Entropy and singular-value moments of products of truncated random unitary matrices

Abstract

Products of truncated unitary matrices, independently and uniformly drawn from the unitary group, can be used to study universal aspects of monitored quantum circuits. The von Neumann entropy of the corresponding density matrix decreases with increasing length $L$ of the product chain, in a way that depends on the matrix dimension $N$ and the truncation depth $\delta N$. Here we study that dependence in the double-scaling limit $L,N\to \infty$, at fixed ratio $\tau =L\delta N/N$. The entropy reduction crosses over from a linear to a logarithmic dependence on $\tau$ when this parameter crosses unity. The central technical result is an expression for the singular-value moments of the matrix product in terms of the Erlang function from queueing theory.

arXiv comment6 pages, 4 figures

References in corpus (17)

• Measurement-Induced Phase Transitions in the Dynamics of Entanglement
• Quantum Zeno Effect and the Many-body Entanglement Transition
• Unitary-projective entanglement dynamics
• Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition
• Dynamical purification phase transitions induced by quantum measurements
• Entanglement transition from variable-strength weak measurements
• Truncations of random unitary matrices
• Recent exact and asymptotic results for products of independent random matrices
• Systematic Analytical Approach to Correlation Functions of Resonances in Quantum Chaotic Scattering
• Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
• Singular value statistics of matrix products with truncated unitary matrices
• Universal microscopic correlation functions for products of truncated unitary matrices
• Determinantal point processes in the plane from products of random matrices
• Random-matrix models of monitored quantum circuits
• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• A Dyson Brownian motion model for weak measurements in chaotic quantum systems
• A Matrix Contraction Process