Symmetric Laplacians, Quantum Density Matrices and their Von-Neumann Entropy

The Curvature of the Bogoliubov-kubo-mori Scalar Product on Matrices

The state space of a finite quantum system is identified with the set of positive semidefinite matrices of trace 1. The set of all strictly positive definite matrices of trace 1 becomes naturally a differentiable manifold and the Bogoliubov-Kubo-Mori scalar product defines a Riemannian structure on it. Reference [4] tells about the relation of this metric to the von Neumann entropy functional. ...

Quantum Entanglement and Entropy

Entanglement is the fundamental quantum property behind the now popular field of quantum transport of information. This quantum property is incompatible with the separation of a single system into two uncorrelated subsystems. Consequently, it does not require the use of an additive form of entropy. We discuss the problem of the choice of the most convenient entropy indicator, focusing our atten...

Continuity bounds for entanglement

To flesh out the notion that entanglement is a resource, various measures of entanglement have been proposed to quantify the amount of entanglement shared between two or more quantum systems. For a pure state of a twoparty quantum system, Popescu and Rohrlich [6] and Vidal [7] have shown that the measure of entanglement is uniquely specified by certain natural axioms: it is given by the von Neu...

Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices

The von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantummechanics and, from a numerical perspective, can be computed simply by computing all the eigenvalues of a density matrix, an operation that could be prohibitively expensive for large-scale density matrices. We present and analyze two randomized algorithms to approximate t...

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras