# Chaos bound The **chaos bound** is a bound on the growth of [[0482 Out-of-time-order correlator|OTOC]] or more precisely the [[0466 Lyapunov exponent|Lyapunov exponent]] that characterises the [[0008 Quantum chaos|chaotic]] properties of the quantum system. It follows from unitarity and analyticity. There is no such analogue for classical chaos. Defining the OTOC as $ F(t)=\operatorname{tr}[y V(0) y W(t) y V(0) y W(t)], \quad y^4=\frac{e^{-\beta H}}{\operatorname{tr}\left[e^{-\beta H}\right]}, $[[2015#Maldacena, Shenker, Stanford|MSS]] started by making some assumptions about this function, $F(t)$, which are believed to be true for quite general physical systems: 1. $F(t)$ is analytic in the half strip $\left\{t \in \mathbb{C} \mid \operatorname{Re} t \geq t_0\right.$ and $|\operatorname{Im} t| \leq \frac{\beta}{4}\}$; 2. $F(t)$ is bounded by the factorised correlator $|F(t)|\le F_d$ on the boundary of the strip; 3. $(F(t))^*=F\left(t^*\right)$ (the Schwarz reflection condition), where the factorised correlator, $F_d$, is defined as $ F_d=\operatorname{tr}\left[y^2 V(0) y^2 V(0)\right] \operatorname{tr}\left[y^2 W(t) y^2 W(t)\right], $and $t_0$ is called the factorisation time, defined to be when $\operatorname{tr}\left[y^2 W(t) V(0) y^2 V(0) W(t)\right] \approx F_d,$i.e., when the OTOC factorises. From these assumptions, it follows that $\left(\frac{2 \pi}{\beta}-\partial_t\right)\left(F_d-F(t)\right) \geq 0,$which leads to the bound on the [[0466 Lyapunov exponent|Lyapunov exponent]] for [[0008 Quantum chaos|chaotic]] systems because $F_d-F(t) \sim e^{\lambda_L t}$ for such systems. See [[2015#Maldacena, Shenker, Stanford|MSS]] for the derivation. According to [[2021#Kundu (Sep, a)|Kundu]], however, that was not the full story: the MSS bound did not fully exploit the properties/assumptions above. Kundu defines a local moment of the OTOC as follows: $\mu_J(t)=e^{\frac{4 \pi J}{\beta} t} \int_{t-i \frac{\beta}{4}}^{t+i \frac{\beta}{4}} d t^{\prime} e^{-\frac{2 \pi}{\beta}\left(t^{\prime}-i \frac{\beta}{4}\right)(2 J+1)}\left(F\left(t^{\prime}\right)-F_d\right).$Then a stronger set of constraints follow from the assumptions above: 1. (Positivity and boundedness): $0<\mu_J(t)<\frac{2 \beta F_d}{\pi(2 J+1)} e^{-\frac{2 \pi}{\beta} t}$; 2. (Monotonicity): $\mu_{J+1}(t)<\mu_J(t)$; 3. (Log-convexity): $\mu_{J+1}(t)^2 \leq \mu_J(t) \mu_{J+2}(t)$. ## Refs - leading bound [[2015#Maldacena, Shenker, Stanford]] (main ref) - subleading (infinitely many) [[2021#Kundu (Sep, a)]][](https://arxiv.org/pdf/2109.03826.pdf) - extremal chaos: saturate all bounds [[Kundu202109b]] - [[2021#Kundu (Apr)]]: constraining bulk higher derivative couplings using chaos bound - causality bounds [[HashimotoSugiura2022]][](https://arxiv.org/pdf/2205.13818.pdf) - velocity dependent bound [[2019#Mezei, Sarosi]] (and from a [[0179 Pole skipping|pole-skipping]] perspective: [[2020#Choi, Mezei, Sarosi]]) - from stability: [[Yoon2019]][](https://arxiv.org/pdf/1905.08815.pdf) - bounds on the *generalised* [[0466 Lyapunov exponent|Lyapunov exponent]] from fluctuation-dissipation theorem: [[2022#Pappalardi, Kurchan]] - relation to [[0040 Eigenstate thermalisation hypothesis|ETH]]: [[2019#Murthy, Srednicki (Jun, b)]] - relation to [[0091 Boundary causality|boundary causality]]: [[2017#Afkhami-Jeddi, Hartman, Kundu, Tajdini]] ## Saturation - holographic systems in the classical gravity limit - SYK-like systems at low temperatures