Witnessing superpositions of causal orders before the process is completed

Abstract

The questions we raise in this letter are as follows: What is the most general representation of a quantum state at a single point in time? Can we adapt the current formalisms to situations where the order of quantum operations is coherently or incoherently superposed? If so, what are the relations between the state at a given time and the uncertainty in the order of events before and after it? Establishing the relationship between two-state vector formalism and pseudo-density operators, we introduce the notion of a single-time pseudo-state. The tomographic construction of single-time pseudo-states is possible by ideal or weak measurements. We demonstrate that the eigenspectrum obtained from weak measurements enables us to discriminate between some coherent and incoherent superpositions of causal orders in pre- and post-selected systems before the process is completed. Finally, we discuss some possible experimental realizations in existing photonic setups.