Recycling qubits for the generation of Bell nonlocality between independent sequential observers

There is currently much interest in the recycling of entangled systems, for use in quantum information protocols by sequential observers. In this work, we study the sequential generation of Bell nonlocality via recycling one or both components of two-qubit states. We first give a description of two-valued qubit measurements in terms of measurement bias, strength, and reversibility, and derive useful tradeoff relations between them. Then, we derive one-sided monogamy relations for unbiased observables that strengthen the recent conjecture by Cheng et al. [Phys. Rev. A 104, L060201 (2021)] that if the first pair of observers violate Bell nonlocality then a subsequent independent pair cannot, and give semianalytic results for the best possible monogamy relation. We also extend the construction by Brown and Colbeck [Phys. Rev. Lett. 125, 090401 (2020)] to obtain (i) a broader class of two-qubit states that allow the recycling of one qubit by a given number of observers on one side and (ii) a scheme for generating Bell nonlocality between arbitrarily many independent observers on each side, via the two-sided recycling of multiqubit states. Our results are based on a formalism that is applicable to more general problems in recycling entanglement and hence is expected to aid progress in this field.

Figure 1

Sequential sharing with multiple observers on each side. A source S generates two qubits on each run, which are received by observers Alice 1 and Bob 1 ($A_1$ and $B_1$ in the main text). They each make one of two local measurements on their qubit with equal probabilities; record their result; and pass their qubit onto independent observers Alice 2 and Bob 2, respectively ($A_2$ and $B_2$ in the main text). It is known that Alice 1 can demonstrate Bell nonlocality with each of an arbitrary number of Bobs in this way, by recycling the second qubit [13]. However, we have recently given strong analytic and numerical evidence for the conjecture that Alice 1 and Bob 1 can demonstrate Bell nonlocality in this manner only if Alice 2 and Bob 2 cannot, and vice versa [18]. A similar result holds for the pairs (Alice 1, Bob 2) and (Alice 2, Bob 1).

Figure 2

One-sided monogamy relations for biased and unbiased observables. For general biased observables, a global optimization algorithm was employed in Ref. [18] to determine the joint range of possible values of $S(A_1,B_1)$ and $S^{*}(A_2,B_2)$, the results of which are reproduced here as the blue dots. When restricted to unbiased observables, new numerical results are displayed as the solid orange curve, which lies strictly below the dashed red line corresponding to the conjectured monogamy relation (41), and which completely coincides with the blue dots for the case $\left|S\right(A_1,B_1\left)\right|\ge 2$. We have further verified that the solid orange curve can be achieved by measuring unbiased observables on a singlet state. The detailed form of the solid orange curve is discussed in Sec. 4c.

Figure 3

Semianalytic monogamy relations for unbiased observables. The orange solid curve reproduces the numerically generated optimal curve in Fig. 2. The black dotted and dashed curves match the optimal curve in the regions $|S(A_1,B_1)\in [0,2]$ and $\left|S\right(A_1,B_1)\gtrsim 2.72$, respectively. These curves are defined in Eqs. (52) and (54), found by numerically motivated ansatzes for the optimal observables measured by $A_1$ and $B_1$ (see main text).