Dual unitaries as maximizers of the distance to local product gates

Abstract

TThe problem of finding the resource free, closest local unitary, to any bipartite unitary gate $U$ is addressed. Previously discussed as a measure of nonlocality, the distance $K_D(U)$ to the nearest product unitary has implications for circuit complexity and related quantities. Dual unitaries, currently of great interest in models of complex quantum many-body systems, are shown to have a preferred role as these are maximally and equally away from the set of local unitaries. This is proved here for the case of qubits and we present strong numerical and analytical evidence that it is true in general. An analytical evaluation of $K_D(U)$ is presented for general two-qubit gates. For arbitrary local dimensions, that $K_D(U)$ is largest for dual unitaries, is substantiated by its analytical evaluations for an important family of dual-unitary and for certain non-dual gates. A closely allied result concerns, for any bipartite unitary, the existence of a pair of maximally entangled states that it connects. We give efficient numerical algorithms to find such states and to find $K_D(U)$ in general.

Figures (5)

• Figure 1: A caricature of the geometry of $K_D(U)$. For a given bipartite unitary operator, the projection to the subset of local unitary products is found, and the distance is calculated.
• Figure 2: The distance to local gates, $K_D(U)$, plotted as a color map in the space of entangling properties of the gate. We observe that the highest values are attained on the dual-unitary boundary line.
• Figure 3: UBB demonstrated for a sample of random bipartite unitaries $V$ with $d=3$. The inset shows the approach to the maximum entropy $2/3$ via the difference $\Delta_n=2/3-(1-\tr\rho_n^2)$.
• Figure 4: $K_D(U)$ obtained from the algorithm vs the corresponding analytical lower bound $K_D^*(U)$. This is displayed for random unitaries from CUE of size $d^2=9$, and random unitaries in the neighborhood of dual unitaries (also magnified in the inset), number of samples taken in each case is $10^3$. The horizontal solid line is at $K_D^2(U)=K_D^{*2}=12$, and the dashed line is the equality $K_D^2(U)=K_D^{*2}$.
• Figure 5: $K_D(U)$ obtained from the algorithm vs the corresponding analytical lower bound $K_D^*(U)$. This is displayed for random diagonal unitaries of size $d^2=9$, and random unitaries in the neighborhood of $\mathcal{D}_H$ (also magnified in the inset), number of samples taken in each case is $10^3$. The horizontal solid line is at $K_D^{2}(\mathcal{D}_H)=18-9\sqrt{3}\approx 7.6$, and the dashed line is the equality $K_D^2(U)=K_D^{*2}$.