Restrictions on realizable unitary operations imposed by symmetry and locality

According to a fundamental result in quantum computing, any unitary transformation on composite system can be generated using so-called 2-local unitaries that act only two subsystems. Beyond its importance this also regarded as statement about the dynamics of systems with local Hamiltonians: although locality puts various constraints short-term dynamics, it does not restrict possible evolutions general Hamiltonian experience after sufficiently long time. Here we show universality remain valid presence conservation laws and global continuous symmetries such U(1) SU(2). In particular, generic symmetric cannot implemented, even approximately, unitaries. Based no-go theorem, propose method for experimentally probing interactions nature. context thermodynamics, our results mean energy-conserving transformations realized solely by combining components. We how circumvented via catalysis. no symmetry, operations combine make across whole system, but if limit allowed operations, they all transformations.