The evolution of complex correlated quantum systems such as random circuit networks is governed by the dynamical buildup of both entanglement and entropy. We here introduce a real-time field theory approach—essentially a fusion of the G Σ functional of the SYK model and the field theory of disordered systems—engineered to microscopically describe the full range of such crossover dynamics: from initial product states to a maximum entropy ergodic state. To showcase this approach in the simplest nontrivial setting, we consider a tensor product of coupled random matrices, and compare to exact diagonalization. Published by the American Physical Society 2024
In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum $\bsA+\bsB$ of two independent random Hermitian matrices $\bsA$ and $\bsB$. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of
n random density matrices chosen from a random state ensemble (specified in the text) increases with the number
n. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components
n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.
Abstract In this paper, we present a general framework to solve a fundamental problem in random matrix theory (RMT), i.e. the problem of describing the joint distribution of eigenvalues of the sum of two independent random Hermitian matrices and . Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of the Duistermaat–Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n . Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.