Device-independent self-testing of unsharp measurements

Measurement plays a pivotal role in quantum theory which is in stark contrast to classical theory. The textbook version of a quantum measurement is modeled by a set of orthogonal projectors belonging to the Hilbert space. However, there exist more general measurements defined in terms of positive-operator-valued measures (POVMs) satisfying the completeness relation. Note that the POVMs can be defined in many different ways. However, in this paper, we are concerned about those POVMs which are noisy or unsharp variants of the projective measurements.

In a sharp projective measurement [1], the system collapses to one of the eigenstates of the measured observable, and the system is completely disturbed by the process of sharp measurement, so that, no residual coherence remains in the system. On the other hand, in the case of unsharp measurements that are characterized by the POVMs, the system is less disturbed compared to the sharp projective measurement. Since a projective measurement maximally disturbs the quantum system and hence extracts more information from the system compared to POVMs, one may think that a sharp measurement is more advantageous in information processing tasks. However, there exist certain information processing tasks where POVMs are proven to be more useful compared to sharp measurement.

Advantage of POVMs over sharp measurement has been explored in the context of quantum state discrimination [2, 3], randomness certification [4, 7, 6, 5], quantum tomography[8], state estimation [9], quantum cryptography [10], information acquisition from a quantum source [11], quantum violation of certain Bell inequalities [12], and many more. There is one more advantage that is particularly relevant in the present work is the sequential sharing of various forms of quantum correlations [13, 14, 15, 16, 17, 18]. For example, a sequential Bell test by multiple independent observers inevitably requires the prior observers to perform the unsharp quantum measurements.

In this work, we aim to provide device-independent (DI) self-testing of the unsharp measurements, which are noisy variants of the sharp projective measurements. Self-testing [19, 20, 21, 22, 23] is the strongest form of certification protocol where the devices are treated as black boxes. Also, the dimension of the quantum system is assumed to be finite but unbounded. In that, observed experimental statistics uniquely certify the state and measurement observables of an unknown dimension. DI self-testing is advantageous over the standard certification methods such as those based on quantum tomography. Essentially, a self-testing protocol requires optimal quantum violation of a suitable Bell’s inequality [24]. For example, optimal violation of Clauser-Horne-Shimony-Holt (CHSH) inequality self-tests the maximally entangled state and mutually anticommuting local observables in any arbitrary dimension. Note that the DI certification encounters practical challenges arising from the requirement of a loophole-free Bell test. Such tests have recently been realized [25, 28, 26, 27, 29] enabling experimental demonstrations of DI certification of randomness [30, 31]. Of late, the DI certification is used as a resource for secure quantum key distribution [32, 33, 34, 35], randomness certification [36, 38, 37], witnessing Hilbert space dimension [39, 40, 41, 42, 43, 44, 45] and for achieving advantages in communication complexity tasks [46].

We provide DI self-testing schemes to certify the unsharpness parameters through the quantum violation of two well-known Bell inequalities viz., the CHSH inequality [24] and the elegant Bell inequality [47]. Note that optimal quantum violations of such Bell inequalities can be obtained only for sharp measurement. Any value less than the optimal quantum value may originate due to various reasons. It may come from the unsharp measurements of local observables but may also come from the nonideal preparation of the state or the inappropriate choices of the local observables than the ones required for the optimal quantum violation. However, a more serious issue regarding DI self-testing of unsharp measurement through a Bell test arises due to Naimark’s theorem. It states that every non-projective measurement can be modeled as a projective measurement in a larger Hilbert space by introducing suitable ancilla in a higher dimension. Since in the DI Bell test, there is no bound on the dimension, a stubborn physicist may argue that the sub-optimal quantum violation is arising due to the inappropriate choice of observables in a higher dimension and not due to the unsharpness of the measurement. Hence, to self-test unsharp measurements, one needs to introduce a protocol that certifies the state, the observables, and the unsharpness parameters without referring to the dimension of the system.

Against this backdrop, we demonstrate that the sequential Bell test has such potential where the sub-optimal sequential quantum violations of a Bell inequality by multiple independent observers on one side can enable such a self-testing. Such a scheme successfully bypasses the constraints that would arise from Naimark’s theorem as the optimization of sequential Bell expression is performed without imposing any constraint on the dimension of Hilbert space. As far as our knowledge goes, the DI self-testing of non-projective measurements through the Bell test has not hitherto been demonstrated. However, semi-DI certification of non-projective measurements has been demonstrated by using the qubit system. In [48, 49, 50], the extremal qubit POVMs were experimentally certified based on the theoretical proposal [4]. Those experiments [48, 49, 50, 51] do certify the non-projective character of measurement, but not how it relates to a specific target POVM. The semi-DI certification of qubit unsharp measurements (noisy variants of projective measurement) in the prepare-measure scenario has recently been proposed [52, 53, 54, 55]. The proposal of [52] has been experimentally verified [51, 56, 57].

Specifically, in our self-testing protocol, two sequential sub-optimal quantum violations of a Bell inequality form an optimal pair powering the DI certifications of the state, the local observables, and the unsharpness parameter of one of the two parties. We also note here that all the previous works that demonstrated the sharing of various quantum correlations [17, 60, 16, 59, 61, 58, 62, 55, 63, 64, 65], the dimension of the system was assumed. In contrast, throughout this work, we impose no bound on the dimension of the Hilbert space, and we consider that the measurement devices are black boxes. We first demonstrate that, at most, two independent sequential observers on one side can violate the CHSH inequality. We simultaneously maximize the quantum violations of CHSH inequality for two sequential observers on one side and demonstrate that there is a trade-off between the two sub-optimal quantum violations. This, in turn, certifies the state, and the observables for both the sequential observers and the unsharpness parameter of the first observer. Since in a practical implementation there remains imperfection, we show how a range of an unsharpness parameter can be self-tested in that scenario.

The protocol developed for the CHSH inequality is further extended to the case of elegant Bell inequality, where we demonstrate that, at most, two observers can share the quantum advantage when considering the inequality’s local bound. However, we argue that the elegant Bell inequality has two classical bounds, the local and the preparation non-contextual bound, and the latter is smaller than the former. We show that if we consider the preparation non-contextual bound of the elegant Bell inequality, then at most three observers can share the quantum advantage. Further, we demonstrate that if the quantum advantage is extended to a third sequential observer, then the range of the values of the unsharpness parameter for the first observer can be more restricted, thereby self-testing a narrow range of the values of the unsharpness parameter in the sub-optimal scenario.

This paper is organized as follows. In Sec. II, we demonstrate the optimal quantum violation of CHSH inequality using an elegant sum-of-squares approach. In section III, we explicitly show the sequential violation of CHSH inequality and self-testing of the unsharpness parameter. In Sec. IV, we briefly introduce the notion of the preparation non-contextuality in an ontological model and provide the preparation non-contextual bound of elegant Bell inequality. In Sec. V, we extend the sharing of preparation contextuality by three sequential observers based on the quantum violation of elegant Bell inequality and demonstrate the self-testing of two unsharpness parameters along with the states and the observables. Finally, we discuss our results in Sec. VI.

The CHSH scenario consists of two space-like separated parties (say, Alice and Bob) who share a common physical system. Alice (Bob) performs local measurements on her (his) subsystem upon receiving inputs $x\in\{1,2\}(y\in\{1,2\})$, and returns outputs $a\in\{0,1\}(b\in\{0,1\})$. Representing $M_{x}$ and $N_{y}$ as respective dichotomic observables of Alice and Bob, the CHSH form of Bell’s inequality can be written as

$$\mathcal{B}=\left(M_{1}+M_{2}\right)N_{1}+\left(M_{1}-M_{2}\right)N_{2}\leq 2.$$

(1)

The optimal quantum value of the CHSH expression is $\left(\mathcal{B}\right)^{opt}_{Q}=2\sqrt{2}$, commonly known as Tirelson bound [66]. The optimal value can be obtained when the shared state is maximally entangled in a two-qubit system and the local qubit observables are mutually anticommuting. However, one does not need to impose the bound on the dimension to obtain the optimal value. Also, the measurement of Alice and Bob remains uncharacterized. In other words, the optimal value $\left(\mathcal{B}\right)^{opt}_{Q}$ can be achieved if Alice and Bob perform dichotomic measurements on a maximally entangled state $\ket{\psi}_{AB}\in\mathcal{C}^{d}\otimes\ \mathcal{C}^{d}$ where $d\geq 2$ is arbitrary.

For our purpose, we provide a derivation of the $\left(\mathcal{B}\right)^{opt}_{Q}$ without imposing a bound on the Hilbert space dimension by introducing an elegant sum-of-squares (SOS) approach. Let us assume that $(\mathcal{B})_{Q}\leq\Omega_{2}$, where $\Omega_{2}$ is a positive quantity and the upper bound of $(\mathcal{B})_{Q}$. Equivalently, one can argue that there is a positive semi-definite operator $\eta\geq 0$, that can be expressed as

$\displaystyle\langle\eta\rangle_{Q}=\Omega_{2}-(\mathcal{B})_{Q}$

(2)

for a quantum state $|\psi\rangle_{AB}$. This can be proven by considering two suitable positive operators, $L_{1}$ and $L_{2}$, which are polynomial functions of $M_{x}$ and $N_{y}$, so that

$\displaystyle\eta=\frac{1}{2}\left(\omega_{1}L_{1}^{\dagger}L_{1}+\omega_{2}L_{2}^{\dagger}L_{2}\right).$

(3)

For our purpose, we suitably choose $L_{1}$ and $L_{2}$ as

	$\displaystyle L_{1}|\psi\rangle_{AB}=\frac{M_{1}+M_{2}}{\omega_{1}}|\psi\rangle_{AB}-N_{1}|\psi\rangle_{AB}$		(4)
	$\displaystyle L_{2}|\psi\rangle_{AB}=\frac{M_{1}-M_{2}}{\omega_{1}}|\psi\rangle_{AB}-N_{2}|\psi\rangle_{AB}$

where $\omega_{1}=||\left(M_{1}+M_{2}\right)|\psi\rangle_{AB}||_{2}$ an $\omega_{2}=||\left(M_{1}-M_{2}\right)|\psi\rangle_{AB}||_{2}$. Here $||.||_{2}$ is the Frobenius norm of a vector, $||\ \mathcal{O}\ ||_{2}=\sqrt{Tr[\mathcal{O}^{\dagger}\mathcal{O}\rho]}$. Plugging Eq. (4) into Eq. (3) and by noting that $M_{x}^{\dagger}M_{x}=N_{y}^{\dagger}N_{y}=\mathbb{I}$, we get

$\displaystyle(\mathcal{B})_{Q}=\left(\omega_{1}+\omega_{2}\right)-\langle\eta\rangle_{Q}.$

(5)

Optimal value of $(\mathcal{B})_{Q}$ can be obtained when $\langle\eta\rangle_{Q}=0$, i.e.,

	$\displaystyle(\mathcal{B})_{Q}^{opt}$	$\displaystyle=$	$\displaystyle\underset{{}}{max}\left(\omega_{1}+\omega_{2}\right)$
		$\displaystyle=$	$\displaystyle\underset{{}}{max}\left(\sqrt{2+\langle\{M_{1},M_{2}\}\rangle}+\sqrt{2-\langle\{M_{1},M_{2}\}\rangle}\right).$

Thus, the maximization requires $\{M_{1},M_{2}\}=0$ implying Alice’s observables have to be anticommuting. In turn, we find the values $\omega_{1}=\omega_{2}=\sqrt{2}$, and consequently the optimal value $(\mathcal{B})_{Q}^{opt}=2\sqrt{2}$. The explicit conditions for the optimization are $L_{1}|\psi\rangle_{AB}=0$ and $L_{2}|\psi\rangle_{AB}=0$, i.e., $N_{1}=(M_{1}+M_{2})/\sqrt{2}$ and $N_{2}=(M_{1}-M_{2})/\sqrt{2}$. It can be easily checked that $\{N_{1},N_{2}\}=0$, i.e., Bob’s observables are also anticommuting.

Note also that for the state $\rho_{AB}=\ket{\psi_{AB}}\bra{\psi_{AB}}\in\mathcal{C}^{d}\otimes\mathcal{C}^{d}$, optimal violation requires $Tr[N_{1}\otimes N_{1}\rho_{AB}]=Tr[N_{2}\otimes N_{2}\rho_{AB}]=1$, which again confirms that $\rho_{AB}$ has to be a pure state. Let us choose the state in Hilbert-Schmidt form as

$\displaystyle\rho_{AB}=\frac{1}{d^{2}}\Big{[}\mathbb{I}\otimes\mathbb{I}+\sum_{i=1}^{d^{2}-1}N_{i}\otimes N_{i}\Big{]}$

(7)

where $\{N_{i},N_{j}\}=0$ and consequently $[N_{i}\otimes N_{i},N_{j}\otimes N_{j}]=0$ for any arbitrary dimension $d$. For a density matrix $\rho_{AB}$, $Tr[\rho_{AB}]=1$ has to be satisfied. This in turn provides $Tr[N_{1}]=Tr[N_{2}]=0$. Also, $Tr[\rho_{AB}^{2}]=1$ ensures that the observables in the summation in Eq. (II) contains full set of mutually anticommuting observables $\{N_{i}\otimes N_{i}\}$. Consequently, $Tr_{A}[\rho_{AB}]=Tr_{B}[\rho_{AB}]=\frac{\mathbb{I}}{d}$, i.e., partial trace of $\rho_{AB}$ is maximally mixed state for both Alice and Bob. Thus, the optimal violation of the CHSH inequality is achieved for the maximally entangled state $\ket{\psi_{AB}}$. We thus derive the optimal quantum value $(\mathcal{B})_{Q}^{opt}$ which uniquely certifies the state and observables. The entire derivation is done without assuming the dimension of the system.

The sequential Bell-CHSH test in the DI scenario is depicted in Figure 1. There is only one Alice, who always performs sharp measurement and arbitrary $k$ number of sequential Bobs (say, Bob_k). Alice and Bob₁ share an entangled state $\rho_{AB_{1}}$. Our aim is to demonstrate the sharing of nonlocality by multiple sequential observers. Since a projective measurement inevitably disturbs the system maximally; hence, in the sequential Bell test, if the first Bob (Bob₁) performs a sharp measurement, the entanglement is lost after the measurement. In such a case, no residual entanglement remains for the second sequential observer (Bob₂); consequently, the Bell inequality cannot be violated. If both the sequential observers obtain the sub-optimal quantum violations, then the first observer must have to perform an unsharp measurement.

In this work, the unsharp measurement corresponds to the noisy variant of projective measurements, i.e., the number of measurement operators is restricted to two. After performing the unsharp measurement, Bob₁ relays the post-measurement state to Bob₂, who performs an unsharp measurement intending to violate the Bell inequality. The chain runs up to arbitrary $k^{th}$ Bob (Bob_k) until the quantum violation of Bell’s inequality is obtained. The $k^{\text{th}}$ Bob may perform the sharp measurement. It is quite known that in the standard scenario at most, two Bobs can sequentially demonstrate nonlocality through the violation of CHSH inequality [13]. We stress again that, before our work all the studies that demonstrated the sharing of nonlocality, the dimension of the system was assumed.

We consider that Alice, upon receiving input $x=\{1,2\}$, always performs the projective measurements of observables $A_{1}$ and $A_{2}$. The CHSH form of Bell’s inequality can be written as

$$\mathcal{B}=\left(A_{1}+A_{2}\right)B_{1}+\left(A_{1}-A_{2}\right)B_{2}\leq 2.$$

(8)

There are arbitrary $k$ number of sequential Bobs (say, Bob_k), who upon receiving input $y_{k}\in\{1,2\}$ perform measurements of $B_{1}$ and $B_{2}$ producing outputs $b_{k}\in\{0,1\}$. We demonstrate how sequential quantum violations of CHSH inequality by multiple independent Bobs enable the DI certification of the unsharpness parameter.

Now, if Bob_k’s instrument is represented by measurement operators $\{K_{b_{k}|y_{k}}\}$ then after $(k-1)^{th}$ Bob’s measurement, the average state shared between Alice and Bob_k is

$\displaystyle\rho_{AB_{k}}=\frac{1}{2}\sum_{b_{k}\in\pm}\sum_{y_{k}=1}^{2}K_{b_{k}|y_{k}}^{\dagger}\ \rho_{AB_{(k-1)}}\ K_{b_{k}|y_{k}}$

(9)

where $\forall k$, $\sum_{b_{k}=\pm 1}K_{b_{k}|y_{k}}^{\dagger}K_{b_{k}|y_{k}}=\mathbb{I}$. Also, $K_{b_{k}|y_{k}}=\sqrt{E_{b_{k}|y_{k}}}$ where $E_{b_{k}|y_{k}}$ is the POVM. We consider that $\forall k$ and $\forall y_{k}$ the POVM [67, 68] is of the form

$\displaystyle E_{\pm|y_{k}}=\frac{1\pm\lambda_{k}}{2}\Pi_{y_{k}}^{+}+\frac{1\mp\lambda_{k}}{2}\Pi_{y_{k}}^{-}$

(10)

where $\{\Pi_{y_{k}}^{\pm}\}$ are the projectors corresponding to the observable $B_{y_{k}}$ satisfying $\Pi_{y_{k}}^{+}+\Pi_{y_{k}}^{-}=\mathbb{I}$, and $\lambda_{k}$ is the unsharpness parameter for $k^{\text{th}}$ Bob. The measurement operators can then be written as, $K{\pm|y_{k}}=\alpha_{k}\mathbb{I}\pm\beta_{k}B_{y_{k}}$ where

	$\displaystyle\alpha_{k}=\frac{1}{2}\Big{(}\sqrt{\frac{(1+\lambda_{k})}{2}}+\sqrt{\frac{(1-\lambda_{k})}{2}}\Big{)}$		(11)
	$\displaystyle\beta_{k}=\frac{1}{2}\Big{(}\sqrt{\frac{(1+\lambda_{k})}{2}}-\sqrt{\frac{(1-\lambda_{k})}{2}}\Big{)},$

satisfying $\alpha_{k}^{2}+\beta_{k}^{2}=1/2$ with $\alpha_{k}\geq\beta_{k}$.

We derive the maximum quantum value of CHSH expression for Alice and Bob₁ as

$\displaystyle(\mathcal{B}^{1})_{Q}=\lambda_{1}(\mathcal{B})_{Q}^{opt}$

(12)

which is independent of assuming the dimension. Here, $\lambda_{1}$ is the unsharpness parameter of Bob₁.

After his unsharp measurement, Bob₁ relays the average state to Bob₂. From Eq. (9), the reduced state for Alice and Bob₂ can be written as

	$\displaystyle\rho_{AB_{2}}$	$\displaystyle=$	$\displaystyle\dfrac{1}{2}\sum_{b_{1}\in\{+,-\}}\sum_{y_{1}=1}^{2}\left(\mathbb{I}\otimes K_{b_{1}|y_{1}}\right)\rho_{AB_{1}}\left(\mathbb{I}\otimes K_{b_{1}|y_{1}}\right)$		(13)
		$\displaystyle=$	$\displaystyle 2\alpha^{2}_{1}\rho_{AB_{1}}+\beta^{2}_{1}\sum_{y_{1}=1}^{2}(\mathbb{I}\otimes B_{y_{1}})\rho_{AB_{1}}(\mathbb{I}\otimes B_{y_{1}}).$

For the time being, let us consider that each sequential Bob measures same set of observables $B_{1}$ and $B_{2}$, i.e., $\forall k,\ B_{y_{k=1}}\equiv B_{1}$ and $B_{y_{k=2}}\equiv B_{2}$. By using Eq. (13), the maximum quantum value of CHSH expression for Alice and Bob₂ can be written as

	$\displaystyle(\mathcal{B}^{2})_{Q}$	$\displaystyle=$	$\displaystyle max\Big{(}Tr[\rho_{AB_{2}}\mathcal{B}]\Big{)}$		(14)
		$\displaystyle=$	$\displaystyle max\Bigg{(}Tr\Big{[}\rho_{AB_{1}}\Big{(}(A_{1}+A_{2})\tilde{B}_{1}+(A_{1}-A_{2})\tilde{B}_{2}\Big{)}\Big{]}\Bigg{)}$

where we assume Bob₂ performs sharp measurement. We derive the explicit forms of $\tilde{B}_{1}$ and $\tilde{B}_{2}$ as

	$\displaystyle\tilde{B}_{1}$	$\displaystyle=$	$\displaystyle(2\alpha_{1}^{2}+\beta_{1}^{2})B_{1}+\beta_{1}^{2}B_{2}B_{1}B_{2}$
	$\displaystyle\tilde{B}_{2}$	$\displaystyle=$	$\displaystyle(2\alpha_{1}^{2}+\beta_{1}^{2})B_{2}+\beta_{1}^{2}B_{1}B_{2}B_{1}.$		(15)

Note that, Eq. (14) has a similar form of CHSH expression as in Eq. (8), if $\tilde{B}_{1}$ and $\tilde{B}_{2}$ are considered to be effective observables of Bob₂. We can use the earlier SOS approach to obtain the maximum quantum value. However, $(\tilde{B}_{1})^{2}\neq\mathbb{I}$ and $(\tilde{B}_{2})^{2}\neq\mathbb{I}$, and hence they need to be properly normalized. By considering $\tilde{\omega}_{1}=||\tilde{B}_{1}||$ and $\tilde{\omega}_{2}=||\tilde{B}_{2}||$, and by using the SOS approach we have

$\displaystyle(\mathcal{B}^{2})_{Q}=max\left(\omega_{1}\tilde{\omega}_{1}+\omega_{2}\tilde{\omega}_{2}\right).$

(16)

As we already proved earlier, to obtain the optimal quantum value Alice’s and Bob’s observables have to be mutually anticommuting. Hence, for Bob’s (unnormalized) observables $\tilde{B}_{1}$ and $\tilde{B}_{2}$ we require,

$\displaystyle\{\tilde{B}_{1},\tilde{B}_{2}\}=4\alpha_{1}^{2}(\alpha_{1}^{2}+2\beta_{1}^{2})\{B_{1},B_{2}\}+\beta_{1}^{4}\{B_{1},B_{2}\}^{3}=0.$

(17)

Since $\alpha_{1}>0$ and $\beta_{1}\geq 0$, Eq. (17) gives $\{B_{1},B_{2}\}=0$. In other words, the observables of Bob₂ have to be anticommuting to obtain the maximum quantum value of the Bell expression $(\mathcal{B}^{2})_{Q}$. This, in turn, provides

$\displaystyle\tilde{\omega}_{1}=\sqrt{(2\alpha_{1}^{2})^{2}+(2\alpha_{1}^{2}+\beta_{1}^{2})\beta_{1}^{2}\{B_{1},B_{2}\}^{2}}=2\alpha_{1}^{2},$

(18)

and $\tilde{\omega}_{2}=\tilde{\omega}_{1}=2\alpha_{1}^{2}$. Note that the above result is obtained by considering that Bob₁ and Bob₂ perform the measurements on the same set of observables $B_{1}$ and $B_{2}$. In Appendix A, we prove that to obtain the maximum quantum value $(\mathcal{B}^{2})_{Q}$ the choices of observables of Bob₂ have to be the same as Bob₁.

Using Eqs. (II) and (18), from Eq. (16) we can then write

$\displaystyle(\mathcal{B}^{2})_{Q}=2\alpha_{1}^{2}\ max\Big{(}\omega_{1}+\omega_{2}\Big{)}=2\alpha_{1}^{2}(\mathcal{B})_{Q}^{opt}.$

Putting the value of $\alpha_{1}$, we write $(\mathcal{B}^{2})_{Q}$ in terms of unsharpness parameter $\lambda_{1}$ of Bob₁ as

$$(\mathcal{B}^{2})_{Q}=\dfrac{1}{2}\Big{(}1+\sqrt{1-\lambda_{1}^{2}}\Big{)}(\mathcal{B})_{Q}^{opt}.$$

(19)

Hence, we have derived the maximum quantum values of Bell expressions for two sequential Bobs where $(\mathcal{B})_{Q}^{opt}$ is common, but they differ by the coefficients, which are solely dependent only on $\lambda_{1}$. Using Eq. (12), the quantum value of the CHSH expression $(\mathcal{B}^{2})_{Q}$ in Eq. (19) can be written in terms of $(\mathcal{B}^{1})_{Q}$ as

$$(\mathcal{B}^{2})_{Q}=\sqrt{2}\Big{(}1+\sqrt{1-\Bigg{(}\frac{(\mathcal{B}^{1})_{Q}}{(\mathcal{B})_{Q}^{opt}}\Bigg{)}^{2}}\Big{)}.$$

(20)

This implies that if $(\mathcal{B}^{1})_{Q}$ increases, then $(\mathcal{B}^{2})_{Q}$ decreases, i.e., the more the Bob₁ disturbs the system, the more he gains the information, and consequently the quantum value $(\mathcal{B}^{2})_{Q}$ of Bob₂ decreases. Hence, there is a trade-off between $(\mathcal{B}^{2})_{Q}$ and $(\mathcal{B}^{1})_{Q}$, which eventually form an optimal pair demonstrating the certification of unsharpness parameter.

In Figure 2, we plot the optimal trade-off characteristics between $(\mathcal{B}^{1})_{Q}$ and $(\mathcal{B}^{2})_{Q}$. The green line corresponds to the maximum classical value of the CHSH expression where Bob₂ can get a maximum value independent of Bob₁, i.e., there is no trade-off in classical theory. The blue curve exhibits the trade-off between quantum values of $(\mathcal{B}^{2})_{Q}$ and $(\mathcal{B}^{1})_{Q}$ where each point on it certifies a unique value of unsharpness parameter $\lambda_{1}$. For example, when $(\mathcal{B}^{2})_{Q}=(\mathcal{B}^{1})_{Q}$, the value of $\lambda_{1}=4/5=0.80$ is certified, as shown in the figure by brown dot.

Note that in our protocol we consider Alice always performs sharp measurements. Before making the self-testing statements of our protocol, let us examine whether our protocol certifies sharp measurements of Alice. As mentioned earlier that the blue curve in Figure 2 represents the optimal trade-off between $(\mathcal{B}^{2})_{Q}$ and $(\mathcal{B}^{1})_{Q}$. If Alice performs unsharp measurement then the trade-off curve will always be below the blue curve. For instance, $(\mathcal{B}^{2})_{Q}=(\mathcal{B}^{1})_{Q}=8\sqrt{2}/5$ (represented by the brown dot over the blue line) can never be reached unless Alice performs the sharp measurement of her anti-commuting observables. The same argument holds for any point over the blue curve in Figure 2. We are now in the position to make the DI self-testing statements of our protocol.

We extend the sequential sharing of nonlocality by using another well-known Bell inequality known as Gisin’s elegant Bell inequality [47]. The elegant Bell expression can be written as,

	$\displaystyle\mathbb{E}$	$\displaystyle=$	$\displaystyle(A_{1}+A_{2}+A_{3}-A_{4})\otimes B_{1}+(A_{1}+A_{2}-A_{3}$		(29)
		$\displaystyle+$	$\displaystyle A_{4})\otimes B_{2}+(A_{1}-A_{2}+A_{3}+A_{4})\otimes B_{3}.$

whose local bound is $(\mathbb{E})_{l}\leq 6$. We show that the elegant Bell expression has two bounds, the local bound (which can be considered as trivial preparation non-contextual bound) and a nontrivial preparation non-contextual bound when there exists a relational constraint between the observables of Alice and Bob. We show that the non-trivial preparation non-contextual bound $(\mathbb{E})_{pnc}\leq 4$.

Before proceeding further, we briefly introduce the notion of preparation non-contextuality in an ontological model of quantum theory. Consider that a preparation procedure $P$ prepares a density matrix $\rho$, and a measurement procedure $M$ realizes the measurement of a POVM $E_{k}$. Quantum theory predicts the probability of obtaining a particular outcome $k$ is $p(k|P,M)=Tr[\rho E_{k}]$ - the Born rule. In an ontological model of quantum theory, preparation procedures assign a probability distribution $\mu_{P}(\lambda|\rho)$ on ontic states $\lambda\in\Lambda$ where $\Lambda$ is the ontic state space. Given the measurement procedure, the ontic state $\lambda$ assigns a response function $\xi_{M}(k|\lambda,E_{k})$. A viable ontological model must reproduce the Born rule, i.e., $\forall k,\rho,E_{k}:\ \ p(k|P,M)=\int_{\Lambda}\mu_{P}(\lambda|\rho)\xi_{M}(k|\lambda,E_{k})d\lambda$.

The dependencies of $P$ and $M$ do not appear if the ontological model is preparation and measurement non-contextual, respectively. Two preparation procedures $P$ and $P^{\prime}$ are said to be operationally equivalent if they can not be distinguished by any measurement, implying, $\forall k,M:\hskip 2.84544ptp(k|P,M)=p(k|P^{\prime},M)$. In quantum theory, such preparation procedures are realized by the density matrix $\rho$. Such equivalence in operational theory can be reflected in the ontic state level, assuming preparation non-contextuality [69]. An ontological model of an operational quantum theory is considered to be preparation non-contextual if two preparation procedures $P$ and $P^{\prime}$ prepare the same density matrix $\rho$, and no measurement can operationally distinguish the context by which $\rho$ is prepared, i.e., $\forall k,M:\hskip 2.84544ptp(k|P,M)=p(k|P^{\prime},M)\Rightarrow\forall\lambda:\mu(\lambda|\rho,P)=\mu(\lambda|\rho,P^{\prime})$, implying two ontic state distributions are equivalent irrespective of the contexts $P$ and $P^{\prime}$ [69, 70, 71, 72, 73].

Let us intuitively understand the notion of preparation non-contextuality in the CHSH scenario. Consider that Alice and Bob share an entangled state $\rho_{AB}$ and Alice measures $A_{1}$ and $A_{2}$. Alice’s measurements on her local system produces density matrices $\rho_{A_{1}}$ and $\rho_{A_{2}}$ on Bob’s side corresponding to measurement contexts $A_{1}$ and $A_{2}$, respectively. The non-signaling condition demands that $\rho_{A_{1}}$ and $\rho_{A_{2}}$ cannot be distinguishable by any measurement of Bob, i.e., $\rho_{A_{1}}=\rho_{A_{2}}\equiv\sigma$. Equivalently, in an ontological model we assume that $\mu(\lambda|\sigma,A_{1})=\mu(\lambda|\sigma,A_{2})$, i.e., the distribution of ontic states are preparation non-contextual. Intuitively, preparation non-contextuality implies the locality assumption in the Bell scenario. In other words, every probability distribution that violates a Bell inequality can also be regarded as proof of preparation contextuality as proved in [74]. Here, we provide a modified version of the proof.

For this, by using Bayes’ theorem we write the joint probability distribution in the ontological model as

$$p(a,b|A_{i},B_{j})=\sum_{\lambda}p(a|A_{i},B_{j})p(\lambda|a,A_{i})p(b|B_{j},\lambda).$$

(30)

Now, the no-signaling condition implies the marginal probability of Alice’s side is independent of Bob’s input and hence we can write

$$p(a,b|A_{i},B_{j})=\sum_{\lambda}p(a|A_{i})p(\lambda|a,A_{i})p(b|B_{j},\lambda).$$

(31)

Using Bayes’ theorem we can write $p(a|A_{i})p(\lambda|a,A_{i})=\mu(\lambda|A_{i})p(a|\lambda,A_{i})$ where we specifically denoted the probability distribution $p(\lambda|A_{i})$ as $\mu(\lambda|A_{i})$.

From Eq. (31), we then obtain

$$p(a,b|A_{i},B_{j})=\sum_{\lambda}\mu(\lambda|A_{i})p(a|\lambda,A_{i})p(b|B_{j},\lambda).$$

(32)

Due to the assignment of the same ontic-state distribution for the different preparation procedures on Bob’s side, the assumption of preparation non-contextuality is enforced. Then the preparation non-contextual assumption for Bob’s preparation reads as $\mu(\lambda|\rho,{A_{1}})=\mu(\lambda|\rho,{A_{2}})\equiv\mu(\lambda)$, which in turn provides

$$p(a,b|A_{i},B_{j})=\sum_{\lambda}\mu(\lambda)p(a|\lambda,A_{i})p(b|\lambda,B_{j}).$$

(33)

which is the desired factorizability condition commonly derived for a local hidden variable model. Therefore, we can argue that whenever in an ontological model $p(a,b|A_{i},B_{j})$ satisfies preparation non-contextuality, this, in turn, satisfies locality in a hidden variable model.

Here we go one step further. The above case involving the CHSH scenario is a trivial one. We introduce a nontrivial form of preparation non-contextuality in the Bell experiment involving more than two inputs. Consider a Bell experiment where Alice receives four inputs and performs the measurements of four observables $A_{1}$, $A_{2}$, $A_{3}$ and $A_{4}$. Bob receives three inputs and performs the measurement of three observables $B_{1}$ , $B_{2}$ and $B_{3}$. In quantum theory, when Alice and Bob share an entangled state $\rho_{AB}$

$\displaystyle\rho_{a|A_{i}}+\rho_{a\oplus 1|A_{i}}=\rho_{a|A_{i^{\prime}}}+\rho_{a\oplus 1|A_{i^{\prime}}}\equiv\sigma$

(34)

where $i,i^{\prime}=1,2,3,4$ with $i\neq i^{\prime}$, and $\rho_{a|A_{i}}=Tr_{A}\left[\rho_{AB}\Pi_{A_{i}}\otimes\mathbb{I}\right]$. This is within the premise of preparation non-contextuality in an ontological model as the distribution of ontic states is assumed to be equivalent for those two preparation procedures $i$ and $i^{\prime}$. As argued above, such a trivial preparation non-contextuality can be attributed to the locality in a Bell experiment. In that case, the local bound of the elegant Bell expression $(\mathbb{E})_{l}\leq 6$.

We introduce a nontrivial form of preparation non-contextuality in an ontological model of quantum theory by imposing an additional relational constraint on Alice’s measurement observables. For this, we consider that the joint probability satisfies

	$\displaystyle\forall b,j,\$		$\displaystyle P(a,b|A_{1},B_{j})+\sum\limits_{i=2}^{4}P(a\oplus 1,b|A_{i},B_{j})$
		$\displaystyle=$	$\displaystyle P(a\oplus 1,b|A_{1},B_{j})+\sum\limits_{i=2}^{4}P(a,b|A_{i},B_{j})$

which in quantum theory implies that

			$\displaystyle\rho_{a\oplus 1|A_{1}}+\rho_{a|A_{2}}+\rho_{a|A_{3}}+\rho_{a|A_{4}}$		(36)
		$\displaystyle=$	$\displaystyle\rho_{a|A_{1}}+\rho_{a\oplus 1|A_{2}}+\rho_{a\oplus 1|A_{3}}+\rho_{a\oplus 1|A_{4}}$

Since $\rho_{a|A_{i}}=Tr_{A}\left[\rho_{AB}\Pi_{A_{i}}\otimes\mathbb{I}\right]$ with $\Pi_{A_{i}}=\left(\mathbb{I}+A_{i}\right)/2$, the above Eq. (36) implies that $A_{1}-A_{2}-A_{3}-A_{4}=0$. Along with the equivalence in Eq. (34) a non-trivial constraint Eq. (36) is also imposed on Alice’s preparation procedures. In such a case, the local bound reduces to the preparation non-contextual bound $(\mathbb{E})_{l}\leq 4$ as $A_{1}=A_{2}+A_{3}+A_{4}$. Thus, the quantum violation of it provides a weaker notion of nonlocality, which we call nontrivial preparation contextuality. It is well-known that steering is a weaker form of nonlocality, but the relation between steering and nontrivial preparation contextuality needs to be properly explored. While we plan to do it in another occasion, interested reader may see a relevant work [74]. However, for our present purpose it is not directly relevant and hence we skip such discussion.

Note that the quantum value of $(\mathbb{E})_{Q}$ has also to be calculated by considering this constraint. Below we show that the optimal quantum value $(\mathbb{E})_{Q}^{opt}=4\sqrt{3}$ satisfies the constraint $A_{1}=A_{2}+A_{3}+A_{4}$.

We derive the optimal quantum value $(\mathbb{E})_{Q}^{opt}$ without using the dimension of the system. For this, we again use the SOS approach developed in Sec. II. We define a positive semidefinite operator $\langle\chi\rangle_{Q}\geq 0$ so that $\langle\chi\rangle_{Q}+\mathbb{E}=\Omega_{3}$ where $\Omega_{3}$ is a positive quantity. By considering suitable positive operators $L_{1}$, $L_{2}$ and $L_{3}$ we can write

$\displaystyle\chi=\frac{1}{2}\left(\omega_{1}L_{1}^{\dagger}L_{1}+\omega_{2}L_{2}^{\dagger}L_{2}+\omega_{3}L_{3}^{\dagger}L_{3}\right)$

(37)

where $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ are positive numbers that will be determined soon. For our purpose, we choose

	$\displaystyle L_{1}|\psi\rangle_{AB}=\frac{A_{1}+A_{2}+A_{3}-A_{4}}{\omega_{1}}|\psi\rangle_{AB}-B_{1}|\psi\rangle_{AB}$
	$\displaystyle L_{2}|\psi\rangle_{AB}=\frac{A_{1}+A_{2}-A_{3}+A_{4}}{\omega_{2}}|\psi\rangle_{AB}-B_{2}|\psi\rangle_{AB}$		(38)
	$\displaystyle L_{3}|\psi\rangle_{AB}=\frac{A_{1}-A_{2}+A_{3}+A_{4}}{\omega_{3}}|\psi\rangle_{AB}-B_{3}|\psi\rangle_{AB}$

where $\omega_{1}=||(A_{1}+A_{2}+A_{3}-A_{4})|\psi\rangle_{AB}||_{2},\omega_{2}=||(A_{1}+A_{2}-A_{3}+A_{4})|\psi\rangle_{AB}||_{2}$ and $\omega_{3}=||(A_{1}-A_{2}+A_{3}+A_{4})|\psi\rangle_{AB}||_{2}$. Substituting Eq. (V) in Eq. (37), we get

$$\langle\chi\rangle_{Q}=-(\mathbb{E})_{Q}+\sum_{i=1}^{3}\omega_{i}.$$

(39)

Hence, the optimal value $(\mathbb{E})_{Q}^{opt}$ is obtained if $\langle\chi\rangle_{Q}=0$, i.e.,

$$(\mathbb{E})_{Q}^{opt}=max(\omega_{1}+\omega_{2}+\omega_{3})$$

(40)

where

$\displaystyle\omega_{1}=\sqrt{4+\langle\{A_{1},(A_{2}+A_{3}-A_{4})\}+\{A_{2},(A_{3}-A_{4})\}-\{A_{3},A_{4}\}\rangle}$

and similarly for $\omega_{2}$ and $\omega_{3}$.