Optimized Detection of High-Dimensional Entanglement

Entanglement detection is one of the most conventional tasks in quantum information processing. While most experimental demonstrations of high-dimensional entanglement rely on fidelity-based witnesses, these are powerless to detect entanglement within a large class of entangled quantum states, the so-called unfaithful states. In this Letter, we introduce a highly flexible automated method to construct optimal tests for entanglement detection given a bipartite target state of arbitrary dimension, faithful or unfaithful, and a set of local measurement operators. By restricting the number or complexity of the considered measurement settings, our method outputs the most convenient protocol which can be implemented using a wide range of experimental techniques such as photons, superconducting qudits, cold atoms, or trapped ions. With an experimental quantum optics setup that can prepare and measure arbitrary high-dimensional mixed states, we implement some three-setting protocols generated by our method. These protocols allow us to experimentally certify two- and three-unfaithful entanglement in four-dimensional photonic states, some of which contain well above 50% of noise.

Figure 1

Experimental setup for generating the unfaithful states. (a) Preparation of a four-dimensional entanglement source @ 1550 nm. After two lenses (not shown), the continuous-wave (cw) laser beam (@ 775 nm with power @ 200 mW) is focused into a small beam with waist radius of $\approx 0.7\mathrm{mm}$. With a beam-displacer (BD), the beam is separated into two paths. Then, these two beams are injected into a Sagnac interferometer, which generates a two-photon polarization-based spontaneous parametric down-conversion (SPDC) entangled state $(1/\sqrt{2})(|HV⟩+|VH⟩)$ in each path [24] by a type-II PPKTP crystal [$1\mathrm{mm}\times 7\mathrm{mm}\times 10\mathrm{mm}$, the polling period is $46.2\mu \mathrm{m}$, and the temperature is set at $35°\mathrm{C}$. In our experiment, the full width at half maximum of the down-converted photons is $\approx 2\mathrm{nm}$.]. After a half-wave plate (HWP) @45°, each entangled state is rotated to $(1/\sqrt{2})(|HH⟩+|VV⟩)$, which is encoded into the paths $w_1(w_3)$ and $w_2(w_4)$, respectively. If we define the path $w_1(w_3)$ with $H$-polarized photon as $|0⟩$, $w_1(w_3)$ with $V$-polarized photon as $|1⟩$, $w_2(w_4)$ with $H$-polarized photon as $|2⟩$, and $w_2(w_4)$ with $V$-polarized photon as $|3⟩$, then a four-dimensional maximally entangled state is generated. Afterwards, to prepare the state ${\rho }_{\mathrm{UNF}}^p$, we need to change the angles of first three HWPs and liquid crystals (LCs) together with their voltages. If we only consider the paths $w_1$ and $w_3$, then we obtain the state $|{\mathrm{\Psi }}_2⟩$. An additional light source is introduced as white noise mixed with the pure state $|{\mathrm{\Psi }}_2⟩$ to generate the states ${\rho }_{\mathrm{ISO}2}^p$. (b) and (c) Setups for Alice’s and Bob’s local measurements. Each party has three LCs, three HWPs, a BD, a PBS and a single-photon superconducting detector. By adjusting the voltages of the LCs and the angles of the HWPs, different measurement bases can be implemented.

Figure 2

Experimental results of the optimal $1-p_{\mathtt{I}}-p_{\mathtt{I}\mathtt{I}}$ for states. Experimental values for $p=0$, 0.2, 0.4, 0.6, 0.8, and 1 of state ${\rho }_{\mathrm{UNF}}^p$ in (a) and $p=0.5$, 0.6, 0.7, 0.8, 0.9, and 1 of state ${\rho }_{\mathrm{ISO}}^p$ in (b) are shown in blue. The solid red lines represent optimal values computed by the SDP solver. States in the green regions are three unfaithful in (a) and two unfaithful in (b). The entanglement dimension is certified when $1-p_{\mathtt{I}}-p_{\mathtt{I}\mathtt{I}}>0$.