Abstract
Quantum entanglement is the key resource for quantum information processing. Deviceindependent certification of entangled states is a long standing open question, which arouses the concept of selftesting. The central aim of selftesting is to certify the state and measurements of quantum systems without any knowledge of their inner workings, even when the used devices cannot be trusted. Specifically, utilizing Bell’s theorem, one can infer the appearance of certain entangled state when the maximum violation is observed, e.g., to selftest singlet state using CHSH inequality. In this work, by constructing a versatile entanglement source, we experimentally demonstrate a generalized selftesting proposal for various bipartite entangled states up to four dimensions. We show that the highquality generated states can approach the maximum violations of the utilized Bell inequalities, and thus, their Schmidt coefficients can be precisely inferred by selftesting them into respective target states with nearunity fidelities. Our results indicate the superior completeness and robustness of this method and promote selftesting as a practical tool for developing quantum techniques.
Introduction
In contrast to theoretical schemes of quantum information processing (QIP), where the imperfections of the involved devices are generally not taken into account, practically we often do not have sufficient knowledge of the internal physical structure, or the used devices cannot be trusted. The researches on this topic open a new realm of quantum science, namely, “deviceindependent” science,^{1,2,3,4,5,6,7,8,9} in which no assumptions are made about the states under observation, the experimental measurement devices, or even the dimensionality of the Hilbert spaces where such elements are defined. In this case, the only way to study the system is to perform local measurements and analyze the statistical results. It seems to be an impossible task if we still want to identify the state and measurements under consideration. However, assuming quantum mechanics to be the underlying theory and within the nosignalling constraints, a purely classical user can still infer the degree of sharing entanglement due to the violation of the Bell inequalities,^{10,11,12} simply by querying the devices with classical inputs and observing the correlations in the classical outputs.
Such a deviceindependent certification of quantum systems is titled “selftesting”, which was first proposed by Mayers and Yao to certify the presence of a quantum state and the structure of a set of experimental measurement operators.^{13} In the past decade, although different quantum features, such as the dimension of the underlying Hilbert space,^{14,15} the binary observable^{16} or the overlap between measurements^{17} can also be tested deviceindependently, most previous researches focus on the problem of certifying the quantum entangled state that is shared between the devices.^{18,19,20,21,22} In particular, intensive studies have been devoted to the maximally entangled “singlet” state, which is the cornerstone for QIP. Meanwhile, a large amount of progress has been made overall on the selftesting of other forms of entangled states. For example, Yang and Navascué propose a complete selftesting certification of all partially entangled pure twoqubit states.^{23,24} In addition, the maximally entangled pair of qutrits,^{25} the partially entangled pair of qutrits that violates maximally the CGLMP3 inequality,^{26,27} a small class of higherdimensional partially entangled pairs of qudits,^{28} and multipartite entangled states^{6,29} and graph states^{30} are also shown to be selftestable. Another interesting application is the possibility of selftesting a quantum computation, which consists of selftesting a quantum state and a sequence of operations applied to this state.^{31}
Selftesting aims to deviceindependent certifications of entangled states from measured Bell correlation. A typical selftesting protocol generally contains two steps, namely, selftesting criterion and selftesting bound. A selftesting criterion underlies the entire protocol, with which one can uniquely infer the presence of a particular ensemble of entangled states, when observing the maximal violation of certain Bell inequality. These states should be different from each other by only a local unitary transformation, which does not change the Bell correlations. In a selftesting language, all the states in this ensemble have the same Schmidt coefficients and are identical up to local isometries. The state certification becomes much more complex if a nonmaximal violation occurs in a blackbox scenario. This nonmaximal violation can result from a strongly nonlocal state with a nonoptimal measurement, or a weakly nonlocal state with a (nearly) optimal state. As a result, one cannot infer the Schmidt coefficients of the tested state from the given criterion. It is of practical significance if one can still give a commitment about the lowest possible fidelity to the target state violating the inequality maximally, namely, selftesting bound. Combining the criterion and this lower bound, selftesting can be used to test the quality of entanglement in a deviceindependent way.
Previous selftesting protocols for entangled states are limited to several special maximally entangled states of lowdimension. A further generalization to nonmaximally entangled states^{32} and multidimensional quantum entanglement^{33,34,35} enables selftesting to be a more powerful tool in practical quantum information processes. Recently, A. Coladangelo et al.^{36} have provided a general method to selftest all pure bipartite entangled states by constructing explicit correlations, which can be achieved exclusively by measurements on a unique quantum state (up to local isometries). In other words, this generalized method allows complete certifications not only for singlet state, but also for any pure bipartite entangled state of arbitrary dimensions. The criterion presented above is still a proof of ideal selftesting, which only considers ideal situations in which the correlations are exact. Practically, however, the robustness to statistical noises and experimental imperfections is essential for selftesting proposals. Despite of considerable progresses made in selftesting theory, the relevant experimental work is very rare due to the weak robustness. Recently, selftesting has been used to estimate the quality of a largescale integrated entanglement source.^{37} Nevertheless, it is necessary to verify the reliability of a certain selftesting criterion and relevant lower bound before utilizing them. In this work, under the fair sampling assumption, we implement a proofofconcept experiment of Coladangelo’s complete selftesting protocol with various entangled states up to four dimensions. We show that even in the appearance of practical imperfections, we can give a trustable description of the tested states, i.e., to quantify how far the tested state from the pure target state can be, in terms of fidelity, when the violation is not maximal.
Results
Theoretical framework
The selftesting of arbitrary entangled twoqubit systems was resolved by Yang and Navascué.^{23} In their work, they considered a scenario in which Alice and Bob share a pure twoqubit entangled state φ〉, and one can record the probabilities in a [{2, 2}, {2, 2}] Bell measurement, in which both Alice and Bob have two possible measurement settings with binary outcomes.
For a general twoqubit entangled state
it has been proved that it can maximally violate a particular family of Bell inequalities,^{32} parametrized as
where 0 ≤ α ≤ 2 and the maximum quantum violation of it is \(b(\alpha ) = \sqrt {8 + 2\alpha ^2}\).
If such Bell correlations are duplicated by Alice and Bob on an unknown state φ〉, it is possible to construct an isometry satisfying that follows equations^{23}
Π_{A}, M_{A} and Π_{B}, M_{B} here stand for projective operators on the Alice and Bob sides.
From these relations, it can be concluded that these Bell correlations can selftest φ_{target}(θ)〉. Intuitively, selftesting means proving the existence of local isometries which extract the target state φ_{target}(θ)〉 from the physical state φ〉.
Hereby, Yang and Navascué gave the criterion to selftest all pure twoqubit states: When one observes a Bell correlation causing b(α_{0}) − 〈β(α_{0})〉 = 0, the corresponding quantum state is identical, up to local isometries, to a certain entangled state φ_{target}(θ)〉 and and θ is determined by the following equation:
Hence, it is proved that any pure twoqubit state can be selftested,^{23} since its violation of Bell inequality Eq. (2) is unique, up to local isometry. The authors also show that it is possible to generalize this method to bipartite highdimensional maximally entangled states. However, it is still unclear whether arbitrary pure bipartite entangled state is selftestable.
Recently, A. Coladangelo et al. successfully addressed this longstanding open question by constructing explicit correlations built on the framework outlined by Yang and Navascués.^{36} The concrete process of this selftesting process is illustrated by Fig. 1. The uncharacterized devices are assigned to Alice and Bob, and they share a pure state ψ〉 which is identical to (up to local isometries):
Initially they receive the inputs x and y deciding their choice of measurement settings and return the outcomes a and b, respectively. Consider a [{3, d}, {4, d}] Bell scenario, in which Alice has three possible measurement settings and Bob has four, all of which have d possible outcomes. The measured statistics are recorded in the form of probabilities P(a, bx, y). Similarly to twoqubit scenario, if the [{3, d}, {4, d}] quantum correlations for bipartite entangled state of qudits are reproduced through local measurements on a joint state ψ〉, it is possible to construct a local isometry Φ such that Φ(ψ〉) = auxiliary〉 ⊗ ψ_{target}〉. The central idea to selftest highdimensional states is to decompose ψ〉 into two series of 2 × 2 blocks, and thus, each block can be selftested through the twoqubit criterion described above.^{23} Referring to the fourdimensional states we test in the experiment, this decomposition can be implemented as shown in Fig. 1b. Grouping P(a, bx, y) elements of which x, y ∈ 0, 1, with a, b ∈ 0, 1 and 2, 3, one can selftest c_{0}00〉 + c_{1}11〉 and c_{2}22〉 + c_{3}33〉, respectively. Similarly, using measurement settings x ∈ 0, 2 and y ∈ 2, 3, one can selftest c_{1}11〉 + c_{2}22〉 and c_{0}00〉 + c_{3}33〉. Such a decomposition procedure can determine the relative ratio of the two modes in each block. The weight of each block in the joint state can be further estimated by the total photon counting on this block, and eventually, the form of the tested state can be inferred.
Experimental implementation and results
In order to implement this generalized selftesting criterion, a versatile entangled photonpair source is constructed which can generate bipartite states up to four dimensions, as shown in Fig. 2. Initially entangled photon pairs are generated by pumping a PPKTP crystal in a Sagnac interferometer (SI).^{38} Afterward, both photons are encoded into polarization and path modes, and therefore, the joint twophoton can be flexibly prepared into the product, twoqubit, twoqutrit, and twoqudit entangled states (see Methods for details). Accordingly, the detecting apparatus is also properly designed in order to perform projective measurements on these two modes. Profiting from this setup, we can implement both selftesting and state tomography on the tested unknown states.
As discussed above, the twoqubit selftesting criterion is the foundation for the entire theoretical frame. First, we test various of twoqubit states through [{2, 2}, {2, 2}] Bell measurements. In the experiment, we initially generate four twoqubit entangled states ρ_{j} (j = 0, 1, 2, 3), each of which is close to a pure state φ_{target}(θ_{j})〉. The value of θ_{j} is selected to maximize the fidelity 〈φ_{target}(θ_{j})ρ_{j}φ_{target}(θ_{j})〉.
For each ρ_{j}, three randomly generated local operators are applied, leading to a ensemble three unknown states ρ_{j(k)}(k = 0, 1, 2) that we want to selftest. For a certain ρ_{j(k)}, we perform Bell measurements and measure its violation 〈β(α_{j})〉 with \(\alpha _j = 2{\mathrm{/}}\sqrt {2{\mathrm {tan}}^2\left( {2\theta _j} \right) + 1}\) (the inverse function of Eq. (5)). As b(α_{j}) − 〈β(α_{j})〉 approaches 0, a selftesting conclusion can then be reached that the tested state have the same Schmidt coefficients with φ_{target}(θ_{j})〉. In principle, the three states in each ensemble all violate the inequality β(α_{j}) maximally. In practical experiment, due to the imperfections in state preparing and unitary transformations, each realistic state is slightly mixed and the measured ultimate violation approaches b(α_{j}) with a tiny difference. When these imperfections are negligibly small, we can either assume the tested state to be an ideally pure state and infer the Schmidt coefficients of it from the given selftesting criterion, or give a lowest possible fidelity F_{S} to the target state according to the selftesting bound (blue squares in Fig. 3).
For four target states investigated here, we give the selftesting bounds through a semidefinite program (SDP) following the method of refs ^{22,23}, and the results are shown in Fig. 3. The selftesting bound suggests a nearly unity fidelity when the observed violation approaches the maximum. The tightness of the calculated bound becomes weaker when the tested states tend to be a product state. This is expected because, as α increases, the range of tolerable error gets smaller since the local bound and quantum bound coincide at α = 2.^{22}
These selftesting conclusions can be verified by measuring the the actual fidelity F_{T} of ρ_{j(k)} to its corresponding target state φ_{target}(θ_{j})〉, which can be calculated as F_{T} = 〈φ_{target}(θ_{j})ρ_{j(k)}φ_{target}(θ_{j})〉. For each ρ_{j(k)}, the density matrix is reconstructed through a state tomography process. These actual fidelities are also labeled in Fig. 3 (red circles in Fig. 3), all of which are well above F_{S} on selftesting bound. This result clearly suggests that the given selftesting criterion is valid for deviceindependent certification of twoqubit pure states, and the lower fidelity bound is also reliable for entanglement quality verification in a blackbox scenario.
For highdimensional scenarios, we decompose the joint state into several 2 × 2 blocks, each of which can be selftested into a pure target state with high fidelity, similar to the procedures implemented for twoqubit scenarios. The probabilities needed to selftest these blocks are measured according to the strategy shown in Fig. 1b. When d = 4, in the selftesting process, Alice has 12 possible projective measurements, and Bob has 16, which results in a total of 192 values of P(a, bx, y) (see Supplementary Note 1 for details). For each block, 16 values of P(a, bx, y) are recorded and the parameter α is ascertained to minimize b(α) − 〈β(α)〉, and thus, the pure target state for each block is determined. Totally, 64 measurements are required to selftest the four decomposed blocks to be the closest pure target states. The summation of the counts in each group of 16 projective measurements can be used as the weight of the corresponding block. Therefore, the four twoqubit target states comprise the joint target state ψ_{target}〉 with respective weights. Considering the fact that only a nonmaximum violation is attained in selftesting each block, this inferred joint states should also deviate from the actual tested states. In principle, a highdimensional selftesting bound can also be obtained from an SDP method. However, due to the appearance of significantly higher order moments in the expression of fidelity, the computing complexity is far beyond the limits of a single computer (see Supplementary Note 3 for details). Nevertheless, we can still verify the reliability of the selftesting results by measuring the density matrix of the tested state and calculating its fidelity F_{T} to inferred target state ψ_{target}〉. We investigate totally 10 highdimensional bipartite entangled states (see Supplementary Note 2 for details), and the selftesting results of their pure target states are shown in Fig. 4. The tested states are all approximately identical (up to local isometries) to ψ_{target}〉 with a high fidelity F_{T}.
Deviceindependent certifications require nosignalling constraints^{39} on the devices, which can be tested through the influence of the measurement of Alice (Bob) side on Bob (Alice) side.^{12} Concretely, nosignalling constraints require the following relations to be satisfied:
In experiment, all the 192 probabilities need to be recorded to verify above equations. In Supplementary Note 4, we give the results when the tested state is ψ_{0}〉.
Discussion
Normally, a selftesting criterion is feasible for an ideal case where the involved Bell inequality can be maximally violated by a pure target state. However, due to unavoidable errors in experiment, the realistic states cannot attain the maximum violation, and thus, it becomes difficult to give an exact description about the states. Fortunately, it is still possible to lowerbound the fidelity to the pure target state as the function of Bell violation. Especially, we show that when the utilized Bell inequalities are almost maximally violated, the generated states from our versatile entanglement source can be selftested into respective target states with pretty high fidelities. We shall note that, as an initial experimental work for selftesting, our experiment constitutes to the proof of principle studies of selftesting. How to close the loopholes of Bell measurement in selftesting, such as detection, locality, and freewill loopholes, are interesting open questions for future works.
There are several ways that a purely classical user can certify the quantum state of a system, among which the standard quantum tomography is the most widely used method. However, it depends on characterization of the degrees of freedom under study and the corresponding measurements, thereby, it becomes invalid if the devices cannot be trusted. Furthermore, it requires performing and storing data from an exponentially large number of projection measurements. Recently, Chapman et al. proposed a selfguided method which exhibits better robustness and precision for quantum state certification.^{40} Unsurprisingly, all of these benefits rely on the ability to fully characterize the measurement apparatus. The presence of selftesting suggests that the state certification can also be implemented in a deviceindependent way. Two main obstacles preventing selftesting from practical applications are the weak robustness and an extremely small quantity of selftestable states, and thus, the relevant experimental work is very rare. Thanks to the consecutive efforts by Yang, Navascue′, Coladangelo, Goh and Scarani, a complete selftesting proposal has been raised for all pure bipartite quantum states utilizing a [{3, d}, {4, d}] Bell measurement. We experimentally demonstrate this proposal with our highquality entanglement source and measurement apparatus. For all of the tested scenarios, the obtained results exhibit excellent precision and robustness. Due to this significant progress, selftesting can now be applied to realistic quantum technologies.
Methods
Versatile entanglement source
The versatile entanglement source consists of two main parts. One part is responsible for the generation of polarization entangled photon pairs, and the other part is in charge of entangling the two photons in both polarization and path modes to form highdimensional states. In the first part, a 405 nm continuouswave diode laser with polarization set by a halfwave plate is used to pump a 4 mmlong PPKTP crystal inside an SI to generate polarizationentangled photons. The photon pairs are in the state cos θHH〉 + sin θVV〉 (H and V denote the horizontal and vertical polarized components, respectively) and θ is controlled by the pumping polarization. The visibility of the maximally entangled state is larger than 0.985. These photon pairs are then sent into the second part by two single mode fibers, and the polarization is maintained by two HWPs before and after each fiber. The relative phase difference due to the birefringence in single mode fibers can be compensated when adjusting the interference between the pairs of BDs. The path mode is then created by BD1 and BD2, which causes an ~4 mm separation between the H and V components at 810 nm. At this stage the path mode is entangled with the polarization mode, and therefore, the joint state remains twodimensional. After BD1 and BD2, an HWP is inserted on each path and totally four HWPs are employed for statepreparing. A centering PBS, serving as a Bellstate synthesizer,^{41} leads to a coincidence registration of a single photon at each output and generates polarization entanglement on both path modes (a and b). This is because, there is no way, even in principle, to distinguish between the two possibilities HH〉 and VV〉 if all the other information (such as time, frequency, and spatial mode) is erased. The arriving time of the two photons on PBS is accurately synchronized by adjusting the position of the FC collimator and the indistinguishability is tested by measuring the visibility of the Hong–Ou–Mandel interference. The main limiting factor for this visibility is the bandwidth difference from the type II SPDC process, which results in a best visibility of the polarization entanglement to be ~0.975. The number of dimensions of the outcome states after the PBS can be selected by the alignments of the four statepreparing HWPs after BD1 and BD2. Concretely, when the four HWPs are set to be 0° or 45°, the outcome is a path mode entangled state cos θaa〉 + sin θbb〉. When HWPs rotate single H or V polarized photons to a superposition state αH〉 + βV〉(α, β ≠ 0), the result is a twoqutrit state ψ〉(d = 3) = c_{0}00〉 + c_{1}11〉 + c_{2}22〉. When both H and V polarized photons are rotated to be superposition states, we can finally obtain a twoqudit state ψ〉(d = 4) = c_{0}00〉 + c_{1}11〉 + c_{2}22〉 + c_{3}33〉. All of the coefficients c_{i} are decided by the rotating angle of the four HWPs and the value of θ. The encoding rule uses 0, 1, 2, 3 to denote Ha〉, Va〉, Hb〉, Vb〉 components, respectively. In order to attain a larger state space, local unitary operations can be applied by the waveplate sets after the crossing PBS, which lead to other entangled states of the same dimensions.
Measurement apparatus
The measurement apparatus in this experiment is properly designed so as to perform both highdimensional quantum tomography and selftesting. On both the Alice and Bob sides, the measurement apparatuses have an array of QWPHWP on both the a and b paths, which is in charge of making projective measurements on the polarization mode. A subsequent BD (BD3 and BD4) combines the two separated paths, and a QWPHWP set with a PBS are responsible for the measurements on the path mode. All the waveplates are mounted in the programmable motorized rotation stages. It is obvious that the two pairs of BDs compose Mach–Zehnderlike interferometers. The distance between two BDs in one pair is about 40 cm, and the optical phase is stabilized by isolating the entire setup from the environment by a sealed box. Each BD used here is selected to be most compatible with its partner thus a high interference visibility of above 0.999 can be attained. The used four BDs are selected from a total number of 12 same products. For a fourdimensional state, we perform 256 joint projective measurements for state tomography and 64 for selftesting.
Data availability
The authors declare that all data supporting the findings of this study are available within the article and its Supplementary Information Files or from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the National Key Research and Development Program of China (Nos. 2017YFA0304100, 2016YFA0302700), National Natural Science Foundation of China (Grant Nos. 11874344, 61835004, 61327901, 11774335, 91536219), Key Research Program of Frontier Sciences, CAS (No. QYZDYSSWSLH003), the Fundamental Research Funds for the Central Universities (Grant No. WK2030020019, WK2470000026), Anhui Initiative in Quantum Information Technologies (AHY020100).
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W.H.Z. and P.Y. made the calculations assisted by X.J.Y. and Y.J.H. C.F.L. and G.C. planned and designed the experiment. W.H.Z. carried out the experiment assisted by G.C., X.X.P., X.M.H., Z.B.H., S.Y., Z.Q.Z., B.H.L., and X.Y.X. whereas J.S.T. and J.S.X. designed the computer programs. W.H.Z. and G.C. analyzed the experimental results and wrote the manuscript. G.C.G. and CF.L. supervised the project. All authors discussed the experimental procedures and results.
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Zhang, WH., Chen, G., Yin, P. et al. Experimental demonstration of robust selftesting for bipartite entangled states. npj Quantum Inf 5, 4 (2019). https://doi.org/10.1038/s4153401801200
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