  We propose a definition of infinite dimensional Choi-Jamiolkowski state
associated with a completely positive trace preserving map. We introduce the
notion of Theta-KMS adjoint of a quantum Markov semigroup, which is identified
with the time reversed semigroup. The break down of Theta-KMS symmetry (or
Theta-standard quantum detailed balance in the sense of Fagnola-Umanita) is
measured by means of the von Neumann relative entropy of the Choi-Jamiolkowski
states associated with the semigroup and its Theta-KMS adjoint.
