The intrinsic idea of superdense coding is to find as many gates as possible
such that they can be perfectly discriminated. In this paper, we consider a new
scheme of discrimination of quantum gates, called ancilla-assisted
discrimination, in which a set of quantum gates on a $d-$dimensional system are
perfectly discriminated with assistance from an $r-$dimensional ancilla system.
The main contribution of the present paper is two-fold: (1) The number of
quantum gates that can be discriminated in this scheme is evaluated. We prove
that any $rd+1$ quantum gates cannot be perfectly discriminated with assistance
from the ancilla, and there exist $rd$ quantum gates which can be perfectly
discriminated with assistance from the ancilla. (2) The dimensionality of the
minimal ancilla system is estimated. We prove that there exists a constant
positive number $c$ such that for any $k\leq cr$ quantum gates, if they are
$d$-assisted discriminable, then they are also $r$-assisted discriminable, and
there are $c^{\prime}r\textrm{}(c^{\prime}>c)$ different quantum gates which
can be discriminated with a $d-$dimensional ancilla, but they cannot be
discriminated if the ancilla is reduced to an $r-$dimensional system. Thus, the
order $O(r)$ of the number of quantum gates that can be discriminated with
assistance from an $r-$dimensional ancilla is optimal. The results reported in
this paper represent a preliminary step toward understanding the role ancilla
system plays in discrimination of quantum gates as well as the power and limit
of superdense coding.
