  Macroscopic quantum effects play central roles in the appearance of
inexplicable phenomena in low-temperature quantum many-body physics. Such
macroscopic quantumness is often evaluated using long-range entanglement, i.e.,
entanglement in the macroscopic length scale. The long-range entanglement not
only characterizes the novel quantum phases but also serves as a critical
resource for quantum computation. Thus, the problem that arises is under which
conditions can the long-range entanglement be stable even at room temperatures.
Here, we show that bi-partite long-range entanglement is unstable at arbitrary
temperatures and exponentially decays with distance. Our theorem provides a
no-go theorem on the existence of the long-range entanglement. The obtained
results are consistent with the existing observations that long-range
entanglement at non-zero temperatures can exist in topologically ordered
phases, where tripartite correlations are dominant. In the derivation of our
result, we introduce a quantum correlation defined by the convex roof of the
standard correlation function. We establish an exponential clustering theorem
for generic quantum many-body systems for such a quantum correlation at
arbitrary temperatures, which yields our main result by relating quantum
correlation to quantum entanglement. As a simple application of our analytical
techniques, we derived a general limit on the Wigner-Yanase-Dyson skew
information and the quantum Fisher information, which will attract significant
attention in the field of quantum metrology. Our work reveals novel general
aspects of low-temperature quantum physics and sheds light on the
characterization of long-range entanglement.
