Constraining hidden-state distributions to ensure quantum consistency

Develop constraints or a constructive characterization for hidden-state distributions p(λ) in local hidden-variable models such that the resulting measurement statistics correspond to a valid quantum state, i.e., satisfy all quantum mechanical compatibility relations across projective measurements for single qubits and multipartite systems.

Background

Local hidden-variable models can reproduce sets of measurement probabilities that are incompatible with any quantum state, as illustrated by single-qubit examples where LHV models can yield probability sums violating quantum mechanical bounds. This reveals that general LHV distributions need not correspond to physical quantum states, especially when approximating non-local correlations.

The authors note the desirability of constraining hidden-state distributions to enforce quantum consistency but explicitly acknowledge the absence of a known method to impose such constraints, framing a methodological open question.

References

This means there are LHV models whose correlations do not correspond to any state. One could try to constrain the hidden-state distributions, however it is not clear how to do this.

— Discovering Local Hidden-Variable Models for Arbitrary Multipartite Entangled States and Arbitrary Measurements  (2407.04673 - Selzam et al., 2024) in Appendix E (Remark on Observables)