Delayed Choice Lorentz Transformations on a Qubit

TL;DR

This paper introduces a Lorentz group analogy for monitored qubit dynamics, visualizing quantum states as a four-momentum affected by stochastic electromagnetic forces, revealing a delayed choice effect in measurement backaction.

Contribution

It develops a four-dimensional visualization of monitored qubit states and demonstrates a delayed choice Lorentz transformation effect in quantum measurement dynamics.

Findings

Qubit states can be represented as four-momentum vectors influenced by stochastic forces.

Unitary and non-unitary dynamics correspond to rotations and boosts in the Lorentz group.

Delayed choice effects allow future measurement decisions to influence past measurement backactions.

Abstract

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.