Projective Measurements Are Sufficient for Recycling Nonlocality

Unsharp measurements are widely seen as the key resource for recycling the nonlocality of an entangled state shared between several sequential observers. Contrasting this, we here show that nonlocality can be recycled using only standard, projective, qubit measurements. Focusing on the Clauser-Horne-Shimony-Holt inequality and allowing parties to share classical randomness, we determine the optimal trade-off in the magnitude of Bell violations for a maximally entangled state. We then find that nonmaximally entangled states make possible larger sequential violations, which contrasts the standard Clauser-Horne-Shimony-Holt scenario. Furthermore, we show that nonlocality can be recycled using projective qubit measurements even when no shared classical randomness is available. We discuss the implications of our results for experimental implementations of sequential nonlocality.

Figure 1

A source distributes a two-qubit state $\psi$ to measuring parties $A$ and $B_1$. The qubit of $B_1$ is sequentially relayed, measured, relayed, etc., until measured by a final party $B_n$. Before the experiment begins, all parties may agree on sharing strings of classically correlated data $\lambda$.

Figure 2

Optimal trade-off between $S_2$ and $S_1$ for a maximally entangled qubit pair under projective measurements and shared randomness. The trade-off consists of four parts represented by red, orange, green, and blue solid curves, respectively. The first and third are stochastic (boundary points marked) while the second and fourth are deterministic. The blue and orange dashed curves are the optimal trade-offs for deterministic projective strategies of types (i) and (iii), respectively. The lower (upper) inset illustrates, in solid black, a deterministic (stochastic) model based on partially entangled states that outperforms the maximally entangled state. The dashed black lines are the classical and quantum bounds of the CHSH inequality.