Symmetries, group actions, and entanglement

We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum composite systems we discuss and give examples of measures of entanglement.Comment: 21 page

Segre maps and entanglement for multipartite systems of indistinguishable particles

We elaborate the concept of entanglement for multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. The entanglement is characterized in terms of generalized Segre maps, supplementing thus an algebraic approach to the problem by a more geometric point of view.Comment: 16 pages, the version to appear in J. Phys. A. arXiv admin note: text overlap with arXiv:1012.075

Unitary quantum gates, perfect entanglers and unistochastic maps

Non-local properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads 8/3 \pi \approx 0.85. We establish explicit conditions under which a given one-qubit bistochastic map is unistochastic, so it can be obtained by partial trace over a one--qubit environment initially prepared in the maximally mixed state.Comment: 14 pages including 6 figures in eps, version 4, title changed according to a suggestion of the editor

On the relation between states and maps in infinite dimensions

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators $\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1)$ and the corresponding tensor products $\mathcal{H}_1\otimes\mathcal{H}_2^*$ of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map $\mathcal{C}:\mathcal{L}_1(\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1))\to \mathcal{L}_\infty(\mathcal{L}(\mathcal{H}_2),\mathcal{L}_1(\mathcal{H}_1))$ from trace-class operators on $\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1)$ (with the nuclear norm) into compact operators mapping the space of all bounded operators on $\mathcal{H}_2$ into trace class operators on $\mathcal{H}_1$ (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.Comment: 19 page

Tensor Products of Random Unitary Matrices

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M=2, N become large or M become large and N=2.Comment: 23 pages, 2 figure

Geometry of quantum dynamics in infinite dimension

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning coadjoint orbits of the unitary group in infinite dimension, embedding of the Hilbert projective space of pure states in the unitary group, and an approach to self-adjoint extensions of symmetric relations.Comment: 32 page