Quantum U-statistics

The notion of a U-statistic for an n-tuple of identical quantum systems is introduced inanalogy to the classical (commutative) case: given a selfadjoint ‘kernel’ K acting on (Cd)rwith r < n, we define the symmetric operator Un = ??nrP K() with K() being the kernelacting on the subset of {1, . . . , n}. If the systems are prepared in the i.i.d state n it isshown that the sequence of properly normalised U-statistics converges in moments to a linearcombination of Hermite polynomials in canonical variables of a CCR algebra defined throughthe Quantum Central Limit Theorem. In the special cases of non-degenerate kernels and kernelsof 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 simplehypotheses scenario, and quantum metrology with interacting hamiltonians.