Purification of mixed states with closed timelike curve is not possible

Abstract

In ordinary quantum theory, any mixed state can be purified in an enlarged Hilbert space by adding an ancillary system. The purified state does not depend on the state of any extraneous system with which the mixed state is going to interact or on the physical interaction. Here, we prove that it is not possible to purify a mixed state that traverses a closed timelike curve (CTC) and is allowed to interact in a consistent way with a causality-respecting (CR) quantum system in the same manner. In other words, if a CTC quantum system with a mixed state has to undergo an arbitrary interaction with an arbitrary CR system then it has to be in a “proper” mixture, as it cannot be regarded as a subsystem of a pure entangled state. Thus, in general for arbitrary interactions between CR and CTC systems, there is no universal purification by embedding in a larger Hilbert space for mixed states with CTCs. This shows that in quantum theory with CTCs there can exist proper and improper mixtures.