  We propose new optimality criterion for the estimation of state-dependent
cloning. We call this measure the relative error because the one compares the
errors in the copies with contiguous size taking into account the similarity of
states to be copied. A copying transformation and dimension of state space are
not specified. Only the unitarity of quantum mechanical transformations is
used. The presented approach is based on the notion of the angle between two
states. Firstly, several useful statements simply expressed in terms of angles
are proved. Among them there are the spherical triangle inequality and the
inequality establishing the upper bound on the modulus of difference between
probability distributions generated by two any states for an arbitrary
measurement. The tightest lower bound on the relative error is then obtained.
Hillery and Buzek originally examined an approximate state-dependent copying
and obtained the lower bound on the absolute error. We consider relationship
between the size of error and the corresponding probability distributions and
obtain the tightest lower bound on the absolute error. Thus, the proposed
approach supplements and reinforces the results obtained by Hillery and Buzek.
Finally, the basic findings of investigation for the relative error are
discussed.
