Mutually exclusive aspects of information carried by physical systems: Complementarity between local and nonlocal information

Complex physical systems contain information which, under some well-defined processes can differentiate between local and nonlocal information. Both these fundamental aspects of information are defined operationally. Local information is locally accessible and allows one to perform processes, such as physical work, while nonlocal information allows one to perform processes such as teleportation. It is shown that these two kinds of information are complementary in the sense that two parties can either gain access to the nonlocal information or to the local information but not both. This complementarity has a form similar to that expressed by entropic uncertainty relations. For pure states, the entanglement plays the role of Planck’s constant. We also find another class of complementarity relations which applies to operators and is induced when two parties can only perform local operations and communicate classical (LOCC). In particular, observables such as the parity and phase of two qubits commute but under LOCC, they are complementary observables. It is also found this complementarity is pure in the sense that it can be “decoupled” from the uncertainty principle. It is suggested that these complementarities represent an essential extension of Bohr’s complementarity to complex (distributed) systems which are entangled.