Minimality of one local phase system per party

Determine whether one local two-dimensional phase system per party is necessary for a compiler that is compositional, preserves the party decomposition, and uniformly compiles arbitrary local complex operators.

Background

The paper presents a distributed realification compiler that assigns one two-dimensional real phase tag to each party. The construction preserves compositionality, party ownership, locality, and computational-basis output probabilities for finite pure-state circuits consisting of party-local complex linear operations.

The authors explicitly distinguish this compiler lower-bound question from the weaker question of how many rebits are needed to reproduce a single correlation, such as the GHZ correlation. The unresolved issue concerns the necessity of the per-party phase-tag architecture under simultaneous requirements of compositionality, preservation of the party decomposition, and uniform compilation of arbitrary local complex operators.