Experimental verification of bound and multiparticle entanglement with the randomized measurement toolbox

Quantum entanglement is one of the most prominent non-classical features of quantum mechanics and often viewed as a resource in quantum information processing [1]. Its generation and characterization is of growing interest from both practical and fundamental perspectives. While deciding whether a given quantum state is entangled or not is in general a hard task [2], many experimentally feasible schemes exist that verify entanglement in some states.

A prominent example for such schemes are entanglement witnesses, which allow for rather simple detection of entanglement using few measurements, whereas other schemes detect non-locality by evaluating some Bell-type inequalities [3]. On the experimental side, numerous entangled states have been generated and multi-qubit entanglement [4, 5], high-dimensional entanglement of two particles [6, 7, 8], and also bound entanglement [9, 10, 11, 12, 13] has been characterized.

When applying the standard criteria in a practical experiment, however, one always needs to align the local measurement settings strictly or to make some assumptions on the target state to prepare, e.g., by tailoring a witness specifically for states close to some fixed target state. To remedy this, several schemes based on the moments of randomized correlations have been proposed [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. They provide an efficient way to characterize multi-particle correlations in states without prior knowledge about the state, nor any alignment of measurement directions. Recently, it has been shown that this approach also allows for the detection of bound entanglement [18].

In this paper, we implement in a photonic setup the randomized measurement scheme to detect entanglement in mixtures of three-qubit GHZ and W-states using second moments of the random outcomes. Furthermore, we prepare bound entangled chessboard states of two qutrits and show their entanglement by evaluating an entanglement criterion which is based on the second and fourth moment of a randomized measurement outcome, without implementing the random unitaries explicitly. This demonstrates that the criterion from Ref. [18] is indeed strong enough to capture this weak form of entanglement, even in the presence of noise and experimental imperfections. Our implementation combines the photon’s polarization and path degrees of freedom to generate precisely controlled high-dimensional states and demonstrates the versatility and efficiency of the randomized measurement approach.

In the randomized measurement scheme [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], a subset $S\subset\{1,\ldots,n\}$ of the parties of an $n$-partite quantum state $\rho$ of fixed local dimension $d$ is measuring some fixed, local observables in random directions. The moments of the distribution of measurement results can be written as

where the $\tau_{i}$ denote the local observables, and $\tau_{i}=\mathbb{I}$ whenever $i\notin S$. The integrals are evaluated over the Haar measure of the unitary group $\mathcal{U}(d)$. In case of qubit systems, one usually chooses $\tau_{i}=\sigma_{z}$ for $i\in S$, in which case the second moments ($t=2$) are related to the purities of the reduced states of $\rho$. The sum of second moments for all subsets $S$ of size $|S|=k$ is proportional to what is known as the $k$-sector length of the state [28, 29, 30, 31, 32, 33]. In particular, for three qubits the sector lengths $A_{k}$ are given by

Decomposing $\rho$ in terms of the local Pauli basis $\{\sigma_{0}=\mathbb{I},\sigma_{1}=\sigma_{x},\sigma_{2}=\sigma_{y},\sigma_{3}=\sigma_{z}\}$, yields

and allows to express the sector lengths in terms of the coefficients $\alpha_{ijk}$ as follows: $A_{1}=\sum_{i=1}^{3}(\alpha_{i00}^{2}+\text{perm.})$, $A_{2}=\sum_{i,j=1}^{3}(\alpha_{ij0}^{2}+\text{perm.})$, and $A_{3}=\sum_{i,j,k=1}^{3}\alpha_{ijk}^{2}$.

In terms of the sector lengths, several entanglement criteria exist that detect certain entangled states. To proceed, let us recall that a three-particle state $\rho_{ABC}$ is called biseparable for a partition $A|BC$ if $\rho_{A|BC}=\sum_{k}q_{k}^{A}\rho_{k}^{A}\otimes\rho_{k}^{BC},$ where the positive coefficients $q_{k}^{A}$ form a probability distribution. Similarly, the biseparable states $\rho_{B|CA}$ and $\rho_{C|AB}$ can be defined. Moreover, we can consider the mixture of biseparable states for all partitions as

where $p_{A},p_{B},p_{C}$ are probabilities. A quantum state is called genuinely multipartite entangled (GME) if it cannot be written in the form of $\rho_{\text{bisep}}$.

For three-qubit states, if $A_{3}>3$, the state must be GME (the maximal value being $A_{3}=4$ for the GHZ state $\ket{\text{GHZ}}=\frac{1}{\sqrt{2}}(\ket{000}+\ket{111}$). A stronger version exists, which states that if

the state cannot be biseparable w.r.t. any fixed partition, and strong numerical evidence exists that in that case, even GME states must be present [18]. In this paper, we aim to detect entanglement in a mixture of a GHZ and a W state, given by

where $g\in[0,1]$ denotes the amount of mixing and $|\text{W}\rangle=\frac{1}{\sqrt{3}}(\ket{001}+\ket{010}+\ket{100})$.

The family of states $\rho(g)$ exhibits some interesting properties. First, it is supported in the symmetric subspace. This implies that $F_{XY}\rho(g)=\rho(g)F_{XY}=\rho(g)$, where $F_{XY}=\sum_{i,j}\ket{ij}\!\bra{ji}_{XY}$ is the flip (swap) operator acting on the subsystems $XY\in\{AB,BC,CA\}$. It is known that if a state lives in the symmetric subspace, it is either fully separable or GME [34, 35, 36, 37, 38]. However, the experimentally generated version of the state $\rho(g)$ cannot be assumed to have the symmetry due to experimental imperfections. Accordingly, the generated state can become biseparable, thus, we employ the criterion in Eq. (7) to detect its entanglement. We stress again that the criterion in Eq. (7) has been conjectured to imply the presence of GME from numerical evidence, but its analytical proof has not yet been provided [18]. That is, even if the criterion Eq. (7) is verified experimentally, the state may be entangled for any fixed partition, but it can be a mixture of at least three biseparable states for different bipartitions.

Second, when the parameter $g$ is outside the region of $0.297\leq g\leq 0.612$, the criterion in Eq. (7) is satisfied. This parameter region is very close to other well-known regions using two other entanglement measures [39, 40]. On the one hand, the three-tangle $\tau$ vanishes for $0\leq g\leq g_{\tau}\approx 0.627$, where $\tau$ measures residual (three-partite) entanglement that cannot be expressed as two-body entanglement [41]. Note that the GHZ state maximizes the three-tangle, while it vanishes for the W state. On the other hand, the sum of squared concurrences $C_{A|B}^{2}+C_{A|C}^{2}$ vanishes for $g_{C}\approx 0.292\ldots\leq g\leq 1$, where the concurrence $C_{X|Y}$ measures bipartite entanglement in the reduced state between the parties $X$ and $Y$ [42]. Hence, we can conclude that the criterion in Eq. (7) can detect the multi-partite entanglement of $\rho(g)$ even in regions where the three-tangle and the concurrence vanish, if the parameter $g$ satisfies $g_{C}\leq g<0.297$ or $0.612<g\leq g_{\tau}$.

In contrast to qubit systems, the second moments of higher-dimensional states are not automatically related to sector lengths. In fact, the choice of the local observables influences which local unitary invariants can be extracted from the moments [22]. Let us expand a bipartite quantum state of dimension $d$ in terms of some local, hermitian operator basis $\{\lambda_{i}\}_{i=0}^{d^{2}-1}$ with $\lambda_{0}=\mathbb{I}$, $\operatorname{Tr}(\lambda_{i}\lambda_{j})=d\delta_{ij}$, such as the Gell-Mann basis [43, 44, 45]. Then

is called the generalized Bloch decomposition of $\rho$, where the matrix $T$ is known as the correlation matrix of $\rho$. For this matrix, many entanglement criteria exist, most notably the de Vicente criterion [46], stating that for separable states, $\operatorname{Tr}(|T|)\leq d-1$. While the left-hand side is not directly accessible from the moments of randomized measurements, it is possible to obtain related quantities by carefully choosing the observables $\tau_{i}$ as detailed in Ref. [21], such that

For example, for $d=3$, $\tau_{i}=\operatorname{diag}(\sqrt{3/2},0,-\sqrt{3/2})$. The combined knowledge of these two quantities allows to detect entanglement, whenever it is incompatible with the de Vicente criterion, i.e., if the measured value of $\mathcal{R}^{(4)}_{AB}$ is below the minimum given by

Note that this lower bound can also be calculated analytically [21]. Interestingly, there exist states which have a positive partial transpose, but can be detected to be entangled by these two moments, implying bound entanglement. A $3\times 3$-dimensional state from the chessboard family of bound entangled states described in Ref. [47] (also see Appendix C2 in [18]) has been identified to violate it extremely, which makes it a good candidate to prepare and detect its entanglement experimentally. It is given by

where $N=1/\sum_{i}\langle{V_{i}}|{V_{i}}\rangle^{2}=1/4$ is a normalization factor and

We proceed with a description of the experimental implementation. The GHZ-W mixed states are prepared by resorting to the states entangled in polarization degree of freedom (d.o.f.) and path d.o.f. of the photon (that is, hyper-entangled) and with methods similar to the ones in Refs. [48, 49]. More detailed information about the state preparation of this family of states is given in Appendix A.

When preparing the bound entangled chessboard state, it is important to ensure that all its eigenvalues remain non-negative under partial transposition. However, the chessboard state is not of full rank. Affected by the imperfections of the experiment, slightly negative eigenvalues of the partial transposition are likely to appear. A more robust way is to prepare the state with a level of white noise [10],

First, let us briefly review the state preparation procedure. As depicted in Fig. 1, we generate polarization entangled ($2\times 2$ entangled) photon pairs through a spontaneous parametric down-conversion (SPDC) process. Subsequently, we expand the dimensionality of the system by introducing the path modes $u$ and $l$. This will results in three modes: $H_{u}$, $V_{u}$, and $H_{l}$, where ${H}_{u}$ represents a horizontally polarized photon occupying path $u$, and so on. Finally, specific operations are applied to the system to steer the state to the target ones.

Specifically, a Half-Wave Plate (HWP) H1 with the optic axis placed at $12.05^{\circ}$ is used to rotate a 390 nm horizontally polarized pump laser (with an 80 MHz repetition rate and a 140-fs pulse duration) to state $\ket{\psi_{p}}=\sqrt{5/6}\ket{H}+\sqrt{1/6}\ket{V}$, where $H$ and $V$ represent the horizontal and the vertical polarization, respectively. The pump photon is then split into two photons after pumping two crossed-axis type-I $\beta$-Barium Borate (BBO) crystals in the SPDC process, transforming the state into $\ket{\psi_{p}}\rightarrow\sqrt{5/6}\ket{HH}+\sqrt{1/6}\ket{VV}$. By passing through the Beam Displacers (BDs) BD1 and BD2, the down-converted photons’ $H-$($V-$) components are directed to path $u$ ($l$). And for path mode $u$, we have the mode labeled as $H_{u}$ and $V_{u}$. By re-encoding $\ket{H}_{u}\rightarrow\ket{0}$, $\ket{V}_{l}\rightarrow\ket{1}$, and $\ket{V}_{u}\rightarrow\ket{2}$, we obtain the hyper-entangled state $\ket{\psi_{s}}=\sqrt{5/6}\ket{H_{u}H_{u}}+\sqrt{1/6}\ket{V_{l}V_{l}}\rightarrow\sqrt{5/6}\ket{00}+\sqrt{1/6}\ket{11}$.

It is worth noting that all the four states $\ket{V_{i}}$ in Eq. (12) can be generated by performing local operations on the state $\ket{\psi_{s}}$,

where

For the states $\ket{V_{3}}$ and $\ket{V_{4}}$, it also works by applying the unitary $U_{3}\otimes\mathbb{I}$, and $U_{2}\otimes U_{1}$, respectively, and then exchanging the labels for the two detectors D1 and D2. Therefore, through performing the operator $U_{3}$ or $U_{2}$ on one photon of a pair and the operator $U_{1}$ or $\mathbb{I}$ on the other photon simultaneously, the state $\ket{\psi_{s}}$ will be transformed to each of the four states $\ket{V_{i}}$. The switches between these operators are implemented by the motorized rotating HWPs and Quarter-Wave Plates (QWPs), which are controlled by the pseudo-random numbers generated from a classical computer. Two adjustable LED lights are placed before the detectors to introduce the different levels of white noise into the system.

In the measurement part, a QWP and an HWP located at path $u$ are used to analyze the correlations between basis elements $\ket{0}$ and $\ket{2}$, and now the afterward BD works as a Polarization Beam Splitter (PBS). When measuring the superposition of basis elements $\ket{0}$ and $\ket{1}$, as well as $\ket{2}$ and $\ket{1}$, we first convert the path d.o.f. to the polarization d.o.f. via the wave plates and BDs, and then analyze with the combination of the QWP and the HWP. Detailed settings of the wave plates for standard quantum state tomography are given in Tab. 1 of Appendix B. For each measurement basis, we randomly change the photon states to every one of the four states $\ket{V_{i}}$. The two-photon coincidence counts are recorded per 10 s.

When it comes to measuring the randomized correlations, as elaborated in the theoretical framework, two distinct approaches are considered. The first one involves conducting local randomized measurements, while the second entails the direct application of Pauli operators or Gell-Mann matrices. In this study, we thoroughly examine and contrast these two methodologies for three-qubit states, utilizing a LabVIEW program to facilitate the automation of numerous measurements. Further details regarding the randomized measurement techniques can be found in the Appendix C. For the bound entangled states, we opt to directly measure the 81 combinations of Gell-Mann matrices to avoid the systematic errors that may emerge from the construction of $3\times 3$ random unitaries.

We experimentally produced a variety of genuinely entangled photonic states consisting of entangled photon pairs amended with path degrees of freedom and characterized them using methods based on locally randomized measurements. First, we showed how to generate genuinely entangled states of three parties and verified them using entanglement criteria based only on the second moments of the randomized measurements. The latter enabled the verification of mulitpartite entanglement in regimes where well-known measures of multipartite entanglement, i.e., the three-tangle or the squared concurrence, are zero. Further on, we demonstrated the production of weakly bound entangled chessboard states of two qutrits and used entanglement criteria based on the second and fourth moments of the taken randomized measurements to analyze the produced states. As a result, bound entangled states with mixed-state fidelities beyond $98\%$ were successfully produced and verified.

Our work demonstrates the outstanding control of quantum states in photonic setups and presents an efficient way for preparing a low-rank bound entangled state. By incorporating appropriate white noise, the setup demonstrates increased robustness against transitioning into the free entangled region. Compared with several previous experiments, the precise control allowed us to directly verify bipartite bound entanglement in minimal case of a $3\times 3$ system, without resorting to the various forms of bound entanglement in higher dimensions or in multiparticle systems. This will facilitate further exploration of interesting entanglement effects in experiments.

We thank Xiao-Dong Yu for discussions. The work in USTC is supported by the National Natural Science Foundation of China (Nos. 11821404, 11734015, 62075208), the Fundamental Research Funds for the Central Universities (Nos. WK2030000061, YD2030002015), and the Innovation Program for Quantum Science and Technology (No. 2021ZD0301604). Y.Z. is support by the Major Key Project of PCL. S.I. and O.G. are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, project numbers 447948357 and 440958198), the Sino-German Center for Research Promotion (Project M-0294), the ERC (Consolidator Grant 683107/TempoQ), and the German Ministry of Education and Research (Project QuKuK, BMBF Grant No.  16KIS1618K). S.I. acknowledges the support from the DAAD. N.W. acknowledges support by the QuantERA project QuICHE via the German Ministry of Education and Research (BMBF Grant No. 16KIS1119K).