Device-independent verification of Einstein-Podolsky-Rosen steering

Entanglement lies at the heart of quantum mechanics, and has been identified an essential resource for diverse applications in quantum information. If entanglement could be verified without any trust in the devices of observers, i.e., in a device-independent (DI) way, then unconditional security can be guaranteed for various quantum information tasks. In this work, we propose an experimental-friendly DI protocol to certify the presence of entanglement, based on Einstein-Podolsky-Rosen (EPR) steering. We first establish the DI verification framework, relying on the measurement-device-independent technique and self-testing, and show it is able to verify all EPR-steerable states. In the context of three-measurement settings as per party, it is found to be noise robustness towards inefficient measurements and imperfect self-testing. Finally, a four-photon experiment is implemented to device-independently verify EPR-steering even for Bell local states. Our work paves the way for realistic implementations of secure quantum information tasks.


I. INTRODUCTION
Entanglement is of fundamental importance to understand quantum theory, and also has found wide applications in quantum communication and computation tasks [1].If its presence could be certified without imposing any trust in the involved parties and their devices, then it is likely to guarantee information processing tasks with unconditional security.As a celebrated example, Bell inequalities [2,3] violation offers such a device-independent (DI) protocol.However, the conclusive violation of Bell inequalities typically requires a high efficiency of measurement apparatuses to close the detection loophole.Besides, it also demands the low transmission loss since sufficiently lossy entangled states are unable to violate any Bell inequality [4].Thus, the practical utility of this DI verification based on Bell inequalities is compromised in noisy quantum networks.
Notably, Bowles et al. has shown in [9,10] that combining the measurement-device-independent (MDI) technique [7] with self-testing [7,8,10] yields an alternate way which is able to device-independently verify all entangled states and circumvent the potential detection loophole [11].However, its complete implementation relies on the near-perfect self-testing of a set of prepared states with average fidelity above 99.998%[9], making it unrealistic to implement within current technology.
We first show that all EPR-steerable states can be verified within this DI framework.Particularly, if threemeasurement settings as per party are assumed, we obtain a steering inequality suitable for DI certification under imperfect self-testing with average fidelity 99.7%, which is a significant reduction in comparison to the DI verification based on entanglement.Finally, we implement a proof of principle experiment with four photons to validate the DI steering protocol, and find it can even verify Bell local states with an experimentally attainable self-testing fidelity of around 99.95%.

II. PRELIMINARIES
Suppose that two space-like separated observers, Alice and Bob say, make measurements on a preshared state.Denote Alice's and Bob's measurements x and y respectively, and the corresponding outcomes a and b.EPR-steering from Alice to Bob is demonstrated if the measurement statistics p(a, b|x, y) cannot be explained by any local hidden state model as p(a, b|x, y) where the hidden variable λ specifies some classical probability distribution p(a|x, λ) for Alice and some quantum probability distribution Tr[E B b|y ρ B λ ] for Bob which is generated via performing a positive-operator-valued measurement {E B b|y } b,y on quantum states ρ B λ [2].Note that Alice's side may not obey quantum rules, so EPR-steering is intrinsically an onesided device-independent verification task.For any steerable state, the detection task can be accomplished via violating a linear steering witness of the form [3] W S = j a j B j ≤ 0. ( Here a j corresponds to the outcome of Alice's measurement j, and B j represents Bob's j-th observable. Certifying the presence of EPR-steering can be adapted to the MDI scenario [4][5][6]30] where the trust in Bob's devices required in Eq. ( 1) is transferred to a third observer, Charlie say, who prepares a set of quantum states and sends them at random to Bob.As in Fig. 1, upon receiving these states described by density matrices {τ T b,j } with T being the transpose operation, Bob is required to perform an arbitrary binary measurement B with which the outcomes are modelled as either "Yes" or "No".Denote by P (a, Yes |x, B, τ T b,j ) the probability that Alice obtains a for the measurement x and Bob answers "Yes" when assigned to τ T b,j .Then, arranging the corresponding outcome statistics as Eq. ( 1) leads to a MDI steering witness [5,6] with g b,j being some predetermined weights.Typically, these coefficients can be chosen as the weights of measurement elements for Bob's observable B j = b g b,j E b|j .As the measurement outcome "Yes" is only recorded, Bob's side allows for extremely low measurement efficiency [11].
The optimal measurement strategy for Bob is to perform a partial Bell state measurement (BSM) B ≡ {B 1 , I − B 1 } where d models the answer "Yes" and d is the dimension of the Hilbert space of {τ T b,j } equal to that of Bob's local system.Indeed, given an arbitrary steerable state, its MDI witness (2) can be constructed from the corresponding witness (1), implying all steerable states are detectable in the MDI manner [5,6].The DI protocol is composed of two procedures.One is illustrated in the left side which corresponds to the MDI verification of the state ρ12.In this step, Alice randomly takes measurements x and obtains a, while Bob performs one binary measurement on his local system and a set of states {τ T b,j } assigned from Charlie, and collects the outcome "Yes".The second is described in the right box, corresponding to the self-testing process.Noting τ T b,j can be prepared by Charlie performing local measurements zj = {τ b,j } on Bell state |Φ + d preshared by Bob and Charlie, this measurement strategy can be self-tested via the violation of Bell inequalities, such as the Bell-CHSH one used in the main text.

III. DEVICE-INDEPENDENT VERIFICATION OF EPR STEERING
Within the MDI framework witnessed via Eq.( 2), both Alice's and Bob's sides are already device-independent, except for Charlie who prepares quantum states for Bob.Consequently, eliminating trust in Charlie's devices yields a fully DI steering verification protocol.As discussed below, this can be accomplished via self-testing which aims to identify the states and measurements for completely uncharacterised devices [7,8,35].
Note first that the states {τ T b,j } can be prepared by Charlie making local measurements z j = {τ b,j } on Bell state |Φ + d due to the relation b,j /d.This preparation process, including Bob's additional measurements y, can be uniquely determined or self-tested via the well-chosen Bell inequality of which its maximal violation is only achieved at each party per- forming a certain set of measurements on a specific state, up to some local isometry.Thus, using self-testing to determine the input states τ T b,j in Eq. ( 2), we can obtain a DI steering inequality as [36] W DI = a,c,j g c,j a j P (a, Yes, Here Charlie making measurements z and obtaining outcomes c is equivalent to he sending a state τ T b,j to Bob, and g c,j are close relate to the weights g b,j in Eq. ( 2).We remark that the self-testing process, involving Φ + d and Charlie's measurements, is not explicitly assessed in the above DI witness (3) and requires a detailed analysis case by case.For example, if dichotomic measurements are chosen, the Clauser-Horne-Shimony-Holt (CHSH) inequality [38] can be used.In the following section, we examine this issue in the case of three dichotomic measurements as per party.
As depicted in Fig. 1, we have established a DI framework to verify EPR-steering.As all pure bipartite entangled states and the associated measurements could be self-tested [39,40], together with experimental confirmations [41,42], it is naturally to witness all steerable states via this DI protocol.

IV. THREE MEASUREMENT SETTINGS
If Bob receives τ T c,j = (I + cσ j )/2 with c = ±1 and j = 1, 2, 3 sent from Charlie where σ j represent three Pauli observables as required in Eq. ( 2), then they can be selftested if the following triple Bell-CHSH inequality [8,36] is maximally violated within quantum theory, where E y,z = b,c=±1 b c p(b, c|y, z) refers to the measurement expectations between Bob's dichotomic measurements y = 1, 2, ..., 6 and Charlie's z = 1, 2, 3. Specifically, its maximal quantum violation , up to a local unitary.Note that there is a sign problem in the second measurement σ 2 for Charlie, however, it does not affect its utility in the DI steering protocol just as the DI entanglement certification [10].
Generally, it is impossible to achieve the perfect selftesting with the violation bound 6 √ 2 .To evaluate imperfections of self-testing, we introduce the fidelity f 0 = Φ + 2 | ρ 0 data |Φ + 2 which measures the overlap between a state self-tested from experimental data and the target state.Correspondingly, the fidelity for Charlie's measurements j = 1, 2, 3 can be cast as the state fidelity in a form of All can be computed via a semi-definite program [11][12][13].Incorporating these fidelity into the DI steering inequality (3), we obtain Its tedious derivation is deferred to the Supplementary Material [36].This inequality accounts for the imperfect self-testing in terms of state fidelity, interchangeable with the trace distance used [9,36].It also differs from the one in [5] which is obtained via tomography.

V. EXPERIMENTAL SETUP
The experimental setup for DI verification of EPRsteering is displayed in Fig. 2. In particular, two pairs of entangled photons pairs are first generated via the spontaneous parametric down-conversion process.One pair labelled as ρ 34 in the setup is prepared as the maximally entangled state |Φ + 2 = (|00 + |11 )/ √ 2 where the horizontally polarised direction (H) and vertically polarised direction (V) encode as state basis |0 , |1 , respectively.While, the other pair is generated in a family of Werner states Here, ) and the white noise with 1− v in Eq. ( 6) is simulated by flipping Alice's measurements with probability (1 − v)/2 [15].This class of states will be tested by the noisy steering witness (5).
Then, these photonic states are distributed to three observers.As shown in left side of Fig. 2, ρ 12 is sent to Alice (photon 1: the green ball) and Bob (photon 2: the blue ball) through single-mode fibres while ρ 34 is distributed to Charlie (photon 3: the red ball) and Bob (photon 4: the yellow ball).Detailed rotation parameters adjusted for wave plates (WPs) to realise Alice's and Charlie's three Pauli measurements σ j and Bob's six measurements are given in Tab.I in [36].In the right side of Fig. 2, Bob's partial BSM is implemented via three polarising beam splitters, two 22.5 • rotated HWPs, and four pseudo photon-number-resolving detectors (PP-NRD).In each PPNRD, a balanced beam splitter splits the light into two fibre-coupled single photon detector.
To improve the quality of the partial BSM, an interference filter of 2 nm is inserted for spectral selection so that a Hong-Ou-Mandel interference visibility higher than 30 : 1 is observed in this experiment.We also do tomography to reconstruct the BSM and obtain a fidelity around 0.9831 ± 0.0040 [36].
Finally, the measurement statistics is collected to do the triple Bell-CHSH test (4) to self-test quantum states τ T c,j = (I+cσ j )/2 input to Bob.Combining with the measurement fidelity f j estimated from imperfect self-testing, we can rewrite the DI steering inequality (5) explicitly as [36] 4 j,a,c Here g c,j is either 1 or −1 for the qubit measurements.For Werner states (6), the theoretical prediction of Eq. ( 7) should be 3v 1 − f j ≤ 0. It is found that the average fidelity around 99.7% of selftesting is allowed for Bell local states with v = 0.7 [36], which is a significant reduction in comparison to entanglement verification with fidelity above 99.998%[9].

VI. RESULTS
The entangled photon pairs encoding |Φ + 2 are collected up to 13, 000 per second with a pump power of 30 mW.We observe an extinction ratio over 500 : 1 in the H/V basis and the H+V/H-V basis, indicating it is generated with fidelity higher than 0.998.Under the fair-sampling assumption, we obtain 2.8241, 2.8211, and 2.8189 for three Bell-CHSH tests in (4) and the sum of them is 8.4641 closing to the maximal quantum bound 6 √ 2 ≈ 8.4853.The uncertainty induced by the Poisson oscillation of photons is about 0.0009.Correspondingly, the fidelity of three Pauli measurements self-tested from experimental data is f 1 = 0.9994, f 2 = 0.9999, and f 3 = 0.9992, respectively, and thus the average fidelity 99.95% is attained in our experiment.The standard deviation is around 10 −5 by optimising 100 groups of the Poisson statistics of the experimental data.
The experimental results of DI verification for Werner states are plotted in Fig. 3.We perform quantum state tomography to show that each Werner state is generated with v = 0.6469(4), 0.6742(4), 0.7015(4), 0.8090(4), 0.9239(3), and 0.9951(1) from about 10 7 photon pairs [36].Ideally, there is the theoretical prediction 3v − √ 3 for f j = 1, yielding the EPR-steering bound v = 1/ √ 3 ≈ 0.5774 [3].Otherwise, the steering inequality (7), incorporated with self-testing imperfections, is shown as the red line in Fig. 3 We observe a violation up to 0.1110 ± 0.07 even for a Bell local state with v = 0.6742(4), smaller than the bound v = 0.707 for the CHSH inequality.For the reason of the small amount of data, about three orders of magnitude small, the error bars for DI verification (7) are much larger than the CHSH inequality.To make a comparison with fully DI, we do tomography on entangled sources and partial BSM and obtain a post-processing probability statistics in DI steering (7) where the fidelity of local measurements are estimated by self-testing, the results are shown as the magenta circles.By contrast, the corresponding theoretical and experimental results for the standard CHSH inequality with two measurements per party are given in blue curve and blue dots respectively, where the bounds are translated down by 2. The error bars are about 0.001.The shaded blue region represents the failure of the DI steering inequality.
if v ≥ 0.6742(4), accounting for statistic errors and imperfections of self-testing.Importantly, a violation up to 0.1110 ± 0.07 is achieved at the point v = 0.6742(4) lower than the CHSH bound 1/ √ 2 ≈ 0.707 [38] and the Vétersi bound 0.7056 [48].This implies that even some Bell local states can be verified via this DI protocol.The error bars for Werner states with v = 0.6469(4) fall into the failure region, so their steerability are not conclusively detected.
We further test the system errors on the performance of DI steering verification.We do quantum tomography on the entangled sources and partial BSM.This process is given in [36], and the calibrated violations of DI steering inequality (7) are shown as the magenta circles in Fig. 3.By contrast, we also perform the CHSH test to verify steerability.In Fig. 3, the blue line describes the theoretical results while blue dots are the experimental observations for these Werner states.

VII. CONCLUSION AND DISCUSSION
We have studied the DI verification of EPR steering and implemented an optical experiment to validate our DI protocol.In principle, we prove that all steerable states, including Bell local states, can be verified deviceindependently.In practice, we analyse noise robustness towards imperfections of self-testing in the implementation process, and derive a steering inequality as per Eq. ( 5) for the three-measurement setting case.Finally, we perform a proof of principle experiment to successfully validate our DI steering protocol.We believe that our work paves the way for realistic implementations of secure quantum information processing tasks based on EPR-steering and also finds practical applications of selftesting.
There are many interesting open questions left for the future work.For example, the methods in [21,49] may be used to tolerate more transmission loss and lower measurement efficiency.The resource efficient approach in [50] could also improve the success probability of the partial BSM, and self-testing could be more noise robust by adopting other techniques [7].Moreover, an alternate DI framework [51] may be possibly used to verify quantum steering.It is also interesting to device-independently certify genuine high-dimensional steering [52] and steering networks [53].
As discussed in the main text, in the fully DI verification of EPR-steering framework, we need to collect the measurement statistics to check whether it violates the DI steering inequality for the ideal case, given any quantum state ρ AB to be tested.For example, suppose that Bob is input τ T b,j = 1 2 (I + bσ j ) with b = ±1, j = 1, 2, 3 randomly from Charlie.Alternate, Bob's input states could be generated by Charlie performing local measurements described by {E b,j = τ b,j } on the Bell state |Φ + d shared by Bob and Charlie.Thus, the DI steering inequality (C1) could be expressed in a more explicit form of Here B 0 represents the Bob's subsystem that τ b,j = 1 2 (I + bσ j ).In particular, these trust input states τ b,j = 1 2 (I + bσ j ) for Bob in MDI steering scenario can be replaced by these untrusted observables via self-testing which refers to a device-independent way to uniquely identify the state and the measurement for uncharacterized quantum devices.The virtual protocol that one considers is described as following.
Consider the scenario in which involves two non-communicating parties Bob and Charlie.Each has access to a black box with an underlying state |ψ .It is accomplished with three Bell-CHSH tests and thus Bob needs to perform the six dichotomic measurements y = 1, 2, ..., 6 and Charlie performs z = 1, 2, 3. Bob's inputs and outputs are denoted respectively by y and b; Charlie's by z and c.After a large number of rounds of experiments, the joint probability distribution p(b, c|y, z) could be reconstructed.Then we are able to construct the triple Bell operator defined in ref. [8] Further, it was proven by Bowles et al. [9] that if the maximal quantum violation B = 6 √ 2 is observed, then there exists a local auxiliary state ) and a local isometry U (see Fig. 4) such that where It means that we can extract the exact information of the maximally entangled state of two-qubit ) and Charlie's three measurements Although there exists the sign problem of σ y to be distinguished, it does not pose any constraint to verify entanglement [10] and EPR steering to be discussed.Note that the measurement set {σ x , −σ y , σ z } could be transformed from the set {σ x , σ y , σ z } on which is acted the transpose operation T because of σ T y = −σ y .It is easy to verify that the state ρ AB has a local hidden state (LHS) model with respect to one measurement if and only if it holds for the other measurement set, since the partial gates evolves the system to be where |± = |0 ± |1 .To extract the information of the trusted auxiliary systems B and C , we take the partial trace of the whole system which be left where ρ j BC = M C j |ψ ψ| M C j † describes the density matrix of untrusted operator M C j acting on the uncharacterised state |ψ and C j is the coefficient matrix of ρ j data with Looking into these single terms, it can be found that for each target Pauli observable, ρ data is a 4 × 4 matrix whose entries are linear combinations of expectation values such as Then the closeness of ρ j data to the target state σ C j |Φ + 2 can be then captured by the fidelity Here f j is a linear function of two types of operator expectations: some observed behavior and some non-observable correlations which involve different measurements on the same party which are left as variables.We define an average fidelity to evaluate the performance of self-testing.It is worth noting that σ y and −σ y have the same fidelity function and thus f for two measurement settings {σ x , σ y , σ z } and {σ x , −σ y , σ z } are identical.Finally, the fidelity f j , j = 1, 2, 3 are calculated with the aid of the NPA hierarchy characterization of the quantum behaviors [11][12][13], and their lower bound can be computed via a SDP: the CHSH operators (D7) (C3a) = 2.8241, (C3b) = 2.8211, (C3c) = 2.8189, where Γ is so-called NPA moment matrix whose rows and columns are numbered by products belonging to Q l , i.e., Γ ij = ψ| Q i l † Q j l |ψ , and Q l is the set of product of B y and C z and defined as outer approximations of the quantum set (the level of the hierarchy l is the number of measurements in the product).In our problem, the moment matrix corresponding to Q 2 , that is to say the products set is with at most operators per party.To improve the precision of fidelity, we increased the size of the Γ matrix by adding terms such as to contain all the average values • that appear in the expression of fidelity.It results in the Γ matrix having a size of 101 × 101 whose elements are divided into two kinds that observed behavior variables are real and non-observable variables are complex.The total number of constrains is K = 2167 (28 variables are real and the left are complex).We used the MATLAB modeling language YALMIP and MOSEK as a solver to solve the SDP.According to our experimental results about the violation of the triple Bell-CHSH test, we obtain the average fidelity f = 0.9995 and f 1 = 0.9994, f 2 = 0.9999, f 3 = 0.9992 for each Pauli observable.

Robust verification of EPR steering
It easily follows from Eq. (D3) that the distance between the pure state estimated from the experimental data via a SDP and the target state satisfies where with ξ 0 |ξ 0 + ξ 1 |ξ 1 = 1 and || • || denotes the trace distance.Thus, these fidelity of Pauli observables f j and the Bell state f 0 give us the error estimate when we use the experimental data to do the verification task.Further, the self-tested pure states in Eq. (D8) could be decomposed as Here the state vector |φ ⊥ j is orthogonal to σ C j |Φ + 2 and it is easy to check that α j = f j .For each Pauli observable σ j , the deviation from the density matrices output from the swap circuit is This matrix has two eigenvalues λ j = ± 1 − α 2 j = ± 1 − f j by solving the following matrix in the basis of {σ C j |Φ + 2 , |φ ⊥ j }.Instead of Charlie's local measurements σ j for the ideal case, σ j + ∆ j represents the real measurements performed on the Bell state |Φ + 2 .To estimate the lower value of the witness when evaluated on a separable state ρ AB = λ p(λ) a j λ ρ B λ , accounting for the imperfections of self-testing, we are able to derive a steering inequality Here E BB0 models the answer "Yes" from Bob's arbitrary joint measurement B, and σ denotes the second term in Eq. (S27).The third equality results from the relation g c,j = c = ±1 and τ c,j = 1 2 (I + c(σ j + ∆ j )).If self-testing is perfect, i.e., f j = α j = 1 and thus ∆ j = 0, then the above quantity recovers the ideal one W DI .
Next, we analyse the noise range induced by imperfection of self-testing.Note first that

FIG. 1 .
FIG. 1. DI verification framework of EPR-steering.The DI protocol is composed of two procedures.One is illustrated in the left side which corresponds to the MDI verification of the state ρ12.In this step, Alice randomly takes measurements x and obtains a, while Bob performs one binary measurement on his local system and a set of states {τ T b,j } assigned from Charlie, and collects the outcome "Yes".The second is described in the right box, corresponding to the self-testing process.Noting τ T b,j can be prepared by Charlie performing local measurements zj = {τ b,j } on Bell state |Φ +

FIG. 2 .
FIG. 2. Experimental setup.Two pairs of entangled photons are generated via the spontaneous parametric down-conversion process.A sandwich-like β-barium borate (BBO) crystal is configured to prepare entangled photons with high fidelity and high brightness.One pair labelled as 1 and 2 distributed to Alice and Bob, is generated as Werner states (6), while the other labelled as 3 and 4 is produced as the Bell state |Φ + 2 sent to Charlie and Bob respectively.For the three-measurement case, a complete implementation of DI steering verification requires a triple Bell-CHSH test (4) and the noisy DI steering test (5).Alice and Charlie perform three Pauli measurements σj on their respect photons, while Bob makes 6 measurements described by (σi + σj)/ √ 2 and (σi − σj)/ √ 2 on the photon 4 and an additional partial BSM B on his photons 2 and 4. Abbreviations of the components: HWP, half wave plate; QWP, quarter wave plate; LiNbO3, Lithium niobate crystal; YVO4, Yttrium vanadate crystal; IF, interference filter; FC, fibre coupler; PBS, polarising beam splitter; BSM, Bell state measurement; BS, beam splitter; D1-D8, single photon detector.

FIG. 6 .
FIG. 5. State tomography for Werner states.The real parts of the Werner states are shown as the colorful bars, and the correspondingly theoretical values are as the transparent bars.Each state is constructed from about 9, 800, 000 photon pairs.
, while the experimental results are displayed in red dots.It is evident that we have successfully witnessed steerability device-independently