Delayed Choice Lorentz Transformations on a Qubit

A continuously monitored quantum bit (qubit) exhibits competition between unitary Hamiltonian dynamics and non-unitary measurement-collapse dynamics, which for diffusive measurements form an enlarged transformation group equivalent to the Lorentz group of spacetime. We leverage this equivalence to develop a four-dimensional generalization of the three-dimensional Bloch ball to visualize the state of a monitored qubit as the four-momentum of an effective classical charge affected by a stochastic electromagnetic force field. Unitary qubit dynamics generated by Hermitian Hamiltonians correspond to elliptic spatial rotations of this effective charge while non-unitary qubit dynamics generated by non-Hermitian Hamiltonians or stochastic measurement collapse correspond to hyperbolic Lorentz boosts. Notably, to faithfully emulate the stochastic qubit dynamics arising from continuous qubit measurement, the stochastic electromagnetic fields must depend on the velocity of the charge they are acting on. Moreover, continuous qubit measurements admit a dynamical delayed choice effect where a future experimental choice can appear to retroactively determine the type of past measurement backaction, so the corresponding point charge dynamics can also exhibit delayed choice Lorentz transformations in which a future experimental choice determines whether stochastic force fields are electric or magnetic in character long after they interact with the particle.