Entanglement Structure: Entanglement Partitioning in Multipartite Systems and Its Experimental Detection Using Optimizable Witnesses

Creating large-scale entanglement lies at the heart of many quantum information processing protocols and the investigation of fundamental physics. For multipartite quantum systems, it is crucial to identify not only the presence of entanglement but also its detailed structure. This is because in a generic experimental situation with sufficiently many subsystems involved, the production of so-called genuine multipartite entanglement remains a formidable challenge. Consequently, focusing exclusively on the identification of this strongest type of entanglement may result in an all or nothing situation where some inherently quantum aspects of the resource are overlooked. On the contrary, even if the system is not genuinely multipartite entangled, there may still be many-body entanglement present in the system. An identification of the entanglement structure may thus provide us with a hint about where imperfections in the setup may occur, as well as where we can identify groups of subsystems that can still exhibit strong quantum-information-processing capabilities. However, there is no known efficient methods to identify the underlying entanglement structure. Here, we propose two complementary families of witnesses for the identification of such structures. They are based, respectively, on the detection of entanglement intactness and entanglement depth, each applicable to an arbitrary number of subsystems and whose evaluation requires only the implementation of solely two local measurements. Our method is also robust against noises and other imperfections, as reflected by our experimental implementation of these tools to verify the entanglement structure of five different eight-photon entangled states. In particular, we demonstrate how their entanglement structure can be precisely and systematically inferred from the experimental measurement of these witnesses. In achieving this goal, we also illustrate how the same set of data can be classically postprocessed to learn the most about the measured system.

Popular Summary

Quantum entanglement is of paramount importance not only to our understanding of nature but also to the implementation of quantum information processing tasks. While many experiments attempt to create large-scale entanglement among hundreds of atoms, they instead end up with a mixture of groups containing much fewer entangled atoms. The ability to precisely characterize the entanglement among all the components in a large ensemble could provide not only a useful diagnostic of where the imperfections in our devices lie but also a clear benchmark of our technological progress towards the ultimate goal of creating true large-scale entanglement. We report here a novel means of identifying entanglement that can be used for this characterization.

Our tools consist of two families of “entanglement witnesses”: One identifies the extent to which entanglement is segregated among groups of subsystems; the other characterizes the minimal number of particles that are genuinely entangled. Each of our tools can be applied to a system involving arbitrarily many subsystems. They require the measurement of only two local observables, and can further be optimized classically after the measurement is completed. These tools can generally return nontrivial information on entanglement structure. We further report a proof-of-principle experiment illustrating how our tools can be applied in a systematic manner to reveal the entanglement structure of various eight-photon entangled states.

An adaptation of our tools to quantum states that are useful for measurement-based quantum computation would help us move towards the demonstration of quantum supremacy, the potential advantage of quantum computers to solve problems that classical computers cannot.

Figure 1

Schematic showing the experimental setup. (a) A global view of the experimental setup used to generate different eight-photon entangled states. A pulse from a pulsed Ti-sapphire laser (with a central wavelength of 780 nm, a duration of 130 fs, and an average power of 3.8 W) passes through a frequency doubler, by which it is changed to an UV pulse with a central wavelength of 390 nm and an average power of 1.6 W. Then, the UV pulse is directed by reflective mirrors to shine on four BiBO crystals successively. Shining a UV pulse on a BiBO crystal will generate (probabilistically) a photon pair maximally entangled in the polarization degree of freedom via spontaneous parametric down-conversion (SPDC). Photons in path modes 2, 3, 6, and 7 are then injected into an interferometric network to generate $|G_8⟩$, $|G_{62}⟩$, $|G_{44}⟩$, $|G_{422}⟩$, and $|G_{2222}⟩$ by the corresponding interferometric geometry settings. Finally, eight photons are analyzed by witness analyzers via single-photon detectors. To suppress the higher-order emission in SPDC, we attenuate the average power of the UV pulse to 500 mW. The eight-fold coincidences we observed in creating $|G_8⟩$, $|G_{62}⟩$, $|G_{44}⟩$, $|G_{422}⟩$, and $|G_{2222}⟩$ are $8/\mathrm{h}$, $20/\mathrm{h}$, $20/\mathrm{h}$, $36/\mathrm{h}$, and $70/\mathrm{h}$, respectively, due to different postselection probabilities (see Fig. 2 for further explanation and Appendix pp2-s1 for the actual number of eight-fold coincidences registered in each case). (b) The experimental setup used to generate maximally entangled photon pairs. More details concerning the generation of entangled photons can be found in Appendix pp2. (c) The witness analyzer. An arbitrary observable $\mathcal{O}$ can be represented as $\mathcal{O}=|i_{+1}⟩⟨i_{+1}|-|i_{-1}⟩⟨i_{-1}|$, where $|i_{\pm 1}⟩$ is the eigenstate of $\mathcal{O}$ with an eigenvalue of $\pm 1$. The combination of QWP and HWP, as well as a PBS, makes $|i_{+1}⟩$ click on the transmissive detector and $|i_{-1}⟩$ click on the reflective detector.

Figure 2

Interferometric geometries leadings to different entanglements. Here, $|{\mathrm{GHZ}}_n⟩$ is an example of a graph state and can be represented by a star graph. Such a representation makes its preparation procedure evident: Each node represents a photon prepared in the state $|+⟩=(1/\sqrt{2})(|H⟩+|V⟩)$, and each edge joining two nodes represents a controlled-$Z$ operation performed between the corresponding photons. The number inscribed in each node labels the path mode of the photons prepared in our experiment. The PBS is fixed on a lifting platform. By adjusting the lift height, we can switch the state of the PBS between up and down. For each PBS set to up, since we postselect the cases where photons exit from both output ports of the PBS, the count rate reduces approximately by half compared with the case when the PBS is set to down.

Figure 3

Experimental results for witnesses certifying genuine multipartite entanglement (GME), entanglement intactness, and entanglement depth. (a) GME revealed by the measurement of the witness of Eq. (3) via $⟨{\mathcal{M}}_Z⟩$ and $⟨{\mathcal{M}}_X⟩$. (b) Entanglement intactness revealed by the measurement of $⟨{\mathcal{W}}_8^{se}(\alpha )⟩$ via $⟨{\mathcal{M}}_Z⟩$ and $⟨{\mathcal{M}}_X⟩$ for a judicious choice of $\alpha$. (c) Entanglement depth revealed by the measurement of $⟨{\mathcal{W}}_8^{de}(\gamma )⟩$ via $\mathcal{A}$ and ${\mathcal{A}}^{\prime }$ for a judicious choice of $\gamma$. For comparison, we have also included here with the red (blue) dot the theoretical value of the witness for an ideal $|G_{53}⟩$ ($|G_{71}⟩$) assuming the same set of measurements.

Figure 4

Experimental results leading to the determination of entanglement structure. (a) Expectation value of the four-partite GME witness ${\mathcal{W}}_4=2[{(|H⟩⟨H|)}^{\otimes 4}+{(|V⟩⟨V|)}^{\otimes 4}]+{\sigma }_X^{\otimes 4}$ for all four-party subsystems of ${\rho }_{422}$. (b) Expectation value of the two-party GME witness ${\mathcal{W}}_2=2[{(|H⟩⟨H|)}^{\otimes 2}+{(|V⟩⟨V|)}^{\otimes 2}]+{\sigma }_X^{\otimes 2}$ for all two-party subsystems among the path modes $\{1,2^{\prime },3^{\prime },4\}$ of ${\rho }_{422}$. (c) Expectation value of the two-party GME witness ${\mathcal{W}}_2$ for all two-party subsystems of ${\rho }_{2222}$.

Figure 5

Numerically determined $k$-producible bounds ${\beta }_{8,k}(\gamma )$ for $k=1,2,\dots ,7$ and for all $\gamma \in (0,2]$.

Figure 6

The detailed experimental setup.

Figure 7

Calculation of the influence of the visibilities for our measurements of $⟨{\mathcal{W}}_{se}^8(\alpha =2)⟩$ (top four plots) and $⟨{\mathcal{W}}_{de}^8(\gamma =2)⟩$ (bottom four plots) with respect to ${\rho }_8$, ${\rho }_{62}$, ${\rho }_{44}$, and ${\rho }_{422}$ (from left to right). The straight white line represents combinations of $v_1$ and $v_2$, where the values of the witnesses for $m$-separability and $k$-producibility (for appropriate values of $m$ and $k$, respectively) are saturated [see Eq. (b4)]. The black circle marks the visibilities observed in our experiment, $v_1=0.967$ and $v_2=0.867$.

Figure 8

Calculation of the noise tolerance for our measurement of $⟨{\mathcal{W}}_{se}^8(\alpha =2)⟩$ (top four plots) and $⟨{\mathcal{W}}_{de}^8(\gamma =2)⟩$ (bottom four plots) with respect to ${\rho }_8$, ${\rho }_{62}$, ${\rho }_{44}$, and ${\rho }_{422}$ (from left to right). The straight line represents combinations of noise parameters where the values of the witnesses for $m$-separability and $k$-producibility (for appropriate values of $m$ and $k$, respectively) are saturated [see Eq. (b5)]. Here, ${\gamma }_w^{(8)}({\gamma }_d^{(8)})$, ${\gamma }_w^{(6)}({\gamma }_d^{(6)})$, ${\gamma }_w^{(4)}({\gamma }_d^{(4)})$ denote the probability of white noise (decoherence noise) in preparing $|GH{Z}_8⟩$, $|GH{Z}_6⟩$, and $|GH{Z}_4⟩$. The ${\gamma }_w^{(2)}$ are negligible compare to ${\gamma }_w^{(4)}$ and ${\gamma }_w^{(6)}$, so we do not consider them in the calculation on ${\rho }_{422}$, ${\rho }_{44}$, and ${\rho }_{62}$. For example, in the rightmost plots (for ${\rho }_{422}$), the straight line corresponds to the combination of ${\gamma }_d^{(4)}$ and ${\gamma }_w^{(4)}$ such that the 4-separable bound (upper plot) and the 3-producible (lower plot) bound are saturated. The black circle marks the noise parameters estimated from our measured value of $⟨{\mathcal{M}}_X⟩$ and $⟨{\mathcal{M}}_Z⟩$.

Figure 9

Calculation of the noise tolerance for our measurement of $⟨{\mathcal{W}}_{se}^8(\alpha =2)⟩$ and $⟨{\mathcal{W}}_{de}^8(\gamma =2)⟩$ on ${\rho }_{2222}$.

Figure 10

Procedure to deduce the underlying entanglement structure. Each orange box represents steps that only involve classical calculation (i.e., no consumption of any quantum resource is required). The blue box represents steps where both quantum resource and classical calculations are needed.

Figure 11

Estimated value of the three-body entanglement witness $⟨{\mathcal{W}}_{se}^3(\alpha )⟩$ for $\{1,2^{\prime },3^{\prime },4\}$ of ${\rho }_{422}$ based on the data acquired during the measurements of $⟨{\mathcal{W}}_{se}^8(\alpha )⟩$.

Figure 12

Results pertinent to the entanglement structure deduction for ${\rho }_{62}$.

Figure 13

Results pertinent to the entanglement structure deduction for ${\rho }_{44}$.