  A distinct feature of Hermitian quantum chaotic dynamics is the exponential
increase of certain out-of-time-order-correlation (OTOC) functions around the
Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs
describe the growth of quantum uncertainty that crucially depends on the nature
of the quantum motion. Here, we employ the OTOC in order to provide a precise
definition of dissipative quantum chaos. For this purpose, we compute
analytically the Lyapunov exponent for the vectorized formulation of the large
$q$-limit of a $q$-body Sachdev-Ye-Kitaev model coupled to a Markovian bath.
These analytic results are confirmed by an explicit numerical calculation of
the Lyapunov exponent for several values of $q \geq 4$ based on the solutions
of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov
exponent decreases monotonically as the coupling to the bath increases and
eventually becomes negative at a critical value of the coupling signaling a
transition to a dynamics which is no longer quantum chaotic. Therefore, a
positive Lyapunov exponent is a defining feature of dissipative many-body
quantum chaos. The observation of the breaking of the exponential growth for
sufficiently strong coupling suggests that dissipative quantum chaos may
require in certain cases a sufficiently weak coupling to the environment.
