Does bound entanglement naturally appear in quantum many-body systems? We address this question by showing the existence of bound-entangled thermal states for harmonic oscillator systems consisting of an arbitrary number of particles. By explicit calculations of the negativity for different partitions, we find a range of temperatures for which no entanglement can be distilled by means of local operations, despite the system being globally entangled. We offer an interpretation of this result in terms of entanglement-area laws, typical of these systems. Finally, we discuss generalizations of this result to other systems, including spin chains.
We address the presence of non-distillable (bound) entanglement in natural many-body systems. In particular, we consider standard harmonic and spin-1/2 chains, at thermal equilibrium and characterized by few interaction parameters. The existence of bound entanglement is addressed by calculating explicitly the negativity of entanglement for different partitions. This allows to individuate a range of temperatures for which no entanglement can be distilled by means of local operations, despite the system being globally entangled. We discuss how the appearance of bound entanglement can be linked to entanglement-area laws, typical of these systems. Various types of interactions and topologies are explored, showing that the presence of bound entanglement is an intrinsic feature of these systems. In the harmonic case, we analytically prove that thermal bound entanglement persists for systems composed by an arbitrary number of particles. Our results strongly suggest the existence of bound entangled states in the macroscopic limit also for spin-1/2 systems.
Entanglement is a ubiquitous feature of low temperature systems and believed to be highly relevant for the dynamics of condensed matter properties and quantum computation even at higher temperatures. The experimental certification of this paradigmatic quantum effect in macroscopic high temperature systems is constrained by the limited access to the quantum state of the system. In this paper we show how macroscopic observables beyond the mean energy of the system can be exploited as proxy witnesses for entanglement detection. Using linear and semi-definite relaxations we show that all previous approaches to this problem can be outperformed by our proxies, i.e. entanglement can be certified at higher temperatures without access to any local observable. For an efficient computation of proxy witnesses one can resort to a generalised grand canonical ensemble, enabling entanglement certification even in complex systems with macroscopic particle numbers.
Why do we not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic. The coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty of observeing a large-scale entanglement. As the number of particles
n grows we need an ever more precise knowledge of the state and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness
W distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness
W to satisfy $\ensuremath{\mid}\mathrm{tr}(W\ensuremath{\rho})\ensuremath{\mid}\ensuremath{\leqslant}1$ for all separable states $\ensuremath{\rho}$, then the efficiency of
W depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm $\ensuremath{\parallel}W\ensuremath{\parallel}$. It is known that there are witnesses on the space of
n qubits for which $\ensuremath{\parallel}W\ensuremath{\parallel}$ is exponential in
n. However, we conjecture that for a large majority of
n-qubit witnesses $\ensuremath{\parallel}W\ensuremath{\parallel}\ensuremath{\leqslant}O(\sqrt{n\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}n})$. Thus, in a nonideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf [Phys. Rev. A 64, 032112 (2001)].
A sufficient condition for entanglement in two-mode continuous systems is constructed with the interference visibility and the uncertainty of the total particle number. The observables to be measured (particle numbers and particle number variances) are relatively easily accessible experimentally. The method may be used to detect entanglement in light fields or in Bose-Einstein condensates. In contrast to the standard approach based on entanglement witnesses, the condition is expressed in terms of an inequality which is nonlinear in expectation values. The condition is constructed using uncertainty relations with the particle number and the destruction operators.
[EN] In this Bachelor Thesis, I present some of the fundaments of entanglement detection. I introduce basic entanglement criteria namely the PPT and CCNR criteria. Entanglement witnesses are defined and used in illustrative examples. I also explain Bell inequalities and their relation with other entanglement witnesses. Lastly, in the last section of the theoretical background I introduce some collective observables and show how we can use them to detect entanglement in our experiments. Once the basics have been explained, I present a new criterion that gives us a lower bound in the amount of entanglement particles of a system. This caracterizes the quantum system from a new perspective and can be of great interest in modern multipartite entanglement experiments. Some results will be used to validate our approach, demostrating that it complements previous methods.