  We consider a situation when evolution of an entangled
Einstein-Podolsky-Rosen (EPR) pair takes place in a regime of quantum chaos
being chaotic in the classical limit. This situation is studied on an example
of chaotic pair dynamics described by the quantum Chirikov standard map. The
time evolution is reversible even if a presence of small errors breaks time
reversal of classical dynamics due to exponential growth of errors induced by
exponential chaos instability. However, the quantum evolution remains
reversible since a quantum dynamics instability exists only on a
logarithmically short Ehrenfest time scale. We show that due to EPR pair
entanglement a measurement of one particle at the moment of time reversal
breaks exact time reversal of another particle which demonstrates only an
approximate time reversibility. This result is interpreted in the framework of
the Schmidt decomposition and Feynman path integral formulation of quantum
mechanics. The time reversal in this system has already been realized with cold
atoms in kicked optical lattices in absence of entanglement and measurements.
On the basis of the obtained results we argue that the experimental
investigations of time reversal of chaotic EPR pairs is within reach of present
cold atom capabilities.
