Bipartite entanglement of large quantum systems

TL;DR

This thesis applies classical statistical mechanics to analyze bipartite entanglement in large quantum systems, revealing phase transitions and metastable states through a novel canonical approach based on the distribution of Schmidt coefficients.

Contribution

It introduces a canonical statistical mechanics framework to study bipartite entanglement, characterizing purity distributions and phase transitions for large quantum systems.

Findings

Identified phase transitions and metastable states in entanglement distributions.

Derived exact expressions for moments of local purity in pure and mixed states.

Connected entanglement properties with quantum channel symmetries and twirling transformations.

Abstract

In this thesis we study the behavior of bipartite entanglement of a large quantum system, by analyzing the distribution of the Schmidt coefficients of the reduced density matrix. Applying the general methods of classical statistical mechanics, we develop a canonical approach for the study of the distribution of these coefficients for a fixed value of the average entanglement. We introduce a partition function depending on a fictitious temperature, which localizes the measure on the set of states with higher and lower entanglement, if compared to typical (random) states with respect to the Haar measure. The purity of one subsystem, which is our entanglement measure/indicator, plays the role of energy in the partition function. This thesis consists of two parts. In the first part, we completely characterize the distribution of the purity and of the eigenvalues for pure states. The global picture unveils several locally stable solutions exchange stabilities, through the presence of first and second order phase transitions. We also detect the presence of metastable states. In the second part, we focus on mixed states. Through the same statistical approach, we determine the exact expression of the first two moments of the local purity and the high temperature expansion of the first moment. We also bridge our problem with the theory of quantum channels, more precisely we exploit the symmetry properties of the twirling transformations in order to compute the exact expression for the first moment of the local purity.