Statistical bounds on the dynamical production of entanglement

We present a random-matrix analysis of the entangling power of a unitary operator as a function of the number of times it is iterated. We consider unitaries belonging to the circular ensembles of random matrices [the circular unitary (CUE) or circular orthogonal ensemble] applied to random (real or complex) nonentangled states. We verify numerically that the average entangling power is a monotonically decreasing function of time. The same behavior is observed for the “operator entanglement”—an alternative measure of the entangling strength of a unitary operator. On the analytical side we calculate the CUE operator entanglement and asymptotic values for the entangling power. We also provide a theoretical explanation of the time dependence in the CUE cases.

Figure 1

(Color online) (a) Linear entropy of states evolved from a fixed initial separable state chosen arbitrarily and then propagated by $U^n$, with $U$ belonging to the CUE averaged over ${10}^8$ CUE matrices. (b) Ensemble of ${10}^3$ random, complex, separable initial states propagated with ${10}^5$ COE matrices. Shown is the linear entropy averaged over matrices and states. (c) As in (a) but for ${10}^8$ COE matrices and one real state. In all cases subsystem dimensions are $d_A=4$ and $d_B=5$. Statistical error bars are smaller than symbol diameters. Horizontal lines correspond to analytical predictions; see Sec. 4.

Figure 2

(Color online) Linear entropy of the random operator $U^n$ averaged over ${10}^7$ realizations. $U$ belongs to (a) the CUE, (b) the COE. In both cases subsystem dimensions are $d_A=4$ and $d_B=5$. Statistical error bars are smaller than symbol diameters. The horizontal line corresponds to the analytical prediction; see Sec. 4.