Entanglement transitions in random definite particle states

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.Comment: Substantially added content (now 24 pages, 5 figures) with a discussion of the possible mechanism for the transition. One additional author in this version that is accepted for publication in Phys. Rev.

Entanglement between two subsystems, the Wigner semicircle and extreme-value statistics

“…(2), and for L 1 + L 2 > L/2, the density matrix is rank deficient.Whereas the critical case as far as this transition goes is at L 1 + L 2 = L/2 − 1 when the density of states of ρ 12 does not diverge at zero. While the rank of the density matrix mattered in the case of an entanglement transition observed for definite-particle states recently [46], it seems to be not exactly the case here, as there is a case when the density of states of ρ 12 is bounded away from zero, but its partial transpose has a significant measure of negative eigenvalues and is predominantly NPT.…”

Signatures of bifurcation on quantum correlations: Case of the quantum kicked top

“…Another example is that of definite particle states. This shows algebraic to exponential decay of entanglement when the number of particles exceed the size of two subsystems [63]. For both these cases, discord and geometric discord between two qubits in a tripartite system goes to zero as the size of the third subsystem is increased.…”

Persistent entanglement in a class of eigenstates of quantum Heisenberg spin glasses

“…This follows since a i ∼ 1/ √ L. As v = O(1), the concurrence (which is proportional to |z| − √ vy) in promoted two-particle states is always in a fine balance between the two competing terms |z| and √ y. In contrast, for generic two-particle states, the order of √ y is 1/L which is much larger than the order of |z| which is 1/L 3/2 , resulting in the probability of nonzero concurrence scaling as 1/ √ L [17]. The probability of finding any two spins entangled when the system is in a promoted two-particle state, i.e., P (C > 0) is now estimated.…”

“…For generic random two-particle states, v = O(1), y ∼ 1/L 2 , |z| 2 ∼ 4/L 3 and |z| 2 = (2/π) |z| 2 . Thus, the negative term in the pre-concurrence 2(|z| − √ vy) is typically larger than the positive term, thus resulting in the probability of a positive concurrence decreasing with increasing L as 1/ √ L (see [17] for calculations). However, for two-particle states promoted from oneparticle states obeying i a i = 0, it is straightforward to show using a ij = (a i + a j )/ √ L − 2 that, up to the leading order, y ≈ (a 1 +a 2 ) 2 /L, z ≈ (1+La 1 a 2 )/L, v ≈ 1.…”

“…To study this in the simplest statistical context, random states of definite particle-number were considered using an ensemble that was uniformly distributed in such subspaces [17]. It was found that while the expected entanglement between two spins for one-particle states having L spins scales as 1/L and that of two-particle states scale as 1/L 2 , in the case of three or more particle states, entanglement is practically absent and is exponentially small in L (exp(−L ln L), to be precise).…”