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arXiv:quant-ph/0204150 · Physical Review A

Discrete Wigner functions and the phase-space representation of quantum teleportation

Juan Pablo Paz

Published Jun 10, 2002• 65 citations
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Abstract

We present a phase-space description of the process of quantum teleportation for a system with an N-dimensional space of states. For this purpose we define a discrete Wigner function which is a minor variation of previously existing ones. This function is useful to represent a composite quantum system in phase space and to analyze situations where entanglement between subsystems is relevant (dimensionality of the space of states of each subsystem is arbitrary). We also describe how a direct tomographic measurement of this Wigner function can be performed.

DOI
10.1103/physreva.65.062311

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We show how to represent the state and the evolution of a quantum computer (or any system with an NNN--dimensional Hilbert space) in phase space. For this purpose we use a discrete version of the Wigner function which, for arbitrary NNN, is defined in a phase space grid of 2Ntimes2N2N\\times 2N2Ntimes2N points. We compute such Wigner function for states which are relevant for quantum computation. Finally, we discuss properties of quantum algorithms in phase space and present the phase space representation of Grover's quantum search algorithm.
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By means of a mod(N)\mathrm{mod}(N)mod(N)-invariant operator basis, sss-parametrized phase-space functions associated with bounded operators in a finite-dimensional Hilbert space are introduced in the context of the extended Cahill-Glauber formalism, and their properties are discussed in details. The discrete Glauber-Sudarshan, Wigner, and Husimi functions emerge from this formalism as specific cases of sss-parametrized phase-space functions where, in particular, a hierarchical process among them is promptly established. In addition, a phase-space description of quantum tomography and quantum teleportation is presented and new results are obtained.
Abstract We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group . This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group and apply this to the study of a particular three-level system. Moreover, we also discuss possible extensions to the discrete Husimi and Glauber–Sudarshan functions, as well as future investigations on multipartite quantum states.