Quantum change point and entanglement distillation

In a quantum change point problem, a source emitting particles in a fixed quantum state (default) switches to a different state at some stage, and the objective is to identify when the change happened by measuring a sequence of particles emitted from such a source. Motivated by entanglement-sharing protocols in quantum information, we study this problem within the paradigm of local operations and classical communication (LOCC). Here, we consider a source that emits entangled pairs in a default state, but starts producing another entangled state (mutation) at a later stage. Then, a sequence of entangled pairs prepared from such a source and shared between distant observers cannot be used for quantum information processing tasks as the identity of each entangled pair remains unknown. We show that identifying the change point using LOCC leads to the distillation of free entangled pairs. In particular, if the default and the mutation are mutually orthogonal, there exists an efficient LOCC protocol that identifies the change point without fail and distills a sufficiently large number of pairs. However, if they are nonorthogonal, there is a probability of failure. In this case, we compute the number of entangled pairs that may be obtained on average. We also consider a relaxation of the two-state problem where the mutation is not known a priori, but instead belongs to a known set. Here we show that local distinguishability plays a crucial role: if the default and the possible mutations are locally distinguishable, the problem reduces to the two-state problem with orthogonal states, but if not, one may still identify the mutation, the change point, and distill entanglement, as we illustrate with a concrete example.

Figure 1

LOCC quantum change point problem: Two remote observers, Alice and Bob, share a sequence of entangled pairs emitted by a possibly faulty source. The green pairs are in the default state $|\mathrm{\Psi }\rangle$ and the red ones are in the mutated state $|\mathrm{\Phi }\rangle$. As explained in the text, the task is to identify the change point and distill free entangled pairs using LOCC.

Figure 2

The two-state protocol with orthogonal states for $n=16$. The numbers inside the squares indicate the change points, and those inside the circles indicate the positions of the entangled pairs on which measurements are performed. The green and red arrows indicate the outcomes $\mathrm{\Psi }$ and ${\mathrm{\Psi }}^{\perp }$, respectively. The tree depicts all possible measurement sequences detecting the change points, including when no change occurs. Here, the worst-case scenario requires five measurements.