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AIP conference proceedings

Dualistic properties of the manifold of quantum states

Anna Jenčová

Published Jan 1, 2001• 4 citations
Publisher

Abstract

In the finite dimensional case, we introduce a family of torsion-free affine connections in the manifold of all invertible density matrices. These are the quantum version of Amari’s α-connections, based on the α-representation of the tangent space. The dual connections with respect to a monotone metric are, in general, not torsion-free. We compute the torsion and the Riemannian curvature. Finally, geodesics are used to define a divergence.

DOI
10.1063/1.1358176

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In the finite dimensional case, we introduce a family of torsion-free affine connections in the manifold of all invertible density matrices. These are the quantum version of Amari’s α-connections, based on the α-representation of the tangent space. The dual connections with respect to a monotone metric are, in general, not torsion-free. We compute the torsion and the Riemannian curvature. Finally, geodesics are used to define a divergence.
Quantum connections are defined by parallel transport operators acting on a Hilbert space. They transport tangent operators along paths in parameter space. The metric tensor of a Riemannian manifold is replaced by an inner product of pairs of operator fields, similar to the inner product of the Kubo-Mori formalism of Linear Response Theory. The metric is used to define the dual of a quantum connection.The gradient of the parallel transport operators is the quantum vector potential. It defines the covariant derivative of operator fields. The covariant derivatives are used to quantify the holonomy of the quantum connection. It is shown that a quantum connection is holonomic if and only if its dual is holonomic. If the parallel transport operators are unitary then an alpha-family of quantum connections can be defined in a way similar to Amari's alpha family of connections in Information Geometry. The minus alpha connection is the dual of the alpha connection. In particular, the alpha equal zero connection is self-dual. An operator field can be combined with a path in parameter space to produce a path in operator space. A definition is given for such a path in operator space to be autoparallel. The path in parameter space is then a geodesic for the induced connection.
We consider the quantum information manifold whose underlying set M consists of density operators rho with the extra property that some fractional power of rho is of trace class. The topology is defined by defining a neighbourhood of a point rho to be all density operators that dominate rho and are dominated by rho. We show that this is the same set as that of all states whose relative Hamiltonian X in the sense of Araki is bounded, and such that X(t) is holomorphic in the circle |t| less than 1/2. Here, X(t) is the time evolution of X determined by the modular automorphism defined by rho. We show that M is a Banach manifold in Araki's norm, and that both the canonical and the mixture affine connections can be defined. These are dual relative to the Kubo-Mori metric, and so generalise Amari's dual theory to quantum theory in infinite dimensions.
In this paper we study the autoparallelity w.r.t. the e-connection for an information-geometric structure called the SLD structure, which consists of a Riemannian metric and mutually dual e- and m-connections, induced on the manifold of strictly positive density operators. Unlike the classical information geometry, the e-connection has non-vanishing torsion, which brings various mathematical difficulties. The notion of e-autoparallel submanifolds is regarded as a quantum version of exponential families in classical statistics, which is known to be characterized as statistical models having efficient estimators (unbiased estimators uniformly achieving the equality in the Cramér-Rao inequality). As quantum extensions of this classical result, we present two different forms of estimation-theoretical characterizations of the e-autoparallel submanifolds. We also give several results on the e-autoparallelity, some of which are valid for the autoparallelity w.r.t. an affine connection in a more general geometric situation.