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arXiv:1112.3164 · European Journal of Physics

Classical and quantum-mechanical state reconstruction

F C Khanna, P A Mello, M Revzen

Published May 14, 2012• 6 citations
arXivHTMLPDF

Abstract

We review the problem of state reconstruction in classical and in quantum physics, which is rarely considered at the textbook level. We review a method for retrieving a classical state in phase space, similar to that used in medical imaging known as Computer Aided Tomography. We explain how this method can be taken over to quantum mechanics, where it leads to a description of the quantum state in terms of the Wigner function which, although may take on negative values, plays the role of the probability density in phase space in classical physics. We explain another approach to quantum state reconstruction based on the notion of Mutually Unbiased Bases, and indicate the relation between these two approaches. Both are for a continuous, infinite-dimensional Hilbert space. We then study the finite-dimensional case and show how the second method, based on Mutually Unbiased Bases, can be used for state reconstruction.

DOI
10.1088/0143-0807/33/4/921

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We review the problem of state reconstruction in classical and in quantum physics, which is rarely considered at the textbook level. We review a method for retrieving a classical state in phase space, similar to that used in medical imaging known as Computer Aided Tomography. We explain how this method can be taken over to quantum mechanics, where it leads to a description of the quantum state in terms of the Wigner function which, although may take on negative values, plays the role of the probability density in phase space in classical physics. We explain another approach to quantum state reconstruction based on the notion of Mutually Unbiased Bases, and indicate the relation between these two approaches. Both are for a continuous, infinite-dimensional Hilbert space. We then study the finite-dimensional case and show how the second method, based on Mutually Unbiased Bases, can be used for state reconstruction.
Classical phase spaces have been widely applied in physics, engineering, economics or biology. In this thesis, phase spaces of quantum systems are considered, which have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states. We provide a complete phase-space description of (coupled) spin systems including their time evolution, tomography, large-spin approximations and their infinite-dimensional limit, which recovers the well-known case of quantum optics. Finally, Born-Jordan distributions of infinite-dimensional quantum systems are described.
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Continuous phase spaces have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states in quantum optics and beyond. A plethora of these phase-space techniques are known, however a thorough understanding of their relations is still lacking for finite-dimensional quantum states. We present a unified approach to continuous phase-space representations which highlights their relations and tomography. The infinite-dimensional case from quantum optics is then recovered in the large-spin limit.
Starting from the famous Pauli problem on the possibility of associating quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference between those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to non-negativity of probability density on phase space in the classical domain. The intersection of such sets is studied. The mathematical mechanism that allows us to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc is clarified and a connection with abstract Hilbert space properties is established. The superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equations like quantum evolution and energy spectra equations are given in an explicit form. A method to check experimentally the uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two-qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.
Finite Geometry is used to underpin operators acting in finite, d, dimensional Hilbert space. Quasi distribution and Radon transform underpinned with finite dual affine plane geometry (DAPG) are defined in analogy with the continuous (d→∞d \rightarrow \inftyd→∞) Hilbert space case. An essntial role in these definitions play the projectors of states of mutual unbiased bases (MUB) and their Wigner function-like mapping onto the generalized phase space that lines and points of DAPG constitutes.