R. Tarrach

Robustness of entanglement

In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurance of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of two-party systems, and analytical bounds for mixed states of two-party systems. Specific results are obtained mainly for the qubit-qubit system (qubit denotes quantum bit). As by-products local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced, and arguments for considering convexity a necessary condition of any entanglement measure are put forward.

Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States

We prove for any pure three-quantum-bit state the existence of local bases which allow one to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the three-quantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties.

Local description of quantum inseparability

We show how to decompose any density matrix of the simplest binary composite systems, whether separable or not, in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a decomposition. Separable states correspond to mixing from one to four pure product states. Inseparable states can be described as pseudomixtures of four or five pure product states, and can be made separable by mixing them with one or two pure product states.

Higher dimensional renormalization group invariant vacuum condensates in quantum chromodynamics

The renormalization of the five- and six-dimensional scalar gauge invariant composite operators is performed. The corresponding renormalization group invariant vacuum condensates are studied. Our main results are that the six-dimensional gluon condensate does not mix with the quartic quark condensates nor the other way round. We also show that the factorization hypothesis of the four-quark operator does not lead to renormalization group invariant quark vacuum condensates. So, one has to be careful when keeping the six-dimensional operators in the QCD sum rule analysis.

Speed of light in non-trivial vacua

The usual vacuum in a Quantum Field Theory is a state characterized by the absence of real particles and classical fields and by its Minkowskian geometry. Electromagnetic and gravitational fields as well as massless particles propagate though it with the same constant, Lorentz invariant speed, c. When the vacuum is modified, so is the speed of propagation of particles and fields. This quantum field theoretical effect has been analyzed within QED for low—energy photons in several cases, resulting in a universal formula 6

BARYON MASSES AND CHIRAL SYMMETRY BREAKING

v = 1 - \frac{{44}}{{135}}{\alpha ^2}\frac{\rho}{{m_e^4}}

Three-qubit pure-state canonical forms

where ρ is the energy density relative to the standard vacuum and where, if the vacuum is a gravitational one, one α has to be substituted by \( m_e^2{G_N} \) . It follows automatically that if the vacuum has a lower energy density than the standard vacuum, ρ 1, and viceversa. Therefore, whether photons move faster or slower than c depends on the lower or higher energy density of the modified vacuum, respectively. Physically, a higher energy density is characterized by the presence of real particles in the vacuum whereas a lower one stems from the absence of some virtual modes.

Perturbative renormalization in quantum mechanics

Abstract Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like quantum gravity, string theories, etc. concern quantum theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in particular. Two of these problems will be discussed in this article: (a) the separability problem, i.e. the question when the state of a composite quantum system does not contain any quantum correlations or entanglement; and (b) the distillability problem, i.e. the question when the state of a composite quantum system can be transformed to an entangled pure state using local operations (local refers here to component subsystems of a given system). Although many results concerning the above mentioned problems have been obtained (in particular in the last few years in the framework of quantum information theory), both problems remain until now essentially open. We will present a primer on the current state of knowledge concerning these problems, and discuss the relation of these problems to one of the most challenging questions of linear algebra: the classification and characterization of positive operator maps.