Spacetime formulation for time crystals and continuous variables

In ordinary, non-relativistic quantum theory, especially in quantum information, space and time are treated differently. For example, time is a parameter rather than an operator; states are defined across the whole space but only at one time. These go against our intuition from relativity as well as classical probability theory. Thus, space-time density matrices have been introduced and here we discuss one of these formulations called pseudo-density matrix (PDM) which treats space and time indiscriminately. As a first application, we use this formulation to analyse time crystals via quantum error correction, which is the first time to discuss time crystals in the language of quantum information. We achieve the same result as the absence of continuous time crystals and formulate discrete time crystals in terms of quantum error correction. NMR experiments have been conducted to verify the results. We also extend this space-time formulation to the continuous variables via a generalisation for Wigner function, and discuss Gaussian case in particular. This is the first step we take to understand temporal vs spatial modes in quantum fields and black holes. We want to analyse temporal correlations in black hole information paradox.