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arXiv:1004.2452 · Journal of Mathematical Physics

Quantum U-statistics

Mădălin Guţǎ, Cristina Butucea

Published Oct 1, 2010• 5 citations
arXivHTMLPDF

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 (Cd)⊗r with r<n, we define the symmetric operator Un=(nr)∑βK(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.

DOI
10.1063/1.3476776

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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 (Cd)⊗r with r<n, we define the symmetric operator Un=(nr)∑βK(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.
The fact that any classical random variable with all moments has a quantum decomposition allows to associate to it a family of quantum moments. On the other hand, a classical random variable may have several inequivalent quantum decompositions, which lead to the same classical, but different quantum moments. Even in the simplest Central Limit Theorems (CLT), i.e. those of Bernoulli type, there are examples in which the corresponding quantum moments converge to the canonical quantum moments of the associated classical random variable, and examples in which this is not the case. This poses the problem to find a constructive criterium that characterizes the quantum moments associated to the canonical quantum decomposition (which is unique) with respect to the other ones. Theorem [Formula: see text] of the present paper provides such a criterium. Theorem [Formula: see text] deals with the case when one knows a priori that the quantum moments come from a central limit theorem (the motivation of the present paper arose in this context). It gives only a sufficient condition, but simpler to verify than the necessary and sufficient conditions of Theorem [Formula: see text]. Theorem [Formula: see text] naturally leads to a classification of Interacting Fock Spaces (IFS) into three types. We construct examples showing that all these possibilities can effectively take place. On the way, we prove that all the best known deformations of Heisenberg commutation relations can be obtained as special cases of a general construction within the algebraic approach to the theory of orthogonal polynomials.
Cabana-Duvillard and lonescu11have proved that any symmetric probability measure with moments of any order can be obtained as central limit theorem of self-adjoint, weakly independent and symmetrically distributed (in a quantum souse) random variables. Results of this type will be called "universal central limit theorem". Using Interacting Fock Space (IFS) techniques we extend this result in two directions: (i) we prove that the random variables can be taken to be generalized Gaussian in the sense of Accardi and Bożejko3and we give a realization of such random variables as sums of creation, annihilation and preservation operators acting on an appropriate IFS; (ii) we extend the above-mentioned result to the nonsymmetric case. The nontrivial difference between the symmetric and the nonsymmetric case is explained at the end of the introduction below.
The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum-type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss-Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.
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Abstract A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.