IQI Weekly Seminar
Abstract: According to an elementary result in quantum computing, any unitary transformation on a composite system can be generated using 2-local unitaries, i.e., those that act only on two subsystems. Does this universality remain valid in the presence of conservation laws and global symmetries? In particular, do k-local symmetric unitaries on a composite system generate all symmetric unitaries on the system? I show that the answer is negative in the case of continuous symmetries, such as U(1) and SU(2): generic symmetric unitaries cannot be implemented, even approximately, using local symmetric unitaries. In the context of quantum thermodynamics this means that generic energy-conserving unitary transformations on a composite system cannot be implemented by applying local energy-conserving unitary transformations on the components. I also show how this no-go theorem can be circumvented via catalysis: any globally energy-conserving unitary can be implemented using a sequence of 2-local energy-conserving unitaries, provided that one can use a single ancillary qubit (catalyst).
In the second part of the talk, I will discuss some recent results for systems of qudits with SU(d) symmetry. These results show an important distinction between the case of d=2 and d>2.