Understanding the Born Rule in Weak Quantum Measurements

Abstract

Quantum measurements are described as instantaneous projections in textbooks. They can be stretched out in time using weak measurements, whereby one can observe the evolution of a quantum state towards one of the eigenstates of the measured operator. This evolution is a continuous nonlinear stochastic process, generating an ensemble of quantum trajectories. In particular, the Born rule can be interpreted as a fluctuation-dissipation relation. We experimentally observe the entire quantum trajectory distribution for weak measurements of a superconducting transmon qubit in circuit QED architecture, quantify it, and demonstrate that it agrees very well with the predictions of a single-parameter white-noise stochastic process. This characterisation of quantum trajectories is a powerful clue to unraveling the dynamics of quantum measurement, beyond the conventional axiomatic quantum theory. We emphasise the key quantum features of this framework, and their implications.