  We consider the problem of discriminating between two different states of a
finite quantum system in the setting of large numbers of copies, and find a
closed form expression for the asymptotic exponential rate at which the
specified error probability tends to zero. This leads to the identification of
the quantum generalisation of the classical Chernoff distance.
  The proof relies on two new techniques that have been introduced in
[quant-ph/0610027] and [quant-ph/0607216], respectively, and that are also well
suited to prove the quantum generalisation of the Hoeffding bound, which is a
modification of the Chernoff distance and specifies the optimal achievable
asymptotic error rate in the context of asymmetric hypothesis testing. This has
been done subsequently by Hayashi [quant-ph/0611013] and Nagaoka
[quant-ph/0611289] for the special case where both hypotheses have full
support.
  Moreover, quantum Stein's Lemma and quantum Sanov's theorem may be derived
directly from the quantum Hoeffding bound combining it with a result obtained
recently in [math/0703772].
  The goal of this paper is to present the proofs of the above mentioned
results in a unified way and in full generality (allowing hypothetic states
with different supports). Additionally, we give an in-depth treatment of the
properties of the quantum Chernoff distance. We argue that, although it is not
a metric, it is a natural distance measure on the set of density operators, due
to its clear operational meaning.
