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arXiv:1302.1333 · Journal of Mathematical Physics

Geometric approach to non-relativistic quantum dynamics of mixed states

Vicent Gimeno, Jose M. Sotoca

Published May 1, 2013• 1 citation
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Abstract

In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincaré quantum recurrence in a physical system with finite energy levels.

DOI
10.1063/1.4807096

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In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincaré quantum recurrence in a physical system with finite energy levels.
The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert–Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework.
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After a review of the pure state case, we discuss from a geometrical point of view the meaning of the quantum Fisher metric in the case of mixed states for a two-level system, i.e. for a q-bit, by examining the structure of the fiber bundle of states, whose base space can be identified with a co-adjoint orbit of the unitary group. We show that the Fisher information metric coincides with the one induced by the metric of the manifold of , i.e. the three-dimensional sphere S 3 , when the mixing coefficients are varied. We define the notion of Fisher tensor and show that its anti-symmetric part is intrinsically related to the Kostant–Kirillov–Souriau symplectic form that is naturally defined on co-adjoint orbits, while the symmetric part is non-trivially again represented by the Fubini–Study metric on the space of mixed states, weighted by the mixing coefficients.
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