Quantum U-statistics

Abstract

The notion of a $U$-statistic for an $n$-tuple of identical quantum systems is introduced in analogy to the classical (commutative) case: given a selfadjoint `kernel' $K$ acting on $({\mathbb{C}}^d{)}^{\otimes r}$ with $r<n$, we define the symmetric operator $U_n=\left(\genfrac{}{}{0ex}{}{n}{r}\right){\sum }_{\beta }{K}^{(\beta )}$ with $K^{(\beta )}$ being the kernel acting on the subset $\beta$ of $\{1,\dots ,n\}$. If the systems are prepared in the i.i.d state ${\rho }^{\otimes n}$ it is shown that the sequence of properly normalised $U$-statistics converges in moments to a linear combination of Hermite polynomials in canonical variables of a CCR algebra defined through the Quantum Central Limit Theorem. In the special cases of non-degenerate kernels and kernels of order $2$ it is shown that the convergence holds in the stronger distribution sense. Two types of applications in quantum statistics are described: testing beyond the two simple hypotheses scenario, and quantum metrology with interacting hamiltonians.

arXiv comment30 pages, added section on quantum metrology

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Cited by in corpus (3)

• Quantum local asymptotic normality based on a new quantum likelihood ratio
• Operational meanings of a generalized conditional expectation in quantum metrology
• Quantum hypothesis testing for quantum Gaussian states: Quantum analogues of chi-square, t and F tests