Abstract:
We show that entanglement of pure multi-party states can be quantified by means of quantum uncertainties of certain basic observables through the use of measure that has been initially proposed in [10] for bipartite systems.

Abstract:
Observations show that maximally-steep breaking water wave crest speeds are much slower than expected. We report a wave-crest slowdown mechanism generic to unsteady propagating deep water wave groups. Our fully nonlinear computations show that just prior to reaching its maximum height, each wave crest slows down significantly and either breaks at this reduced speed, or accelerates forward unbroken. This finding is validated in our extensive laboratory and field observations. This behavior appears to be generic to unsteady dispersive wave groups in other natural systems.

Abstract:
The intrinsic fluctuations of the underlying, immutable quantum fields that fill all space and time can support the element of reality of a wave function in quantum mechanics. The mysterious non-locality of quantum entanglement may also be understood in terms of these inherent quantum fluctuations, ever-present at the most fundamental level of the universe.

Abstract:
Two wave fronts $W_1$ and $W_2$ that originated at some points of the manifold $M^n$ are said to be causally related if one of them passed through the origin of the other before the other appeared. We define the causality relation invariant $CR (W_1, W_2)$ to be the algebraic number of times the earlier born front passed through the origin of the other front before the other front appeared. Clearly, if $CR(W_1, W_2)\neq 0$, then $W_1$ and $W_2$ are causally related. If $CR(W_1, W_2)= 0$, then we generally can not make any conclusion about fronts being causally related. However we show that for front propagation given by a complete Riemannian metric of non-positive sectional curvature, $CR(W_1, W_2)\neq 0$ if and only if the two fronts are causally related. The models where the law of propagation is given by a metric of constant sectional curvature are the famous Friedmann Cosmology models. The classical linking number $lk$ is a $Z$-valued invariant of two zero homologous submanifolds. We construct the affine linking number generalization $AL$ of the $lk$ invariant to the case of linked $(n-1) $-spheres in the total space of the unit sphere tangent bundle $(STM)^{2n-1}\to M^n$. For all $M$, except of odd-dimensional rational homology spheres, $AL$ allows one to calculate the value of $CR(W_1, W_2)$ from the picture of the two wave fronts at a certain moment. This calculation is done without the knowledge of the front propagation law and of their points and times of birth. Moreover, in fact we even do not need to know the topology of $M$ outside of a part $\bar M$ of $M$ such that $W_1$ and $W_2$ are null-homotopic in $\bar M$.

Abstract:
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.

Abstract:
We present a novel, universal description of quantum entanglement using group theory and generalized characteristic functions. It leads to new reformulations of the separability problem, and the positivity of partial transpose (PPT) criterion. The latter turns out to be intimately related to a certain reality condition for group representations. Within our formalism, we also show a connection between existence of entanglement and group non-commutativity.

Abstract:
An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglement of two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.

Abstract:
Density functional theory (DFT) is shown to provide a novel conceptual and computational framework for entanglement in interacting many-body quantum systems. DFT can, in particular, shed light on the intriguing relationship between quantum phase transitions and entanglement. We use DFT concepts to express entanglement measures in terms of the first or second derivative of the ground state energy. We illustrate the versatility of the DFT approach via a variety of analytically solvable models. As a further application we discuss entanglement and quantum phase transitions in the case of mean field approximations for realistic models of many-body systems.

Abstract:
We introduce one-way LOCC protocols for quantum state merging for compound sources, which have asymptotically optimal entanglement as well as classical communication resource costs. For the arbitrarily varying quantum source (AVQS) model, we determine the one-way entanglement distillation capacity, where we utilize the robustification and elimination techniques, well-known from classical as well as quantum channel coding under assumption of arbitrarily varying noise. Investigating quantum state merging for AVQS, we demonstrate by example, that the usual robustification procedure leads to suboptimal resource costs in this case.

Abstract:
Traditional mean-field theory is a simple generic approach for understanding various phases. But that approach only applies to symmetry breaking states with short-range entanglement. In this paper, we describe a generic approach for studying 2D quantum phases with long-range entanglement (such as topological phases). Our approach is a variational method that uses tensor product states (also known as projected entangled pair states) as trial wave functions. We use a 2D real space RG algorithm to evaluate expectation values in these wave functions. We demonstrate our algorithm by studying several simple 2D quantum spin models.