The First Law of Complexity

Abstract

We investigate the variation of holographic complexity for two nearby target states. Based on Nielsen's geometric approach, we find the variation only depends on the end point of the optimal trajectory, a result which we designate the first law of complexity. As an example, we examine the complexity=action conjecture when the AdS vacuum is perturbed by a scalar field excitation, which corresponds to a coherent state. Remarkably, the gravitational contributions completely cancel and the final variation reduces to a boundary term coming entirely from the scalar field action. Hence the null boundary of Wheeler-DeWitt patch appears to act like the "end of the quantum circuit".

Figures (2)

• Figure 1: The variation of the Nielsen circuit due to a perturbation ${\left\vert{\Psi_{\textrm{\tiny T}} +\delta\Psi}\right\rangle}$ of the target state ${\left\vert{\Psi_{\textrm{\tiny T}}}\right\rangle}$.
• Figure 2: The plot for $C_{j_1,j_2}$ with fixed $j_2$. Each curve has peaks at $j_1=1$ and $j_1=j_2$. The envelope of the latter is shown with the dashed gray line. Although we draw continuous curves to help guide the eye, one should only think of $j_1$ as taking integer values, i.e.,$j_1=0,1,2,\cdots$.