Download Entanglement Spectroscopy in Quantum Many

Ph.D. Project 2017-2020
Entanglement Spectroscopy in Quantum
Many-Body Systems
Project supervisor: Gabriele De Chiara
Contact: [email protected]
Background
Entanglement, discovered by E. Schrödinger in 1935, is
one of the peculiar phenomena predicted by quantum theory. Dubbed “spooky action at a distance” by A. Einstein,
it was only in the Nineties that the value of entanglement
as a resource was recognised after the discovery of quantum
information science. Entanglement is in fact at the basis
of many quantum information protocols: quantum teleportation, dense coding, cryptography. Although it is still an
open problem, entanglement is conjectured to play a role
in the speed-up of quantum algorithms, such as the famous
Shor’s algorithm for factoring integer numbers. Producing
and maintaining entanglement in quantum systems is therefore of paramount importance.
Fig. 1: Positrons annihilate with
In the early 2000’s, it was realised that many-body quan- electrons in atoms producing two detum systems, for example magnetic materials, when cooled tectable γ-rays whose energies are
to low temperatures are naturally endowed with entangle- Doppler shifted by an amount charment. Thus one can think of these systems as entanglement acteristic of the electron state inreservoirs from which entanglement can be extracted for free. volved. Positron annihilation is a
However when a system is close to a phase transition, sep- unique probe for material characterarating two macroscopically distinct configurations, all its isation, medical imaging and astroconstituents develop strong correlations that increase dra- physics.
matically its entanglement content.
This distinctive increase has, since then, been used, not only to detect the location of phase
transitions, but also the so-called universal scaling of thermodynamic properties such as the magnetisation of a quantum material. The word “universal” refers to the fact that distinct systems
exhibiting the same kind of transition show scaling close to the phase transition with the same
critical exponents. Entanglement, easy to compute numerically and amenable of experimental
measures, gives therefore precious insight of the properties of a quantum many-body system.
Objectives & Methodology
The goal of the project will be to analyse entanglement in one- and two-dimensional lattice
spin systems close to a quantum phase transition induced, for example, by an external magnetic
field. Entanglement has been normally characterised by a single number called the entanglement
entropy. This is the von Neumann entropy of the reduced density matrix of a subsystem. Here we
go beyond this scenario and look at the whole spectrum, called the entanglement spectrum, of
the reduced density matrix. Such entanglement spectroscopy is essential for the characterisation
of one-dimensional spin chains, the Kondo model, the quantum Hall effect and other topological
states of matter.
In this project we will tackle important open questions,
such as the effects of impurities and disorder on the entanglement spectrum. Another area of investigation will be
the time-dependent dynamics of entanglement spectrum for
which very little is known. The study of strongly-correlated
many-body systems can be developed using exact solutions
and numerical methods. In the former case, we will consider
models, including the one-dimensional XX-Ising chain and
the two-dimensional Kitaev model, that admit an analytical
solution. In the latter, we will employ state-of-the-art nu- Fig. 2: Positrons annihilate with
merical methods employing tensor network theory and ma- electrons in atoms producing two detrix product states. The student will be able to use existing tectable γ-rays whose energies are
codes in the group or to develop new codes. These are very Doppler shifted by an amount charpowerful techniques that can be easily transferred for future acteristic of the electron state involved. Positron annihilation is a
applications and research.
unique probe for material characterisation, medical imaging and astrophysics.
Required skills
An excellent knowledge of quantum theory and ideally of quantum information processing.
Knowledge of at least one programming language, e.g. Matlab, Fortran, C++, Python, Mathematica.
Further information
The student will be a member of the Quantum Technology group at Queen’s University Belfast
and will participate to its activities (group meetings, seminars, meetings with guest scientists)
and it is expected the occurrence of a wide participation with all the group members.
For further information, please contact Dr. G. De Chiara [email protected].
References
[1] G. De Chiara, L. Lepori, M. Lewenstein, A. Sanpera: Entanglement Spectrum, Critical
Exponents and Order Parameters in Quantum Spin Chains, Phys. Rev. Lett. 109, 237208
(2012), Available at this link: http://arxiv.org/pdf/1104.1331v4.pdf
[2] G. Torlai, L. Tagliacozzo, G. De Chiara: Dynamics of the entanglement spectrum in spin
chains, J. Stat. Mech. (2014) P06001, Available at this link: http://arxiv.org/pdf/1311.
5509.pdf
[3] F. Verstraete, V. Murg, J. I. Cirac: Matrix product states, projected entangled pair states,
and variational renormalization group methods for quantum spin systems, Advances in
Physics 57 (2): 143 (2008), Available at this link: http://arxiv.org/pdf/0907.2796.pdf
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