OTOC and Scrambling

Swingle, B. (2018). Unscrambling the physics of out-of-time-order correlators. Nature Physics, 14(10), 988–990. https://doi.org/10.1038/s41567-018-0295-5
一篇简单的说明性文章，很好地介绍了什么是OTOC以及scrambling。不过看完还需要具体读一些其他文章。

Out-of-Time-Order Correlators

Out-of-Time-Order Correlators can describe the relation between two operators. The OTOC between two operators and can be defined as:

$$F(t) = \langle \psi | e^{iHt} W e^{-iHt} V e^{iHt} W e^{-iHt} V| \psi \rangle \tag{1}$$

It is actually the overlap of and . In some way it describes the inability of rewinding the time. And according to the paper, in the context of thermalization the OTOC can illuminate many useful aspects: when and how thermalization happens, how rapidly it can occur, and the role played by entanglement and non-classicality.

For short we can write as:

$$F(t) = \langle W(t)^\dagger V^\dagger W(t)V\rangle\tag{2}$$

And it can be related with the squre mean calue of the commutator

$$C(t) = \langle [W(t), V]^\dagger [W(t), V]\rangle$$

If and are hermitian and unitary, then . Thus the OTOC is related to the commutator as .

Information Scrambling

This phenomenon of the spread of quantum information is sometimes called information scrambling. It can be usefully characterized in terms of the growth of Heisenberg operators, and is connected with the rewinding of time in OTOC.

Given some operators , the t-dependent Heisenburg operator with each e-term representing time forward/backward.