Ultrafine Entanglement Witnessing

Entanglement witnesses are invaluable for efficient quantum entanglement certification without the need for expensive quantum state tomography. Yet, standard entanglement witnessing requires multiple measurements and its bounds can be elusive as a result of experimental imperfections. Here, we introduce and demonstrate a novel procedure for entanglement detection which simply and seamlessly improves any standard witnessing procedure by using additional available information to tighten the witnessing bounds. Moreover, by relaxing the requirements on the witness operators, our method removes the general need for the difficult task of witness decomposition into local observables. We experimentally demonstrate entanglement detection with our approach using a separable test operator and a simple fixed measurement device for each agent. Finally, we show that the method can be generalized to higher-dimensional and multipartite cases with a complexity that scales linearly with the number of parties.

Figure 1

In a standard witnessing, the finest entanglement witness (EW) is obtained by shifting a test operator so that its corresponding hyperplane becomes tangent to the set of separable states, and thus, optimal with respect to this set. However, additional information or constraints on the quantum states under investigation can effectively reduce the size of the set of candidate separable states (the hashed subset). Our technique takes this into account to provide an ultrafine EW that is tangent to this reduced set of separable states. This, in general, leads to an advantage over the standard procedure (the yellow region). By varying the constraint, one can scan a large range of entangled states.

Figure 2

Experimental results for ultrafine entanglement witnessing using the single measurement device defined in Eqs. (3) and (4) with $x=2/3$ and $\theta =0$. The separability curve, which represents the largest expectation values of the test operator obtainable from separable states, is shown in orange, while the maximal values obtainable with entangled states are represented by the blue curve. The blue data points correspond to 21 equispaced entangled states and include $3\sigma$ error bars. The black dots, obtained from randomly sampled pure product states with uniform distributions over the local Bloch spheres, illustrate the density of separable states with respect to the test and constraint operators. Note that the point where the orange and blue curves meet is the optimal point one would obtain for $\widehat{L}$ in SEW.

Figure 3

(a) Experimental implementation of the three-outcome qubit measurement of Eq. (3). The first POVM element ${\widehat{\mathrm{\Pi }}}_1$ is implemented directly by a partially polarizing beam splitter (PPBS) with reflection coefficients $r_H=0$ for horizontal and $r_V=2/3$ for vertical polarization. The other POVM elements ${\widehat{\mathrm{\Pi }}}_2$ and ${\widehat{\mathrm{\Pi }}}_3$ are implemented using a set of quarter-wave plate (QWP), half-wave plate (HWP), and a polarizing beam splitter (PBS). (b) Visualization of our three-outcome POVM in the $xz$ plane of the Bloch sphere.

Figure 4

The bounds $g(N,M_k)$ versus the cardinality of the largest party $M_k$ for $2\le N\le 6$, where each $N$ is shown in a different color. For a state of interest with $c=0$, a violation of the bound $g(N,1)$ proves its (partial) entanglement, while a violation $g(N,N-1)$ proves its genuine $N$-partite entanglement. The inset shows the case $N=4$ with the four black dots representing the four subsystem and the purple shaded boxes visualize the maximally allowed entangled subset.