The notion of locality of quantum systems is typically modeled using tensor products of Hilbert spaces, where each Hilbert space represents a local quantum system. While this tensor product model is good at capturing properties of finite dimensional quantum systems, it fails to describe certain situations if the dimension of the Hilbert space is infinite. An example of such situation is the task of embezzlement, where Alice and Bob are asked to create a Bell state using only local operations with the help of a shared state (catalyst state) that must remain unchanged at the end of the protocol. To achieve embezzlement, a different model describing locality, the commuting operator model, is needed. In the commuting operator framework, Alice and Bob share a single Hilbert space, and all of Alice’s operators must mutually commute with Bob’s operators. Our most recent work tackles the problem of self-embezzlement, where Alice and Bob are tasked to create a copy of some entangled catalyst state using only local operations. While it is possible to construct an explicit protocol using commuting operators directly, we introduce a more intuitive new model using C*-algebras to formulate the self-embezzlement protocol. Interestingly, we show that there exists a constant gap in fidelity between what can be achieved using the tensor product model and that of the commuting operator model. This propert is in spirit similar to the Connes’ embedding conjecture, which is equivalent to showing the (non-)existence of a constant gap between tensor product and commuting operator protocol in the context of non-local games.
This work is presented at QIP 2019.