Quantum Change Point

Sudden changes are ubiquitous in nature. Identifying them is crucial for a number of applications in biology, medicine, and social sciences. Here we take the problem of detecting sudden changes to the quantum domain. We consider a source that emits quantum particles in a default state, until a point where a mutation occurs that causes the source to switch to another state. The problem is then to find out where the change occurred. We determine the maximum probability of correctly identifying the change point, allowing for collective measurements on the whole sequence of particles emitted by the source. Then, we devise online strategies where the particles are measured individually and an answer is provided as soon as a new particle is received. We show that these online strategies substantially underperform the optimal quantum measurement, indicating that quantum sudden changes, although happening locally, are better detected globally.

Figure 1

The quantum change point problem. A quantum source emits particles in a default state $|0⟩$, until the point where a mutation occurs, causing the source to emit particles in a different state $|\varphi ⟩$. A detector receives the stream of particles emitted from the source and measures them, producing an estimate of the point where the mutation occurred.

Figure 2

Probability of correct identification of the change point as a function of $c^2=\mathbf{|}⟨0|\varphi ⟩{\mathbf{|}}^2$. The (blue) dashed line is the asymptotic result in Eq. (1). For a sequence of $n=50$ states, we also plot the maximum probability, obtained by SDP optimization, (blue) solid dots, and the results corresponding to the square root measurement, (black) crosses (which lie virtually on top of the dots); the basic local strategy, (red) straight line; and the greedy strategy, (orange) triangles.