Abstract
Within the framework of quantum refereed steering games, quantum steerability can be certified without any assumption on the underlying state nor the measurements involved. Such a scheme is termed the measurementdeviceindependent (MDI) scenario. Here, we introduce a measure of steerability in an MDI scenario, i.e., the result merely depends on the observed statistics and the quantum inputs. We prove that such a measure satisfies the convex steering monotone. Moreover, it is robust against not only measurement biases but also losses. We also experimentally estimate the amount of the measure with an entangled photon source. As two byproducts, our experimental results provide lower bounds on an entanglement measure of the underlying state and an incompatible measure of the involved measurement. Our research paves a way for exploring oneside deviceindependent quantum information processing within an MDI framework.
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Introduction
Entanglement^{1}, steerability^{2}, and Bell nonlocality^{3} are three types of quantum correlations which play essential roles in quantum cryptography, quantum teleportation, and quantum information processing^{4,5,6}. The fact that quantum steering is treated as an intermediate quantum correlation between entanglement and nonlocality leads to a hierarchical relation among them. That is, all nonlocal states are steerable, and all steerable states are entangled, but not vice versa^{7,8,9}. During the past decade, there have been many significant experimental works^{10,11,12,13,14,15,16} and various theoretical results on quantum steering^{17,18,19,20,21,22}, including the correspondence with measurement incompatibility^{23,24,25,26,27}, oneway steering^{28,29}, temporal steering^{30,31,32,33,34}, continuousvariable steering^{35,36,37}, and measures of steering^{38,39,40,41,42,43}.
Bell nonlocality enables one to perform the socalled deviceindependent (DI) quantum information processing^{5,44,45,46,47}, i.e., one makes no assumption on the underlying state nor the measurements performed. From the hierarchical relation^{7}, it naturally leads to the fact that a Bell inequality can be treated as a DI entanglement witness. Nevertheless, not all entangled states can be detected by using a Bell inequality violation^{48}. Recently, based on Buscemi’s semiquantum nonlocal games^{49}, Branciard et al.^{50} proposed a collection of entanglement witnesses in the socalled measurementdeviceindependent (MDI) scenario. Compared with the standard DI scenario, there is one more assumption in an MDI scenario: the input of each detector has to be a set of tomographically complete quantum states instead of real numbers. Such a simple relaxation leads to that all entangled states can be certified by the proposed MDI entanglement witnesses^{49,50}. This characterization gives rise to the recent works providing frameworks for MDI measures of entanglement^{51,52,53,54}, nonclassical teleportation^{55}, and nonentanglementbreaking channel verification^{56,57,58}.
Recently, Cavalcanti et al.^{59} introduced another type of nonlocal game, dubbed as quantum refereed steering games (QRSGs). In each of such games, one player, denoted as Alice, is questioned and answers with real numbers, while the other player, saying Bob, is questioned with (isolated) quantum states but still answers with real numbers. They showed that there always exists a QRSG with a higher winning probability when the players are correlated by a steerable state^{59}. Later, Kocsis et al.^{60} experimentally proposed a QRSG and verified the steerability for the family of twoqubit Werner states in such a scenario, which is also referred to as an MDI scenario. Moreover, such a QRSG scenario can be used to generate the private random number by maximal violation of the higher dimensional steering inequality under the MDI framework^{61,62}.
Here we consider a variant of QRSGs, by which we propose the MDI steering measure (MDISM) of the underlying unknown steerable resource without accessing any knowledge of the involved measurements. We show that the MDISM is a standard measure of steerability, i.e., a convex steering monotone^{41}, by proving that it is equivalent to the previously proposed measures: the steering robustness^{38} and the steering fraction^{40}. Therefore, our proposed measure not only coincides with the degree of steerability of the underlying steerable resource, but also quantifies the degree of entanglement of the shared quantum state^{38} and incompatibility of the measurements involved^{23,26,63}. Furthermore, MDISM can be computed via a semidefinite program. We also show the MDISM is robust, in the sense that it can detect steerability in the presence of detection losses and biases^{50,51,52,53}.
Finally, we experimentally estimate the degree of steerability of the family of twoqubit Werner states in an MDI scenario. We consider that Alice performs three qubitmeasurements in the mutually unbiased bases (MUBs) since they can be used to demonstrate the strongest steerability to Bob when Alice has three measurement settings^{39}. On the other hand, Bob performs the Bellstate measurement (BSM) on his part of the state and the quantum inputs. Based on the observed correlations, the steerability of the family of twoqubit Werner states are quantified by solving a semidefinite program. As mentioned before, the experimental data naturally bounds the degree of entanglement of the underlying state, and the amount of measurement incompatibility of Alice’s measurements. Compared with the previous experimental works^{53,60,64,65,66} in the MDI scenarios, our method not only certifies the existence of entanglement and measurement incompatibility, but also bounds these quantities. Moreover, our experimental result roughly relates with the probabilities of successful subchannel discrimination in the MDI scenario^{38,67}.
Results
MDI measure of steerability
Through this work, we assume that all quantum states act on a finite dimensional Hilbert space \({\mathcal{H}}\). The sets of density matrices and operators acting on \({\mathcal{H}}\) are denoted by \({\mathsf{D}}({\mathcal{H}})\) and \({\mathsf{L}}({\mathcal{H}})\), respectively. We denote the index sets of a finite number of elements by \({\mathcal{A}}\), \({\mathcal{B}}\), \({\mathcal{X}}\), and \({\mathcal{Y}}\). The probability of a specific index, say \(a\in {\mathcal{A}}\), is denoted by p(a).
In the MDI steering scenario, we consider two spatially separated parties, Alice and Bob, sharing a quantum state \({\rho }_{\text{AB}}\in {\mathsf{D}}({{\mathcal{H}}}_{\text{A}}\otimes {{\mathcal{H}}}_{\text{B}})\) (see Fig. 1). During each round of the experiment, Alice receives a classical input \(x\in {\mathcal{X}}\) and performs the corresponding measurement on her system with an outcome \(a\in {\mathcal{A}}\). On the other hand, Bob performs a joint measurement on his system and a trusted input quantum state \({\tau }_{y}\in {\mathsf{D}}({{\mathcal{H}}}_{{\text{B}}_{0}})\), with \(y\in {\mathcal{Y}}\). We note that the trustiness represents the state is well prepared and there is no side channel to transmit the state information. Their joint probability distributions can be expressed as: \(p(a,b x,{\tau }_{y})=\,\text{Tr}\,\left[\left({E}_{a x}\otimes {E}_{b}\right)\left({\rho }_{\text{AB}}\otimes {\tau }_{y}\right)\right]\quad \forall a,b,x,y,\) where \({\{{E}_{a x}\}}_{a}\in {\mathsf{L}}({{\mathcal{H}}}_{\text{A}})\) and \({\{{E}_{b}\}}_{b}\in {\mathsf{L}}({{\mathcal{H}}}_{\text{B}}\otimes {{\mathcal{H}}}_{{\text{B}}_{0}})\) are the positiveoperator valued measurements (POVM) (i.e., the general quantum measurements) describing Alice’s and Bob’s measurement with the corresponding outcomes {a} and {b}, respectively.
Within the framework of the resource theory of quantum steering^{41}, we concern more about the underlying assemblage^{18} Bob receives rather than the shared quantum state. That is, we describe the obtained correlation by Bob’s joint measurement {E_{b}} on the quantum inputs {τ_{y}} and the assemblage {σ_{a∣x}}:
An assemblage {σ_{a∣x}} is a set of subnormalized quantum states defined by σ_{a∣x} = Tr_{A}(ρ_{AB}E_{a∣x} ⊗ id)^{18}, which includes both the information of Alice’s marginal statistics p(a∣x) = tr(σ_{a∣x}) and the normalized states \({\hat{\sigma }}_{a x}={\sigma }_{a x}/p(a x)\in {\mathsf{D}}({{\mathcal{H}}}_{\text{B}})\) Bob receives. Here, id is the identity operator. The free state of the quantum steering (denoted as unsteerable assemblage) is the assemblage admitting a localhiddenstate (LHS) model^{7}, described by a deterministic strategy D(a∣x, λ) and preexisting (subnormalized) quantum states {σ_{λ}}, such that \({\sigma }_{a x}={\sigma }_{a x}^{\,\text{US}\,}={\sum }_{\lambda }D(a x,\lambda ){\sigma }_{\lambda }\,\forall \,a,x\). In particular, the set of all unsteerable assemblages LHS forms a convex set; consequently, for a given steerable assemblage \(\{{\sigma }_{a x}^{{\rm{S}}}\}\), there always exists a set of positive semidefinite operators {F_{a∣x} ≽ 0}, called a steering witness, such that \({\rm{Tr}}{\sum }_{a,x}{F}_{a x}{\sigma }_{a x}^{{\rm{S}}}>\alpha\), while \({\rm{Tr}}{\sum }_{a,x}{F}_{a x}{\sigma }_{a x}^{{\rm{US}}}\le \alpha \quad \forall \{{\sigma }_{a x}^{{\rm{US}}}\}\in {\mathsf{LHS}}\)^{16,18,38,39,42}, where \(\alpha :=\mathop{\max }\nolimits_{\{{\sigma }_{a x}^{{\rm{US}}}\}\in {\mathsf{LHS}}}{\rm{Tr}}{\sum }_{a,x}{F}_{a x}{\sigma }_{a x}^{{\rm{US}}}\) is the local bound of the steering witness.
In what follows, we will construct the MDISM by using the aforementioned existence of a steering witness for any steerable assemblage. We start by considering a variant of QRSGs. Indeed, Eq. (1) can be treated as correlations obtained in a variant of QRSGs with steerable assemblages being a resource. We stress that, in the standard QRSGs, one instead treats a set of steerable states as a resource in such a game. These two resources are inequivalent because one can obtain the same assemblage from different states and measurements. With this, we define a payoff associated exclusively to a single Bob’s outcome (b = 1) as
where P: = {p(a, 1∣x, τ_{y})} is the experimentally observed statistics from an assemblage {σ_{a∣x}} based on Eq. (1) and \(\beta :=\{{\beta }_{a,1}^{x,y}\}\) is a set of real coefficients.
With the above definition, we prove that, given any steerable assemblage, there always exists a set of real coefficients β, such that the payoff \({\mathcal{W}}\left({\bf{P}},\beta \right)\) is strictly higher than those obtained from unsteerable assemblages. Details of the proof are given in Supplementary Note 1. In other words, the payoff \({\mathcal{W}}\left({\bf{P}},\beta \right)\) is effectively the same as the standard steering witness, in the sense that all steerable assemblages can be faithfully verified by a properly chosen \({\mathcal{W}}\left({\bf{P}},\beta \right)\). We note that the witness \({\mathcal{W}}\left({\bf{P}},\beta \right)\) can be seen as a generalization of a standard Bell inequality (see ref. ^{50} for a similar formulation in the entanglement scenario), and is used to generalize the result of ref. ^{60}, wherein the family of twoqubit Werner states is explicitly considered.
Now we stand in the position to introduce the MDISM for an unknown assemblage {σ_{a∣x}}, denoted by
with
where \({{\mathcal{W}}}_{\text{LHS}}(\beta )={\sup }_{{\bf{P}}\in {\mathsf{LHS}}}{\mathcal{W}}({\bf{P}},\beta )\) is the local bound for a given β. The physical meaning of the proposed measure is simple and the idea is very similar to that of the nonlocality fraction^{68}: if the given correlation is unsteerable (i.e., it admits an LHS model), then \({\mathcal{W}}({\bf{P}},\beta )\le {{\mathcal{W}}}_{\text{LHS}}(\beta )\), and therefore \({{\mathcal{S}}}_{1}=0\). On the other hand, if the correlation is steerable, \({\mathcal{W}}({\bf{P}},\beta )\,>\,{{\mathcal{W}}}_{\text{LHS}}(\beta )\), then \({{\mathcal{S}}}_{1}\,>\,0\). In Supplementary Notes 2 and 3, we further prove that:

\({{\mathcal{S}}}_{1}\) is a steering monotone since it is equivalent to the steering fraction and the steering robustness.

The optimal \({\bf{P}}:=\{p(a,1 x,{\tau }_{y})={\rm{Tr}}[{E}_{1}({\sigma }_{a x}\otimes {\tau }_{y})]\}\) in Eq. (4) is obtained when Bob’s measurement is the projection onto the maximally entangled state. That is, \({E}_{1}=\left{\Phi }_{+}^{{\text{BB}}_{0}}\right\rangle \left\langle {\Phi }_{+}^{{\text{BB}}_{0}}\right\), with \(\left{\Phi }_{+}^{{\text{BB}}_{0}}\right\rangle =1/\sqrt{{d}_{\text{B}}}\mathop{\sum }\nolimits_{i = 1}^{{d}_{\text{B}}}\lefti\right\rangle \otimes \lefti\right\rangle\).
After introducing our measure of the steerability in an MDI scenario, we proceed by considering the following two practical circumstances. First, one would like to estimate the degree of steerability of a given data table without any a priori knowledge about the experimental setup. Second, as the experimental apparatuses are inevitably erroneous in practical situations, how can one estimate the degree of steerability in the absence of the optimization of Bob’s measurement? These two circumstances give rise to the attempt to estimate the degree of steerability of an experimentally observed correlation P when lacking the knowledge about the underlying assemblage.
In the case of an inaccessible assemblage, the optimization over P in Eq. (4) becomes not feasible. Consequently, the alternative quantity \({{\mathcal{W}}}_{1}^{\,\text{LB}\,}({\bf{P}}):={\sup }_{\beta }\frac{{\mathcal{W}}({\bf{P}},\beta )}{{{\mathcal{W}}}_{\text{LHS}}(\beta )}\) is a lower bound on \({{\mathcal{W}}}_{1}\), and
provides a lower bound on \({{\mathcal{S}}}_{1}\). Trivially, the bound becomes tight when Bob’s measurement is the projection onto the maximally entangled state \({E}_{1}=\left{\Phi }_{+}^{{\text{BB}}_{0}}\right\rangle \left\langle {\Phi }_{+}^{{\text{BB}}_{0}}\right\). Note that even if Bob’s inputs do not form a complete set, Eq. (5) still provides a valid lower bound^{51}. This can be understood from the fact that the set of tomographically complete inputs is a resource for Bob to demonstrate steerability in an MDI scenario. The lack of a completeness of quantum inputs can only decrease the degree of steerability.
Furthermore, to underpin the practical viability of our measure, we stress that the maximal value of Eq. (5) is computable via a semidefinite program (see Supplementary Note 4 for details):
In the above equation, d = d_{B} is the dimension of Bob’s system. This program can be performed for a given experimentally observed correlation P. Therefore, it works well particularly when Bob’s measurement is the optimal one, i.e., the projection onto the maximally entangled state. In this case, the solution of Eq. (6) gives the exact value of the MDISM defined in Eq. (3).
Finally, we would like to show that the MDISM is robust against detection losses. To see this, we consider the average loss rate of Bob’s measurement η ∈ [0, 1]. The observed correlation in this case is p_{η}(a, 1∣x, τ_{y}) = η ⋅ p(a, 1∣x, τ_{y}), shrinking the MDISM by η, i.e., \(\eta \cdot {{\mathcal{S}}}_{1}\). As can be seen above, the shrinking quantity \(\eta \cdot {{\mathcal{S}}}_{1}\) is still able to detect steerability in an MDI scenario with arbitrary detection losses and provide a lower bound on the steerability of the underlying assemblage (see refs. ^{50,53} for similar discussions in the MDI entanglement scenario).
Experimental results
In the following, we will experimentally demonstrate how to estimate, in an MDI manner, the degree of steerability of the underlying steerable resource given by Alice’s three measurement settings with the two dimensional MUBs acting on the twoqubit Werner states, namely \({\rho }_{\text{AB}}=v\left{\psi }^{}\right\rangle \left\langle {\psi }^{}\right+\left(\frac{1v}{4}\right)\,\text{id}\,\), with visibility 0 ≤ v ≤ 1, singlet state \(\left{\psi }^{}\right\rangle =\frac{1}{\sqrt{2}}(\leftHV\right\rangle \leftVH\right\rangle )\), and id being the identity operator.
The experimental setup is schematically shown in Fig. 2. Further details are given in “Methods”. Specifically, after sending the twoqubit Werner state ρ_{AB} to Alice and Bob, we obtain the set of probability distributions {p(a, b∣x, τ_{y})} [described in Eq. (1)] by which Alice performs measurements in the Pauli bases X, Y, and Z, on her part of the system, while Bob performs the joint measurement on his part of the system and his quantum inputs τ_{y}. Bob’s tomographically complete set of quantum inputs is composed of eigenstates of the three Pauli matrices. The joint measurement performed by Bob is the BSM, i.e., the optimal measurement, so that the value of the measure \({{\mathcal{S}}}_{1}\) can be achieved.
Due to our experimental setup, we further show that for the underlying assemblage {σ_{a∣x}} being a qubit, all of the four measurement operators \({\{{E}_{b}\}}_{b = 1,2,3,4}\) of the BSM are optimal for Bob, i.e., the produced correlation for each b leads to the maximum value of Eq. (4). Further discussions on the twoqubit case are given in Supplementary Note 5. Therefore, Eq. (5) can be modified into the following form:
where \({{\mathcal{W}}}_{b}^{\,\text{LB}\,}({\bf{P}}):={\sup }_{\beta ^{\prime} }\frac{{\mathcal{W}}({\bf{P}},\beta ^{\prime} )}{{{\mathcal{W}}}_{\text{LHS}}(\beta ^{\prime} )}\) with \(\beta ^{\prime} :={\{{\beta }_{a,b}^{x,y}\}}_{a,x,y}\) for each b. When there is a detection bias between the four detectors of the BSM, Eq. (7) also provide a valid lower bound on the proposed measure. More specifically, consider that we have four detectors with the biased detection rates of ξ_{1}, ξ_{2}, ξ_{3}, and ξ_{4}, respectively, with ∑_{b}ξ_{b} = 4 and ξ_{b} ≥ 0 ∀b. For the ideal case, ξ_{b} = 1 for all b. When there exists some bias, the observed correlation will be ξ_{b} ⋅ p(a, b∣x, τ_{y}). Obviously, this correlation also reveals the steerability of the underlying resource, i.e.,
where \({\beta }_{a,b}^{* ,x,y}\) is the optimal set of coefficients for the biased correlation ξ_{b} ⋅ p(a, b∣x, τ_{y}).
Our experimental estimation on \({{\mathcal{S}}}_{1}\) is plotted in Fig. 3a. As can be seen, although \({{\mathcal{S}}}^{\text{LB}}({\bf{P}})\) in Eq. (7) may not perform the best among the other finegrained terms \({{\mathcal{S}}}_{b}^{\,\text{LB}\,}({\bf{P}}):={{\mathcal{W}}}_{b}^{\,\text{LB}\,}({\bf{P}})1\), it is the most suitable one in the sense that the variance from the theoretical prediction is the smallest. Besides, some finegrained terms wrongly detect the existence of steerability due to the overestimation caused by the detection bias (i.e., the estimation of steerability in Fig. 3a when the visibility is lower than \(1/\sqrt{3}\)). With Eq. (8), such overestimation will not occur when we use the quantity \({{\mathcal{S}}}^{\text{LB}}({\bf{P}})\). Therefore, our estimation on the MDISM is robust against not only detection biases but also losses.
Except for estimating the degree of steerability of the underlying assemblage in an MDI scenario, here we show that our experimental results directly bound the degree of entanglement ER(ρ_{AB}) of the underlying state and the degree of measurement incompatibility IR({E_{a∣x}}) of Alice’s measurements. We briefly recall these two quantities in Supplementary Note 6. The result is shown in Fig. 3b. The detail of the quantum state tomography to access these two quantities are also shown in Supplementary Note 7. Our results are based on the fact that the steering robustness of the assemblage SR({σ_{a∣x}}) is a lower bound on the entanglement robustness ER(ρ_{AB})^{38} and incompatibility robustness IR({E_{a∣x}})^{23,26,63}. Therefore, as \({{\mathcal{S}}}^{\text{LB}}({\bf{P}})\) is a lower bound on the steering robustness, \({{\mathcal{S}}}^{\text{LB}}({\bf{P}})\) is also used to provide a lower bound on ER(ρ_{AB}) and IR({E_{a∣x}}).
Discussion
In this work, we consider a variant of QRSGs, by which we introduce a measure of steerability in a MDI scenario, i.e., without making assumptions on the involved measurements nor the underlying assemblage. The only characterized quantities are the observed statistics and a tomographically complete set of quantum states for Bob. Through this, all steerable assemblages can be witnessed, in contrast to the fact that only a subset of steerable assemblages can be detected in the standard DI scenario. We further show that it is a convex steering monotone by proving the equivalence to the steering fraction as well as the steering robustness. Therefore, the MDISM provides a lower bound on the degree of entanglement of the unknown quantum state and measurement incompatibility of the involved measurements. Besides, our approach is able to detect steerability in an MDI scenario with arbitrary detection losses and provide a lower bound on the steerability of the underlying assemblage.
Moreover, we tackle two optimization problems in Eq. (4). That is, the optimal measurement and MDI steering witness used for MDISM are obtained, or equivalently, we obtain the optimal strategies for the variant of QRSGs. At first glance, it seems to be a difficult problem to obtain the optimal measurement, since Bob has to optimize over all possible measurements. However, we show that the projection onto the maximally entangled state is always an optimal one for any steerable resource. The optimal MDI steering witness (the variant QRSGs), on the other hand, can be efficiently computed by semidefinite programming. Finally, we provide an experimental demonstration of estimating the degree of steerability. The result also bounds the degree of entanglement, and incompatibility in an MDI scenario. We have also proposed an improved MDISM which decreased the effect of some detection biases between Bob’s detectors.
This work also reveals some open questions: It is interesting to investigate whether our method can be modified to all steerable assemblages in a standard DI scenario with the approach recently proposed in refs. ^{69,70}. More recently, the DI certification of all steerable states has experimentally been implemented by selftesting an ancilla entangled pair^{71}. It is also interesting to propose practical applications with the MDI scenario (or even a fully DI scheme following the work of refs. ^{69,70,71}). Since the formulation of the standard steering scenario can be applied to certify the security of quantum keys^{72}, one can ask if this is also the case in the MDI scenario.
Methods
Experimental estimation of MDISM
The system state is encoded on the polarization (H, V) where H(V) represents the horizontally (vertically) polarized direction of the photon. Through a spontaneous parametric downconversion process, we generate pairs of maximally entangled photons’ state \(\frac{1}{\sqrt{2}}(\leftHV\right\rangle \leftVH\right\rangle )\). The Werner state is prepared by dephasing the photons to a completely mixed state with probability (1 − v)^{73,74,75}. On Bob’s side, a trusted device shown in Fig. 2b prepares the auxiliary qubit τ_{y} on the path degree of freedom of his owned photon. Note that, although we encode Bob’s shared state (that with Allice) and his quantum input in the same photon, these two states are indeed in different degrees of freedom. More specifically, these two states are prepared by different preparation devices, one for creating the bipartite quantum state ρ_{AB} while the other for generating τ_{y}. That is to say, in our MDI scenario under consideration, the former preparation device is not trusted while the latter is trusted.
On Alice’s side, she uses the quarterwave plate Q1, the halfwave plate H1 combined with a polarization beam splitter to perform a measurement according to the value of x, and returns the outcome a to the referee. While Bob needs to implement the optimal joint measurement, i.e., BSM on two degrees of freedom of the same particle (the polarization and the path degree of freedom), similar to the former works^{53,76,77}. This method avoids the entangled measurement on two particles, which is a tough task with 50% efficiency in linear optics^{78,79}. All the experimental details can be found in Supplementary Note 7. Moreover, a jointmeasurement apparatus does not receive any information of the input quantum state before performing the measurement. More specifically, there is no side channel which transmits any information of the state to the measurement apparatus. Such protocol is physically and realistically more reliable than a situation where a referee prepares a trust quantum input to Bob. See Supplementary Note 7 for more experimental details.
Data availability
All data not included in the paper are available upon reasonable request from the corresponding authors.
Code availability
All code not included in the paper are available upon reasonable request from the corresponding authors.
References
Einstein, A., Podolsky, B. & Rosen, N. Can quantummechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935).
Schrödinger, E. Discussion of probability relations between separated systems. Proc. Cambridge Phil. Soc. 31, 555 (1935).
Bell, J. S. On the EinsteinPodolskyRosen paradox. Physics 1, 195 (1964).
Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009).
Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V. & Wehner, S. Bell nonlocality. Rev. Mod. Phys. 86, 419 (2014).
Uola, R., Costa, A. C. S., Nguyen, H. C. & Gühne, O. Quantum steering. Rev. Mod. Phys. 92, 015001 (2020).
Wiseman, H. M., Jones, S. J. & Doherty, A. C. Steering, entanglement, nonlocality, and the EinsteinPodolskyRosen paradox. Phys. Rev. Lett. 98, 140402 (2007).
Jones, S. J., Wiseman, H. M. & Doherty, A. C. Entanglement, EinsteinPodolskyRosen correlations, Bell nonlocality, and steering. Phys. Rev. A 76, 052116 (2007).
Quintino, M. T. et al. Inequivalence of entanglement, steering, and Bell nonlocality for general measurements. Phys. Rev. A 92, 032107 (2015).
Saunders, D. J., Jones, S. J., Wiseman, H. M. & Pryde, G. J. Experimental EPRsteering using Belllocal states. Nat. Phys 6, 845 (2010).
Bennet, A. J. et al. Arbitrarily losstolerant EinsteinPodolskyRosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole. Phys. Rev. X 2, 031003 (2012).
Händchen, V. et al. Observation of oneway EinsteinPodolskyRosen steering. Nat. Photonics 6, 596 (2012).
Smith, D. H. et al. Conclusive quantum steering with superconducting transitionedge sensors. Nat. Commun. 3, 845 (2012).
Schneeloch, J., Dixon, P. B., Howland, G. A., Broadbent, C. J. & Howell, J. C. Violation of continuousvariable EinsteinPodolskyRosen steering with discrete measurements. Phys. Rev. Lett. 110, 130407 (2013).
Sun, K. et al. Experimental quantification of asymmetric EinsteinPodolskyRosen steering. Phys. Rev. Lett. 116, 160404 (2016).
Cavalcanti, E. G., Jones, S. J., Wiseman, H. M. & Reid, M. D. Experimental criteria for steering and the EinsteinPodolskyRosen paradox. Phys. Rev. A 80, 032112 (2009).
Reid, M. D. Demonstration of the EinsteinPodolskyRosen paradox using nondegenerate parametric amplification. Phys. Rev. A 40, 913 (1989).
Pusey, M. F. Negativity and steering: a stronger Peres conjecture. Phys. Rev. A 88, 032313 (2013).
Walborn, S. P., Salles, A., Gomes, R. M., Toscano, F. & Souto Ribeiro, P. H. Revealing hidden EinsteinPodolskyRosen nonlocality. Phys. Rev. Lett. 106, 130402 (2011).
Kogias, I., Lee, A. R., Ragy, S. & Adesso, G. Quantification of Gaussian quantum steering. Phys. Rev. Lett. 114, 060403 (2015).
Costa, A. C. S. & Angelo, R. M. Quantification of EinsteinPodolskiRosen steering for twoqubit states. Phys. Rev. A 93, 020103 (2016).
Chiu, C.Y., Lambert, N., Liao, T.L., Nori, F. & Li, C.M. Nocloning of quantum steering. npj Quantum Inf. 2, 16020 (2016).
Cavalcanti, D. & Skrzypczyk, P. Quantitative relations between measurement incompatibility, quantum steering, and nonlocality. Phys. Rev. A 93, 052112 (2016).
Uola, R., Moroder, T. & Gühne, O. Joint measurability of generalized measurements implies classicality. Phys. Rev. Lett. 113, 160403 (2014).
Quintino, M. T., Vértesi, T. & Brunner, N. Joint measurability, EinsteinPodolskyRosen steering, and Bell nonlocality. Phys. Rev. Lett. 113, 160402 (2014).
Chen, S.L., Budroni, C., Liang, Y.C. & Chen, Y.N. Natural framework for deviceindependent quantification of quantum steerability, measurement incompatibility, and selftesting. Phys. Rev. Lett. 116, 240401 (2016).
Uola, R., Budroni, C., Gühne, O. & Pellonpää, J. Onetoone mapping between steering and joint measurability problems. Phys. Rev. Lett. 115, 230402 (2015).
Wollmann, S., Walk, N., Bennet, A. J., Wiseman, H. M. & Pryde, G. J. Observation of genuine oneway EinsteinPodolskyRosen steering. Phys. Rev. Lett. 116, 160403 (2016).
Bowles, J., Vértesi, T., Quintino, M. T. & Brunner, N. Oneway EinsteinPodolskyRosen steering. Phys. Rev. Lett. 112, 200402 (2014).
Chen, Y.N. et al. Temporal steering inequality. Phys. Rev. A 89, 032112 (2014).
Chen, S.L. et al. Quantifying nonMarkovianity with temporal steering. Phys. Rev. Lett. 116, 020503 (2016).
Ku, H.Y. et al. Temporal steering in four dimensions with applications to coupled qubits and magnetoreception. Phys. Rev. A 94, 062126 (2016).
Li, C.M., Chen, Y.N., Lambert, N., Chiu, C. & Nori, F. Certifying singlesystem steering for quantuminformation processing. Phys. Rev. A 92, 062310 (2015).
Ku, H.Y., Chen, S.L., Lambert, N., Chen, Y.N. & Nori, F. Hierarchy in temporal quantum correlations. Phys. Rev. A 98, 022104 (2018).
Tatham, R., Mišta, L., Adesso, G. & Korolkova, N. Nonclassical correlations in continuousvariable nonGaussian Werner states. Phys. Rev. A 85, 022326 (2012).
He, Q., RosalesZárate, L., Adesso, G. & Reid, M. D. Secure continuous variable teleportation and EinsteinPodolskyRosen steering. Phys. Rev. Lett. 115, 180502 (2015).
Xiang, Y., Kogias, I., Adesso, G. & He, Q. Multipartite Gaussian steering: monogamy constraints and quantum cryptography applications. Phys. Rev. A 95, 010101 (2017).
Piani, M. & Watrous, J. Necessary and sufficient quantum information characterization of EinsteinPodolskyRosen steering. Phys. Rev. Lett. 114, 060404 (2015).
Skrzypczyk, P., Navascués, M. & Cavalcanti, D. Quantifying EinsteinPodolskyRosen steering. Phys. Rev. Lett. 112, 180404 (2014).
Hsieh, C.Y., Liang, Y.C. & Lee, R.K. Quantum steerability: characterization, quantification, superactivation, and unbounded amplification. Phys. Rev. A 94, 062120 (2016).
Gallego, R. & Aolita, L. Resource theory of steering. Phys. Rev. X 5, 041008 (2015).
Cavalcanti, D. & Skrzypczyk, P. Quantum steering: a review with focus on semidefinite programming. Rep. Prog. Phys. 80, 024001 (2017).
Ku, H.Y. et al. EinsteinPodolskyRosen steering: its geometric quantification and witness. Phys. Rev. A 97, 022338 (2018).
Gallego, R., Brunner, N., Hadley, C. & Acín, A. Deviceindependent tests of classical and quantum dimensions. Phys. Rev. Lett. 105, 230501 (2010).
Bancal, J.D., Gisin, N., Liang, Y.C. & Pironio, S. Deviceindependent witnesses of genuine multipartite entanglement. Phys. Rev. Lett. 106, 250404 (2011).
Cavalcanti, D., Rabelo, R. & Scarani, V. Nonlocality tests enhanced by a third observer. Phys. Rev. Lett. 108, 040402 (2012).
Acín, A. et al. Deviceindependent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007).
Werner, R. F. Quantum states with EinsteinPodolskyRosen correlations admitting a hiddenvariable model. Phys. Rev. A 40, 4277 (1989).
Buscemi, F. All entangled quantum states are nonlocal. Phys. Rev. Lett. 108, 200401 (2012).
Branciard, C., Rosset, D., Liang, Y.C. & Gisin, N. Measurementdeviceindependent entanglement witnesses for all entangled quantum states. Phys. Rev. Lett. 110, 060405 (2013).
Rosset, D., Martin, A., Verbanis, E., Lim, C. C. W. & Thew, R. Practical measurementdeviceindependent entanglement quantification. Phys. Rev. A 98, 052332 (2018).
Shahandeh, F., Hall, M. J. W. & Ralph, T. C. Measurementdeviceindependent approach to entanglement measures. Phys. Rev. Lett. 118, 150505 (2017).
Verbanis, E. et al. Resourceefficient measurementdeviceindependent entanglement witness. Phys. Rev. Lett. 116, 190501 (2016).
Guo, Y. et al. Measurementdeviceindependent quantification of irreducible highdimensional entanglement. npj Quantum Inf. 6, 52 (2020).
Cavalcanti, D., Skrzypczyk, P. & Šupić, I. All entangled states can demonstrate nonclassical teleportation. Phys. Rev. Lett. 119, 110501 (2017).
Rosset, D., Buscemi, F. & Liang, Y.C. Resource theory of quantum memories and their faithful verification with minimal assumptions. Phys. Rev. X 8, 021033 (2018).
Uola, R., Kraft, T. & Abbott, A. A. Quantification of quantum dynamics with inputoutput games. Phys. Rev. A 101, 052306 (2020).
Yuan, X. et al. Robustness of quantum memories: an operational resourcetheoretic approach. Preprint at https://arxiv.org/abs/1907.02521 (2020).
Cavalcanti, E. G., Hall, M. J. W. & Wiseman, H. M. Entanglement verification and steering when Alice and Bob cannot be trusted. Phys. Rev. A 87, 032306 (2013).
Kocsis, S., Hall, M. J. W., Bennet, A. J., Saunders, D. J. & Pryde, G. J. Experimental measurementdeviceindependent verification of quantum steering. Nat. Commun. 6, 5886 (2015).
Skrzypczyk, P. & Cavalcanti, D. Maximal randomness generation from steering inequality violations using qudits. Phys. Rev. Lett. 120, 260401 (2018).
Guo, Y. et al. Experimental measurementdeviceindependent quantum steering and randomness generation beyond qubits. Phys. Rev. Lett. 123, 170402 (2019).
Chen, S.L., Budroni, C., Liang, Y.C. & Chen, Y.N. Exploring the framework of assemblage moment matrices and its applications in deviceindependent characterizations. Phys. Rev. A 98, 042127 (2018).
Xu, P. et al. Implementation of a measurementdeviceindependent entanglement witness. Phys. Rev. Lett. 112, 140506 (2014).
Wollmann, S., Hall, M. J. W., Patel, R. B., Wiseman, H. M. & Pryde, G. J. Referenceframeindependent EinsteinPodolskyRosen steering. Phys. Rev. A 98, 022333 (2018).
Wollmann, S., Uola, R. & Costa, A. C. S. Experimental demonstration of robust quantum steering. Phys. Rev. Lett. 125, 020404 (2020).
Sun, K. et al. Demonstration of EinsteinPodolskyRosen steering with enhanced subchannel discrimination. npj Quantum Inf. 4, 12 (2018).
Cavalcanti, D., Acín, A., Brunner, N. & Vértesi, T. All quantum states useful for teleportation are nonlocal resources. Phys. Rev. A 87, 042104 (2013).
Bowles, J., Šupić, I., Cavalcanti, D. & Acín, A. Deviceindependent entanglement certification of all entangled states. Phys. Rev. Lett. 121, 180503 (2018).
Chen, S.L., Ku, H.Y., Zhou, W., Tura, J. & Chen, Y.N. Robust selftesting of steerable quantum assemblages and its applications on deviceindependent quantum certification. Preprint at https://arxiv.org/abs/2002.02823 (2020).
Zhao, Y.Y. et al. Deviceindependent verification of EinsteinPodolskyRosen steering. Preprint at https://arxiv.org/abs/1909.13432 (2019).
Branciard, C., Cavalcanti, E. G., Walborn, S. P., Scarani, V. & Wiseman, H. M. Onesided deviceindependent quantum key distribution: Security, feasibility, and the connection with steering. Phys. Rev. A 85, 010301 (2012).
Laine, E.M., Breuer, H.P., Piilo, J., Li, C.F. & Guo, G.C. Nonlocal memory effects in the dynamics of open quantum systems. Phys. Rev. Lett 108, 210402 (2012).
Qi, B. et al. Adaptive quantum state tomography via linear regression estimation: theory and twoqubit experiment. npj Quantum Inf. 3, 19 (2017).
White, A. G., James, D. F. V., Munro, W. J. & Kwiat, P. G. Exploring hilbert space: accurate characterization of quantum information. Phys. Rev. A 65, 012301 (2001).
Popescu, S. An optical method for teleportation. Preprint at https://arxiv.org/abs/quantph/9501020 (1995).
Boschi, D., Branca, S., De Martini, F., Hardy, L. & Popescu, S. Experimental realization of teleporting an unknown pure quantum state via dual classical and EinsteinPodolskyRosen channels. Phys. Rev. Lett 80, 1121 (1998).
Lütkenhaus, N., Calsamiglia, J. & Suominen, K.A. Bell measurements for teleportation. Phys. Rev. A 59, 3295 (1999).
Bouwmeester, D. et al. Experimental quantum teleportation. Nature 390, 575 (1997).
Acknowledgements
The authors acknowledge fruitful discussions with Francesco Buscemi, Ana Cristina Sprotte Costa, YeongCherng Liang, Chau Nguyen, Paul Skrzypczyk, Roope Uola and KangDa Wu. The authors acknowledge the support of the Graduate Student Study Abroad Program (Grant No. MOST 1072917I006002) for H.Y.K.; the Postdoctoral Research Abroad Program (Grant No. MOST 1072917I564 007) for S.L.C.; the National Center for Theoretical Sciences and Ministry of Science and Technology, Taiwan (Grants No. MOST 1072628M006002MY3, 1082627E006001, and 1082811M006536), and Army Research Office (Grant No. W911NF1910081) for Y.N.C.; the National Center for Theoretical Sciences and Ministry of Science and Technology, Taiwan (Grant No. MOST 1082112M006020MY2) for H.B.C.; the National Natural Science Foundation of China (Grants No. 11574291 and 11774334) for G.Y.X.; Y.Y.Z. is supported by the National Natural Science Foundation for the Youth of China (No. 11804410); F.N. is supported in part by: NTT Research, Army Research Office (ARO) (Grant No. W911NF1810358), Japan Science and Technology Agency (JST) (the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (via the KAKENHI Grant No. JP20H00134, and the grant JSPSRFBR Grant No. JPJSBP120194828), and the Foundational Questions Institute (FQXi) (Grant No. FQXiIAF1906).
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H.Y.K. and S.L.C. contributed equally to the development of the theoretical analysis and conceived the project; G.Y.X. supervised the experiment; G.Y.X. and Y.Y.Z. designed the experiment; Y.Y.Z. conducted the experiment and collected data with the help from G.Y.X.; Y.Y.Z. and H.Y.K. analyzed the experimental data with the help from G.Y.X., C.F.L., and G.C.G.; H.Y.K., S.L.C., and H.B.C. proved the theoretical results; Y.N.C. and F.N. supervised the research. All authors contributed to the writing of the manuscript.
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Zhao, YY., Ku, HY., Chen, SL. et al. Experimental demonstration of measurementdeviceindependent measure of quantum steering. npj Quantum Inf 6, 77 (2020). https://doi.org/10.1038/s41534020003079
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DOI: https://doi.org/10.1038/s41534020003079
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