For two observable V and W in
Heisenberg picture, the out-of-time-order correlator (OTOC) is typically defined in two different but physically closely related ways: • Based on commutator of W(t) and V(0):C(t)=\langle [W(t),V(0)]^{\dagger} [W(t),V(0)]\rangledirect calculation gives C(t) = \langle V(0)^{\dagger} W(t)^{\dagger} W(t) V(0) \rangle + \langle W(t)^{\dagger} V(0)^{\dagger} V(0) W(t) \rangle - \langle V(0)^{\dagger} W(t)^{\dagger} V(0) W(t) \rangle - \langle W(t)^{\dagger} V(0)^{\dagger} W(t) V(0) \rangle . • More directly F(t)=\langle W(t)^{\dagger} V(0)^{\dagger}W(t)V(0)\rangleGenerally C(t) = \langle V(0)^{\dagger} W(t)^{\dagger} W(t) V(0) \rangle + \langle W(t)^{\dagger} V(0)^{\dagger} V(0) W(t) \rangle - 2\,\mathrm{Re}\,F(t) . When V and W are unitaries, we have C(t) = 2 \big( 1 - \mathrm{Re}\,F(t) \big) . where the expectation value \langle \bullet \rangle =\operatorname{Tr} [\rho\, \bullet] is usually taken over some thermal state \rho=\exp(-\beta H)/Z with \beta=1/k_BT (k_B is
Boltzmann constant, T is temperature) and H is Hamiltonian, Z=\operatorname{Tr} \exp(-\beta H) is canonical partition function. Physically, the growth of this commutator measured by C(t) tracks scrambling. And from chaos theory perspective, we have C(t)\simeq e^{\lambda_L t} where \lambda_L is the quantum
Lyapunov exponent. This has a similar form as the classical dependence of initial pertuvation in classical chaos theory. Thus OTOC can be regarded as an indicator of quantum chaos. == See also ==