CSCO Criterion for Entanglement and Heisenberg Uncertainty Principle

We show that quantum entanglement and the Heisenberg uncertainty principle are inextricably connected. Toward this end, a complete set of commuting observables (CSCO) criterion for the entanglement is developed. Assuming (A1,A2,...) and (B1,B2,...) being two CSCO's for a given system, and C being the matrix, Cij = i [Bi,Aj], for each given row i (i=1,2,...) if at least one matrix element Cij (j=1,2,...) is nonzero, then for the simultaneous eigenstates |\psi)=|A1',A2',...) of (A1,A2,...), the simultaneous measurements of (B1,B2,...) are, in general,entangled. The only exception is when all the simultaneous eigenstates |\psi)= A1', A2',...), (\psi|C|\psi)=0. This CSCO criterion may be considered as an extension of the Heisenberg uncertainty principle to quantum systems with either two (or more) particles or multi-degrees of freedom (MDF).