  Fidelity plays an important role in measuring distances between pairs of
quantum states, of single as well as multiparty systems. Based on the concept
of fidelity, we introduce a physical quantity, shared purity, for arbitrary
pure or mixed quantum states of shared systems of an arbitrary number of
parties in arbitrary dimensions. We find that it is different from quantum
correlations. However, we prove that a maximal shared purity between two
parties excludes any shared purity of these parties with a third party, thus
ensuring its quantum nature. Moreover, we show that all generalized GHZ states
are monogamous, while all generalized W states are non-monogamous with respect
to this measure. We apply the quantity to investigate the quantum XY spin
models, and observe that it can faithfully detect the quantum phase transition
present in these models. We perform a finite-size scaling analysis and find the
scaling exponent for this quantity.
