  Symmetry is among the most fundamental and powerful concepts in nature, whose
existence is usually taken as given, without explanation. We explore whether
symmetry can be derived from more fundamental principles from the perspective
of quantum information. Starting with a two-qubit system, we show there are
only two minimally entangling logic gates: the Identity and the SWAP. We
further demonstrate that, when viewed as an entanglement operator in the
spin-space, the $S$-matrix in the two-body scattering of fermions in the
$s$-wave channel is uniquely determined by unitarity and rotational invariance
to be a linear combination of the Identity and the SWAP. Realizing a minimally
entangling $S$-matrix would give rise to global symmetries, as exemplified in
Wigner's spin-flavor symmetry and Schr\"odinger's conformal invariance in low
energy Quantum Chromodynamics. For $N_q$ species of qubit, the Identity gate is
associated with an $[SU(2)]^{N_q}$ symmetry, which is enlarged to $SU(2N_q)$
when there is a species-universal coupling constant.
