Abstract
The structure of the Heisenberg evolution of operators plays a key role in explaining diverse processes in quantum many-body systems. In this paper, we discuss a new universal feature of operator evolution: an operator can develop a void during its evolution, where its nontrivial parts become separated by a region of identity operators. Such processes are present in both integrable and chaotic systems, and are required by unitarity. We show that void formation has important implications for unitarity of entanglement growth and generation of mutual information and multipartite entanglement. We study explicitly the probability distributions of void formation in a number of unitary circuit models, and conjecture that in a quantum chaotic system the distribution is given by the one we find in random unitary circuits, which we refer to as the random void distribution. We also show that random unitary circuits lead to the same pattern of entanglement growth for multiple intervals as in (1 + 1)-dimensional holographic CFTs after a global quench, which can be used to argue that the random void distribution leads to maximal entanglement growth.
Highlights
Where 1A denotes the identity operator in A, OAis some operator in A,1 and O2(t) is an operator whose projection onto A is orthogonal to 1A
To develop intuition for probability distributions of void formation, we study three types of unitary circuit models in one spatial dimension: (i) the random unitary model of [8,9,10], which can be considered a minimal model for quantum chaotic systems; (ii) a “free propagating” model [11] in which entanglement can only be spread, but not created, which may be considered a proxy for free theories; (iii) a circuit built from perfect tensors [12], which may be considered a model for non-chaotic systems
As contrasts to the random void distribution, we consider the void formation structure of two other examples of unitary circuits: (i) a free propagating model in which entanglement can only be spread, but not created, which may be considered a proxy for free theories; (ii) a circuit built from perfect tensors, which may be considered a model for non-chaotic systems, as while it can generate entanglement in certain initial product states, like all Clifford circuits it does not lead to growth of operator entanglement, and does not have the form of the out-of-time-ordered correlator (OTOC) expected in chaotic systems [8, 9]
Summary
We first describe our general setup, and derive some simple constraints from unitarity on void formation during Heisenberg evolution. O0 is the identity operator for the full Hilbert space H, and all other Oα’s are traceless. For the purpose of not obscuring the conceptual picture with technicalities, throughout this paper, we will consider a simple form of operator evolution in which the. The statement about light-cone growth does not say anything about the internal structure of an operator under time evolution. The probability for a basis operator Oα to develop a void in region A at time t is PO(Aα)(t) =. Where we have used (2.4), and the fact that due to tracelessness of all nontrivial basis operators, only operators of the form Oβ ⊗ 1Awith Oβ an operator in region A (denoted by β ∈ A) contribute to TrAOα(t). NA(t) has a simple physical interpretation: it is the expected number of operators in the set I contained within region A at time t. Throughout the paper, we will denote the union of two regions A1 ∪A2 as A1A2
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