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Kluwer Academic Publishers eBooks

Reconstructing the Discrete Wigner Function through Complementary Measurements

Roberth Asplund, Gunnar G. Bjork

Published Feb 24, 2006• 0 citations
Publisher

Abstract

We report a novel algorithm to reconstruct the discrete Wigner function of aquantum system from a set of non-redundant measurements on an ensemble of identically prepared systems. Not surprisingly, the necessary von Neumann measurements form a complementary set. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/0-306-47114-0_18

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We report a novel algorithm to reconstruct the discrete Wigner function of aquantum system from a set of non-redundant measurements on an ensemble of identically prepared systems. Not surprisingly, the necessary von Neumann measurements form a complementary set. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We derive a direct reconstruction algorithm for the discrete Wigner function through different types of measurements. For a system described in a Hilbert space of dimension N=N1…Np,{N=N}_{1}\dots{}{N}_{p},N=N1​…Np​, where the numbers Ni{N}_{i}Ni​ are prime, the reconstruction is accomplished with (N1+1)…(Np+1){(N}_{1}+1)\dots{}{(N}_{p}+1)(N1​+1)…(Np​+1) factorable (local) von Neumann measurements. For the special case where the dimension is a power of a prime, the reconstruction can be performed in a much more efficient way using N+1N+1N+1 complementary measurements. If the system is composed of a number of smaller subsystems, these measurements will then in general be nonseparable.
In 1932, E.Wigner [1] introduced a phase-space method of computing physical observables in quantum (statistical) mechanics; some surveys have been published recently [2], [3]. See also a review article by Carruthers [4]. Here we give a brief synopsis of the method to make this paper self-contained, referring the reader to the references for details, including proofs. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The notion of mixed strategy, as originally introduced by Von Neumann and Morgenstern ( 1944 ) is a basic ingredient of classical game theory. Yet, as pointed out by von Neumann and Morgenstern themselves, the idea that a rational player may have to use a randomizing device, such as a coin flip, to decide on their actions poses some insuperable conceptual difficulties. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This chapter gives an introduction to a chain of results attempting to exclude deeper layers underneath quantum mechanics that restore some form of classical physics: ‘[Such results] more or less illustrate the ways along which some opponents might hope to escape Bohr’s reasonings and von Neumann’s proof and the places where they are dangerously near breaking their necks.’ (Groenewold, 1946, p. 454). In so far as they are mathematically precise, such no-go results have their roots in von Neumann’s 1932 book, which gave rise to two traditions that were often in polemical opposition to each other. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.