Creating Ensembles of Dual Unitary and Maximally Entangling Quantum Evolutions

Maximally entangled bipartite unitary operators or gates find various applications from quantum information to many-body physics wherein they are building blocks of minimal models of quantum chaos. In the latter case, they are referred to as “dual unitaries.” Dual unitary operators that can create the maximum average entanglement when acting on product states have to satisfy additional constraints. These have been called “2-unitaries” and are examples of perfect tensors that can be used to construct absolutely maximally entangled states of four parties. Hitherto, no systematic method exists in any local dimension, which results in the formation of such special classes of unitary operators. We outline an iterative protocol, a nonlinear map on the space of unitary operators, that creates ensembles whose members are arbitrarily close to being dual unitaries. For qutrits and ququads we find that a slightly modified protocol yields a plethora of 2-unitaries.

Figure 1

Evolution of the operator entanglement $E(U_n)$ under the $M_R$ map for local dimensions $d=2$ (left panel) and $d=3$ (right panel) for two random initial operators $U_0$ in each case. The $E(U_n)$ saturates to the corresponding maximum possible value $(d^2-1)/d^2$, clearly distinguishable from the average value, $\overline{E(U)}=(d^2-1)/(d^2+1)$ indicated in the boxes (see [5] for a derivation of the average value).

Figure 2

The distribution of the entangling power $e_p(U)$ for both the CUE ensemble and the one obtained by iterating it with the $M_R$ map $n$ times, indicated as the “dual-CUE” ensemble. The inset shows the distribution of the operator entanglements $E(U)$, notice the different scales on the vertical axes, left for the CUE and right for the dual CUE. Here $n=10$, 10, 30, 50 for $d=2$, 3, 4, 5, respectively, and 10 000 realizations from the CUE are chosen in each case for the seed unitaries.

Figure 3

Trajectories under the nonlinear $M_R$ map in half of the Weyl chamber of 4 representative initial unitaries as they limit to the edge filled with dual unitary operators. All operators starting from the base reach the dual dcnot gate in one iteration.

Figure 4

The distribution of entangling power $e_p(U_n)$ under the $M_{TR}$ map, after $n=2000$ iterations and 10000 realizations. For the cases of $d=3$, 4 2-unitaries or perfect tensors achieving the maximum value of $e_p$ are realized. The insets show the distributions of $e_p(U)$ for the corresponding CUE for comparisons.