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Journal of Physics A Mathematical and Theoretical

Symplectic transformations and quantum tomography in finite quantum systems

Apostol Vourdas, Cyril Banderier

Published Mar 14, 2011• 12 citations
Publisher

Abstract

Quantum systems where the position and momentum are in the ring Zd (d is an odd integer) are considered. Symplectic transformations are studied, and the order of Sp(2, Zd ) is calculated. Quantum tomography is also discussed. It is shown that measurements (used in the inverse Radon transform) need to be made on J2(d) lines (where J2(d) is the Jordan totient function).

DOI
10.1088/1751-8113/44/14/149501

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Quantum systems where the position and momentum are in the ring Zd (d is an odd integer) are considered. Symplectic transformations are studied, and the order of Sp(2, Zd ) is calculated. Quantum tomography is also discussed. It is shown that measurements (used in the inverse Radon transform) need to be made on J2(d) lines (where J2(d) is the Jordan totient function).
The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.
Quantum systems with finite Hilbert space are considered. Position and mo- \nmentum states and their relation through a Fourier transform, displacement \nin the position-momentum phase-space, and symplectic transformations are \nintroduced and their properties are studied. Symplectic Sp(2l;Zp) trans- \nformations in l-partite finite system are explicit constructed. The general \nmethod is applied to bi-partite and tri-partite systems. The effect of these \ntransformations on the correlations is discussed. Entanglement calculations \nbetween the subsystems in a bi-partite system and a tri-partite system are \npresented. The effect of measurements is also studied.
Quantum systems with finite Hilbert space where position x and momentum \np take values in Z(d) (integers modulo d) are considered. Symplectic tranformations \nS(2¿,Z(p)) in ¿-partite finite quantum systems are studied and \nconstructed explicitly. Examples of applying such simple method is given \nfor the case of bi-partite and tri-partite systems. The quantum correlations \nbetween the sub-systems after applying these transformations are discussed \nand quantified using various methods. An extended phase-space x¿p¿X¿P \nwhere X, P ¿ Z(d) are position increment and momentum increment, is introduced. \nIn this phase space the extended Wigner and Weyl functions are \ndefined and their marginal properties are studied. The fourth order interference \nin the extended phase space is studied and verified using the extended \nWigner function. It is seen that for both pure and mixed states the fourth \norder interference can be obtained.
Quantum systems where the position and momentum are in the ring ( d is an odd integer) are considered. A tomographically complete set of bases, i.e. a set of bases such that probabilities from tomography experiments corresponding to these bases, can be used for the calculation of an arbitrary density matrix, is considered. Such a set of bases is constructed by considering a factorization of a system with dimension d = p 1 p 2 (where p 1 , p 2 are prime numbers) in terms of two subsystems with dimensions p 1 and p 2 . Appropriate combination of the mutually unbiased bases in the subsystems, leads to a tomographically complete set of bases in the full system. This set leads to probabilities along all the lines through the origin (0, 0) in the phase space, with exactly d points. It is shown that the number of these lines is ψ( d ) (the Dedekind function). The general theory is exemplified with examples.