Maximal Violation of a Broad Class of Bell Inequalities and Its Implications on Self-Testing

TL;DR

This paper demonstrates that lifting procedures preserve the maximal violation of Bell inequalities and their ability to self-test quantum states, broadening the scope of device-independent quantum information protocols.

Contribution

It shows that lifted Bell inequalities maintain maximal violations and self-testing capabilities, extending their applicability in complex quantum scenarios.

Findings

Maximal violation is preserved under lifting for both nonsignaling and quantum correlations.

Self-testing of quantum states is preserved under lifting operations.

Partially robust self-testing can be achieved from lifted Bell inequalities.

Abstract

In quantum information, lifting is a systematic procedure that can be used to derive---when provided with a seed Bell inequality---other Bell inequalities applicable in more complicated Bell scenarios. It is known that the procedure of lifting introduced by Pironio [J. Math. Phys. A 46, 062112 (2005)] preserves the facet-defining property of a Bell inequality. Lifted Bell inequalities therefore represent a broad class of Bell inequalities that can be found in {\em all} Bell scenarios. Here, we show that the maximal value of {\em any} lifted Bell inequality is preserved for both the set of nonsignaling correlations and quantum correlations. Despite the degeneracy in the maximizers of such inequalities, we show that the ability to self-test a quantum state is preserved under these lifting operations. In addition, except for outcome-lifting, local measurements that are self-testable using the original Bell inequality---despite the degeneracy---can also be self-tested using {\em any} lifted Bell inequality derived therefrom. While it is not possible to self-test {\em all} the positive-operator-valued measure elements using an outcome-lifted Bell inequality, we show that partial, but robust self-testing statements on the underlying measurements can nonetheless be made from the quantum violation of these lifted inequalities. We also highlight the implication of our observations on the usefulness of using lifted Bell-like inequalities as a device-independent witnesses for entanglement depth. The impact of the aforementioned degeneracy on the geometry of the quantum set of correlations is briefly discussed.

Equations

$$\sum _{k_i}P(k_1\cdots k_i\cdots k_n|j_1\cdots j_i\cdots j_n)$$

$$=\sum _{k_i}P(k_1\cdots k_i\cdots k_n|j_1\cdots j_i^{\prime }\cdots j_n)$$

$$P(k_1\cdots k_n|j_1\cdots j_n)=\sum _{\lambda }{p}_{\lambda }P(k_1|j_1,\lambda )\cdots P(k_n|j_n,\lambda )$$

$$I_n(\vec{P}):=\vec{B}\cdot \vec{P}=\sum _{k_1\cdots k_n,j_1\cdots j_n}{B}_{\vec{k},\vec{j}}P(\vec{k}|\vec{j})\stackrel{\mathcal{L}}{\le }{\beta }_{\mathcal{L}},$$

$$P(k_1\cdots k_n|j_1\cdots j_n)=\mathrm{tr}\left({\otimes }_{i=1}^n{M}_{k_i|j_i}^{(i)}{\rho }_{12\cdots n}\right)$$

$$P(ab|xy)=\mathrm{tr}\left[M_{a|x}^{(1)}\otimes M_{b|y}^{(2)}{\rho }_{12}\right],$$

$$\Phi {\rho }_{12}{\Phi }^{†}=|{\tilde{\psi }}_{12}⟩⟨{\tilde{\psi }}_{12}|\otimes {\rho }_{aux}$$

$$\Phi [M_{a|x}^{(1)}\otimes M_{b|y}^{(2)}{\rho }_{12}]{\Phi }^{†}=[{\tilde{M}}_{a|x}^{(1)}\otimes {\tilde{M}}_{b|y}^{(2)}|{\tilde{\psi }}_{12}⟩⟨{\tilde{\psi }}_{12}|]\otimes {\rho }_{aux}.$$

$$I_2(\vec{P}):=\sum _{a,b,x,y}{B}_{a,b,x,y}P(ab|xy)\stackrel{\mathcal{L}}{\le }{\beta }_{\mathcal{L}},$$

$${\beta }_{\mathcal{Q}}=\underset{\vec{P}\in \mathcal{Q}}{\max }{I}_2(\vec{P})>{\beta }_{\mathcal{L}}.$$

$$I_2^{\text{LO}}=$$

$$+\sum _{a,x}{B}_{a,b^{\prime },x,y^{\prime }}\left[P(a{b}^{\prime }|x{y}^{\prime })+P(au|x{y}^{\prime })\right]\stackrel{\mathcal{L}}{\le }{\beta }_L$$

$$\begin{array}{rl}\tilde{P}(ab|xy)& =P(ab|xy),y\ne y^{\prime },\\ \tilde{P}(ab|x{y}^{\prime })& =P(ab|x{y}^{\prime }),b\notin \{b^{\prime },u\},\\ \tilde{P}(a{b}^{\prime }|x{y}^{\prime })& =P(a{b}^{\prime }|x{y}^{\prime })+P(au|x{y}^{\prime })\end{array}$$

$${\beta }_{\mathcal{N}}\ge {\beta }_{\mathcal{N}}^{\text{LO}}{\beta }_{\mathcal{Q}}\ge {\beta }_{\mathcal{Q}}^{\text{LO}}.$$

$$\begin{array}{rl}\hat{P}(ab|xy)=P(ab|xy),& y\ne y^{\prime }\text{ or }y=y^{\prime },b\ne b^{\prime },u\\ 0\le \hat{P}(a{b}^{\prime }|x{y}^{\prime })& ,\hat{P}(au|x{y}^{\prime })\le 1,\\ \hat{P}(a{b}^{\prime }|x{y}^{\prime })+\hat{P}& (au|x{y}^{\prime })=P(a{b}^{\prime }|x{y}^{\prime })\end{array}$$

$${\beta }_{\mathcal{Q}}\le {\beta }_{\mathcal{Q}}^{\text{LO}};{\beta }_{\mathcal{N}}\le {\beta }_{\mathcal{N}}^{\text{LO}}.$$

$${\beta }_{\mathcal{Q}}={\beta }_{\mathcal{Q}}^{\text{LO}};{\beta }_{\mathcal{N}}={\beta }_{\mathcal{N}}^{\text{LO}}.$$

$$\Phi {\rho }_{12}^{\ast }{\Phi }^{†}=|{\tilde{\psi }}_{12}⟩⟨{\tilde{\psi }}_{12}|\otimes {\rho }_{aux}$$

$$I_2:=\sum _{a,b,x,y}{B}_{a,b,x,y}P(ab|xy)\stackrel{\mathcal{L}}{\le }0,$$

$$I_2^{\text{LP}}:=\sum _{a,b,x,y}{B}_{a,b,x,y}P(ab{c}^{\prime }|xy{z}^{\prime })\stackrel{\mathcal{L}}{\le }0.$$

$$P(c^{\prime }|z^{\prime })=\sum _{a,b}P(ab{c}^{\prime }|xy{z}^{\prime }).$$

$$P_{c^{\prime }|z^{\prime }}(ab|xy):=P(ab{c}^{\prime }|xy{z}^{\prime })/P(c^{\prime }|z^{\prime }),$$

$${\beta }_{\mathcal{Q}}^{\text{LP}}$$

$$=\underset{\{P(ab{c}^{\prime }|xy{z}^{\prime })\}\in {\mathcal{Q}}_3}{\max }\sum _{a,b,x,y}{B}_{a,b,x,y}{P}_{c^{\prime }|z^{\prime }}(ab|xy)P(c^{\prime }|z^{\prime })$$

$$\le \underset{{\vec{P}}_{c^{\prime }|z^{\prime }}\in {\mathcal{Q}}_2}{\max }\sum _{a,b,x,y}{B}_{a,b,x,y}{P}_{c^{\prime }|z^{\prime }}(ab|xy)\max P(c^{\prime }|z^{\prime })$$

$$=\underset{{\vec{P}}_{c^{\prime }|z^{\prime }}\in {\mathcal{Q}}_2}{\max }\sum _{a,b,x,y}{B}_{a,b,x,y}{P}_{c^{\prime }|z^{\prime }}(ab|xy)={\beta }_{\mathcal{Q}},$$

$$P(ab{c}^{\prime }|xy{z}^{\prime })=P^{\ast }(ab|xy)P(c^{\prime }|z^{\prime })\forall a,b,x,y,$$

$$P(c^{\prime }|z^{\prime })=1,$$

$$\sum _{a,b,x,y}{B}_{a,b,x,y}{P}^{\ast }(ab|xy)={\beta }_{\mathcal{Q}}.$$

$$M_{a|x}^{(1)}=M_{a|x}^{(1\ast )},M_{b|y}^{(2)}=M_{b|y}^{(2\ast )},M_{c|z^{\prime }}^{(3)}=1{\delta }_{c,c^{\prime }},$$

$$\begin{array}{rl}{\beta }_{\mathcal{Q}}& =\underset{\vec{P}\in {\mathcal{Q}}_2}{\max }\sum _{a,b,x,y}{B}_{a,b,x,y}P(ab|xy)\\ & =\sum _{a,b,x,y}{B}_{a,b,x,y}{P}^{\ast }(ab|xy)\\ & =\sum _{a,b,x,y}{B}_{a,b,x,y}⟨{\psi }_{12}^{\ast }|M_{a|x}^{(1\ast )}\otimes M_{b|y}^{(2\ast )}|{\psi }_{12}^{\ast }⟩.\end{array}$$

$$P(ab{c}^{\prime }|xy{z}^{\prime })=P(ab|xy)P(c^{\prime }|z^{\prime })\forall a,b,x,y.$$

$$\begin{array}{rl}& \Phi {\mathrm{tr}}_3\left({\rho }_{123}\right){\Phi }^{†}=|\tilde{\psi }⟩⟨\tilde{\psi }|\otimes {\rho }_{aux},\\ & \Phi \left[M_{a|x}^{(1)}\otimes M_{b|y}^{(2)}{\mathrm{tr}}_3\left({\rho }_{123}\right)\right]{\Phi }^{†}\\ =& \left({\tilde{M}}_{a|x}^{(1)}\otimes {\tilde{M}}_{b|y}^{(2)}|\tilde{\psi }⟩⟨\tilde{\psi }|\right)\otimes {\rho }_{aux}.\end{array}$$

$$\sum _{k_1\cdots k_n,j_1\cdots j_n}{B}_{\vec{k},\vec{j}}P(\vec{k}|\vec{j})\stackrel{{\mathcal{R}}_{n,\ell }}{\le }0,$$

Full text

I Introduction

Inspired by the thought-provoking paper of Einstein, Podolsky, and Rosen Einstein et al. (1935), Bell derived Bell (1964)—based on well-accepted classical intuitions—an inequality constraining the correlations between local measurement outcomes on two distant systems. He further showed that the so-called Bell inequality can be violated by quantum theory using local, but incompatible measurements on entangled states. Since then, various Bell inequalities, such as the one due to Clauser, Horne, Shimony and Holt (CHSH) Clauser et al. (1969) have been derived to investigate the intriguing nature of quantum theory and also the information-processing power enabled by these nonclassical, Bell-nonlocal Brunner et al. (2014) correlations.

For example, Ekert Ekert (1991) showed in 1991 that the quantum violation of Bell inequalities offers an unprecedented level of security for quantum key distribution protocols. Independently, Mayers and Yao Mayers and Yao (1998, 2004) showed that certain extremal quantum correlation enables the possibility to self-test quantum devices. These discoveries prompted the paradigm of device-independent quantum information Scarani (2012); Brunner et al. (2014) in which the physical properties of quantum devices are certified without making any assumption on the internal working of the devices, but rather through the observation of a Bell-nonlocal correlation.

Interestingly, an observation of the maximal quantum violation of certain Bell inequalities, such as the CHSH inequality, is already sufficient to self-test the underlying quantum state and the measurements that give rise to the observed violation Popescu and Rohrlich (1992). To date, numerous Bell inequalities have been derived (see, e.g., Braunstein and Caves (1990); Werner and Wolf (2001); Collins et al. (2002); Kaszlikowski et al. (2002); Żukowski and Brukner (2002); Śliwa (2003); Collins and Gisin (2004); Buhrman and Massar (2005); Gühne et al. (2005); Avis et al. (2006); Liang et al. (2009); Gisin et al. (2009); Acín et al. (2012); Bancal et al. (2012a); Grandjean et al. (2012); Brunner et al. (2014); Mančinska (2014); Liang et al. (2015); Schwarz et al. (2016); Salavrakos et al. (2017); Oudot et al. ; Cope and Colbeck ; Bacarri et al. ; Cruzeiro and Gisin (2019) and references therein). However, beyond the CHSH Bell inequality, only a handful of them Miller and Shi (2013); Yang and Navascués (2013); Mančinska (2014); Bamps and Pironio (2015); Šupić et al. (2016); Kaniewski (2016); Andersson et al. (2017); Salavrakos et al. (2017); Kaniewski et al. ; Bacarri et al. have been identified as relevant for the purpose of self-testing (see Šupić and Bowles for a recent review on the topic of self-testing). Is it possible to make some general statements regarding the self-testing property of Bell inequalities defined for an arbitrary Bell scenario?

To answer the above question, we consider, in this work, Bell inequalities that may be obtained from the procedure of Pironio’s lifting Pironio (2005). Importantly, such inequalities exist in all Bell scenarios beyond the simplest one for two parties, two settings and two outcomes. If a Bell inequality is facet-defining Collins and Gisin (2004), the same holds for its liftings Pironio (2005). What about their quantum violation? In Rosset et al. (2014) (see also Curchod et al. (2015)), it was shown that in addition to the bound satisfied by Bell-local correlations, the maximal quantum (nonsignaling Barrett et al. (2005)) value of Bell inequalities is preserved for party-lifting. In this work, we give an alternative proof of this fact and show, in addition, that the maximal quantum (non-signaling) value of a Bell inequality is also preserved for other types of Pironio’s lifting.

As a corollary of our results, we further show that the self-testing properties of a Bell inequality is largely preserved through the procedure of lifting. In other words, if a Bell inequality can be used to self-test some quantum state $\ket{\psi}$, so can its liftings. Moreover, except for the case of outcome-lifting, the possibility to self-test the underlying measurement operators using a Bell inequality remains intact upon the application of Pironio’s lifting. As we illustrate in this work, the maximizers of lifted Bell inequalities are not unique. There is thus no hope (see, e.g., Goh et al. (2018)) of providing a complete self-testing of the employed quantum device using a lifted Bell inequality. Nonetheless, we provide numerical evidence suggesting that lifted Bell inequalities provide the same level of robustness for self-testing the relevant parts of the devices.

The rest of this paper is organized as follows. In Sec. II, we introduce basic notions of a Bell scenario and remind the definitions of self-testing. After that, we investigate and compare the maximal violation of lifted Bell inequalities against that of the original Bell inequalities, assuming quantum, or general nonsignaling correlations Barrett et al. (2005). In the same section, we also discuss the self-testing property of lifted Bell inequalities, and the usefulness of party-lifted Bell-like inequalities as device-independent witnesses for entanglement depth Liang et al. (2015). In Sec. IV, we present some concluding remarks and possibilities for future research. Examples illustrating the non-uniqueness of the maximizers of lifted Bell inequalities, as well as their implications on the geometry of the quantum set of correlations are provided in the Appendices.

II Preliminaries

II.1 Bell scenario

Consider a Bell scenario involving $n$ spatially separated parties labeled by $i\in\{1,2,\cdots,n\}$ and let the $i$-th party performs a measurement labeled by $j_{i}$,with outcomes labeled by $k_{i}$. In any given Bell scenario, we may appreciate the strength of correlation between measurement outcomes via a collection of joint conditional probabilities. Following the literature (see, e.g., Brunner et al. (2014)), we represent these conditional probabilities—dubbed a correlation—of getting the outcome combination $\vec{k}=(k_{1}k_{2}\cdots k_{n})$ conditioned on performing the measurements $\vec{j}=(j_{1}j_{2}\cdots j_{n})$ by the vector $\vec{P}:=\{P(\vec{k}|\vec{j})\}$. In addition to the normalization condition and the positivity constraint $P(\vec{k}|\vec{j})\geq 0$ for all $\vec{k},\vec{j}$, each correlation is required to satisfy the nonsignaling constraints Barrett et al. (2005) (see also Popescu and Rohrlich (1994))

[TABLE]

for all $i,j_{1},\cdots j_{i-1},j_{i},j^{\prime}_{i},j_{i+1}\cdots j_{n}$, and $k_{\ell}$ (with $\ell\neq i$). In any given Bell scenario, the set of correlations satisfying these constraints forms the so-called nonsignaling polytope $\mathcal{N}$ Barrett et al. (2005).

A correlation is called Bell-local Brunner et al. (2014) if it can be explained by a local-hidden-variable model Bell (1964),

[TABLE]

for all $k_{1}\cdots k_{n},j_{1}\cdots j_{n}$, where $\lambda$ is the hidden variable which occurs with probability $p_{\lambda}$, $\sum_{\lambda}p_{\lambda}=1$ and $P(k_{i}|j_{i},\lambda)$ is the probability of obtaining the measurement outcome $k_{i}$ given the setting $j_{i}$ and the hidden variable $\lambda$. In a given Bell scenario, the set of Bell-local correlations forms a polytope called a Bell polytope, or more frequently a local polytope $\mathcal{L}$, which is a subset of the nonsignaling polytope.

A correlation which cannot be explained by a local-hidden-variable model is said to be Bell-nonlocal and must necessarily violate a Bell inequality Bell (1964) — a constraint satisfied by all $\vec{P}\in\mathcal{L}$. A linear Bell inequality has the generic form:

[TABLE]

where $\vec{B}:=\{B_{\vec{k},\vec{j}}\}$ denotes the vector of Bell coefficients, $\vec{B}\cdot\vec{P}$ and $\beta_{\mathcal{L}}$ are, respectively, the Bell expression and the local bound of a Bell inequality.

A correlation $\vec{P}$ is called quantum if the joint probabilities can be written as

[TABLE]

where $\rho_{12\cdots n}$ is an $n$-partite density matrix and $\{M^{(i)}_{k_{i}|j_{i}}\}_{k_{i}}$ is the positive-operator-valued measure (POVM) describing the $j_{i}$-th measurement of the $i$-party. By definition, POVM elements satisfy the constraints of being positive semidefinite, $M^{(i)}_{k_{i}|j_{i}}\succeq 0$ for all $k_{i}$ and $j_{i}$, as well as the normalization requirement $\sum_{k_{i}}M^{(i)}_{k_{i}|j_{i}}={\bf{1}}$ for all $j_{i}$. Thus, a correlation is quantum if and only if the joint probabilities of such a correlation can be realized experimentally by performing local measurements on an $n$-partite quantum system. The set of quantum correlations $\mathcal{Q}$ forms a convex set satisfying $\mathcal{L}\subset\mathcal{Q}\subset\mathcal{N}$. It is, however, not a polytope Werner and Wolf (2001) (see also Goh et al. (2018)). When necessary, we will use $\mathcal{Q}_{n}$ to denote the set of quantum correlations arising in an $n$-partite Bell scenario.

II.2 Self-testing

Certain nonlocal correlations $\vec{P}\in\mathcal{Q}$ have the appealing feature of being able to reveal (essentially unambiguously) the quantum strategy, i.e., the underlying state and/or the measurement operators leading to these correlations Popescu and Rohrlich (1992); Werner and Wolf (2001); Mayers and Yao (2004); Šupić and Bowles . Following Mayers and Yao (2004), we say that such a $\vec{P}\in\mathcal{Q}$ self-tests the underlying quantum strategy. To this end, it is worth noting that all pure bipartite entangled states can be self-tested Coladangelo et al. (2017).

To facilitate subsequent discussions, we remind here the formal definition of self-testing in a bipartite Bell scenario following the approach of Goh et al. (2018) (see also Šupić and Bowles ). Specifically, consider two spatially separated parties Alice and Bob who each performs measurements labeled by $x$, $y$ and, respectively, observes the outcomes $a$, $b$. We say that a bipartite correlation $\vec{P}:=\{P(ab|xy)\}$ satisfying

[TABLE]

for all $a$, $b$, $x$ and $y$ self-tests the reference (entangled) state $\ket{\tilde{\psi}_{12}}$ if there exists a local isometry $\Phi=\Phi_{1}\otimes\Phi_{2}$ such that

[TABLE]

where $\rho_{12}$ is the measured quantum state [acting on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$], $\rho_{aux}$ is an auxiliary state acting on $\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}}$, and $\mathcal{H}_{A^{\prime}}$ and $\mathcal{H}_{B^{\prime}}$ are the Hilbert spaces associated with the other degrees of freedom of Alice and Bob’s subsystem respectively Coopmans et al. (2019).

Often, a $\vec{P}\in\mathcal{Q}$ that self-tests some reference quantum state can also be used to certify the measurements as well. In such cases, we say that a bipartite correlation $\vec{P}$ obtained from Eq. (4) self-tests the reference quantum state $\ket{\tilde{\psi}_{12}}$ and the reference POVM $\{\tilde{M}^{(1)}_{a|x}\}_{a}$, $\{\tilde{M}^{(2)}_{b|y}\}_{b}$ if there exists a local isometry $\Phi=\Phi_{1}\otimes\Phi_{2}$ such that Eq. (5) holds and

[TABLE]

for all $a$, $b$, $x$ and $y$. By summing over $a,b$, and using the normalization of POVM, one recovers Eq. (5) from Eq. (6).

Interestingly, there are Bell inequalities whose maximal quantum violation alone is sufficient to self-test the quantum state (and the measurement operators) McKague et al. (2012); Yang and Navascués (2013); Pál et al. (2014); Šupić et al. (2016); Kaniewski (2016); Salavrakos et al. (2017); Natarajan and Vidick (2017); Coladangelo (2018). Since then, identifying Bell inequalities which can be used for the task of self-testing has received considerable attention. To this end, note that if the maximal quantum violation of a Bell inequality self-tests some quantum state as well as the underlying measurements, then this maximal quantum violation must be achieved by a unique correlation Goh et al. (2018).

More formally, consider a bipartite Bell inequality,

[TABLE]

with a quantum bound (maximal quantum violation):

[TABLE]

We say that an observation of the quantum violation $I_{2}(\vec{P})=\beta_{\mathcal{Q}}$ self-tests the reference (entangled) state $\ket{\tilde{\psi}_{12}}$ and the reference POVM $\{\tilde{M}^{(1)}_{a|x}\}_{a}$, $\{\tilde{M}^{(2)}_{b|y}\}_{b}$ if there exists a local isometry $\Phi=\Phi_{1}\otimes\Phi_{2}$ such that Eq. (5), Eq. (6) hold for all $\vec{P}\in\mathcal{Q}$ satisfying Eq. (4) and $I_{2}(\vec{P})=\beta_{\mathcal{Q}}$.

III Maximal violation of lifted Bell inequalities and its implications

In this section, we show that the maximal quantum (nonsignaling) violation of a lifted Bell inequality must be the same as that of the Bell inequality from which the lifting is applied. We then discuss the implication of these observations in the context of self-testing, on the geometry of the quantum set of correlations, as well as on the device-independent certification of entanglement depth. For ease of presentation, our discussion will often be carried out assuming a bipartite Bell scenario (for the original Bell inequality). However, it should be obvious from the presentation that our results also hold for any Bell scenario with more parties, and also for a Bell inequality that is not necessarily facet-defining.

III.1 More inputs

Let us begin with the simplest kind of lifting, namely, one that allows additional measurement settings. Applying Pironio’s input-lifting Pironio (2005) to a Bell inequality means to consider the very same Bell inequality in a Bell scenario with more measurement settings for at least one of the parties. At first glance, it may seem rather unusual to make use of only the data collected for a subset of the input combinations, but in certain cases (see, e.g., Shadbolt et al. (2012)), the consideration of all input-lifted facets is already sufficient to identify the non-Bell-local nature of the observed correlations.

Since an input-lifted Bell inequality is exactly the same as the original Bell inequality, its maximal quantum (non-signaling) violation is obviously the same as that of the original Bell inequality. Similarly, it is evident that if the maximal quantum violation of the original Bell inequality self-tests some reference quantum state $\ket{\psi}$ and POVMs $\{M^{(1)}_{a|x}\}_{a,x}$,$\{M^{(2)}_{b|y}\}_{b,y}$,…, so does the maximal quantum violation of the input-lifted Bell inequality.

However, since no constraint is imposed on the additional inputs that do not appear in the Bell expression, it is clear that even if we impose the constraint that the maximal quantum violation of an input-lifted Bell inequality is attained, these other local POVMs can be completely arbitrary. Thus, the subset of quantum correlation attaining the maximal quantum violation of any input-lifted Bell inequality is not unique, and has a degeneracy that increases with the number of these “free" inputs. In other words, the set of quantum maximizers of any input-lifted Bell inequality define a flat region of the boundary of the quantum set of correlations, cf. Goh et al. (2018). In particular, it could lead to completely flat boundaries of $\mathcal{Q}$ on specific two-dimensional slices in the correlation space (see Fig. 1). For some explicit examples illustrating the aforementioned non-uniqueness, see Appendix A.

III.2 More outcomes

Instead of the trivial input-lifting, one may also lift a Bell inequality to a scenario with more measurement outcomes. Specifically, consider a bipartite Bell scenario where the $y^{\prime}$-th measurement of Bob has $v\geq 2$ possible outcomes. The simplest outcome-lifting à la Pironio Pironio (2005) then consists of two steps: (1) choose an outcome, say, $b=b^{\prime}$ from Bob’s $y^{\prime}$-th measurement, and (2) replaces in the sum of Eq. (7) all terms of the form $P(ab^{\prime}|xy^{\prime})$ by $P(ab^{\prime}|xy^{\prime})+P(au|xy^{\prime})$.

The resulting outcome-lifted Bell inequality reads as:

[TABLE]

where the local bound $\beta_{\mathcal{L}}$ is provably the same Pironio (2005) as that of the original Bell inequality, Eq. (7). It is worth noting that outcome-lifted Bell inequalities arise naturally in the study of detection loopholes in Bell experiments, see, e.g., Branciard (2011); Cope and Colbeck .

III.2.1 Preservation of quantum and nonsignaling violation

As with input-lifting, we now proceed to demonstrate the invariance of maximal Bell violation with outcome-lifting.

Proposition 1**.**

Lifting of outcomes preserves the quantum bound and the nonsignaling bound of any Bell inequality, i.e., $\beta^{\text{\tiny LO}}_{\mathcal{Q}}=\beta_{\mathcal{Q}}$ and $\beta^{\text{\tiny LO}}_{\mathcal{N}}=\beta_{\mathcal{N}}$, where $\beta_{\mathcal{Q}}$ ($\beta^{\text{\tiny LO}}_{\mathcal{Q}}$) and $\beta_{\mathcal{N}}$ ($\beta^{\text{\tiny LO}}_{\mathcal{N}}$) are, respectively, the quantum and the nonsignaling bounds of the original (outcome-lifted) Bell inequality.

Proof.

From Eq. (III.2), one clearly sees that the $b^{\prime}$-th outcome and the $u$-th outcome of Bob’s $y^{\prime}$-th measurement are treated on equal footing. So, we may as well consider Bob’s $y^{\prime}$-th measurement as an effective $v$-outcome measurement by considering its $b^{\prime}$-th outcome and its $u$-th outcome together as one outcome. Hence, if we define

[TABLE]

and substitute it back into Eq. (III.2), we recover the Bell expression of the original Bell inequality [left-hand-side of Eq. (7)] by identifying $\tilde{P}(ab|xy)$ in $I^{\text{\tiny LO}}_{2}$ as ${P}(ab|xy)$ in $I_{2}$. Moreover, if $\vec{P}$ defined for this more-outcome Bell scenario is quantum (nonsignaling), the resulting correlation obtained with the coarse-graining procedure of Eq. (10) is still quantum (nonsignaling). A proof of this for quantum correlations is provided in Appendix B (see, e.g., de Vicente (2014) for the case of nonsignaling correlation).

This implies that for any violation of the outcome-lifted Bell inequality (III.2) by a quantum (nonsignaling) correlation, there always exists another quantum (nonsignaling) correlation that gives the same amount of violation for the original Bell inequality (7). In particular, the maximal quantum (nonsignaling) violation of these inequalities must satisfy

[TABLE]

On the other hand, instead of grouping the outcomes in the outcome-lifted Bell scenario, one could also start from the original Bell scenario and (arbitrarily) split the $b^{\prime}$-th outcome of Bob’s $y^{\prime}$-th measurement into two outcomes labeled by $b=b^{\prime}$ and $b=u$. Hence, if we define $\widehat{P}(ab^{\prime}|xy^{\prime})$, $\widehat{P}(au|xy^{\prime})$ in the outcome-lifted Bell scenario such that

[TABLE]

and substitute it into Eq. (7), we recover the outcome-lifted Bell expression [Eq. (III.2)] by identifying $\widehat{P}(ab^{\prime}|xy^{\prime})$ and $\widehat{P}(au|xy^{\prime})$, respectively, as $P(ab^{\prime}|xy^{\prime})$ and $P(au|xy^{\prime})$ in $I^{\text{\tiny LO}}_{2}$, cf Eq. (III.2).

Moreover, the correlation obtained by locally splitting the outcomes, as required in Eq. (12), is realizable quantum-mechanically (see Appendix B) or in general nonsignaling theory (see de Vicente (2014)) if the original correlations are, respectively, quantum and nonsignaling. Hence, for any violation of the original Bell inequality [Eq. (7)] by a quantum (nonsignaling) correlation, there always exists a quantum (nonsignaling) correlation giving the same amount of violation for the outcome-lifted Bell inequality (III.2), i.e.,

[TABLE]

Combining Eqs. (11) and (13), it then follows that the maximal quantum (nonsignaling) violation of any Bell inequality is preserved through the procedure of outcome-lifting, i.e.,

[TABLE]

This completes the proof when only one of the outcomes ($b=u$) of one of the measurements ($y=y^{\prime}$) of one of the parties (Bob) is lifted. However, since more complicated outcome-lifting can be achieved by concatenating the simplest outcome-lifting presented above, the proof for the general scenarios can also be obtained by concatenating the proof given above, thus completing the proof for the general scenario. ∎

III.2.2 Implications on self-testing

As an implication of the above Proposition, we obtain the following result in the context of quantum theory.

Corollary 1**.**

If the maximal quantum violation of a Bell inequality self-tests a quantum state $\ket{\tilde{\psi}}$, then any Bell inequality obtained therefrom by outcome-lifting also self-tests $\ket{\tilde{\psi}}$.

Proof.

For definiteness, we prove this for the specific case of $n=2$, the general proof is completely analogous. To this end, let $\rho^{*}_{12}$ denote an optimal quantum state that maximally violates the outcome-lifted Bell inequality (III.2) with appropriate choice of POVMs $\{M^{(1)}_{a|x}\}_{a,x}$,$\{M^{(2)}_{b|y}\}_{b,y}$. As shown in Appendix B, this quantum state $\rho^{*}_{12}$ can also be used to realize an effective $v$-outcome distribution for Bob’s $y^{\prime}$-th measurement by combining Bob’s relevant POVM elements for this measurement into a single POVM element, thereby implementing the local coarse graining given in Eq. (10) to give the maximal quantum violation of the original Bell inequality, Eq. (7). Suppose that the maximal quantum violation of inequality (III.2) does not self-test the reference state $\ket{\tilde{\psi}_{12}}$, i.e., there does not exist any local isometry $\Phi=\Phi_{1}\otimes\Phi_{2}$ such that

[TABLE]

for some $\rho_{aux}$. Then, we see that the maximal quantum violation of inequality (7) (attainable using $\rho^{*}_{12}$) also cannot self-test the reference state $\ket{\tilde{\psi}_{12}}$. The desired conclusion follows by taking the contrapositive of the above implication. ∎

A few remarks are now in order. As with any other Bell inequality, in examining the quantum violation of an outcome-lifted Bell inequality, one may consider arbitrary local POVMs having the right number of outcomes (acting on some given Hilbert space). A priori, they do not have to be related to the optimal POVM of the original Bell inequality. However, from the proof of Proposition 1, one notices that POVMs arising from splitting the outcomes of the original optimal POVM do play an important role in attaining the maximal quantum violation of the outcome-lifted Bell inequality.

The arbitrariness in this splitting, nonetheless, implies that $\vec{P}\in\mathcal{Q}$ maximally violating an outcome-lifted Bell inequality are not unique (see Appendix A.2 for some explicit examples). Since this invalidates a necessary requirement to self-test both the state and all the local POVMs (see Proposition $C.1.$ of Ref. Goh et al. (2018)), we must thus conclude—given that such an inequality preserves the ability to self-test the underlying state—that its maximal violation cannot be used to completely self-test the underlying measurements. Using the swap method of Yang et al. (2014), we nevertheless show in Appendix C that the quantum violation of an outcome-lifted Bell inequality may still provide robust self-testing of some of the underlying POVM elements, as well as the nature of the merged POVM elements.

III.3 More parties

Finally, let us consider the party-lifting of Pironio (2005). Again, for simplicity, we provide hereafter explicit constructions and proofs only for the bipartite scenario, with the multipartite generalizations proceed analogously. To this end, it is expedient to write a generic bipartite Bell inequality such that

[TABLE]

i.e., with its local bound being zero.111This can always be achieved by (repeatedly) applying identity of the form given in Eq. (II.1) to both sides of Eq. (7). For any fixed but arbitrary input-output pair $c^{\prime},z^{\prime}$ of the additional party (Charlie), applying the party-lifting of Pironio (2005) to inequality (16) gives rise to the tripartite Bell inequality:

[TABLE]

It is worth noting that such Bell inequalities has found applications in the foundations of quantum theory Bancal et al. (2012b); Barnea et al. (2013), as well as in the systematic generation Curchod et al. (2015) of device-independent witnesses for entanglement depth Liang et al. (2015).

III.3.1 Preservation of quantum and nonsignaling violation

That the maximal quantum and nonsignaling violation remain unchanged under Pironio’s party-lifting operation Pironio (2005) follows directly from the results shown in Section 2.4 of Rosset et al. (2014), as well as a special case (with $n=k$) of Theorem 2 of Curchod et al. (2015). For the convenience of subsequent discussions, however, we provide below an alternative proof of this observation.

Observation 1**.**

Lifting of parties preserves the quantum bound and the nonsignaling bound of any Bell inequality, i.e., $\beta^{\text{\tiny LP}}_{\mathcal{Q}}=\beta_{\mathcal{Q}}$ and $\beta^{\text{\tiny LP}}_{\mathcal{N}}=\beta_{\mathcal{N}}$, where $\beta_{\mathcal{Q}}$ ($\beta^{\text{\tiny LP}}_{\mathcal{Q}}$) and $\beta_{\mathcal{N}}$ ($\beta^{\text{\tiny LP}}_{\mathcal{N}}$) are, respectively, the quantum and the nonsignaling bounds of the original (party-lifted) Bell inequality.

Proof.

For a tripartite Bell scenario relevant to inequality (17), the marginal probability of Charlie getting the outcome $c^{\prime}$ conditioned on him performing the measurement labeled by $z^{\prime}$ is:

[TABLE]

Since the party-lifted inequality of Eq. (17) is saturated with the choice of $P(c^{\prime}|z^{\prime})=0$, thereby making $P(abc^{\prime}|xyz^{\prime})=0$ for all $a,b,x,y$, the observation holds trivially if inequality (17) cannot be violated by general nonsignaling correlations.

Conversely, if inequality (17) can be violated by some quantum, or general nonsignaling correlation, the corresponding $P(c^{\prime}|z^{\prime})$ must be nonvanishing. Henceafter, we thus assume that $P(c^{\prime}|z^{\prime})>0$. To this end, note that

[TABLE]

gives the probabilities of Alice and Bob obtaining the outcomes $a$ and $b$ conditioned on her (him) choosing measurement $x$ ($y$), Charlie measuring $z^{\prime}$ and obtaining the outcome $c^{\prime}$. Note that the vector of probabilities $\vec{P}_{c^{\prime}|z^{\prime}}:=\{{P}_{c^{\prime}|z^{\prime}}(ab|xy)\}$ is a legitimate correlation in the (original) Bell scenario corresponding to inequality (16).

To prove the observation, we now focus on the case of finding the quantum bound, i.e., the maximum value of the left-hand-side of Eq. (17) for quantum correlations — the proof for the nonsignaling case is completely analogous. To this end, note that the quantum bound of inequality (17)—given the above remarks—satisfies:

[TABLE]

where the first inequality follows from the fact that an independent maximization over $\vec{P}_{c^{\prime}|z^{\prime}}\in\mathcal{Q}_{2}$ and $P(c^{\prime}|z^{\prime})$ is, in principle, less constraining than a maximization over all tripartite quantum distributions $\{P(abc^{\prime}|xyz^{\prime})\}$, the second-last equality follows from the fact that $P(c^{\prime}|z^{\prime})\leq 1$ for legitimate marginal probability distributions, and the last equality follows from the fact that any bipartite quantum correlation can be seen as the marginalization of a tripartite one.

To complete the proof, note that the inequality $\beta^{\text{\tiny LP}}_{\mathcal{Q}}\leq\beta_{\mathcal{Q}}$ can indeed be saturated if

[TABLE]

Moreover, these equations can be simultaneously satisfied with the three parties sharing a state of the form $\ket{\psi_{123}}=\ket{\psi^{*}_{12}}\otimes\ket{\psi_{3}}$ while employing the local measurements:

[TABLE]

where $\ket{\psi^{*}_{12}},\{M^{(1*)}_{a|x}\}_{a,x},\{M^{(2*)}_{b|y}\}_{b,y}$ consistute a maximizer for the (original) Bell inequality of Eq. (16), i.e.,

[TABLE]

∎

III.3.2 Implications on self-testing

As an implication of the above observation, we obtain the following result in the context of quantum theory.

Corollary 2**.**

If a Bell inequality self-tests $\ket{\tilde{\psi}}$ and some reference POVMs $\{\tilde{M}^{(1)}_{a|x}\}_{a,x},\{\tilde{M}^{(2)}_{b|y}\}_{b,y}$ etc., then the maximal quantum violation of any Bell inequality obtained therefrom via party-lifting also self-tests the same state and the same local POVMs for an appropriate subset of parties.

Proof.

In the following, we use inequality (17) to illustrate how the proof works in the tripartite case. Note from Eq. (III.3.1) that when the party-lifted Bell inequality of Eq. (17) is violated to its quantum maximum $\beta_{\mathcal{Q}}$, the marginal distribution of Charlie necessarily satisfies $P(c^{\prime}|z^{\prime})=1$. It then follows from Eq. (19) that

[TABLE]

Furthermore, from Eq. (III.3.1), this tripartite distribution gives the quantum bound of inequality (17) if and only if the marginal distributions $P(ab|xy)$ of Eq. (24) violates the original Bell inequality of Eq. (16) to its quantum bound. Therefore, if the original Bell inequality self-tests the $2$-partite entangled state $\ket{\tilde{\psi}_{12}}$, then for any tripartite density matrix $\rho_{123}$ leading to the quantum maximum of inequality Eq. (17), there must exist a local isometry $\Phi=\Phi_{1}\otimes\Phi_{2}$ such that

[TABLE]

where $\rho_{aux}$ is some auxiliary density matrix acting on other degrees of freedom of Alice and Bob’s subsystem. In other words, if the quantum maximum of the original Bell inequality can be used to self-test $\ket{\tilde{\psi}}$ and reference POVMs $\{\tilde{M}^{(1)}_{a|x}\}_{a,x}$, $\{\tilde{M}^{(2)}_{b|y}\}_{b,y}$, so does the quantum maximum of the party-lifted Bell inequality. ∎

III.3.3 Implications on device-independent certification of entanglement depth

In Theorem 2 of Curchod et al. (2015), it was shown that if

[TABLE]

is satisfied by an $n$-partite resource $\mathcal{R}$ (quantum or nonsignaling) that has a group size of $\ell$, its lifting to $(n+h)$ parties also holds for the same kind of resource of group size $\ell$:

[TABLE]

where $\vec{o}$ ($\vec{s}$) is any fixed, but arbitrary string of outputs (inputs) for the $h$ additional parties. For the case of $\ell=n$, the above result reduces to Observation 1 discussed in Sec. III.3.1.

When the considered resource is restricted to shared quantum correlations, inequality (26) and inequality (27) are instances of so-called device-independent witnesses for entanglement depth Liang et al. (2015), i.e., Bell-like inequalities capable of certifying—directly from the observed correlation—a lower bound on the entanglement depth Sørensen and Mølmer (2001) of the measured system. More specifically, if the observed quantum value of the left-hand-side of Eq. (26) or Eq. (27) is greater than 0, then one can certify that the locally measured quantum state must have an entanglement depth of at least $\ell+1$.

Although the above result of Curchod et al. (2015) can be applied to an arbitrary number of $(n+h)$ parties, Observation 1 implies that if the seed inequality is that applicable to an $n$-partite Bell scenario, the extended scenario can never be used to certify an entanglement depth that is larger than $n$. This follows from the fact that the maximal quantum value of these party-lifted Bell-like inequalities is the same as the original Bell-like inequality [cf. Eq. (26)], and is already attainable using a quantum state of entanglement depth $n$.

IV Concluding Remarks

Lifting, as introduced by Pironio Pironio (2005), is a procedure that allows one to systematically construct Bell inequalities for all Bell scenarios starting from a Bell inequality applicable to a simpler scenario. It is known that Pironio’s lifting preserves the facet-defining property of Bell inequalities, and thus lifted Bell inequalities (in particular, those lifted from the CHSH Bell inequality) can be found in all nontrivial Bell scenarios. In this work, we show that lifting leaves the maximal quantum and non-signaling value of Bell inequalities unchanged.

Naturally, one may ask whether the quantum state and local measurements maximally violating a lifted Bell inequality are related to that of the original Bell inequality. Indeed, we show that Pironio’s lifting also preserves the self-testability of a quantum state. Hence, the quantum state maximally violating a lifted Bell inequality is—modulo irrelevant local degrees of freedom—the same as that of the original inequality. Likewise, the self-testability of given local measurements is preserved using any but the outcome-lifting procedure.

The maximizers of lifted Bell inequalities are, as we show, generally not unique. Consequently, it is impossible to use the observed quantum value of such an inequality to self-test both the underlying state and all the local measurements: in the case of an input-lifted Bell inequality, no conclusions can be drawn regarding the additional measurements that do not appear in the inequality; in the case of a party-lifted Bell inequality, nothing can be said about the measurements of the additional party; in the case of an output-lifted Bell inequality, the self-testing of all the local POVM elements is impossible, but the self-testing of their combined effect seems possible. In fact, our numerical results (see Appendix C) suggest that such a self-testing is just as robust as the original Bell inequality. Thus, Bell inequalities lifted from CHSH serve as generic examples whose maximal quantum violation can be used to self-test a state, but not the underlying measurements in its entirety.

Notice also that the non-uniqueness mentioned above evidently becomes more and more pronounced as the number of “irrelevant" degrees of freedom increases, for example, by repeatedly applying lifting to a given Bell inequality. Since only correlation $\vec{P}$ belonging to the boundary of $\mathcal{Q}$ could violate a linear Bell inequality maximally, as the complexity of the Bell scenario (say, in terms of the number of measurements, outcomes, or parties) increases, it is conceivable that one can always find a flat boundary of $\mathcal{Q}$ (corresponding to those of the lifted Bell inequality) with increasing dimension. Proving this statement rigorously and finding the exact scaling of the dimension of these flat boundaries would be an interesting direction to pursue for future research.

Besides, it will be interesting to see—in comparison with the original Bell inequality—whether the robustness in self-testing that we have observed for a particular version of the outcome-lifted CHSH Bell inequality is generic. From our example for the outcome-lifted CHSH inequality, it becomes clear that self-testing of the combined POVM elements is (sometimes) possible even if the self-testing of all individual POVM elements is not. This possibility opens another direction of research in the context of self-testing. In addition, our results also prompted the following question: does there exist a physical situation (say, the observation of the maximal quantum value of some Bell inequality) where the underlying measurements can be self-tested, but not the underlying state? Since the self-testing of measurements is seemingly more demanding than that for a quantum state, it is conceivable that no such examples can be constructed. Proving that this is indeed the case, however, clearly lies beyond the scope of the present paper.

Note added: While completing this manuscript, we became aware of the work of Rosset et al. which also discusses, among others, extensively the properties of liftings, as well as the work of Kaniewski , which exhibits examples of quantum correlations that can only be used to self-test the measured quantum state but not the underlying measurements.

Acknowledgements.

We thank Jean-Daniel Bancal, Jędrzej Kaniewski, Denis Rosset, and Ivan Šupić for useful discussions. This work is supported by the Foundation for the Advancement of Outstanding Scholarship, Taiwan as well as the Ministry of Science and Technology, Taiwan (Grants No. 104-2112-M-006-021-MY3, No. 107-2112-M-006-005-MY2, 107-2627-E-006-001, 108-2811-M-006-501, and 108-2811-M-006-515).

Appendix A Examples of $\vec{P}\in\mathcal{Q}$ violating lifted Bell inequalities maximally

To illustrate the non-unique nature of the maximizers of lifted Bell inequalities, consider the CHSH Bell inequality Clauser et al. (1969):

[TABLE]

as our seed inequality. Since this is a Bell inequality defined in the simplest Bell scenario (with two binary-output per party), its liftings can be found in all nontrivial Bell scenarios.

The quantum bound and nonsignaling bound of the above Bell inequality are given, respectively, by $\beta_{\mathcal{Q}}=2\sqrt{2}$ and $\beta_{\mathcal{N}}=4$. It is known Popescu and Rohrlich (1992) that the maximal quantum violation of the CHSH inequality can be used to self-test (up to local isometry) the two-qubit maximally entangled state $\ket{\phi^{+}}=\left(\ket{00}+\ket{11}\right)/\sqrt{2}$ and the Pauli observables $\{\sigma_{z},\sigma_{x}\}$ on one side and the Pauli observables $\{(\sigma_{x}+\sigma_{z})/\sqrt{2},(\sigma_{x}-\sigma_{z})/\sqrt{2}\}$ on the other. Thus, the correlation that gives the quantum maximum of the inequality (28) is unique and is given by

[TABLE]

where the $+1$-outcome of the observables is identified with the 0-th outcome in the conditional outcome probability distributions.

A.1 Lifting of Inputs

For input-lifting, consider now a bipartite Bell scenario where Bob has instead three binary inputs $(y=0,1,2)$. In this new Bell scenario, the following Bell inequality:

[TABLE]

with $\{\tilde{P}(ab|xy)\}_{a,b,x=0,1,y=0,1,2}$ can be obtained by applying input-lifting to inequality (28).

To illustrate the non-uniqueness of its maximizers, one may employ, e.g., either of the two trivial measurements for the third measurement $(y=2)$:

[TABLE]

Correspondingly, one obtains, in addition to [cf. Eq. (29)]

[TABLE]

for $a,b,x=0,1$ but $y=2$.222More abstractly, these correlations can also be obtained from Eq. (29) by applying input operations as given by Eq. (10) in de Vicente (2014).

It is then easily verified that both these correlations violate inequality (30) to its quantum maximum of $2\sqrt{2}$. In fact, since the Bell expression of Eq. (30) is linear in $\tilde{P}$, it follows that an arbitrary convex combination of $\tilde{P}^{\text{\tiny LI}}_{Q_{1}}$ and $\tilde{P}^{\text{\tiny LI}}_{Q_{2}}$

[TABLE]

where $0\leq p\leq 1$, must also give the maximal quantum value of Bell inequality (30). Moreover, from the convexity of the set of quantum correlations $\mathcal{Q}$, we know that an arbitrary convex combination of the two correlations given above is also quantum realizable. Geometrically, this means that the set of $\tilde{P}\in\mathcal{Q}$ defined by Eq. (33) forms a one-dimensional flat region of the quantum boundary. In Fig. 1, we show a two-dimensional slice on the space of correlations spanned by $\tilde{P}^{\text{\tiny LO}}_{Q_{1}}$, $\tilde{P}^{\text{\tiny LO}}_{Q_{2}}$, and the uniform distribution $\tilde{P}_{0}$. Note that on this peculiar slice, even $\mathcal{Q}$ appears to be a polytope.

A.2 Lifting of Outcomes

For output-lifting, consider a bipartite Bell scenario where Bob’s measurements have instead three outcomes $(b=0,1,2)$. In this Bell scenario, the following Bell inequality:

[TABLE]

can be obtained by applying outcome-lifting to the $b=0$ outcome of Bob’s measurements in inequality (28). Here, $b=2$ is the new outcome.

Indeed, it is readily seen that the two correlations:

[TABLE]

as well as

[TABLE]

all give rise to the quantum maximum of $2\sqrt{2}$ for the Bell inequality of Eq. (34).

To see that they are indeed quantum realizable, we note that the correlation given in Eq. (35) can be produced by employing the quantum strategy used to produce the correlation given in Eq. (29). By construction, Bob’s measurement only produces two outcomes labeled by $b=0$ and $b=1$, and thus the $b=2$ outcome never appears, as required in Eq. (35). To obtain the correlation given in Eq. (36), one may start from the correlation of Eq. (35) and apply the classical relabeling of $b=0\leftrightarrow b=2$.

Similarly, the two correlations of Eq. (37) can be realized by first implementing the quantum strategy that realizes $\vec{P}^{\text{\tiny LO}}_{Q_{1}}$, followed by applying the classical relabeling of $b=0\leftrightarrow b=2$ depending on whether $y=0$ or $y=1$ 333This belongs to the class of outcome operation given by Eq. (9) in de Vicente (2014).. Since $\{\vec{P}^{\text{\tiny LO}}_{Q_{i}}\}_{i=1}^{4}$ forms a linearly independent set, and an arbitrary convex combination of them also gives the quantum bound, we thus see that the quantum face444The set of quantum correlations that gives the quantum bound of a Bell inequality is called quantum face, see Goh et al. (2018). of the outcome-lifted inequality of (34) is (at least) three-dimensional.

A.3 Lifting of Party

Geometrically, party-lifting also introduces degeneracy in the maximizers of a Bell inequality. For example, a possible party-lifting of the CHSH Bell inequality of Eq. (28) to the three-party, two-input, two-output Bell scenario reads as:

[TABLE]

As mentioned in the proof of Corollary 2, maximizers of this Bell inequality must be such that the bipartite marginal distribution $P(ab|xy)$ violates CHSH Bell inequality maximally while the marginal distribution for the third party must satisfy $P(0|0)=1$. However, these conditions do not impose any constraint on the other marginal distribution $P(c|z)$ for $z\neq 0$. In particular, both the choice of $P(c|1)=\delta_{c,0}$ and $P(c|1)=\delta_{c,1}$ would fulfill the above requirement. As such, although the quantum face of the CHSH Bell inequality is a point in the correlation space, the quantum face of the party-lifted Bell inequality of Eq. (38) has become one-dimensional.

Appendix B Quantum realizability of distributions obtained by grouping and splitting outcomes

In this Appendix, we provide the details showing how one can—while preserving between the quantum violation of a Bell inequality and its outcome-lifted version—realize quantum-mechanically a fewer-outcome (more-outcome) correlation if the original more-outcome (fewer-outcome) correlation is quantum.

B.1 Grouping of outcomes

Suppose that the joint probabilities $P(ab^{\prime}|xy^{\prime})$ and $P(au|xy^{\prime})$ on the right-hand-side of Eq. (10) are realized by a quantum state $\rho_{12}$ with the POVM $\{M^{(1)}_{a|x}\}_{a,x}$ on Alice’s side and $\{M^{(2)}_{b|y}\}_{b,y}$ on Bob’s side, cf. Eq. (4). Then the joint probabilities $\tilde{P}(ab^{\prime}|xy^{\prime})$ appearing on the left-hand-side of Eq. (10), which corresponds to an effective $v$-outcome distribution, is realizable by the same quantum state $\rho_{12}$ with the same POVM $\{M^{(1)}_{a|x}\}_{a,x}$ on Alice’s side and the following POVM $\tilde{M}^{(2)}_{b|y}$ on Bob’s side:555That $\tilde{M}^{(2)}_{b|y}$ satisfies both the positivity constraints and the normalization constraints is evident from Eq. (39).

[TABLE]

With this choice, it follows from

[TABLE]

Eq. (39), and

[TABLE]

that Eq. (10) is satisfied, and hence that the violation-preserving fewer-outcome correlation is indeed attainable by coarse graining, i.e., the grouping of outcomes.

B.2 Splitting of outcomes

On the other hand, if we instead start from the original (fewer-outcome) Bell scenario, then the joint probabilities appearing on the right-hand-side of Eq. (12), which corresponds to a $v+1$-outcome distribution, can be realized by, e.g., employing the same quantum state $\rho_{12}$ with the same POVM $\{M^{(1)}_{a|x}\}_{a,x}$ on Alice’s side and the following POVM $\tilde{M}^{(2)}_{b|y}$ on Bob’s side:

[TABLE]

for arbitrary $0\leq p\leq 1$. The positivity of the left-hand-side of Eq. (42) and their normalization are evident from their definition. Moreover, using Eq. (B.1) and

[TABLE]

it is easy to see that Eq. (12) holds with the assignment given in Eq. (42). Hence, the POVM given by the left-hand-side of Eq. (42) indeed realizes the required violation-preserving more-outcome correlation by splitting the $b^{\prime}$-outcome of Bob’s $y^{\prime}$-th measurement.

Appendix C Robust self-testing based on the quantum violation of the outcome-lifted CHSH inequality

We show in this Appendix that self-testing via the quantum violation of the outcome-lifted inequality of Eq. (34) is robust. In this regard, note that the maximal quantum violation of inequality (34) can also be achieved by Alice and Bob sharing the following two-qubit maximally entangled state

[TABLE]

while performing the optimal qubit measurements for Alice:

[TABLE]

and for Bob:

[TABLE]

where $\{E_{b|y}\}_{b=0,2}$ are any valid POVM elements satisfying $\sum_{b=0,2}E_{b|y}=\openone-M_{1|y}^{(2)}$ for all $y$.

Notice that Eq. (5) and Eq. (6) only hold for the case of perfect self-testing of the reference state and reference measurements. To demonstrate robust self-testing for the above reference state $\ket{\tilde{\psi}}$ and reference measurements $\{\tilde{M}^{(1)}_{a|x}\}_{a,x}$, $\{\tilde{M}^{(2)}_{b|y}\}_{b,y}$, we follow the approach of Yang et al. (2014) to arrive at statements saying that if the observed quantum violation of inequality (34) is close to its maximal value, then (1) the measured system contains some degrees of freedom that has a high fidelity with respect to the reference state $\ket{\tilde{\psi}}$, and (2) with high probability, the uncharacterized measurement devices function like $\{\tilde{M}^{(1)}_{a|x}\}_{a,x}$, $\{\tilde{M}^{(2)}_{b|y}\}_{b,y}$ acting on the same degrees of freedom.

C.1 Robust self-testing of the reference state

To this end, we shall make use of the swap method proposed in Yang et al. (2014). The key idea is to introduce local swap operators $\Phi_{1},\Phi_{2}$ so that the state acting on Alice’s and Bob’s Hilbert space (of unknown dimension) gets swapped locally with some auxiliary states of trusted Hilbert space dimension (qubit in our case). To better understand how this works, let us first consider an example with characterized devices before proceeding to the case where the devices are uncharacterized.

For this purpose, let us concatenate the following controlled-not (CNOT) gates

[TABLE]

to obtain the (two-qubit) swap gate $\Phi_{1}=U_{1}V_{1}U_{1}$ acting on $\mathcal{H}_{A}\otimes\mathcal{H}_{A^{\prime}}$ (see Sec. II.2). We may define a swap operator $\Phi_{2}=U_{2}V_{2}U_{2}$ acting on Bob’s systems in exactly the same way. Importantly, one notices from Eq. (45) and Eq. (46) that it is possible to express the individual unitaries in terms of the POVM elements leading to the maximal quantum violation of inequality (34). For example, one may take

[TABLE]

Moreover, if we define the global swap gate by $\Phi=\Phi_{1}\otimes\Phi_{2}$ and denote the state acting on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ by $\rho_{12}$, then the “swapped” state is:

[TABLE]

where $\operatorname{tr}_{12}$ represents a partial trace over the Hilbert space of $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. When $\Phi$ is exactly the swap gate defined above via Eq. (47), $\rho^{\text{\tiny SWAP}}$ is exactly $\rho_{12}$. Thus, the fidelity between $\rho^{\text{\tiny SWAP}}$ and the reference state $\ket{\tilde{\psi}}$: $F=\langle\tilde{\psi}|\rho^{\text{\tiny SWAP}}|\tilde{\psi}\rangle$ provides a figure of merit on the similarity between (some relevant parts of) the shared state $\rho_{12}$ and the reference state $\ket{\tilde{\psi}}$.

To perform a device-independent characterization, the assumption of $\tilde{M}_{1|x}^{(1)}$ and $\tilde{M}_{1|y}^{(2)}$ is relaxed to unknown projectors $M_{1|x}^{(1)}$ and $M_{1|y}^{(2)}$ (acting on Hilbert space of arbitrary dimensions), and the corresponding “CNOT” gates becomes

[TABLE]

for $i=1,2$. One can verify that the fidelity $F=\langle\tilde{\psi}|\rho^{\text{\tiny SWAP}}|\tilde{\psi}\rangle$ is then a linear function of the moments such as $\langle M_{a|x}^{(1)}\otimes M_{b|y}^{(2)}\rangle$, $\langle M_{a|x}^{(1)}\otimes M_{b|y}^{(2)}M_{b^{\prime}|y^{\prime}}^{(2)}\rangle$ etc., where $\langle\cdot\rangle:=\operatorname{tr}(\cdot\rho_{12})$.

Thus, a lower bound on $F$ for any observed value of Bell inequality violation (without assuming the shared state or the measurements performed) can be obtained by solving the following semidefinite program:

[TABLE]

where $\Gamma^{S}$ is any Navascués-Pironio-Acín-type Navascués et al. (2007) moment matrix that contains all the moments appearing in $F$. In our computation, we employed a moment matrix that is built from a sequence of operators $S$ that contains all operators from level 1+AB (or equivalently, level 1 from the hierarchy of Ref. Moroder et al. (2013)) and some additional operators from level 3.

Our results (see Fig. 2) clearly shows that the self-testing property of $I_{2}^{\text{\tiny LO-CHSH}}$ with respect to the reference maximally entangled state $\ket{\tilde{\psi}}$ of Eq. (44) is indeed robust. In other words, as long as the observed violation of $I_{2}^{\text{\tiny LO-CHSH}}$ is greater than $\approx 2.4$, one can still obtain a non-trivial lower bound on the fidelity ($>1/2$) with respect to $\ket{\tilde{\psi}}$. Moreover, a separate computation using the original CHSH Bell inequality of Eq. (28) (and the same level of approximation of $\mathcal{Q}$) gives—within the numerical precision of the solver—the same curve, thereby suggesting that the outcome-lifted Bell inequality of (34) offers the same level of robustness as compared with its seed inequality.

C.2 Robust self-testing of Alice’s POVM

Even though it is impossible to completely self-test all local measurements, robust self-testing of Alice’s POVM—as one would intuitively expect—can still be achieved. In particular, when the observed violation of $I_{2}^{\text{\tiny LO-CHSH}}$ is close to the quantum bound of $2\sqrt{2}$, it must be the case that Alice’s measurements (on the relevant degrees of freedom) indeed behave like measurements in the $\sigma_{z}$ and $\sigma_{x}$ basis, respectively, for $x=0,1$.

To that end, we again make use the swap method proposed in Ref. Yang et al. (2014). The idea is that if these measurements behave as expected, then their measurements on the auxiliary states swapped into the uncharacterized device, i.e., $\Phi_{1}(|\varphi\rangle)$—with $\ket{\varphi}$ being eigenstates of $\sigma_{z}$ and $\sigma_{x}$—should produce outcomes $a$ with statistics $\{P(a|x,|\varphi\rangle)\}$ satisfying

[TABLE]

Using the same swap operator defined via Eq. (50), we get

[TABLE]

where we have, for simplicity, omitted the identity operator acting on Bob’s system. Notice that the left-hand-side of Eq. (53) is again some linear combination of moments. Likewise for the following figure of merit Yang et al. (2014):

[TABLE]

which takes value between $-1$ and $+1$. The maximum of $+1$, in particular, happens only when Alice’s POVM $M^{(1)}_{a|x}$ correspond to measurements in $\sigma_{z}$ and $\sigma_{x}$, respectively, for $x=0,1$. $\tau$ therefore quantifies the extent to which the measurement devices function like the reference measurements. Given a violation of $I_{2}^{\text{\tiny LO-CHSH}}$, a lower bound on $\tau$ can thus be obtained by solving the following semidefinite program:

[TABLE]

The resulting lower bounds on $\tau$ are shown in Figure 3. We see that the value of $1$ is obtained when the maximal quantum value of $I_{2}^{\text{\tiny LO-CHSH}}$ is observed, which means that the reference measurements of Eq. (45) are correctly certified in this case. For nonmaximal values of $I_{2}^{\text{\tiny LO-CHSH}}$, we see that $\tau$ decreases accordingly. Importantly, as pointed out in Ref. Yang et al. (2014), the procedure of sending the prepared eigenstates into the swap gate is a virtual process that allows us to interpret the figure of merit operationally, but the result still holds without any assumption on the devices of interest.

C.3 Partial but robust self-testing of Bob’s POVMs

Finally, we would like to show that the outcome-lifted CHSH inequality of Eq. (34) can also be used for a “partial” self-testing of Bob’s optimal measurements. The steps are the same as those described in the self-testing of Alice’s measurements. That is, the eigenstates of $\sigma_{z}$ and $\sigma_{x}$ are sent to the swap gate before Bob performs his measurements $\{M_{b|y}^{(2)}\}$.

To this end, we define the analog of Eq. (54) as:

[TABLE]

and introduce a further figure of merit

[TABLE]

to self-test only the POVM element corresponding to Bob’s outcome 1 for both measurements.

Thus, $\tau_{3}$ takes into account of Bob’s all measurements outcomes while $\tau_{1}$ only involves the second measurement outcome. All these figures of merit range from $-1$ to $+1$, and $+1$ is recovered for

• •

$\tau_{3}$ if Bob’s measurement device acts on the swapped eigenstate according to Eq. (46);

• •

$\tau_{1}$ if Bob’s measurement device acts on the swapped eigenstate in such a way that the 2nd POVM element for each measurement functions according to Eq. (46)

In other words, the value of $\tau_{1}$ measures the extent to which $M_{1|y}^{(2)}$ behaves according to that prescribed in Eq. (46), while the value of $\tau_{3}$ further indicates if the combined effect of $M_{0|y}^{(2)}+M_{2|y}^{(2)}$ also behaves according to that prescribed in Eq. (46).

By solving the semidefinite program of Eq. (55) using the appropriate objective functions, we obtained lower bounds on each figure of merit as a function of the quantum violation of $I_{2}^{\text{\tiny LO-CHSH}}$. As shown in Fig. 4, the bounds on $\tau_{3}$ and $\tau_{1}$ when $I_{2}^{\text{\tiny LO-CHSH}}$ takes its maximal value successfully self-tests, respectively, the combined effect of $M_{0|y}^{(2)}+M_{2|y}^{(2)}$ as well as that of $M_{1|y}^{(2)}$. In summary, for the outcome-lifted CHSH inequality of Eq. (34), where the first outcome is lifted in each of Bob’s measurement, it is still possible to self-test Alice’s optimal measurements and the overall behavior of Bob’s measurements.