Theories with indefinite causal structure

Abstract

To describe observed phenomena in the lab and to apply superposition principle to gravity, quantum theory needs to be generalized to incorporate indefinite causal structure. Practically, indefinite causal structure offers advantage in communication and computation. Fundamentally, superposing causal structure is one approach to quantize gravity (spacetime metric is equivalent to causal structure plus conformal factor, so quantizing causal structure effectively quantizes gravity). 

We develop a framework to do Operational Probabilistic Theories (OPT) with indefinite causal structure. For the interest of quantum gravity, this framework gives a general prescription to quantize causal structure, assuming linearity is intact. For the interest of quantum foundations, this framework can support new experimental tests about the validity of quantum theory in complex Hilbert space. It also offers opportunities for constructing new OPT models to substitute ordinary quantum theory. Along this direction, we identify principles that single out the complex Hilbert space theory within the general framework.