Generalized entanglement as a framework for complex quantum systems: purity versus delocalization measures

Abstract

We establish contact between the delocalization properties of pure quantum states, as quantified by their number of principal components, and the average generalized entanglement properties, as quantified by purity measures relative to different observable sets. We find that correlations between products of state vector components with respect to Hamming distance play an important role in the structure of subsystem-based purity measures. In particular, we derive general conditions under which the amount of global multipartite entanglement relates to the inverse participation ratio averaged over a maximal set of mutually unbiased product bases. Furthermore, we provide a method for computing the expected amount of generalized entanglement with respect to an arbitrary observable set for random pure states. Specific examples and an explicit application to a disordered quantum spin chain are discussed.

Quantum chaos, delocalization, and entanglement in disordered Heisenberg models

“…Finally, from a technical standpoint, the general method suggested in [93] and explicitly illustrated here for computing typical entanglement properties of random pure states localized to a proper subspace of states in Hilbert space is likely to find broader applications within both quantum chaos and QIS.…”

“…Thanks to the average over many blocks, P n is less sensitive to edge effects. Apart from that, a main advantage with respect to von Neumann block entropy is its mathematical simplicity, which allows analytic calculations of the expected GE for random pure states to be and also reveals a quantitative relationship with delocalization as measured by NPC in a local basis [93]. In particular, for random pure states with purely real components, as expected for GOE eigenvector statistics, the average value P n with respect to the appropriate Haar measure may be exactly computed for arbitrary lattice size L. While we defer the details of the derivation to Appendix A, the results for states in the S z = 0-subspace with L = 12 are summarized in Table I.…”

Entanglement and Localization of Wavefunctions

“…It is interesting to compare this formula with a similar one obtained in [16] using different assumptions, in particular without average over random phases. The formula obtained relates entanglement to the mean inverse participation ratio calculated in three different bases, a quantity that is more general but often delicate to evaluate.…”

Quantum pseudorandomness from cluster-state quantum computation

“…Mn, 03.67.Lx, 03.67.Bg, Methods for characterizing and efficiently generating random quantum states and unitary operators have broad conceptual and practical significance across quantum physics. From a fundamental standpoint, a main motivation stems from the challenge of modeling complex quantum behavior, including quantum chaos [1] and typical entanglement in many-body systems [2,3,4,5,6,7,8]. Within quantum information science, states and unitaries sampled from the appropriate uniform (Haar) distribution provide the enabling resource in a growing number of algorithms and protocols.…”

“…For a random pure state, the expected value of Q is (Q) ≡ Q R = (2 n − 2)/(2 n + 1). Within the approach of generalized entanglement [22], Q is a representative of a class of quadratic measures of state delocalization and generalized purities [8], for which a similar analysis may be developed. Pseudo-Random Cluster-State Computation.− Cluster states are highly entangled states which serve as the basic resource for measurement-based QC [17].…”