Conserved quantities in parity-time symmetric systems

Conserved quantities such as energy or the electric charge of a closed system, or the Runge-Lenz vector in Kepler dynamics, are determined by its global, local, or accidental symmetries. They were instrumental in advances such as the prediction of neutrinos in the (inverse) beta decay process and the development of self-consistent approximate methods for isolated or thermal many-body systems. In contrast, little is known about conservation laws and their consequences in open systems. Recently, a special class of these systems, called parity-time (PT) symmetric systems, has been intensely explored for their remarkable properties that are absent in their closed counterparts. A complete characterization and observation of conserved quantities in these systems and their consequences is still lacking. Here, we present a complete set of conserved observables for a broad class of PT-symmetric Hamiltonians and experimentally demonstrate their properties using a single-photon linear optical circuit. By simulating the dynamics of a four-site system across a fourth-order exceptional point, we measure its four conserved quantities and demonstrate their consequences. Our results spell out nonlocal conservation laws in nonunitary dynamics and provide key elements that will underpin the self-consistent analyses of non-Hermitian quantum many-body systems that are forthcoming.