Experimental demonstration of measurement-device-independent measure of quantum steering

TL;DR

This paper introduces and experimentally demonstrates a measurement-device-independent measure of quantum steering, which relies solely on observed statistics and quantum inputs, and is robust against measurement biases and losses.

Contribution

The paper presents a new MDI measure of quantum steerability, proves its properties, and experimentally estimates it using entangled photons, advancing one-sided device-independent quantum information processing.

Findings

Successfully estimated the steerability measure experimentally.

Provided lower bounds on entanglement and measurement incompatibility.

Demonstrated robustness against measurement biases and losses.

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 measurement-device-independent (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 by-products, 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 one-side device-independent quantum information processing within an MDI framework.

Equations

$$p(a,b|x,{\tau }_y)=\text{Tr}\left[E_b({\sigma }_{a|x}\otimes {\tau }_y)\right].$$

$$\mathcal{W}\left(\mathbf{P},\beta \right)=\sum _{a,x,y}{\beta }_{a,1}^{x,y}p(a,1|x,{\tau }_y),$$

$${\mathcal{S}}_1:=\max \left\{{\mathcal{W}}_1-1,0\right\},$$

$${\mathcal{W}}_1:=\underset{\beta ,\mathbf{P}}{\sup }\frac{\mathcal{W}(\mathbf{P},\beta )}{{\mathcal{W}}_{\text{LHS}}(\beta )},$$

$${\mathcal{S}_{1}^{\text{\tiny LB}}(\mathbf{P})}:=\max\Big{\{}{\mathcal{W}_{1}^{\text{\tiny LB}}(\mathbf{P})}-1,0\Big{\}}$$

$$\{p(a,1|x,{\tau }_y)\}\text{and}\{{\tau }_y\}$$

$$\underset{\tilde{\beta }}{\max }$$

$$d\text{openone}-\sum _{a,x,y}D(a|x,\lambda ){\tilde{\beta }}_{a,1}^{x,y}{\tau }_y⪰0\forall \lambda$$

$$\sum _y{\tilde{\beta }}_{a,1}^{x,y}{\tau }_y⪰0\forall a,x.$$

$${\mathcal{S}}^{\text{LB}}(\mathbf{P}):=\max \left\{\frac{1}{4}\sum _{b=1}^4{\mathcal{W}}_b^{\text{LB}}(\mathbf{P})-1,0\right\},$$

$${\mathcal{S}}_{\xi }^{\text{LB}}(\mathbf{P},\{{\xi }_b\}):=\max \left\{\frac{1}{4}\sum _{b=1}^4{\mathcal{W}}_{b,\xi }^{\text{LB}}(\mathbf{P},\{{\xi }_b\})-1,0\right\}$$

$$:=\max \left\{\frac{1}{4}\sum _{b=1}^4{\xi }_b\cdot \sum _{axy}{\beta }_{a,b}^{\ast ,x,y}p(a,b|x,{\tau }_y)-{\xi }_b,0\right\}$$

$$\le \max \left\{\frac{1}{4}\sum _{b=1}^4{\xi }_b{\mathcal{W}}_b^{\text{LB}}(\mathbf{P})-1,0\right\}$$

$$={\mathcal{S}}^{\text{LB}}(\mathbf{P}),$$

Full text

I Introduction

Entanglement Einstein et al. (1935), steerability Schrödinger (1935), and Bell nonlocality Bell (1964) are three types of quantum correlations which play essential roles in quantum cryptography, quantum teleportation, and quantum information processing Horodecki et al. (2009); Brunner et al. (2014); Uola et al. (2020a). 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 Wiseman et al. (2007); Jones et al. (2007); Quintino et al. (2015) During the past decade, there have been many significant experimental works Saunders et al. (2010); Bennet et al. (2012); Händchen et al. (2012); Smith et al. (2012); Schneeloch et al. (2013); Su et al. (2013); Sun et al. (2016); Cavalcanti et al. (2009) and various theoretical results on quantum steering Reid (1989); Pusey (2013); Walborn et al. (2011); Kogias et al. (2015); Costa and Angelo (2016); Chiu et al. (2016), including the correspondence with measurement incompatibility Cavalcanti and Skrzypczyk (2016); Uola et al. (2014); Quintino et al. (2014); Chen et al. (2016a); Uola et al. (2015), one-way steering Wollmann et al. (2016); Bowles et al. (2014), temporal steering Chen et al. (2014, 2016b); Ku et al. (2016); Li et al. (2015); Ku et al. (2018a), continuous-variable steering Tatham et al. (2012); He et al. (2015); Xiang et al. (2017), and measures of steering Piani and Watrous (2015); Skrzypczyk et al. (2014); Hsieh et al. (2016); Gallego and Aolita (2015); Cavalcanti and Skrzypczyk (2017); Ku et al. (2018b).

Bell nonlocality enables one to perform the so-called device-independent (DI) quantum information processing Brunner et al. (2014); Gallego et al. (2010); Bancal et al. (2011); Cavalcanti et al. (2012); Acín et al. (2007), i.e., one makes no assumption on the underlying state nor the measurements performed. From the hierarchical relation Wiseman et al. (2007), 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 Werner (1989). Recently, based on Buscemi’s semi-quantum nonlocal games Buscemi (2012), Branciard et al. Branciard et al. (2013) proposed a collection of entanglement witnesses in the so-called measurement-device-independent (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 Buscemi (2012); Branciard et al. (2013). This characterization gives rise to the recent works providing frameworks for MDI measures of entanglement Rosset et al. (2018a); Shahandeh et al. (2017); Verbanis et al. (2016), non-classical teleportation Cavalcanti et al. (2017), and non-entanglement-breaking channel verification Rosset et al. (2018b); Uola et al. (2020b); Yuan et al. .

Recently, Cavalcanti et al. Cavalcanti et al. (2013a) 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 Cavalcanti et al. (2013a). Later, Kocsis et al. Kocsis et al. (2015) experimentally proposed a QRSG and verified the steerability for the family of two-qubit 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 Skrzypczyk and Cavalcanti (2018); Guo et al. (2019).

Here we consider a variant of QRSGs, by which we propose the first MDI steering measure (MDI-SM) of the underlying unknown steerable resource without accessing any knowledge of the involved measurements. We show that the MDI-SM is a standard measure of steerability, i.e., a convex steering monotone Gallego and Aolita (2015), by proving that it is equivalent to the previously proposed measures: the steering robustness Piani and Watrous (2015) and the steering fraction Hsieh et al. (2016). 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 Piani and Watrous (2015) and incompatibility of the measurements involved Cavalcanti and Skrzypczyk (2016); Chen et al. (2016a, 2018). Furthermore, MDI-SM can be computed via a semidefinite program. We also show the MDI-SM is robust, in the sense that it can detect steerability in the presence of detection losses and biases Branciard et al. (2013); Rosset et al. (2018a); Shahandeh et al. (2017); Verbanis et al. (2016).

Finally, we experimentally estimate the degree of steerability of the family of two-qubit Werner states in an MDI scenario. We consider that Alice performs three qubit-measurements in the mutually unbiased bases (MUBs) since they can be used to demonstrate the strongest steerability to Bob when Alice has three measurement settings Skrzypczyk et al. (2014). On the other hand, Bob performs the Bell-state measurement (BSM) on his part of the state and the quantum inputs. Based on the observed correlations, the steerability of the family of two-qubit 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 Xu et al. (2014); Kocsis et al. (2015); Verbanis et al. (2016); Wollmann et al. (2018, 2019) 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 Piani and Watrous (2015); Sun et al. (2018).

II Measurement-device-independent 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 $\rho_{AB}\in\mathsf{D}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$ (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 $\tau_{y}\in\mathsf{D}(\mathcal{H}_{B_{0}})$, 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: $p(a,b|x,\tau_{y})=\text{Tr}\left[\left(E_{a|x}\otimes E_{b}\right)\left(\rho_{AB}\otimes\tau_{y}\right)\right]\quad\forall a,b,x,y,$ where $\{E_{a|x}\}_{a}\in\mathsf{L}(\mathcal{H}_{A})$ and $\{E_{b}\}_{b}\in\mathsf{L}(\mathcal{H}_{B}\otimes\mathcal{H}_{B_{0}})$ are the positive-operator 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 Gallego and Aolita (2015), we concern more about the underlying assemblage Pusey (2013) 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 $\{\tau_{y}\}$ and the assemblage $\{\sigma_{a|x}\}$:

[TABLE]

An assemblage $\{\sigma_{a|x}\}$ is a set of subnormalized quantum states defined by $\sigma_{a|x}=\text{Tr}_{A}(\rho_{AB}~{}E_{a|x}\otimes\openone)$ Pusey (2013), which includes both the information of Alice’s marginal statistics $p(a|x)=\text{tr}(\sigma_{a|x})$ and the normalized states $\hat{\sigma}_{a|x}=\sigma_{a|x}/p(a|x)\in\mathsf{D}(\mathcal{H}_{B})$ Bob receives. The free state of the quantum steering (denoted as unsteerable assemblage) is the assemblage admitting a local-hidden-state (LHS) model Wiseman et al. (2007), described by a deterministic strategy $D(a|x,\lambda)$ and pre-existing (subnormalized) quantum states $\{\sigma_{\lambda}\}$, such that $\sigma_{a|x}=\sigma^{\text{US}}_{a|x}=\sum_{\lambda}D(a|x,\lambda)\sigma_{\lambda}~{}\forall~{}a,x$. In particular, the set of all unsteerable assemblages $\mathsf{LHS}$ forms a convex set; consequently, for a given steerable assemblage $\{\sigma_{a|x}^{\rm{S}}\}$, there always exists a set of positive semidefinite operators $\{F_{a|x}\succeq 0\}$, called a steering witness, such that $\operatorname{Tr}\sum_{a,x}F_{a|x}\sigma_{a|x}^{\rm{S}}>\alpha$, while $\operatorname{Tr}\sum_{a,x}F_{a|x}\sigma_{a|x}^{\rm{US}}\leq\alpha\quad\forall\{\sigma_{a|x}^{\rm{US}}\}\in\mathsf{LHS}$ Cavalcanti et al. (2009); Pusey (2013); Cavalcanti and Skrzypczyk (2017); Skrzypczyk et al. (2014); Piani and Watrous (2015), where $\alpha:=\max_{\{\sigma_{a|x}^{\rm{US}}\}\in\mathsf{LHS}}\operatorname{Tr}\sum_{a,x}F_{a|x}\sigma_{a|x}^{\rm{US}}$ is the local bound of the steering witness.

In what follows, we will construct the MDI-SM 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

[TABLE]

where $\mathbf{P}:=\{p(a,1|x,\tau_{y})\}$ is the experimentally observed statistics from an assemblage $\{\sigma_{a|x}\}$ 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 $\beta$, such that the payoff $\mathcal{W}\left(\mathbf{P},\beta\right)$ is strictly higher than those obtained from unsteerable assemblages. Details of the proof are given in the Supplementary Material. In other words, the payoff $\mathcal{W}\left(\mathbf{P},\beta\right)$ is effectively the same as the standard steering witness, in the sense that all steerable assemblages can be faithfully verified by a properly chosen $\mathcal{W}\left(\mathbf{P},\beta\right)$. We note that the witness $\mathcal{W}\left(\mathbf{P},\beta\right)$ can be seen as a generalization of a standard Bell inequality (see Ref. Branciard et al. (2013) for a similar formulation in the entanglement scenario), and is used to generalize the result of Ref. Kocsis et al. (2015), wherein the family of two-qubit Werner states is explicitly considered.

Now we stand in the position to introduce the MDI-SM for an unknown assemblage $\{\sigma_{a|x}\}$, denoted by

[TABLE]

with

[TABLE]

where $\mathcal{W}_{\text{LHS}}(\beta)=\sup_{\mathbf{P}\in\mathsf{LHS}}\mathcal{W}(\mathbf{P},\beta)$ is the local bound for a given $\beta$. The physical meaning of the proposed measure is simple and the idea is very similar to that of the nonlocality fraction Cavalcanti et al. (2013b): if the given correlation is unsteerable (i.e., it admits an LHS model), then $\mathcal{W}(\mathbf{P},\beta)\leq\mathcal{W}_{\text{LHS}}(\beta)$, and therefore ${\mathcal{S}_{1}}=0$. On the other hand, if the correlation is steerable, $\mathcal{W}(\mathbf{P},\beta)>\mathcal{W}_{\text{LHS}}(\beta)$, then ${\mathcal{S}_{1}}>0$. In the Supplementary Material, 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 $\mathbf{P}:=\{p(a,1|x,\tau_{y})=\operatorname{Tr}[E_{1}(\sigma_{a|x}\otimes\tau_{y})]\}$ in Eq. (4) is obtained when Bob’s measurement is the projection onto the maximally entangled state. That is, $E_{1}=|{\Phi^{BB_{0}}_{+}}\rangle\langle{\Phi^{BB_{0}}_{+}}|$, with $|{\Phi^{BB_{0}}_{+}}\rangle=1/\sqrt{d_{B}}\sum_{i=1}^{d_{B}}|i\rangle\otimes|i\rangle$.

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 $\mathbf{P}$ when lacking the knowledge about the underlying assemblage.

In the case of an inaccessible assemblage, the optimization over $\mathbf{P}$ in Eq. (4) becomes not feasible. Consequently, the alternative quantity ${\mathcal{W}_{1}^{\text{\tiny LB}}(\mathbf{P})}:=\sup_{\beta}\frac{\mathcal{W}(\mathbf{P},\beta)}{\mathcal{W}_{\text{LHS}}(\beta)}$ is a lower bound on ${\mathcal{W}_{1}}$, and

[TABLE]

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 ${E_{1}}=|{\Phi^{BB_{0}}_{+}}\rangle\langle{\Phi^{BB_{0}}_{+}}|$. Note that even if Bob’s inputs do not form a complete set, Eq. (5) still provides a valid lower bound Rosset et al. (2018a). 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 the Supplementary Material for details):

[TABLE]

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 MDI-SM defined in Eq. (3).

Finally, we would like to show that the MDI-SM is robust against detection losses. To see this, we consider the average loss rate of Bob’s measurement $\eta\in[0,1]$. The observed correlation in this case is $p_{\eta}(a,1|x,\tau_{y})=\eta\cdot p(a,1|x,\tau_{y})$, shrinking the MDI-SM by $\eta$, 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. Verbanis et al. (2016); Branciard et al. (2013) for similar discussions in the MDI entanglement scenario.)

III Experimental estimation of MDI-SM

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 two-qubit Werner states, namely $\rho_{AB}=v|\psi^{-}\rangle\langle\psi^{-}|+\left(\frac{1-v}{4}\right)\openone$, with visibility $0\leq v\leq 1$, singlet state $|\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|HV\rangle-|VH\rangle)$, and $\openone$ being the identity operator.

The schematic diagram of our experimental setup is given in Fig. 2. 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 down-conversion process, we generate pairs of maximally entangled photons’ state $\frac{1}{\sqrt{2}}(|HV\rangle-|VH\rangle)$. The Werner state is prepared by dephasing the photons to a completely mixed state with probability $(1-v)$ Laine et al. (2012); Qi et al. (2017). On Bob’s side, a trusted device shown in Fig. 2(b) prepares the auxiliary qubit $\tau_{y}$ 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 ${\rho_{AB}}$ while the other for generating $\tau_{y}$. 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 quarter-wave plate Q1, the half-wave 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 Popescu (1995); Boschi et al. (1998); Verbanis et al. (2016). This method avoids the entangled measurement on two particles, which is a tough task with 50% efficiency in linear optics Lütkenhaus et al. (1999); Bouwmeester et al. (1997). All the experimental details can be found in the Supplementary Material. Moreover, a joint-measurement 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 the Supplementary Material for more experimental details.

IV Experimental results

To measure the steerability, we use the above experimental setup. More specifically, after sending the two-qubit Werner state $\rho_{AB}$ to Alice and Bob, we obtain the set of probability distributions $\{p(a,b|x,\tau_{y})\}$ [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 $\tau_{y}$. 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 $\{\sigma_{a|x}\}$ 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). Details of the discussion of the two-qubit case is given in the Supplementary Material. Therefore, Eq. (5) can be modified into the following form:

[TABLE]

where ${\mathcal{W}}_{b}^{\text{\tiny LB}}(\mathbf{P}):=\sup_{\beta^{\prime}}\frac{\mathcal{W}(\mathbf{P},\beta^{\prime})}{\mathcal{W}_{\text{LHS}}(\beta^{\prime})}$ 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 $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$, respectively, with $\sum_{b}\xi_{b}=4$ and $\xi_{b}\geq 0$ $\forall b$. For the ideal case, $\xi_{b}=1$ for all $b$. When there exists some bias, the observed correlation will be $\xi_{b}\cdot p(a,b|x,\tau_{y})$. Obviously, this correlation also reveals the steerability of the underlying resource, i.e.,

[TABLE]

where $\beta_{a,b}^{*,x,y}$ is the optimal set of coefficients for the biased correlation $\xi_{b}\cdot p(a,b|x,\tau_{y})$.

Our experimental estimation on ${\mathcal{S}_{1}}$ is plotted in Fig. 3 (a). As can be seen, although ${\mathcal{S}^{\text{\tiny LB}}(\mathbf{P})}$ in Eq. (7) may not perform the best among the other fine-grained terms ${\mathcal{S}_{b}^{\text{\tiny LB}}(\mathbf{P})}:={\mathcal{W}}_{b}^{\text{\tiny LB}}(\mathbf{P})-1$, it is the most suitable one in the sense that the variance from the theoretical prediction is the smallest. Besides, some fine-grained terms wrongly detect the existence of steerability due to the overestimation caused by the detection bias (i.e., the estimation of steerability in Fig. 3(a) when the visibility is lower than $1/\sqrt{3}$). With Eq. (8), such overestimation will not occur when we use the quantity $\mathcal{S}^{\text{\tiny LB}}(\mathbb{P})$. Therefore, our estimation on the MDI-SM 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 $\text{ER}({\rho_{AB}})$ of the underlying state and the degree of measurement incompatibility $\text{IR}(\{E_{a|x}\})$ of Alice’s measurements. We briefly recall these two quantities in the Supplementary Material. The result is shown in Fig. 3 (b). The detail of the quantum state tomography to access these two quantities are also shown in the Supplementary Material. Our results are based on the fact that the steering robustness of the assemblage $\text{SR}(\{\sigma_{a|x}\})$ is a lower bound on the entanglement robustness $\text{ER}({\rho_{AB}})$ Piani and Watrous (2015) and incompatibility robustness $\text{IR}(\{E_{a|x}\})$ Cavalcanti and Skrzypczyk (2016); Chen et al. (2016a, 2018). Therefore, as ${\mathcal{S}^{\text{\tiny LB}}(\mathbf{P})}$ is a lower bound on the steering robustness, ${\mathcal{S}^{\text{\tiny LB}}(\mathbf{P})}$ is also used to provide a lower bound on $\text{ER}({\rho_{AB}})$ and $\text{IR}(\{E_{a|x}\})$.

V Concluding Remarks

In this work, we consider a variant of quantum refereed steering games (QRSGs), by which we introduce a measure of steerability in a measurement-device-independent (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 device-independent (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 MDI-SM 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 MDI-SM 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 MDI-SM 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 novel approach recently proposed in Refs. Bowles et al. (2018); Chen et al. . More recently, the DI certification of all steerable states has experimentally been implemented by self-testing an ancilla entangled pair Zhao et al. . It is also interesting to propose practical applications with the MDI scenario (or even a fully DI scheme following the work of Refs. Bowles et al. (2018); Chen et al. ; Zhao et al. ). Since the formulation of the standard steering scenario can be applied to certify the security of quantum keys Branciard et al. (2012), one can ask if this is also the case in the MDI scenario.

Data and code availability

The main data and code supporting the findings of this study are available within the manuscript. Additional data can be provided upon request from the corresponding author.

Acknowledgements.

The authors acknowledge fruitful discussions with Francesco Buscemi, Ana Cristina Sprotte Costa, Yeong-Cherng Liang, Chau Nguyen, Paul Skrzypczyk, Roope Uola and Kang-Da Wu. The authors acknowledge the support of the Graduate Student Study Abroad Program (Grant No. MOST 107-2917-I-006-002) for HYK; the Postdoctoral Research Abroad Program (Grant No. MOST 107-2917-I-564 -007) for SLC; the National Center for Theoretical Sciences and Ministry of Science and Technology, Taiwan (Grants Nos. MOST 107-2628-M-006-002-MY3, 108-2627-E-006-001, and 108-2811-M-006-536), and Army Research Office (Grant No. W911NF-19-1-0081) for YNC; the National Center for Theoretical Sciences and Ministry of Science and Technology, Taiwan (Grant No. MOST 108-2112-M-006-020-MY2) for HBC; the National Natural Science Foundation of China (Grants No. 11574291 and No. 11774334) for GYX; YYZ 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. W911NF-18-1-0358), 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 JSPS-RFBR Grant No. JPJSBP120194828), and the Foundational Questions Institute (FQXi) (Grant No. FQXi-IAF19-06).

VI CONTRIBUTIONS:

HYK and YYZ contributed equally to this work. HYK and SLC contributed equally to the development of the theoretical analysis and conceived the project; GYX supervised the experiment; GYX and YYZ designed the experiment; YYZ conducted the experiment and collected data with the help from GYX; YYZ and HYK analyzed the experimental data with the help from GYX, CFL and GCG; HYK, SLC and HBC proved the theoretical results; YNC and FN supervised the research. All authors contributed to the writing of the manuscript.

VII Competing interests:

The authors declare no competing interests.

VIII Figure legends

Fig1: Schematic illustration of the entanglement, quantum steering, Bell nonlocality, and MDI steering scenarios. A pair of entangled photons ${\rho_{AB}}$ (pink balls) are shared between two spatially separated parties: Alice and Bob. They verify whether they share the entanglement, steering, and nonlocal resource by violating the entanglement witness, steering inequality, and Bell inequality, respectively. (a) In the entanglement certification task, Alice and Bob both perform characterized measurements (transparent box). (b) In the quantum steering scenario, one party performs uncharacterized measurements (black box) according to the classical input $\{x\}$, while the other party performs a set of characterized measurements. (c) In Bell nonlocality, Alice (Bob) receives the classical input $\{x\}$ ($\{y\}$) and returns the outcomes $\{a\}$ ($\{b\}$) with uncharacterised measurements. (d) In the MDI steering scenario, Bob’s classical input $\{y\}$ of the steering scenario is replaced with quantum inputs $\{\tau_{y}\}$, removing the necessity of trustiness of the measurement device.

Fig2: Schematic drawing of the experimental setup. (a) The singlet state of a pair of photons $|HV\rangle-|VH\rangle$ is generated by a spontaneous parametric down-conversion process, where $H$ ($V$) represents the horizontally (vertically) polarized direction. The Werner state is prepared by adding white noise (denoted by $\Omega$) to the system. Then one of the photons is sent to Alice, who uses Q1, H1, and PBS to perform the measurement $x$. The other photon is sent to Bob with an additional qubit system $\tau_{y}$ encoded on the photon’s path degree of freedom ‘0’ and ‘1’. We emphasize the preparation of the trusted quantum system in panel (b). Now Bob performs a complete Bell-state measurement on the equivalent two-qubit systems, i.e., measuring the polarization directions and the spatial paths of the single particle, and returns an outcome $b$. At the end, a set of probability distributions $\{p(a,b|x,\tau_{y})\}$ is obtained to quantify the degree of steerability of the steerable resource. Abbreviations of the components are: BBO, barium borate crystal; HWP(H), half-wave plate; IF, interference filter; Att, attenuator; Mir, mirror; QP, quartz plate; QWP(Q), quarter-wave plate; PBS, polarizing beam splitter; BS, beam splitter; BD, beam displacer. The star represents that the HWP’s axis is oriented at $45^{\circ}$.

Fig3: Results of the MDI-SM and the estimation of entanglement and measurement incompatibility. (a) The MDI experimental demonstration of estimating steerability of the family of two-qubit Werner states when considering Alice has three measurement settings. The theoretical prediction of the MDI-SM is plotted in the black line. The tailored estimator ${\mathcal{S}^{\text{\tiny LB}}(\mathbf{P})}$ described in Eq. (7) for this experiment is marked as diamonds ( $\blacklozenge$). The MDI-SM in Eq. (3) are marked using circles ( $\bigcirc$), crosses ( $\times$), stars ( $*$), and triangles ( $\triangledown$). (b) MDI lower bounds on the degree of entanglement and incompatibility. The diamond symbols ( $\blacklozenge$) in (a) and (b) represent the same quantity. We use the tailored estimator ${\mathcal{S}^{\text{\tiny LB}}(\mathbf{P})}$ as lower bounds on the entanglement robustness (ER) of the underlying state and the incompatibility robustness (IR) of Alice’s measurements. The actual values of these two quantities are represented by triangles ( $\blacktriangledown$) and squares ( $\blacksquare$), respectively. By using the Monte Carlo algorithm, we obtain the standard deviations of ${\mathcal{S}_{b}^{\text{\tiny LB}}(\mathbf{P})}$ in the value around $0.007$ and the standard deviations of ${\mathcal{S}^{\text{\tiny LB}}(\mathbf{P})}$ in the value around $0.004$ for three measurement settings by error propagation.