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 thatPrED[|Tr(Eσ)Tr(Eρ)|>γ]εwith a probability of at least 1 − δ over E and B, provided thatmKγ4ε2(nγ4ε2log21γε+log1δ)

Here, rather than working with single-measurement outcomes bi, we are concerned with estimated expected valuesTr(Eiρ)j=1Sbi(j)/Swhere 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 definef=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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