Quantum Stein’s lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states (ρ or σ). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, n) identical copies of the state (ρ⊗n or σ⊗n) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability αn of erroneously inferring the state to be σ, the probability βn of erroneously inferring the state to be ρ decays exponentially in n, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures, and quasi-free states of fermionic lattice gases.
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We use the notation to denote a normal distribution of mean μ and variance σ2.
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(−Xn) is known as the observed log-likelihood ratio or information content random variable and can be used to define the classical Neyman-Pearson tests.
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