Packaged Quantum States and Symmetry: A Group-Theoretic Approach to Gauge-Invariant Packaged Entanglements

Abstract: A packaged quantum state refers to a quantum state that includes an inseparable block of internal quantum numbers. These types of quantum states are the result of gauge invariance and superselection, and therefore have particular symmetric structures. Here we show that, in multiparticle quantum systems, any nontrivial representation of a finite or compact group inherently induces packaged entanglement that inseparably entangles every internal quantum number (IQN). In this theory, every single-particle excitation carries an inseparable IQN block controlled by the irreducible representation of the group. The local gauge constraints or superselection rules forbid the occurrence of any partial charges or partial IQN entanglement. We demonstrate this principle using various specific symmetries, such as gauge symmetries ($U(1)$, $SU(2)$, and $SU(3)$), discrete symmetries (charge conjugation $C$, parity $P$, time reversal $T$, and their combinations), and $p$-form symmetries. In each case, gauge invariance and superselection rules ensure that the resulting quantum states cannot be factorized. We then used these ideas to explain phenomena like Bell-type structures, color confinement, and hybrid gauge-invariant configurations. The packaging principle connects concepts from gauge theory, topological classifications, and quantum information. These results may be useful for applications in exotic hadron spectroscopy, extended symmetries in quantum field theory, and quantum technologies.