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Laser Physics

Factorization of the Wigner distribution in prime power dimensions

Thomas Durt

Published Oct 30, 2006• 5 citations
Publisher

Abstract

There are different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g., to perform full tomography of this state. For instance, quasi-distributions such as the Weyl or Wigner distributions provide complete information about a quantum state which is equivalent to the information contained in the density matrix. In the case of composite systems, of which the subsystems are not necessarily located at the same place, the experimental feasibility of the tomographic process is considerably simplified whenever it can be realized through local operations and classical communications between local observers. This brings us naturally to study the possibility to factorize the (discrete) Wigner distribution of a composite system into the product of local Wigner distributions, which is the subject of the present paper. The discrete Heisenberg-Weyl group is an essential ingredient of our approach.

DOI
10.1134/s1054660x06110090

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There are different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g., to perform full tomography of this state. For instance, quasi-distributions such as the Weyl or Wigner distributions provide complete information about a quantum state which is equivalent to the information contained in the density matrix. In the case of composite systems, of which the subsystems are not necessarily located at the same place, the experimental feasibility of the tomographic process is considerably simplified whenever it can be realized through local operations and classical communications between local observers. This brings us naturally to study the possibility to factorize the (discrete) Wigner distribution of a composite system into the product of local Wigner distributions, which is the subject of the present paper. The discrete Heisenberg-Weyl group is an essential ingredient of our approach.
We study different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g. to perform full tomography of this state. We focus on two apparently simple cases, full tomography of one and two qubit systems. We analyze and compare those techniques according to two figures of merit. Our first criterion is the minimisation of the redundancy of the data acquired during the tomographic process. In the case of two-qubits tomography, we also analyze this process from the point of view of factorisability, so to say we analyze the possibility to realise the tomographic process through local operations and classical communications between local observers. This brings us naturally to study the possibility to factorize the (discrete) Wigner distribution of a composite system into the product of local Wigner distributions. The discrete Heisenberg-Weyl group is an essential ingredient of our approach. Possible extensions of our results to higher dimensions are discussed in the last section and in the conclusions.
We study different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g. to perform full tomography of this state. In a first time, we focus on two simple cases, full tomography of one- and two-qubit systems. We analyze and compare those techniques according to two criteria. Our first criterion is the minimisation of the redundancy of the data acquired during the tomographic process. In the case of two-qubits tomography, we also analyze this process from the point of view of factorisability, so to say we analyze the possibility to realise the tomographic process through local operations and classical communications between local observers. Finally, we present new results that concern the extension of the one- and two-qubit cases to higher dimensions.
Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view. For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups. This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N. The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra. The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.