  We use strong sub-additivity of entanglement entropy, Lorentz invariance, and
the Markov property of the vacuum state of a conformal field theory to give a
new proof of the irreversibility of the renormalization group in d=4 space-time
dimensions -- the a-theorem. This extends the proofs of the c and F theorems in
dimensions d=2 and d=3 based on vacuum entanglement entropy, and gives a
unified picture of all known irreversibility theorems in relativistic quantum
field theory.
