Impact of local dynamics on entangling power

Abstract

It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate-typicality and study its properties. Analyzing multiple applications of any entangling operator interlaced with random local gates, we prove that both investigated quantities approach their asymptotic values in a simple exponential form. This rapid convergence to equilibrium, valid for subsystems of arbitrary size, is illustrated by studying multiple actions of diagonal unitary gates and controlled unitary gates.Introduction: The uniquely nonclassical phenomenon of entanglement is a well-known resource for quantum information [1]. It is increasingly used to characterize complex states, from many-body ground states [2], to infinite temperature quantum phase transitions such as the ergodic to localized phase in strongly interacting many-body systems [3]. Simple coupled quantum chaotic models have been studied [4,5] and experimentally realized [6,7] to demonstrate the large entanglement growth wherein the subsystems are nearly maximally mixed. In general the dynamics of entanglement in a non-equilibrium context is responsible for thermalization [8]. More recently in a different setting, black-holes are conjectured to scramble quantum information in a time that is logarithmic in the entropy via entanglement [9].While much work has centered on properties of states, quantum operators have also been studied as a physical resource for creating entanglement [10][11][12][13][14][15]. Studying directly the operators, such as propagators in time, frees us from the arbitrariness of initial states or the choice of eigenvectors. The entangling power [10,16] while referring to an inherent property of operators on a bipartite composite system is also related to how much state entanglement can be created, on the average, using one application of the unitary on product states. Investigations on quantum transport in light harvesting complexes [17], quantum chaos [18] and thermalization [19]

Bipartite unitary gates and billiard dynamics in the Weyl chamber

“…We have studied the time evolution of a two-qubit unitary gate initially in the Cartan form (2) showing that it corresponds to ergodic dynamics inside the 3D Weyl chamber, so that the space average over the set of all non-equivalent orbits coincides with the average over a single generic trajectory. Due to a non-intuitive influence of local transformations for nonlocality of iterated gates [15] this statement does not hold for generic random unitary gates of size four. However, a stronger property is true in the case of a large system size: randomization of nonlocality of bipartite N × N gates can be achieved by averaging over a single trajectory U t 0 stemming from a generic random gate U 0 , which asymptotically yields the same results as averaging over the ensemble of Haar random matrices from U(N 2 ).…”

“…To demonstrate influence of local unitaries on the nonlocality of a sequence of bipartite gates [15] we compare the information content α(V t ) of iterated generic gate V in the Cartan form with the content α(U t ) of a locally equivalent gate U = VY loc . As the trajectory α(V t ) corresponds to a billiard dynamics in the tetrahedron, the evolution of α[(VY loc ) t ] can be related to a dynamics of a billiard with a noise which enhances the diffusion in the tetrahedron.…”

Entangling power of time-evolution operators in integrable and nonintegrable many-body systems

“…Thus although we can interpret the results elegantly in terms of entanglement between Majorana fermions of two species (those in A and B), we proceed with the spin language as it provides the Schmidt decompositions of operators written with spin variables. All the measures used in this work, E l,vN (U), E l,vN (US), ep l,vN (U) are local operator invariants( [30]), that is they are the same as for (U A ⊗U B )U (U A ⊗U B ). They can hence be obtained by just considering the nonlocal part of U n ≡ (U n ) nl .…”

“…Then E l,vN (U) (E l,vN (US)) are the linear and von Neumann entropies of the reduced state ρ AA = Tr BB (|Φ U Φ U |) (ρ AB = Tr A B (|Φ U Φ U |)). These reduced states can also be related to reshuffling and partial transpose of U, allowing for their direct evaluation [30]. The central difference therefore is that operator entanglement can be viewed as the entanglement in one particular 4-party state engendered by the action of a bipartite unitary operator, while the entangling power is the average entanglement in an ensemble of states resulting from its action on all 2-party product states.…”

“…(5) is of the form U(τ, h) = (U A ⊗U B )U AB (τ) where U A,B are local to blocks A and B consisting of spins 1, · · · , L/2 and L/2 + 1, · · · , L respectively and U AB (τ) = exp{−iτσ z L/2 σ z L/2+1 } is the nonlocal interaction between the blocks. While the local operators, which contain all the information of the fields, do not contribute to the entangling power of U itself, they play a crucial role for the powers U n we are interested in [30]. The RMT model U RMT (τ) is a hybrid one wherein we replace U A,B by local random unitary matrices and retain the interaction as is.…”