  Motivated by deterministic identification via (classical) channels, where the
encoder is not allowed to use randomization, we revisit the problem of
identification via quantum channels but now with the additional restriction
that the message encoding must use pure quantum states, rather than general
mixed states. Together with the previously considered distinction between
simultaneous and general decoders, this suggests a two-dimensional spectrum of
different identification capacities, whose behaviour could a priori be very
different.
  We demonstrate two new results as our main findings: first, we show that all
four combinations (pure/mixed encoder, simultaneous/general decoder) have a
double-exponentially growing code size, and that indeed the corresponding
identification capacities are lower bounded by the classical transmission
capacity for a general quantum channel, which is given by the
Holevo-Schumacher-Westmoreland Theorem. Secondly, we show that the simultaneous
identification capacity of a quantum channel equals the simultaneous
identification capacity with pure state encodings, thus leaving three linearly
ordered identification capacities. By considering some simple examples, we
finally show that these three are all different: general identification
capacity can be larger than pure-state-encoded identification capacity, which
in turn can be larger than pure-state-encoded simultaneous identification
capacity.
