  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, where
SWAP interchanges the two states of the qubits. 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.
