Reliable experimental quantification of bipartite entanglement without reference frames

Simply and reliably detecting and quantifying entanglement outside laboratory conditions will be essential for future quantum information technologies. Here we address this issue by proposing a method for generating expressions which can perform this task between two parties who do not share a common reference frame. These reference-frame-independent expressions require only simple local measurements, which allows us to experimentally test them using an off-the-shelf entangled photon source. We show that the values of these expressions provide bounds on the concurrence of the state and demonstrate experimentally that these bounds are more reliable than values obtained from state tomography since characterizing experimental errors is easier in our setting. Furthermore, we apply this idea to other quantities, such as the Renyi and von Neumann entropies, which are also more reliably calculated directly from the raw data than from a tomographically reconstructed state. This highlights the relevance of our approach for practical quantum information applications that require entanglement.

Figure 1

Concurrence vs $Q_2$. The proposed upper and lower bounds on $C$ (the former plotted as a dashed line since it is not proven in generality) are respected by several million randomly generated mixed states. Furthermore, categorizing these states by purity lets us further discriminate entanglement. States of purity $\le 0.5$ are shown in blue, $0.5-0.6$ in green, $0.6-0.7$ in red, $0.7-0.8$ in cyan, and $0.8-0.9$ in yellow (in black and white these categories appear as increasingly pale shades of gray). The boundaries suggest that purity can be used to tighten the upper bound on concurrence.

Figure 2

Concurrence vs $Q_3$ (blue), $Q_4$ (green), and $Q_5$ (cyan) (in black and white, $Q_3,Q_4$, and $Q_5$ are shown in increasingly pale shades of gray) for 5 million randomly generated bipartite mixed states. Also shown are the bounds of $Q_2$ (black). All the expressions have been normalized to have a maximum value of 1. The spreads of the values of $Q_3,Q_4$, and $Q_5$ become increasingly narrower, suggesting that $Q_3,Q_4$, and $Q_5$ may give tighter bounds on the concurrence than $Q_2$, especially for highly entangled states.

Figure 3

The experimental values of $Q_2$ (black), $Q_3$ (blue), $Q_4$ (green), and $Q_5$ (cyan) for ten different rotations of the state. Rotation 1 corresponds to the unrotated state. All quantities have been normalized to a maximal value of 1. The horizontal black line indicates the bound for separable states for $Q_2$, which is equal to $1/3$ because of the normalization. We show the efficient lower bounds on $Q_2$ found using three measurements chosen according to the method of [2] in gray. The dark blue regions show possible values of the concurrence $C$, which is plotted on the right-hand axis.