Abstract
Quantum state verification provides an efficient approach to characterize the reliability of quantum devices for generating certain target states. The figure of merit of a specific strategy is the estimated infidelity ϵ of the tested state to the target state, given a certain number of performed measurements n. Entangled measurements constitute the globally optimal strategy and achieve the scaling that ϵ is inversely proportional to n. Recent advances show that it is possible to achieve the same scaling simply with nonadaptive local measurements; however, the performance is still worse than the globally optimal bound up to a constant factor. In this work, by introducing classical communication, we experimentally implement an adaptive quantum state verification. The constant factor is minimized from ~2.5 to 1.5 in this experiment, which means that only 60% measurements are required to achieve a certain value of ϵ compared to optimal nonadaptive local strategy. Our results indicate that classical communication significantly enhances the performance of quantum state verification, and leads to an efficiency that further approaches the globally optimal bound.
Introduction
Quantum information science aims to enhance traditional information techniques by introducing the advantage of ‘quantumness’. To date, the major subfields in quantum information include quantum computation^{1}, quantum cryptography^{2} and quantum metrology^{3,4}, which are respectively in pursuit of more efficient computation, more secure communication, and more precise measurement. To achieve these innovations, one needs to manufacture quantum devices and verify that these devices indeed operate as expected. Various techniques have been developed for the task to inspect the quantum states generated from these devices. Quantum state tomography (QST)^{5} provides full information about an unknown state by reconstructing the density matrix and constitutes a popular point estimation method. However, the conventional tomographic reconstruction of a state is an exponentially timeconsuming and computationally difficult process^{6}. In order to reduce the measurement complexity to certify the quantum states, substantial efforts have been made to formalizing more efficient methods. These improved methods normally require prior information or access partial knowledge about the states. On the one hand, it has been found that with prior information about the category of the tested states, compressed sensing^{7,8} and matrix product state tomography^{9} can be used to simplify the measurement of quantum states. On the other hand, entanglement witnesses can justify the appearance of entanglement with far fewer measurements^{10,11}; in a radical case, it is shown that local measurement on few copies is sufficient to certify the appearance of entanglement for multipartite entangled systems^{12,13}. Furthermore, when the applied measurements are correlated through classical communication, quantum tomography can be implemented in a significantly more efficient way^{14,15,16}.
In quantum information processing, the quantum device is generally designed to generate a specific target state. In this case, the user only needs to confirm that the actual state is sufficiently close to the target state, in the sense that the full knowledge about the exact form of the state is excessive for this requirement. Quantum state verification (QSV) provides an efficient solution applicable to this scenario. As mentioned above, tomography aims to address the following question: What is the state? While QSV addresses a different question: Is the state identical/close to the target state? From a practical point of view, answering this question is sufficient for many quantum information applications. By performing a series of measurements on the output copies of state, QSV reaches a conclusion like ‘the device outputs copies of a state that has at least 1 − ϵ fidelity with the target, with 1 − δ confidence’.
In order to verify a specific quantum state, different kinds of strategies can be constructed, and thus, it is profitable for the user to seek an optimal strategy. Rigorously, this optimization can be achieved by minimizing the number of measurements of n for given values of ϵ and δ. Similar to the realm of quantum metrology^{17,18}, an optimal QSV strategy also strives for a 1/n scaling of ϵ, with a minimum constant factor before. For QSV, if the target state is a pure state, the best strategy is the projection onto the target state and its complementary space, then the 1/n scaling is reached, we call this strategy the globally optimal QSV strategy. Unfortunately, if the target is entangled state, entangled measurements are demanded while they are rare resources and difficult to obtain^{19}. Recently, several works have shown that 1/n scaling can be achieved with a nonadaptive local (LO) strategy^{20,21,22}, the LO here means that the applied measurement operators are separable as oppose to the entangled ones used in globally optimal strategy. However, this nonadaptive LO strategy is still worse than the globally optimal strategy by a constant factor, which represents the number of additional measurements required to compete with the globally optimal strategy.
In this work, we demonstrate adaptive QSV using a photonic apparatus with active bidirectional feedforward of classical communications between entangled photon pairs based on recent theoretical works^{23,24,25}. The achieved efficiency not only attains the 1/n scaling but also further minimizes the constant factor from before. Both bi and unidirectional classical communications are utilized in our experiment, and the results show that these adaptive strategies significantly outperform the nonadaptive LO strategy. Furthermore, the bidirectional strategy achieves higher efficiency than the unidirectional strategy, and the number of required measurements is reduced by ~40% compared to the nonadaptive LO strategy. Our results indicate that classical communication is beneficial resources in QSV, which enhances the performance to a level comparable with the globally optimal strategy.
Results
Theoretical framework
In a QSV task, the verifier is assigned to certify that his onhand quantum device does produce a series of quantum states (σ_{1}, σ_{2}, σ_{3}, …, σ_{n}) satisfying the following inequality:
where \(\left\Psi \right\rangle\) is the target state that the device is supposed to produce. Equation (1) assumes a different scenario from that of QST, for which all σ_{i} are required to be independent and identically distributed.
Typically, with the probability as p_{l}(l = 1, 2, …, m), the verifier randomly performs a twooutcome local measurement M_{l}, which is accepted with certainty when performed on the target state. When all the measurement outcomes are accepted, the verifier can reach a statistical inference that the state from the tested device has a minimum fidelity 1 − ϵ to the target state, with a statistical confidence level of 1 − δ.
For a specific strategy Ω = Σ_{l}p_{l}M_{l}, the minimum number of measurements n required to achieve certain values of ϵ and δ is then given by^{23}
This result indicates that it is possible to achieve the 1/n scaling of ϵ in the QSV of pure entangled states. Furthermore, the verifier can optimize the strategy by minimizing the second largest eigenvalue \({\lambda }_{2}^{\downarrow }(\Omega )\), as well as the constant factor \(\frac{1}{1{\lambda }_{2}^{\downarrow }(\Omega )}\). For LO strategies with nonadaptive local measurements, the optimal strategy to verify \(\left\Psi (\theta )\right\rangle =\cos \theta \leftHV\right\rangle \sin \theta \leftVH\right\rangle\) is identified with a minimum \({\lambda }_{2}^{\downarrow }(\Omega )\) as^{20}
The globally optimal strategy can be realized by projecting σ_{i} to the target state \(\left\Psi \right\rangle\) and its orthogonal state \(\left{\Psi }^{\perp }\right\rangle\), under which \({\lambda }_{2}^{\downarrow }(\Omega )=0\), and thus, the globally optimal bound is calculated as
For QSV of entangled states, entangled measurements are required to implement the globally optimal strategy, which are sophisticated to perform^{26,27,28,29}. Therefore, local measurements are preferred from a practical view of point. This realistic contradiction naturally yields a question that how to further minimize the gap between locally and globally optimal strategies with currently accessible techniques.
Recently, a theoretical work generalizes the nonadaptive LO strategy to adaptive versions by introducing classical communication between the two parties sharing entanglement^{23}. The elementary adaptive strategy utilizes local measurements and unidirectional classical communication (UniLOCC), as diagrammed in Fig. 1. The optimal UniLOCC QSV for \(\left\Psi (\theta )\right\rangle\) can be implemented by randomly choosing M_{1}, M_{2} or M_{3} (see ‘Methods' for details) with prior probabilities \(\{\frac{1}{2+2{\sin }^{2}\theta },\frac{1}{2+2{\sin }^{2}\theta },\frac{{\sin }^{2}\theta }{1+{\sin }^{2}\theta }\}\) (θ ∈ (45^{∘}, 90^{∘})), and the corresponding strategy can be written as^{23}
A bidirectional LOCC (BiLOCC) strategy can be implemented by randomly switching the role between Alice and Bob, which can be denoted as \({\Omega }_{\leftarrow }^{\to }=\left\Psi (\theta )\right\rangle \left\langle \Psi (\theta )\right+\frac{1}{3}(I\left\Psi (\theta )\right\rangle \left\langle \Psi (\theta )\right)\). Although both of these two strategies utilize onestep adaptive measurement, the BiLOCC strategy outperforms the UniLOCC when θ ≠ 45^{∘}.
When verifying entangled states with local measurements, adaptive strategies Ω_{→} and \({\Omega }_{\leftarrow }^{\to }\) achieve higher efficiency compared to the nonadaptive LO strategy^{23,25}. The efficiency of LO, UniLOCC and BiLOCC strategies depend on their respective constantfactors \(\frac{1}{1{\lambda }_{2}^{\downarrow }(\Omega )}\), which are \(2+\sin \theta \cos \theta\), \(1+{\sin }^{2}\theta\) and 3/2. Although the performance of all these strategies coincides with twoqubit maximally entangled states (θ = 45^{∘}), the adaptive strategies are still preferred in most practical scenarios, where the realistic states are always different from the maximally entangled ones and actually closer to target states with θ ≠ 45^{∘}.
Experimental implementation and results
In the above QSV proposals, a valid statement about the tested states is based on the fact that all the outcomes are accepted, while a single appearance of rejection will cease the verification without a quantified conclusion. In practice, the generated states from the quantum devices are unavoidably nonideal with a limited fidelity to the target state; thus, there is always a certain probability to be rejected in each measurement. Even the probability of single rejection is small, it is natural to observe rejection events in an experiment involving a sequence of measurements. As a result, these original proposals are likely to mistakenly characterize qualified quantum devices as unqualified, which is inadequate for experimental implementation.
By considering the proportion of accepted outcomes, a modified strategy is thus developed here, which is robust to a certain proportion of rejection events. Quantitatively, we have the corollary that if 〈Ψ∣σ_{i}∣Ψ〉 ≤ 1 − ϵ for all the measured states, the probability for each outcome to be accepted is smaller than \(1(1{\lambda }_{2}^{\downarrow }(\Omega ))* \epsilon\). As a result, in the case that the verifier observes an accepted probability \(p\ge 1(1{\lambda }_{2}^{\downarrow }(\Omega ))* \epsilon\), it should be concluded that the actual state satisfies Eq. (1) with a confidence level of 1−δ, where ϵ and δ are calculated from the inequality^{12}
with
and m results are accepted when n measurements are performed. As a result of this modification, in the case that the final accepted probability \(p\ge 1(1{\lambda }_{2}^{\downarrow }(\Omega ))* \epsilon\), the verification can eventually reach a conclusion quantifying the distance between the actual and target states.
Benefiting from this modification, QSV can be applied to realistic nonideal states, which allows us to experimentally verify twoqubit entangled states using the above adaptive proposals. With the setup shown in Fig. 2, we can perform adaptive QSV. The setup consists of an entangled photonpair source (see ‘Methods’ for details), two mechanical optical switcher (MOS) and two highspeed triggered polarization analyzer (TPA). For adaptive QSV, Alice can guide her photon towards the MOS and perform a randomly selected projective measurement by TPA. Afterward, through a unidirectional classical communication, Alice’s outcome is sent to Bob to control the measurement performed on the paring photon, which is delayed on Bob’s MOS. An opposite adaptive process can also be realized by switching the role of Alice and Bob; and thus, the symmetric adaptive QSV can be executed by randomly selecting the two communication directions with equal probabilities. Technically, this random adaptive operation can be realized by controlling the MOS with a quantum random number generator (QRG), which outputs a binary signal (0 and 1) to decide which MOS transmits the photon directly while the other MOS delays the passing photon. For both Uni and BiLOCC strategies, we use a QRNG to randomly decide the applied setting among M_{1}, M_{2}, M_{3}; therefore, the settings are unknown to the incident photon pairs in prior to the measurement.
In order to confirm the power of classical communication in QSV, three strategies (LO, UniLOCC and BiLOCC) are utilized to verify a partially entangled state \(\left\Psi (6{0}^{\circ })\right\rangle\) and the results are shown in Fig. 3. In Fig. 3a, the results of 50 trials are averaged, which approximately coincide with the theoretical lines for the first few measurements and deviate from the predicted linearity afterward. This deviation mainly results from the difference between verified states and the ideal target state, which leads to rejection outcomes in QSV. In other words, only if the verified states are perfectly identical to the target state, a persistent 1/n scaling can be observed in a practical QSV. Since the occurrence rates of rejections are in principle equal for different strategies, a distinct gap in the estimated fidelity can be seen between the adaptive and nonadaptive strategies as predicted in the theory part. These results indicate the power of classical communication in boosting the performance of QSV. However, the practical scaling is not only determined by the optimality of the strategy, but also the quality of the actual state. In this sense, we can only access the intrinsic performance of a strategy by testing an ideal state. Although it is impossible to generate an ideal state in experiment, we can circumvent this difficulty by studying the first few measurements, of which the occurrence of rejections is fairly rare. In Fig. 3b, the first 25 measurements of single trials with all the outputs to be accepted are plotted, accompanied by the averaged results in Fig. 3a in the same range. The efficiency can be characterized by the slope of linear fitting lines of these data points. For LO, UniLOCC and BiLOCC strategies, the fitted slope values of the averaged points are 0.13, 0.17 and 0.187, respectively. After eliminating the effects of the state deviations by considering allaccepted single trials, the fitted slope values are 0.135, 0.188 and 0.22 for LO, UniLOCC and BiLOCC strategies, respectively, and these values are exactly the theoretical predictions for ideal states. As a result, the efficiency of the BiLOCC strategy is 1.63 times higher than that of LO and 1.17 times higher than that of UniLOCC. In other words, by introducing bidirectional classical communications, only 60% measurements of the nonadaptive LO scenario are required to verify the states to a certain level of fidelity. The performance gap between optimal local strategy and the globally optimal strategy is further minimized. Concretely, the constant factor \(\frac{1}{1{\lambda }_{2}^{\downarrow }(\Omega )}\) is reduced to approximately 1.5.
A further study of the performance gap between Uni and BiLOCC strategies is made to verify another two entangled states \(\left\Psi (7{0}^{\circ })\right\rangle\) and \(\left\Psi (8{0}^{\circ })\right\rangle\), and the averaged results of 50 trials are shown in Fig. 4. Both results show that BiLOCC significantly outperforms UniLOCC, and the differences of estimated fidelities are 2.1% and 1.3% for \(\left\Psi (7{0}^{\circ })\right\rangle\) and \(\left\Psi (8{0}^{\circ })\right\rangle\), respectively. Classical communication better enhances QSV by transferring the information bidirectionally rather than an ordinary unidirectional configuration.
Discussion
One main motivation to explore the quantum resources, such as entangled states and measurements, is their potential power to surpass the classical approaches. On the other hand, the fact that the quantum resources are generally complicated to produce and control inspires another interesting question: how to use classical resources exhaustively to approach the bound set by quantum resources? In the task to verify an entangled state, the utilization of entangled measurements constitutes a globally optimal strategy that achieves the best possible efficiency. Surprisingly, one can also construct strategies merely with local measurements and achieve the same scaling. In this experiment, we show that by introducing classical communications into QSV, the performance with local measurements can be further enhanced to approach the globally optimal bound. As a result, to verify the states to a certain level of fidelity, the number of required measurements is only 60% of that for nonadaptive local strategy. Meanwhile, the gap between the locally and globally optimal bound is distinctly reduced, with the constant factor minimized to 1.5 before 1/n scaling. Furthermore, recently QSV has been generalized to the adversarial scenario where arbitrary correlated or entangled state preparation is allowed^{30,31}.
Methods
Generation of entangled photon pairs
In the first part of the setup, tunable twoqubit entangled states are prepared by pumping a nonlinear crystal placed into a phasestable Sagnac interferometer (SI). Concretely, a 405.4 nm singlemode laser is used to pump a 5mm long bulk typeII nonlinear periodically poled potassium titanyl phosphate (PPKTP) nonlinear crystal placed into a phasestable SI to produce polarizationentangled photon pairs at 810.8 nm. A polarized beam splitter (PBS) followed by an HWP and a PCP are used to control the polarization mode of the pump beam. These lenses before and after the SI are used to focus the pump light and collimate the entangled photons, respectively. The interferometer is composed of two highly reflective and polarizationmaintaining mirrors, a DiHWP and a DiPBS. ‘Di’ here means it works for both 405.4 and 810.8 nm. The DiHWP flips the polarization of passing photons, such that the typeII PPKTP can be pumped by the same horizontal light from both clockwise and counterclockwise directions. DiIF and LPF (Long pass filter) are used to remove the pump beam light. BPF (bandpass filter) and SMF are used for spectral and spatial filtering, which can significantly increase the fidelity of entangled states. The whole setup, in particular the PPKTP, is sensitive to temperature fluctuations. Placing the PPKTP on a temperature controller (±0.002 °C stability) and sealing the SI with an acrylic box would help improve temperature stability. Polarizationentangled photon pairs are generated in the state \(\left\Psi (\theta )\right\rangle =\cos \theta \leftHV\right\rangle \sin \theta \leftVH\right\rangle\) (H and V denote the horizontally and vertically polarized components, respectively) and θ is controlled by the pumping polarization.
Measurement setting for adaptive QSV
For the QSV of twoqubit pure entangled states, Alice’s measurement Π_{i} (i = 1, 2, 3) are selected to be Pauli X, Y and Z measurements. When the outcome of X, Y and Z is 1(0), Bob performs \({\Pi }_{11}=\left{\upsilon }^{+}\right\rangle \left\langle {\upsilon }^{+}\right\) (\({\Pi }_{10}=\left{\upsilon }^{}\right\rangle \left\langle {\upsilon }^{}\right\)), \({\Pi }_{21}=\left{\omega }^{+}\right\rangle \left\langle {\omega }^{+}\right\) (\({\Pi }_{20}=\left{\omega }^{}\right\rangle \left\langle {\omega }^{}\right\)) and \({\Pi }_{31}=\leftV\right\rangle \left\langle V\right\) (\({\Pi }_{30}=\leftH\right\rangle \left\langle H\right\)), respectively, and the vectors are defined as \(\left{\upsilon }^{\pm }\right\rangle =\sin \theta \leftH\right\rangle \mp \cos \theta \leftV\right\rangle\) and \(\left{\omega }^{\pm }\right\rangle =\sin \theta \leftH\right\rangle \pm i\cos \theta \leftV\right\rangle\). These adaptive measurement settings constitute the optimal UniLQCC strategy which has the form^{23}
where \(\left+\right\rangle \equiv \frac{1}{\sqrt{2}}(\leftH\right\rangle +\leftV\right\rangle )\) and \(\left\right\rangle \equiv \frac{1}{\sqrt{2}}(\leftH\right\rangle \leftV\right\rangle )\) denote the eigenstates of Pauli X operator, \(\leftR\right\rangle \equiv \frac{1}{\sqrt{2}}(\leftH\right\rangle +i\leftV\right\rangle )\) and \(\leftL\right\rangle \equiv \frac{1}{\sqrt{2}}(\leftH\right\rangle i\leftV\right\rangle )\) denote the eigenstates of Pauli Y operator. In each of these three combined local measurement settings, the choice of Bob’s measurement setting is determined by the outcome of Alice’s measurement, which can be achieved by controlling the local operation of Bob’s EOM according to Alice’s outcome.
Data availability
The authors declare that all data supporting the findings of this study are available within the article 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. 2016YFA0302700 and 2017YFA0304100), National Natural Science Foundation of China (Grant Nos. 11874344, 61835004, 61327901, 11774335, 91536219 and 11821404), Key Research Program of Frontier Sciences, CAS (No. QYZDYSSWSLH003), Anhui Initiative in Quantum Information Technologies (AHY020100 and AHY060300), the Fundamental Research Funds for the Central Universities (Grant No. WK2030020019 and WK2470000026), Science Foundation of the CAS (No. ZDRWXH20191).
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W.H.Z. made the calculations assisted by P.Y. and J.S.X. C.F.L. and G.C. planned and designed the experiment. W.H.Z. carried out the experiment assisted by G.C., X.L., G.C.L., X.Y.X., S.Y., Z.B.H., Y.J.H. and Z.Q.Z. whereas W.H.Z. and X.X.P. 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., Liu, X., Yin, P. et al. Classical communication enhanced quantum state verification. npj Quantum Inf 6, 103 (2020). https://doi.org/10.1038/s41534020003284
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