arXiv:0712.0015 · Physical Review Letters

Phase Transitions of Bipartite Entanglement

Paolo Facchi A, Ugo Marzolino, Giorgio Parisi A, Saverio Pascazio, Antonello Scardicchio

Published Jul 30, 2008• 92 citations

Abstract

We analyze the statistical properties of the entanglement of a large bipartite quantum system. By framing the problem in terms of random matrices and a fictitious temperature, we unveil the existence of two phase transitions, characterized by different spectra of the reduced density matrices.

DOI: 10.1103/physrevlett.101.050502

Related papers

Phase transitions of bipartite entanglement

Paolo Facchi A, Ugo Marzolino, Giorgio Parisi A +2 more

We study a random matrix model for the statistical properties of the purity of a bipartite quantum system. This enables us to write the generating function for the cumulants, for both balanced and unbalanced bipartitions. It also unveils an unexpected feature of the system, namely the existence of two phase transitions.

International Conference on Quantum InformationJan 1, 20088 citations

Phase transitions and metastability in the distribution of the bipartite entanglement of a large quantum system

Antonella De Pasquale, Paolo Facchi A, Giorgio Parisi A +2 more

We study the distribution of the Schmidt coefficients of the reduced density matrix of a quantum system in a pure state. By applying general methods of statistical mechanics, we introduce a fictitious temperature and a partition function and translate the problem in terms of the distribution of the eigenvalues of random matrices. We investigate the appearance of two phase transitions, one at a positive temperature, associated with very entangled states, and one at a negative temperature, signaling the appearance of a significant factorization in the many-body wave function. We also focus on the presence of metastable states (related to two-dimensional quantum gravity) and study the finite size corrections to the saddle point solution.

arXiv:0911.3888Physical Review AMay 19, 201052 citations

Entropy-driven phase transitions of entanglement

Paolo Facchi A, Giuseppe Florio, Giorgio Parisi A +2 more

We study the behavior of bipartite entanglement at fixed von Neumann entropy. We look at the distribution of the entanglement spectrum, that is, the eigenvalues of the reduced density matrix of a quantum system in a pure state. We report the presence of two continuous phase transitions, characterized by different entanglement spectra, which are deformations of classical eigenvalue distributions.

arXiv:1302.3383Physical Review AMay 28, 201328 citations

Quantum Phase Transitions and Bipartite Entanglement

Lian-Ao Wu, M. S. Sarandy, Daniel A. Lidar

We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and bipartite entanglement. We derive a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of d-level systems. The ground state energy eigenvalue and its derivatives, whose nonanalyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signaled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.

arXiv:quant-ph/0407056Physical Review LettersDec 15, 2004484 citations

Phase diagram of bipartite entanglement

Paolo Facchi A, Giorgio Parisi A, Saverio Pascazio +2 more

Abstract We investigate the features of the entanglement spectrum (distribution of the eigenvalues of the reduced density matrix) of a large quantum system in a pure state. We consider all Rényi entropies and recover purity and von Neumann entropy as particular cases. We construct the phase diagram of the theory and unveil the presence of two critical lines.

arXiv:1906.01269Journal of Physics A Mathematical and TheoreticalAug 30, 20196 citations

Bipartite entanglement of large quantum systems

Antonella De Pasquale

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.

arXiv:1206.6749Jun 28, 20125 citations