Circuit complexity of coupled four-mode squeezed states in two-field inflation

Abstract

We study the circuit complexity of coupled curvature and entropy perturbations in two-field inflation. We start from the quadratic action and derive the full Hamiltonian. The Hamiltonian includes all diagonal and off-diagonal quadratic terms. We then obtain three coupled equations for the pair amplitudes of the four-mode squeezed state. From these amplitudes, we calculate the two principal squeezing strengths. We use them to determine the Nielsen circuit complexity. For an equal-weight Gaussian circuit, the complexity takes the form $\mathcal{C}_{2,k}=\sqrt{2(r_{1k}^{2}+r_{2k}^{2})}$. It depends on the squeezing strengths but not explicitly on the squeezing angles. Our numerical results show that the complexity grows after horizon crossing in both coupled and uncoupled models. Interfield coupling changes the pair amplitudes and the four-mode squeezed state. However, its effect on the circuit complexity is small for the parameters considered. This result shows that different four-mode pair-amplitude matrices can yield similar values of the chosen equal-weight circuit complexity. The present calculation provides a reference for studying Gaussian circuit complexity in more general two-field inflationary models.