  In this paper with two equivalent representations of the information
contained by a SAT formula, the reason why string generated by succinct SAT
formula can be greatly compressed is firstly presented based on Kolmogorov
complexity theory. Then what strings can be greatly compressed were classified
and discussed. In this way we discovered the SAT problem was composed of a
basic distinguish problem: distinguish two different distributions induced
under the computer with certain SAT formula ensemble. We then tried to map this
problem into quantum mechanics, or the quantum version basic distinguish
problem: this time two different distributions are induced under quantum
mechanics. Based on the equivalence of statistical distance between probability
space and Hilbert space, in the same time this distance is invariant under all
unitary transformations. The quantum version basic problem cannot be
efficiently solved by any quantum computer. In the worst case, any quantum
computer must perform exponential times measurement in order to solve it. In
the end we proposed the main theorem : The statistical distance in program
space and probability space are identical. We tried to prove it using the
relationship of Kolmogorov complexity and entropy. It showed there is no
difference to solve the basic problem in SAT formula space or probability
space. In the worst case, exponential trials must be performed to solve it.
NP!=P.
