Witnessing bipartite entanglement sequentially by multiple observers

We investigate sharing of bipartite entanglement in a scenario where half of an entangled pair is possessed and projectively measured by one observer, called Alice, while the other half is subjected to measurements performed sequentially, independently, and unsharply, by multiple observers, called Bobs. We find that there is a limit on the number of observers in this entanglement distribution scenario. In particular, for a two-qubit maximally entangled initial shared state, no more than 12 Bobs can detect entanglement with a single Alice for arbitrary, possibly unequal, sharpness parameters of the measurements by the Bobs. The number of Bobs remains unaltered for a finite range of near-maximal pure initial entanglement, a feature that also occurs in the case of equal sharpness parameters at the Bobs. Furthermore, we show that for nonmaximally entangled shared pure states, the number of Bobs decreases with the amount of initial entanglement, providing a coarse-grained but operational measure of entanglement.

Figure 1

Different Bobs appear at the same scene (laboratory) to perform measurements on the same quantum particle on the Bob part of the Alice-Bob partition. The laboratory of Alice is spatially separated from that of the Bobs. In the schematic diagram, time separation is depicted along the horizontal axis, while space separation is represented along the vertical one.

Figure 2

Conceptualization of a coarse-grained operational entanglement measure. We consider the scenario where the two separated laboratories share a pure two-qubit state. The horizontal axis represents the entanglement of the initial state, as measured by the von Neumann entropy $\mathcal{E}$ of one of the local states and is measured in ebits. The vertical axis counts the number of Bobs $n$ who can succeed in detecting entanglement with Alice and is dimensionless. The monotonic nature of the function plotted implies that it can act as an entanglement measure, and it is clearly operationally defined. However, the steps in the function points to a coarse-grained nature of the measure. The existence of a step of finite (i.e., nonzero) length on the extreme right implies that the maximal number of Bobs remains fixed for a certain range of $\mathcal{E}$.

Figure 3

Case of equal sharpness for all the observers on one side. The $x$ axis labeled $\lambda$ represents the common sharpness parameter of all the Bobs involved. The $y$ axis stands for the maximal number of Bobs $n$ who are able to detect entanglement with a single Alice. Both axes are dimensionless. Circles and dashes exhibit the cases when the initial shared pure state is maximally entangled $\left[\mathcal{E}\left(\right|\psi \rangle )=1\right]$ and not maximally entangled $\left[\mathcal{E}\left(\right|\psi \rangle )=0.914\right]$, respectively. Red, blue, green, violet, and orange colors correspond to the number of Bobs $n=1,2,3,4,5$, respectively.