerned with estimated expected values$Tr(Eiρ)≈∑j=1Sbi(j)/S$where each $bi(j)$ is the jth measurement outcome corresponding to Ei. To show that the hypothesis σ generated by considering the expected values is equivalent to that obtained by taking the measurements outcome bi, we define$f′=∑i=1m′(Tr(Eiσ)−Tr(Eiρ))2$

If we take m = mS and solve for σ, then in the equations df/dσ = 0 and df′/dσ = 0, it is possible to verify that the hypothesis that minimizes the function f′ is also satisfying f.

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