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A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.
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A recent canonical divergence, which is introduced on a smooth manifold $mathrm{M}$ endowed with a general dualistic structure $(mathrm{g}, abla, abla^*)$, is considered for flat $alpha$-connections. In the classical setting, we compute such a canonical divergence on the manifold of positive measures and prove that it coincides with the classical $alpha$-divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum $alpha$-connections on the manifold of all positive definite Hermitian operators. Also in this case we obtain that the recent canonical divergence is the quantum $alpha$-divergence.
Using the method of canonical group quantization, we construct the angular momentum operators associated to configuration spaces with the topology of (i) a sphere and (ii) a projective plane. In the first case, the obtained angular momentum operators are the quantum version of Poincares vector, i.e., the physically correct angular momentum operators for an electron coupled to the field of a magnetic monopole. In the second case, the obtained operators represent the angular momentum operators of a system of two indistinguishable spin zero quantum particles in three spatial dimensions. We explicitly show how our formalism relates to the one developed by Berry and Robbins. The relevance of the proposed formalism for an advance in our understanding of the spin-statistics connection in non-relativistic quantum mechanics is discussed.
This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs. In the quantum setting, the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to more general realizations of the density operator which have recently appeared in quantum molecular dynamics. Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical dynamics. Uhlmanns density matrix and Koopman-von Neumann wavefunctions are shown to be special cases of this construction.
Invariant operator-valued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n)xU(n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n=2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and anti-symmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a non-linear realization of the Lie algebra of SU(2)xSU(2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables.
The principles of classical mechanics have shown that the inertial quality of mass is characterized by the kinetic energy. This, in turn, establishes the connection between geometry and mechanics. We aim to exploit such a fundamental principle for information geometry entering the realm of mechanics. According to the modification of curve energy stated by Amari and Nagaoka for a smooth manifold $mathrm{M}$ endowed with a dual structure $(mathrm{g}, abla, abla^*)$, we consider $ abla$ and $ abla^*$ kinetic energies. Then, we prove that a recently introduced canonical divergence and its dual function coincide with Hamilton principal functions associated with suitable Lagrangian functions when $(mathrm{M},mathrm{g}, abla, abla^*)$ is dually flat. Corresponding dynamical systems are studied and the tangent dynamics is outlined in terms of the Riemannian gradient of the canonical divergence. Solutions of such dynamics are proved to be $ abla$ and $ abla^*$ geodesics connecting any two points sufficiently close to each other. Application to the standard Gaussian model is also investigated.
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