The Entropy-Based Quantum Metric

The von Neumann entropy S( D ^ ) generates in the space of quantum density matrices D ^ the Riemannian metric ds2 = −d2S( D ^ ), which is physically founded and which characterises the amount of quantum information lost by mixing D ^ and D ^ + d D ^ . A rich geometric structure is thereby implemented in quantum mechanics. It includes a canonical mapping between the spaces of states and of observables, which involves the Legendre transform of S( D ^ ). The Kubo scalar product is recovered within the space of observables. Applications are given to equilibrium and non equilibrium quantum statistical mechanics. There the formalism is specialised to the relevant space of observables and to the associated reduced states issued from the maximum entropy criterion, which result from the exact states through an orthogonal projection. Von Neumann’s entropy specialises into a relevant entropy. Comparison is made with other metrics. The Riemannian properties of the metric ds2 = −d2S( D ^ ) are derived. The curvature arises from the non-Abelian nature of quantum mechanics; its general expression and its explicit form for q-bits are given, as well as geodesics.