Optimized state-independent entanglement detection based on a geometrical threshold criterion

Experimental procedures are presented for the rapid detection of entanglement of unknown arbitrary quantum states. The methods are based on the entanglement criterion using accessible correlations and the principle of correlation complementarity. Our first scheme essentially establishes the Schmidt decomposition for pure states, with few measurements only and without the need for shared reference frames. The second scheme employs a decision tree to speed up entanglement detection. We analyze the performance of the methods using numerical simulations and verify them experimentally for various states of two, three, and four qubits.

Figure 1

The systematic way to experimentally verify entanglement of an arbitrary pure two-qubit state without any a priori knowledge and in the absence of a common reference frame. The steps of this diagram are described in detail in the main text.

Figure 2

Decision strategies to detect entanglement. Start with a measurement of the correlation $T_{zz}$ and proceed with the correlation along the solid (dotted) arrow if the measured correlation is higher (lower) than the chosen threshold value; here of $t=0.4$. Due to correlation complementarity, there is a good chance of detecting entanglement in a small number of steps. The measurements in the blue shaded area suffice to detect all maximally entangled pure states with Schmidt-basis vectors $x$, $y$, or $z$.

Figure 3

One branch of the decision tree for three qubits, starting with a measurement of the correlation $T_{xxx}$ assumed to be big. The best efficiency is obtained for the threshold $t=0.5$.

Figure 4

Efficiency of the decision tree for two-qubit random mixed states. The states were uniformly sampled according to the Haar measure. The efficiency increases with the purity of the state (top panel) as well as with the amount of entanglement in a tested state (bottom panel). Note that all pure entangled states are detected after nine steps, as well as all the states with negativity of more than $0.2$ independently of their purity. Solid lines show the results when using the decision tree; dotted lines show the results when using random choices for the measurements.

Figure 5

Efficiency of one branch of the decision tree for three-qubit random pure states. The states were uniformly sampled according to the Haar measure.

Figure 6

Efficiency of one branch of the decision tree for many qubits compared with a random choice of measurements. The plot shows the gain in the number of measurements provided by the decision tree. Pure states were uniformly sampled according to the Haar measure and the percentage of detected entangled states was calculated for different number of steps (measurements) in the tree as well as for the random order of measurements that start with $X\otimes \cdots \otimes X$ for a fair comparison. We then compare the number of measurements for which the percentage of detected entangled states using the decision tree is the same as using the randomized measurements and plot here the maximal difference between them. The improvement provided by the tree grows exponentially with the number of qubits.

Figure 7

Scheme of the experimental type-I SPDC source used to prepare the state $\frac{1}{\sqrt{2}}\left(\right|HH\rangle +e^{i\varphi }|VV\rangle )$. The phase $\varphi$ can be set by an yttrium-vanadate crystal (YVO${}_4$). Spectral filtering is performed by means of interference filters (F) and spatial filtering by single mode fibers (SM). Half- (HWP) and quarter-wave plates (QWP) are used for state preparation and analysis. Brewster plates (BP) enable the performance of the filter operation and the preparation of asymmetric states.

Figure 8

Schmidt decomposition of a maximally entangled unknown state. The correlation tensor and the density matrix are determined (a) before and (b) after applying local filtering. (c) After removing the filter, the state can be measured in its Schmidt basis.

Figure 9

If Alice wants to measure in the basis ${\sigma }_{i^{\prime }}={|\downarrow \rangle }_{i{}^{\prime }}{\langle \downarrow |}_{i{}^{\prime }}-{|\uparrow \rangle }_{i{}^{\prime }}{\langle \uparrow |}_{i{}^{\prime }}$ ($i=x,y,x$), then the HWP and the QWP of the polarization analysis have to be aligned such that ${|\downarrow \rangle }_{i{}^{\prime }}$ and ${|\uparrow \rangle }_{i{}^{\prime }}$ are detected at different outputs of the PBS. The same holds for Bob.

Figure 10

Schmidt decomposition of a nonmaximally entangled state. (a) The correlation tensor and density matrix are displayed for an unknown asymmetric state. (b) The state has nonzero Bloch vectors enabling one to determine the corresponding Schmidt basis for which the measured correlations are maximal.

Figure 11

Application of the decision tree on a selection of maximally entangled states, allowing one to determine the entanglement of the state by measuring the correlations marked in red. As an alternative, local filtering is applied in order to extract the correlations with maximal value (blue correlations).

Figure 12

Application of the decision tree on a selection of nonmaximally entangled states, allowing one to determine the entanglement of the state by measuring the correlations marked in red. Due to the asymmetry of the states, local filtering is unnecessary, and the information on the Bloch vectors can be used to detect entanglement with a maximal number of three correlation measurements (blue correlations). (f) For a product state, the full set of correlations does not reveal entanglement, as it should be.