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arXiv:quant-ph/0412093 · Journal of Mathematical Physics

Moment operators of the Cartesian margins of the phase space observables

Jukka Kiukas, Pekka Lahti, Kari Ylinen

Published Mar 24, 2005• 4 citations
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Abstract

The theory of operator integrals is used to determine the moment operators of the Cartesian margins of the phase space observables generated by the mixtures of the number states. The moments of the x-margin are polynomials of the position operator and those of the y-margin are polynomials of the momentum operator.
DOI
10.1063/1.1851957

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The theory of operator integrals is used to determine the moment operators of the Cartesian margins of the phase space observables generated by the mixtures of the number states. The moments of the xxx-margin are polynomials of the position operator and those of the yyy-margin are polynomials of the momentum operator.
Abstract The randomized quantum marginal problem asks about the joint distribution of the partial traces (“marginals”) of a uniform random Hermitian operator with fixed spectrum acting on a space of tensors. We introduce a new approach to this problem based on studying the mixed moments of the entries of the marginals. For randomized quantum marginal problems that describe systems of distinguishable particles, bosons, or fermions, we prove formulae for these mixed moments, which determine the joint distribution of the marginals completely. Our main tool is Weingarten calculus, which provides a method for computing integrals of polynomial functions with respect to Haar measure on the unitary group. As an application, in the case of two distinguishable particles, we prove some results on the asymptotic behavior of the marginals as the dimension of one or both Hilbert spaces goes to infinity.