Entanglement transitions in random definite particle states

TL;DR

This paper investigates how entanglement in definite particle states of qubits transitions from algebraic to exponential decay as the number of particles increases, revealing a phase change linked to the density of states.

Contribution

It provides a detailed analysis of entanglement decay transitions in definite particle states, including analytical calculations and statistical validation.

Findings

Entanglement decay shifts from algebraic to exponential with increasing particles.

Probability of non-zero concurrence matches statistical predictions.

Transition correlates with divergence in the density of states at zero eigenvalue.

Abstract

Entanglement within qubits are studied for the subspace of definite particle states or definite number of up spins. A transition from an algebraic decay of entanglement within two qubits with the total number $N$ of qubits, to an exponential one when the number of particles is increased from two to three is studied in detail. In particular the probability that the concurrence is non-zero is calculated using statistical methods and shown to agree with numerical simulations. Further entanglement within a block of $m$ qubits is studied using the log-negativity measure which indicates that a transition from algebraic to exponential decay occurs when the number of particles exceeds $m$. Several algebraic exponents for the decay of the log-negativity are analytically calculated. The transition is shown to be possibly connected with the changes in the density of states of the reduced density matrix, which has a divergence at the zero eigenvalue when the entanglement decays algebraically.