Quantum baker maps with controlled-not coupling

Abstract

The characteristic stretching and squeezing of chaotic motion is linearized within the finite number of phase space domains which subdivide a classical baker map. Tensor products of such maps are also chaotic, but a more interesting generalized baker map arises if the stacking orders for the factor maps are allowed to interact. These maps are readily quantized, in such a way that the stacking interaction is entirely attributed to primary qubits in each map, if each j'th subsystem has Hilbert space dimension Dj = 2 n j . We here study the particular example of two baker maps that interact via a controlled-not interaction, which is a universal gate for quantum computation. Numerical evidence indicates that the control subspace becomes an ideal Markovian environment for the target map in the limit of large Hilbert space dimension.

Quantum baker maps for spiraling chaotic motion

“…Section 3 then presents the loxodromic baker map and its quantization. It is shown in section 4 that, contrary to the generalized baker maps in [6], the quantum eigenvalue spectrum does not obey a generic random matrix ensemble. The reason for this failure of the BGS conjecture is shown to be that the π/2 rotation commutes with the component baker map.…”

“…The analysis of the loxodromic baker map is preceeded by a brief review of the ordinary two-dimensional baker map and its quantization. We point out the possibilities for different pilings that lead to the simplest couplings of baker maps [6]. Section 3 then presents the loxodromic baker map and its quantization.…”

Statistical bounds on the dynamical production of entanglement

“…However, there is a much simpler statistical approach, based on the assumption that a typical initial state submitted to a "chaotic" dynamics must eventually evolve into a random state, uniformly distributed on the sphere, as far as its average properties are concerned. This hypothesis was tested in several finite dimensional quantum maps, with a satisfactory quantitative agreement between theory and simulation [12,14,15,23,24].…”

“…where ρ [28]. For long times, and typical U and |ψ , the system comes into an equilibrium regime, where the linear entropy shows small fluctuations around a stationary average (see, e.g., [12,14,15,16,23]), given by…”

All Pure Two-Qudit Entangled States Can be Generated via a Universal Yang--Baxter Matrix Assisted by Local Unitary Transformations

“…(12) where E(|Ψ ) = 1 − Tr[ρ 2 A ], the overbar stands for the average over all the product states, and it can be simplified as e p (U) = (d/d + 1) 2 [E(U) + E(US) − E(S)], with S = d−1 i,j=0 |ij ij| is the permutation operator of two qudits. The entangling power has been useful for the study of quantum evolutions and Hamiltonians [19,20,21,22,23,24,25], and been also applied to some quantum chaotic systems [26,27,28,29]. Actually, E(|Ψ ) = 1 − Tr[ρ 2 A ] is the entanglement invariant I ′ 1 up to a normalized constant d/(d − 1).…”