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.

We define a coupling of two baker maps through a π/2 rotation both in position and in momentum. The classical trajectories thus exhibit spiraling, or loxodromic motion, which is only possible for conservative maps of at least two degrees of freedom. This loxodromic baker map is still hyperbolic, that is, fully chaotic. Quantization of this map follows on similar lines to other generalized baker maps. It is found that the eigenvalue spectrum for quantum loxodromic baker map is far removed from those of the canonical random matrix ensembles. An investigation of the symmetries of the loxodromic baker map reveals the cause of this deviation from the Bohigas-Giannoni-Schmit conjecture.