  In this paper, we propose a new protocol for a data compression task, blind
quantum data compression, with finite local approximations. The rate of blind
data compression is susceptible to approximations even when the approximations
are diminutive. This instability originates from the sensitivity of a structure
of quantum states against approximations, which makes the analysis of blind
compression in the presence of approximations intractable. In this paper, we
constructed a protocol that takes advantage of the instability to reduce the
compression rate substantially. Our protocol shows a significant reduction in
rate for specific examples we examined. Moreover, we apply our methods to
diagonal states, and propose two types of approximation methods in this special
case. We perform numerical experiments and observe that one of these two
approximation methods performs significantly better than the other. Thus, our
analysis makes a first step toward general investigation of blind quantum data
compression with the allowance of approximations towards further investigation
of approximation-rate trade-off of blind quantum data compression.
