We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signaling function of quantum order-disorder transitions. We discuss the possibility for such transitions to characterize interesting physical phenomena, as quantum phase transitions, or abrupt variations in correlation distributions. We apply our measure on two exactly solvable Hamiltonian models: the 1D-Quantum Ising Model (in the single-particle reduced state), and on Heisenberg XXZ spin-1/2 chain (in the two-particle reduced state). We analyze its behavior across quantum phase transitions for finite system sizes, as well as in the thermodynamic limit by using Bethe Ansatz technique.
We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signaling function of quantum order-disorder transitions. We discuss the possibility for such transitions to characterize interesting physical phenomena, as quantum phase transitions, or abrupt variations in correlation distributions. We apply our measure on two exactly solvable Hamiltonian models: the 1D-Quantum Ising Model (in the single-particle reduced state), and on Heisenberg XXZ spin-1/2 chain (in the two-particle reduced state). We analyze its behavior across quantum phase transitions for finite system sizes, as well as in the thermodynamic limit by using Bethe Ansatz technique.
Statistical complexity measures are distinctive tools that can detect significant physical phenomena. Many of these phenomena occur in such a way that they leave signatures that can be interpreted as changes in the state of the system, which can be read as statistical complexity through these measures, measured on a scale of order and disorder. We investigate the classical statistical complexity measure (CSCM) defined by López-Ruiz et al. and apply it to a simple model of discrete time evolution, characterized by finite Markov chains in dimension 2, for stochastic and bistochastic evolutions. Additionally, we introduce a quantum version for measuring statistical complexity: the quantum statistical complexity measure (QSCM), in the context of quantum information theory, and use it as a signaling function for quantum correlation transitions, measured on a scale of order and disorder. We consider the possibility that such transitions characterize intriguing physical phenomena, such as quantum phase transitions or sudden variations in correlation distributions. Our measure is applied to two analytically solvable Hamiltonian models: the one-dimensional quantum Ising model in the reduced single-particle state, and the Heisenberg chain XXZ of spin-1/2 in the state reduced by two particles. We analyze the measurement behavior in quantum phase transitions for finite system sizes and in the thermodynamic limit using the Bethe ansatz technique. We conclude that the quantum statistical complexity measure can serve as a signaling function of correlation transitions and distinguish quantum phase transitions, due to the abruptness that physical quantities experience at these transition points and the way in which the reduced states of these critical systems change within a scale of order and disorder.
We explore quantum and classical correlations along with coherence in the ground states of spin-1 Heisenberg chains, namely the one-dimensional XXZ model and the one-dimensional bilinear biquadratic model, with the techniques of density matrix renormalization group theory. Exploiting the tools of quantum information theory, that is, by studying quantum discord, quantum mutual information, and three recently introduced coherence measures in the reduced density matrix of two nearest neighbor spins in the bulk, we investigate the quantum phase transitions and special symmetry points in these models. We point out the relative strengths and weaknesses of correlation and coherence measures as figures of merit to witness the quantum phase transitions and symmetry points in the considered spin-1 Heisenberg chains. In particular, we demonstrate that, as none of the studied measures can detect the infinite-order Kosterlitz-Thouless transition in the XXZ model, they appear to be able to signal the existence of the same type of transition in the biliear biquadratic model. However, we argue that what is actually detected by the measures here is the SU(3) symmetry point of the model rather than the infinite-order quantum phase transition. Moreover, we show in the XXZ model that examining even single site coherence can be sufficient to spotlight the second-order phase transition and the SU(2) symmetry point.
The aim of this work is to develop a technique for identifying quantum phase transitions which does not rely on the existence of a thermodynamic limit, for studies in finite systems. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-particle Schrodinger operator. Bifurcations of the minima for the potential of the Schrodinger operator determine critical ground-state couplings. By considering the behaviour of certain ground-state correlation functions, these may be identified as quantum phase transitions in the many-body integrable system with finite particle number. We study two particular examples of bosonic Hamiltonians which admit second-order transitions, and discuss further applications.
Characterizing and quantifying quantum correlations in states of many-particle systems is at the core of a full understanding of phase transitions in matter. In this work, we continue our investigation of the notion of generalized entanglement [Barnum et al., Phys. Rev. A 68, 032308 (2003)] by focusing on a simple Lie-algebraic measure of purity of a quantum state relative to an observable set. For the algebra of local observables on multi-qubit systems, the resulting local purity measure is equivalent to a recently introduced global entanglement measure [Meyer and Wallach, J. Math. Phys. 43, 4273 (2002)]. In the condensed-matter setting, the notion of Lie-algebraic purity is exploited to identify and characterize the quantum phase transitions present in two exactly solvable models, namely the Lipkin-Meshkov-Glick model and the spin-21 anisotropic XY model in a transverse magnetic field. For the latter, we argue that a natural fermionic observable set arising after the Jordan-Wigner transformation better characterizes the transition than alternative measures based on qubits. This illustrates the usefulness of going beyond the standard subsystem-based framework while providing a global disorder parameter for this model. Our results show how generalized entanglement leads to useful tools for distinguishing between the ordered and disordered phases in the case of broken symmetry quantum phase transitions. Additional implications and possible extensions of concepts to other systems of interest in condensed-matter physics are also discussed.
We apply the generalized Wigner function formalism to detect and characterize a range of quantum phase transitions in several cyclic, finite-length, spin-12 one-dimensional spin-chain models, viz., the Ising and anisotropic XY models in a transverse field, and the XXZ anisotropic Heisenberg model. We make use of the finite system size to provide an exhaustive exploration of each system’s single-site, bipartite and multi-partite correlation functions. In turn, we are able to demonstrate the utility of phase-space techniques in detecting and characterizing first-, second- and infinite-order quantum phase transitions, while also enabling an in-depth analysis of the correlations present within critical systems. We also highlight the method’s ability to capture other features of spin systems such as ground-state factorization and critical system scaling. Finally, we demonstrate the generalized Wigner function’s utility for state verification by determining the state of each system and their constituent sub-systems at points of interest across the quantum phase transitions, enabling interesting features of critical systems to be readily analyzed.
We quantify the emergent complexity of quantum states near quantum critical points on regular 1D lattices, via complex network measures based on quantum mutual information as the adjacency matrix, in direct analogy to quantifying the complexity of electroencephalogram or functional magnetic resonance imaging measurements of the brain. Using matrix product state methods, we show that network density, clustering, disparity, and Pearson's correlation obtain the critical point for both quantum Ising and Bose-Hubbard models to a high degree of accuracy in finite-size scaling for three classes of quantum phase transitions, Z_{2}, mean field superfluid to Mott insulator, and a Berzinskii-Kosterlitz-Thouless crossover.