Quantum U-statistics
- 1 October 2010
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 51 (10), 102202
- https://doi.org/10.1063/1.3476776
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 self-adjoint “kernel” K acting on ( C d ) ⊗ r with r < n , we define the symmetric operator U n = ( n r ) ∑ β K ( β ) with K ( β ) being the kernel acting on the subset β of { 1 , … , n } . If the systems are prepared in the product state ρ ⊗ n , it is shown that the sequence of properly normalized U -statistics converges in moments to a linear combination of Hermite polynomials in canonical variables of a canonical commutation relation algebra defined through the quantum central limit theorem. In the special cases of nondegenerate kernels and kernels of order of 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.Other Versions
This publication has 25 references indexed in Scilit:
- On normal approximations to U-statisticsThe Annals of Probability, 2009
- Quantum-limited metrology with product statesPhysical Review A, 2008
- Generalized Limits for Single-Parameter Quantum EstimationPhysical Review Letters, 2007
- A study of LOCC-detection of a maximally entangled state using hypothesis testingJournal of Physics A: General Physics, 2006
- Strong converse and Stein's lemma in quantum hypothesis testingIEEE Transactions on Information Theory, 2000
- Krein Condition in Probabilistic Moment ProblemsBernoulli, 2000
- The Classical Moment Problem as a Self-Adjoint Finite Difference OperatorAdvances in Mathematics, 1998
- The proper formula for relative entropy and its asymptotics in quantum probabilityCommunications in Mathematical Physics, 1991
- Generalized uncertainty relations and efficient measurements in quantum systemsTheoretical and Mathematical Physics, 1976
- A Class of Statistics with Asymptotically Normal DistributionThe Annals of Mathematical Statistics, 1948