Probing the entanglement of operator growth

Abstract

In this work we probe the operator growth for systems with Lie symmetry using tools from quantum information. Namely, we investigate the Krylov complexity, entanglement negativity, von Neumann entropy and capacity of entanglement for systems with SU(1,1) and SU(2) symmetry. Our main tools are two-mode coherent states, whose properties allow us to study the operator growth and its entanglement structure for any system in a discrete series representation of the groups under consideration. Our results verify that the quantities of interest exhibit certain universal features in agreement with the universal operator growth hypothesis. Moreover, we illustrate the utility of this approach relying on symmetry as it significantly facilitates the calculation of quantities probing operator growth. In particular, we argue that the use of the Lanczos algorithm, which has been the most important tool in the study of operator growth so far, can be circumvented and all the essential information can be extracted directly from symmetry arguments.

Quantum chaos and the complexity of spread of states

“…These results match the SL(2,R) Lanczos coefficients (58) if we pick the representation h = 1/2 with Hamiltonian coefficients…”

“…Our approach to state complexity is related to the notion of Krylov operator complexity, which has been put forward in [46], and developed in [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61], based on the Lanczos approach [13] to operator dynamics in manybody systems. This approach starts with a Hamiltonian H and a time-dependent operator O(t), determined by…”

“…In fact, however, Krylov operator complexity has a further ambiguity that is not present in our approach, arising from the choice of inner product. Starting with [46], recent literature [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61] has mainly focused on the Wightman inner product…”

Krylov complexity in saddle-dominated scrambling

“…Interestingly, the validity of this hypothesis goes beyond the semi-classical regime, where OTOC or, more specifically, the Lyapunov exponent is ill-defined. In recent years, the study of operator growth and K-complexity has received significant attention from many-body systems to the conformal field theories and black hole physics [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55].…”

The growth of operator entropy in operator growth

“…The two notions have attracted a lot of attentions in literature [3,4,5,6,7,8,9,10,11,12]. In particular, it was established in [10] that for general irreversible process, the two quantities enjoy a universal logarithmic relation to leading order at long times:…”