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Abstract
Discrete coherent states for a system of n qubits are introduced in terms of eigenstates of the finite Fourier transform. The properties of these states are pictured in phase space by resorting to the discrete Wigner function.
Highlights
Discrete quantum systems were studied originally by Weyl[1] and Schwinger,[2] and later by many authors.[3,4] many concepts that appear sharp for continuous systems become fuzzy when one tries to apply them to discrete ones
Coherent states constitute an archetypical example of the situation: for the standard harmonic oscillator they are well understood, and sensible generalizations have been devised to deal with systems with more general dynamical groups.[6]
A decisive step was given by Mehta,[9] who obtained the eigenstates of the finite Fourier transform
Summary
Discrete quantum systems were studied originally by Weyl[1] and Schwinger,[2] and later by many authors.[3,4] many concepts that appear sharp for continuous systems become fuzzy when one tries to apply them to discrete ones. The reason is that the in the continuum we have only one harmonic oscillator, while for finite systems, there are a number of candidates for that role, each one with its own virtues and drawbacks.[5]. Number states are eigenstates of the Fourier transform, but they are not the only ones: coherent states, with their associated Gaussian wave functions, are. In our opinion, this subtle, yet obvious, observation has not been taken in due consideration in this field. This subtle, yet obvious, observation has not been taken in due consideration in this field From this perspective, a decisive step was given by Mehta,[9] who obtained the eigenstates of the finite Fourier transform. Using the notions presented in the comprehensive review 10, we construct a Wigner function for these coherent states and discuss some of their properties
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