Alladi Ramakrishnan Hall

Entanglement and its role in efficient representation of quantum states

Sudipto Singha Roy

IIT--Dhanbad

Entanglement is one of the most important characteristic traits of the quantum world, and
constitutes one of the main resources in quantum technologies. Over the past few decades, it has
also received much attention in other areas of physics, e.g. quantum many-body systems, highenergy physics, etc. In this talk, I will present an aspect of quantum entanglement, namely, its role
in efficient representation of a generic quantum state. One of the biggest obstacles for studying
physical properties of quantum systems comprising many quantum particles is the curse of
dimensionality - the exponential growth of the dimension of the Hilbert space of quantum states
with increasing number of parties. In this regard, I will first argue that often the knowledge of the
distribution of entanglement in the system state can be the savior. To make the connection clearer, I
will introduce the formalism of tensor network theory which has turned out to be a state-of-the-art
technique to represent quantum states, efficiently, up to considerably large system sizes. With this
background, I will present one of the directions that we are exploring along this line. It is the
formalism that we have introduced in our recent works, namely, the “link-representation” formalism
of entanglement. I will discuss how this formalism helps us obtain a more fine-grained structure of
entanglement and its significance in designing more efficient tensor network states. Next, I will
present examples where the representation is exact along with the cases where it deviates
significantly. Finally, I will conclude the discussion with an overview of the work that we are
currently doing along this line comprising physical systems having more complex structures and in
higher dimensions.
References:
[1] R. Horodecki, P. Horodecki, M.Horodecki, and K. Horodecki, “Quantum entanglement”, Rev. Mod.
Phys. 81, 865 (2009).
[2] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, “Entanglement in many-body systems”, Rev. Mod. Phys.
80, 517 (2008).
[3] R. Orús, “A practical introduction to tensor networks: Matrix product states and projected entangled pair
states”. Annals of Physics 349 117 (2014).
[4] S.Singha Roy, S. N. Santalla, J. Rodríguez-Laguna, and G. Sierra, “Entanglement as geometry and flow”,
Physical Review B 101, 195134 (2020).
[5] S. Singha Roy, S. N.Santalla, G. Sierra, and J. Rodríguez-Laguna, “Link representation of the
entanglement entropies for all bipartitions”, Journal of Physics A: Mathematical and Theoretical 54, 305301
(2021).
[6] S. Singha Roy, G. Ramírez, S. N. Santalla, G. Sierra, and J.r Rodríguez-Laguna, “Exotic correlation
spread in free-fermionic states with initial patterns”, Physical Review B 105, 214306 (2022).
[7] S. N. Santalla, G. Ramírez, S. Singha Roy, G. Sierra, and J. Rodríguez-Laguna, “Entanglement links and
the quasiparticle picture”, Physical Review B (Letter) 107, L121114 (2023)