Self-Testing of a Single Quantum System: Theory and Experiment

We first outline our theoretical contributions (\SecrefTheoreticalFramework) and subsequently delineate the experiment (\SecrefExperiment). A self-testing scheme treats a quantum device like a black-box whose inner working cannot be accessed beyond giving it classical inputs and receiving classical outputs (\SubsecrefTheory_Background). Every non-trivial self-test must make additional assumptions. Informally, our assumption which we termed KCBS orthogonality (LABEL:Assu:KCBSexclusivity, \SubsecrefTheory_robustKCBS) is that the measurements are projective and satisfy orthogonality corresponding to a cyclic graph (vertices represent measurements and edges relations of orthogonality). We denote our self-test by $\mathcal{I}$ (LABEL:Eq:self-test) which is a linear expression in the input/output probabilities associated with the device. $\mathcal{I}$ can be made $0$ by a quantum device but every classical device must yield $\mathcal{I}\geq Q-C>0$, where $Q$ and $C$ are constants, independent of the device once the number of measurements are fixed. It is known (17) that the states and measurements associated with the device, henceforth referred to as a configuration, are unique for any device which yields $\mathcal{I}=0$; the uniqueness is up to a global isometry. Such a configuration is termed the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) configuration. Note that, using global isometry, one can change any state $|\phi\rangle$ to any other state $|\xi\rangle$. This might prompt one to think that assuming the freedom up to global isometries may render the problem trivial. This is not true. If there are two configurations that attain $\mathcal{I}=0$, we require the same global isometry to map states as well as measurement settings from one configuration to another. If one were to use different global isometries for mapping states and measurements, the problem would have been definitely trivial. Further, for a single device, the notion of local isometry does not make sense. Consider a configuration for which $\mathcal{I}$ is close to zero. To quantify the closeness of it to the KCBS configuration, we define total fidelity, $F$ (LABEL:Eq:fidelity). $F$ is six for the KCBS configuration.²²2The number $6$ appears because the KCBS configuration is specified by five measurements and a state; the fidelity to each is summed to obtain total fidelity. We give a lower bound on $F$ as a function of $\mathcal{I}$. The idea is (\SubsecrefProofOverview) to define an isometry in terms of the measurement operators of the device itself, which maps a device’s configuration to the KCBS configuration when $\mathcal{I}=0$. The expression for $F$ when $\mathcal{I}>0$ becomes an optimisation problem which can be relaxed to a hierarchy of semidefinite programs (SDPs). This in turn can be solved numerically to obtain lower bounds on $F$. The details of the proof appear in \SubsecreflowerBoundSDPrelaxation and those of the numerical solution in \Subsecrefnumerics. Obtaining the numerical solution was challenging. This is primarily because the description of the SDP we obtained is implicit. Therefore, we first find this explicit description by performing symbolic computation and subsequently, solve the resulting SDP. After performing symbolic computation for finding the objective and the constraints, we obtain an SDP in $18,336$ variables (Hermitian matrix of size $192\times 192$) with $16,859$ constraints. The size of the matrix is determined by the number and type of terms in the objective function. We emphasise that even the relevant representation of the objective function itself (in terms of variables we optimise over) already turns out to be quite long and would have been difficult to obtain without symbolic computation.

We apply this self-test to an experimental setup based on a single trapped ${}^{40}\rm{Ca}^{+}$ion. The KCBS configuration can be realised using a qutrit and projective measurements. In the experiment, the three levels of the qutrit are formed by the ground state, and the first excited state which is further split by the presence of a magnetic field. The measurements are executed by rotating between the ground state and the excited state using $729$ nm lasers, performing a photon number measurement (via fluorescence detection), followed by an inverse of the rotation (see LABEL:Fig:experiment_result, Left and \Subsecrefsetup_outline). The initial state and the rotation angles can be chosen such that they correspond directly with the KCBS configuration (\SubsecrefKCBSconfig). The actual experimental setup, of course, requires numerous details to be first addressed (\Subsecrefsetup_outline) for this correspondence to work. Using this setup, we perform two types of experiments (\SubsecreftheExp), both of which are deviations from the KCBS configuration, constructed to ensure LABEL:Assu:KCBSexclusivity (discussed above) holds. First, we leave the measurements unchanged but progressively depolarise the initial state. Second, we change the measurement settings, parametrised by an angle $\theta$ and tilt the initial state. In effect, these experiments determine $\mathcal{I}$ and the robustness curve immediately lower bounds the total fidelity of the underlying configuration to the KCBS configuration. By performing tomography, we determine the actual fidelity to the KCBS configuration and find that the lower bound is indeed satisfied, as expected (LABEL:Fig:experiment_result, Right; \Subsecrefthe_data).

The first local self-testing scheme was presented in reference (17), where the scheme relied on violation of non-contextuality inequalities via qutrits. Self-testing schemes for single device has since been a topic of active investigation. The approach based on contextuality was further explored in (31) and extended to arbitrarily high-dimensional quantum systems (18). The primary limitation of these approaches is that they do not yield the complete robustness curve; in particular, while (17) construct a robust self-testing scheme, this robustness is only shown up to multiplicative factors and is thus not directly applicable to real-world experimental scenarios.

Subsequently, in a breakthrough result, a self-test for a single quantum system was introduced in (32) which relied on cryptographic assumptions, as opposed to contextuality. The advantage of this approach over contextuality is that it is in the fully device independent setting, i.e., where one only accesses the input/output behaviour of the device. On the other hand, all contextuality based approaches reported so far make some assumptions, however mild, about the internal working of the device.³³3These assumptions may be dropped but then the robustness curve only holds against “memoryless model”, i.e., the guarantee holds in a very restricted setting. Therefore, theoretically cryptographic self-tests are appealing. However, in practice they are harder to implement since cryptographic functions require coherent control of a large number of qubits (the number of qubits determine the security). Based on a recent works (e.g. (33)), one can estimate that such cryptographic functions would require a circuit with at least $1000$ qubits and a depth of $10000$ to be implemented. This limitation is circumvented by our contextuality based robust self-test.

We now compare our experiment to previous experiments related to contextuality. To this end, we note that for demonstration of contextuality, it is conventional to adopt an operational language. In this work, however, we take a more pragmatic approach and work directly under the assumption that quantum mechanics governs the behaviour of our devices. Consequently, the terminology we use to characterise our experiment is slightly different from prior works. More precisely, the analogue of our KCBS orthogonality requirement, in the operational perspective may be thought of as compatibility and sharpness of measurements (see \SubsecrefnoiseComment). In our experiment, we simultaneously quantify and minimise divergence from the basic assumptions—perfect detection, random selection of measurements, compatibility and sharpness—which, to the best of our knowledge, makes it the most comprehensive contextuality experiment to be reported (see Table 1).

The construction of swap isometry in our paper is motivated by (34). While the construction in (34) works for Bell scenarios, our construction applies for single party setting.

Suppose we have a set of (independent) identical devices each of which takes a string as input and produces a string as output. Informally, we say the device is self-testing if there exists some constraints on the input-output behaviour of the device, which if satisfied, guarantees the integrity of the device. More precisely, let $p_{i}$ denote the probability of event $i$, where an event is a set of input output behaviours. Let $\mathcal{I}:=\sum_{i}q_{i}p_{i}$ denote a linear combination of these probabilities, where $q_{i}\in\mathbb{R}$. We say that $(\mathcal{I},c)$, for some $c\in\mathbb{R}$, self-tests the device if the following holds—if the device satisfies $\mathcal{I}=c$, then the device is uniquely described as containing a specific quantum state, the inputs corresponding to selecting specific measurements, the outputs correspond to the action of the measurement on the aforementioned quantum state, up to a global isometry. Clearly, no such self-test can exist unless one makes additional assumptions.⁴⁴4This is simply because the device could classically store and reproduce whatever statistics the self-test is required to satisfy. Three assumptions have been explored in the literature—spatial separation, computational assumptions and orthogonality of measurements.⁵⁵5First, physical separation between the components of the device which may be termed the “Bell setting”. A self-test in this setup harnesses nonlocal correlations which can be produced using quantum devices but not by classical devices. Second, computational hardness assumptions (such as Learning With Errors), i.e., certain tasks are assumed hard for the device being self-tested. A self-test in this setup harnesses the ability of a quantum device to evaluate cryptographic functions in superpositions and produce correlations which a classical device cannot. In this work, we focus on the latter which requires that the measurements are repeatable and satisfy an orthogonality relation among them—for instance if one obtains the outcome corresponding to $\Pi_{0}$, then the outcome corresponding to $\Pi_{1}$ cannot occur (upon subsequent measurement). This assumption may be motivated physically by considering a form of tomography where the experimentalist is only required to ensure that the measurements performed in the device are reliable and by construction satisfy the orthogonality condition.⁶⁶6A self-test in this setup harnesses the ability of quantum devices to produce “non-contextual” correlations. Bell nonlocality may be seen as a special case of contextuality.

We make the following assumption about the device we wish to self-test. To describe it, we setup some notations. We denote the $i$th binary measurement by $M_{i}:=(\Pi_{0|i},\Pi_{1|i})$, where $\Pi_{0|i}$ denotes the measurement operator corresponding to the zeroth output and $\Pi_{1|i}$ denotes that corresponding to the first output. Being measurement operators, they satisfy the probability conservation condition $\Pi_{0|i}^{\dagger}\Pi_{0|i}+\Pi_{1|i}^{\dagger}\Pi_{1|i}=\mathbb{I}$. We say the binary measurement is repeatable and Hermitian if $\Pi_{b|i}^{2}=\Pi_{b|i}$ for $b\in{0,1}$ and $\Pi_{b|i}^{\dagger}=\Pi_{b|i}$, respectively.

For a device satisfying LABEL:Assu:KCBSexclusivity, we consider the self-test⁷⁷7This particular form is of interest because, informally, any classical (i.e., non-contextual) model satisfying LABEL:Assu:KCBSexclusivity cannot result in $\sum_{i=1}^{n}p_{i}>C_{n}$, where $C_{n}:=(n-1)/2$ while quantumly, for the KCBS configuration (see LABEL:Def:idealKCBSconfig) one obtains $\sum_{i=1}^{n}p_{i}=Q_{n}$, the highest possible (41). The self-test measures how close one is to the maximum quantum value, $Q_{n}$. $\mathcal{I}$

where $p_{i}:=\text{tr}(\Pi_{i}\rho)=\left\langle\Pi_{i}\right\rangle$ and $Q_{n}:=\frac{n\cos(\pi/n)}{1+\cos(\pi/n)}$ is the “quantum value”.⁸⁸8The maximum value of $\sum_{i}p_{i}$ that is classically achievable (i.e., when all measurements commute) is $(n-1)/2$. This quantum value is achieved by $\sum_{i}\left\langle\Pi_{i}\right\rangle$ for the KCBS configuration ($\rho^{\text{KCBS}},\{\Pi_{i}^{\text{KCBS}}\})$, where $\rho^{\text{KCBS}}$ is a qutrit in a pure state and $\Pi_{i}^{\text{KCBS}}$ are projectors (see LABEL:Def:idealKCBSconfig below), yielding $\mathcal{I}=0$. It was shown in (17) that any quantum realisation $(\rho,\{\Pi_{i}\})$ which satisfies LABEL:Assu:KCBSexclusivity and yields $\mathcal{I}=0$, must be the same as the KCBS configuration, up to a global isometry.⁹⁹9In fact, they also showed that if $\mathcal{I}=\epsilon$ is close to zero, then the quantum realisation must be close to the KCBS configuration. However, their analysis did not yield the exact function corresponding to (LABEL:fidelity)—they only had an asymptotic bound (on the fidelity to the KCBS configuration) of the form $\mathcal{O}(\sqrt{\epsilon})$.

For clarity, we restrict to $n=5$ henceforth but the arguments readily generalise. Suppose the device corresponds to a quantum realisation $(\rho,\{\Pi_{i}\})$ for which $\mathcal{I}=\epsilon$ is small but not exactly zero. In this case, we derive a lower bound on the fidelity of $(\rho,\{\Pi_{i}\})$ to $(\rho^{\text{KCBS}},\{\Pi_{i}^{\text{KCBS}}\})$ (up to a global isometry). More precisely, we lower bound the value of the following function

where $(\rho,\{\Pi_{i}\}_{i=1}^{5})$ is a quantum realisation of $\{p_{i}\}_{i=1}^{5}$ satisfying $\mathcal{I}=\epsilon$, $V$ is an isometry from $\mathcal{H}$ to $\mathcal{H}\otimes\mathcal{H}^{\text{KCBS}}$, i.e., from the space on which $\rho,\{\Pi_{i}\}_{i=1}^{5}$ are defined, to itself tensored with the $3$-dimensional space where $\rho^{\text{KCBS}},\{\Pi_{i}^{\text{KCBS}}\}_{i=1}^{5}$ are defined, and $\mathcal{F}(\sigma,\tau):={\rm tr}\sqrt{\left|\sigma^{1/2}\tau^{1/2}\right|}$ is the fidelity of $\rho\geq 0$ to $\sigma\geq 0$. This lower bound can be approximated by a sequence of semi-definite programs and can be evaluated for any¹⁰¹⁰10Instead of the sum, $\sum_{i}p_{i}$, one could impose constraints on the values of $p_{i}$ individually. In fact, this is what we do in our analysis as it yields a better bound. $\mathcal{I}=\epsilon$.

We briefly describe an algorithm for estimating $F$ and outline an argument which shows that this algorithm always yields a lower bound (on $F$). First, we drop the maximisation over $V$ in LABEL:Eq:fidelity and replace it with a particular isometry $V$ which is expressed in terms of $\rho,\{\Pi_{i}\}_{i=1}^{5}$. Then, as we shall see, the expression for the fidelity appears as a sum of terms of the following form. Let $w$ be a word created from the letters $\{\mathbb{I},\Pi_{1},\Pi_{2}\dots\Pi_{5},\hat{P}\}$ with $\hat{P}^{\dagger}\hat{P}=\mathbb{I}$, $\Pi_{i}^{2}=\Pi_{i}$ and $\Pi_{i}\Pi_{j}=0$ if $(i,j)\in E$, i.e., when $i,j$ are exclusive.¹¹¹¹11We introduced $\hat{P}$ for completeness; its role is explained later. The fidelity is a linear combination of these words, i.e., $F=\min_{\left\{\left\langle w\right\rangle\right\}}\sum_{w}\alpha_{w}\left\langle w\right\rangle$, where $\text{tr}[w\rho]=:\left\langle w\right\rangle$, subject to the constraint that $\{\left\langle w\right\rangle\}_{w}$ corresponds to a quantum realisation. The advantage of casting the problem in this form is that one can now relax the problem to a sequence of semi-definite programs.¹²¹²12I.e., construct an NPA-like hierarchy (42). The idea is simple to state. Treat $\left\{\left\langle w\right\rangle\right\}_{w}$ as a vector. Denote by $Q$ the set of all such vectors which correspond to a quantum realisation (of $\left\{p_{i}\right\}_{i=1}^{5}$). It turns out that one can impose constraints on words with $k$ letters, for instance. Under these constraints, denote by $Q_{k}$ the set that is obtained. Note that $Q_{k}\supseteq Q$ for it may contain vectors which don’t correspond to the quantum realisation. In fact, $Q_{k}$ can be characterised using semi-definite programming constraints (which in turn means they are efficiently computable). Intuitively, one expects that $\lim_{k\to\infty}Q_{k}=Q$. Further, it is clear that $F=\min_{\{\left\langle w\right\rangle\}_{w}\in Q}\sum_{w}\alpha_{w}\left\langle w\right\rangle\geq\min_{\left\{\left\langle w\right\rangle\right\}_{w}\in Q_{k}}\sum_{w}\alpha_{w}\left\langle w\right\rangle$ as we are minimising over a larger set on the right hand side.

We begin with defining the ideal KCBS configuration, referred to above.

For concreteness, we first consider a unitary $U_{{\rm SWAP}}$ instead of an isometry, which acts on two spaces $\mathcal{A}$ and $\mathcal{A}^{\prime}$, i.e., $U_{{\rm SWAP}}:\mathcal{A}\otimes\mathcal{A}^{\prime}\to\mathcal{A}\otimes\mathcal{A}^{\prime}$. The space $\mathcal{A^{\prime}}$ is three-dimensional and $\mathcal{A}$ is an arbitrary Hilbert space. Informally, we want to construct the unitary $U_{{\rm SWAP}}$ to be such that it takes a realisation in $\mathcal{A}$ and maps it to a realisation in $\mathcal{A}^{\prime}$ which has a large overlap with the KCBS configuration. To this end, we first assume that $\mathcal{A}$ is three-dimensional. In particular, suppose that the $\mathcal{A}$ register is in the state

Then, at the very least, we want $U_{{\rm SWAP}}$ to map $\sigma_{\mathcal{A}}\otimes\left|0\right\rangle\left\langle 0\right|_{\mathcal{A}^{\prime}}$ to $\left|0\right\rangle\left\langle 0\right|_{\mathcal{A}}\otimes\sigma_{\mathcal{A}^{\prime}}$ (see LABEL:Fig:IllustrationSWAP). Our strategy is to construct $U_{{\rm SWAP}}$ in this seemingly trivial case and then express it in terms of the state and measurement operators on $\mathcal{A}$. The rationale is that by construction, $U_{{\rm SWAP}}$ will work for the ideal case and therefore should also work for cases close to ideal. This should become clear momentarily. Note that any circuit that swaps two qutrits should let us achieve our simplified goal (because all elements of $S^{\text{KCBS}}$ are defined on a three dimensional Hilbert space). One possible qutrit swapping unitary/circuit (a special case of the general qudit swapping unitary/circuit defined in (34)) may be defined as $S_{{\rm SWAP}}^{\prime\prime}:=TUVU$, where

where $P:=\sum_{i=0}^{2}\left|\overline{k+1}\right\rangle\left\langle\bar{k}\right|$ is a translation operator, the arithmetic operations are modulo 3 and $\{\left|\bar{0}\right\rangle,\left|\bar{1}\right\rangle,\left|\bar{2}\right\rangle\}$ is a basis for the qutrit space $\mathcal{A}$. We omit the proof that $S^{\prime\prime}_{{\rm SWAP}}$ indeed performs a swap operation (for a proof see (34)). To generalise this idea and to construct an isometry, we relax the assumption that $\mathcal{A}$ is a three dimensional Hilbert space. We re-express/replace the operations in $T,U,V$ which act on the $\mathcal{A}$ space by linear combinations of monomials in $\{\Pi_{i}\}_{i=1}^{5}$. We obtain the coefficients used in these linear combinations by assuming the space $\mathcal{A}^{\prime}$ is three dimensional. The idea is simply that this map reduces to a swap operation when we re-impose the assumptions and for cases close to it, we expect it to behave appropriately. We describe this procedure more precisely below.

We found an explicit linear combination for step 1 (a) of LABEL:Alg:ConstructingAnIsometry.¹³¹³13Using prior results, we expect one can prove the existence of such a linear combination, but we do not pursue this here. Thus, the algorithm always succeeds at constructing $S^{\prime}_{{\rm SWAP}}$. We must show that $S^{\prime}_{{\rm SWAP}}$ is in fact an isometry. This is important because of the following reason. Recall that our objective was to lower bound LABEL:Eq:fidelity. To this end, we said we drop the maximisation over all possible $V$s, (for a given quantum realisation $(\rho,\left\{\Pi_{i}\right\}_{i=1}^{5})$) and instead insert a specific isometry $S^{\prime}_{{\rm SWAP}}$ (which is a function of $(\rho,\{\Pi_{i}\}_{i=1}^{5})$). Note that this argument for lower bounding LABEL:Eq:fidelity breaks if $S^{\prime}_{{\rm SWAP}}$ is not an isometry. In fact, $P_{\mathcal{A}}$ as produced by the algorithm is not necessarily unitary (viz. $P_{\mathcal{A}}^{\dagger}P_{\mathcal{A}}=\mathbb{I}_{\mathcal{A}}$ may not hold). To address this, we use the localising matrix technique introduced in  (43). Let $\hat{P}_{\mathcal{A}}$ be unitary matrix satisfying $\hat{P}P\geq 0$, where we dropped the subscript for clarity. Consider the case where $P$ is not unitary. In that case, one can use polar decomposition to write $P=\left|P\right|U$ (not to be confused with the $U$ above; where $\left|P\right|\geq 0$ and $U^{\dagger}U=\mathbb{I}$) so choosing $\hat{P}=U^{\dagger}$ satisfies $\hat{P}P\geq 0$. Thus for each $P$, the constraint can be satisfied. Consider the other case, i.e., where $P=U$ is unitary. Then,¹⁴¹⁴14To see this, write the polar decomposition of $\hat{P}=\left|\hat{P}\right|E$ (where $\left|\hat{P}\right|\geq 0$ and $E^{\dagger}E=\mathbb{I}$). Then, $\hat{P}U=\left|\hat{P}\right|EU$, which is a polar decomposition of $\hat{P}U$. The polar decomposition of a positive semi-definite matrix $M$ always has the form $M.\mathbb{I}$. Using $M=\hat{P}U$, and identifying $EU$ with $\mathbb{I}$, we have $E=U^{\dagger}$. $\hat{P}=U^{\dagger}$. Thus, in the ideal case, we recover the same unitary and for the case close to ideal, we are guaranteed that there is some solution (which we expect should also work reasonably). Combining these, we can construct $S_{{\rm SWAP}}$, which is an isometry.

We can now combine all the pieces to write the final optimisation problem we solve. We use $\mathcal{G}(\{a_{i}\})$ to denote the set of all letters

We thus have an algorithm which can calculate the required lower bound on fidelity, given the observed value of the KCBS operator.

As previously stated, obtaining the numerical solution was quite involved, owing to the fact that the description of the semidefinite program we obtain is implicit. Consequently, one first needs to find this explicit description through symbolic computation and then solve the resulting semidefinite program. After performing the symbolic computation to find the objective and constraints, we solve an SDP over a $192\times 192$ matrix with $16859$ constraints. The terms in the objective function must appear in the SDP matrix in order for us to impose appropriate constraints. This is because the optimisation program would otherwise be undefined. The unusually large size of the SDP matrices in our case reflects the complexity of the objective function in our case, which would have been difficult to obtain without symbolic computation. To capture all the terms that appear in the objective, we need words with three letters. Further, we require $769$ localising matrix constraints.

We used Jupyter notebooks (based on python) as our programming environment. We used sympy (44) for symbolic computations and cvxpy (45; 46) for translating the resulting SDP into an instance which MOSEK (an SDP solver) could solve. Our implementation is available on GitHub (47).

We describe our experimental setup which is designed to realise the ideal KCBS configuration. We use a ${}^{40}{\rm Ca}^{+}$ ion trapped in a blade-shaped Paul trap. The qutrit basis states are encoded into three Zeeman sub-levels of the ${}^{40}{\rm Ca}^{+}$ ion, with

as shown in LABEL:Fig:experiment_resulta. Unitary operations on these qutrits are performed by shining a linear polarized narrow-linewidth 729 $\rm nm$ laser beam, propagating along the trap axis, at an angle of 45^∘ with respect to the quantization axis. For $k\in\{1,2\}$, the laser can be frequency modulated to be on resonant with the specific transition between the states $\left|0\right\rangle$ and $\left|k\right\rangle$ by an acousto-optic modulator (AOM). Coupling strength $\Omega_{k}$, duration $t$ and phase $\phi_{k}$ of this laser pulse then can be controlled by this AOM to perform a rotation, $R_{k}(\theta_{k},\phi_{k})$ between the states $|0\rangle$ and $|k\rangle$, where $\theta_{k}=\Omega_{k}t$ and $\phi_{k}$ represent the polar angle and the azimuthal angle respectively, i.e.,

The parameters of the rotation are controlled by varying the duration and phase of the corresponding laser pulse under constant intensity via the acoustic-optic modulator with high fidelity. Using $R_{1}$ and $R_{2}$, one can perform an arbitrary rotation in the qutrit space. Measurements in our setup are performed by fluorescence detection, i.e., counting photons. These operations can be combined to perform various experiments nearly satisfying LABEL:Assu:KCBSexclusivity. We delineate this procedure for realising the KCBS configuration (see LABEL:Def:idealKCBSconfig) at the logical level, followed by experimental details corresponding to each step.

Using the notation introduced in LABEL:Assu:KCBSexclusivity, we denote the binary measurements of interest by $M_{i}:=(\Pi_{0|i},\Pi_{1|i})$ and the quantum state of our system by $\rho$. To realise the KCBS configuration in LABEL:Def:idealKCBSconfig, we require $\Pi_{1|i}=|u_{i}\rangle\langle u_{i}|$ and $\rho=|0\rangle\langle 0|$. To implement the measurement $\Pi_{1|i}$, as a composition of the operations our setup permits, we first compute a qutrit rotation $U_{i}$ which maps $|u_{i}\rangle$ to $|0\rangle$. This, in turn, is used to compute the parameters $\theta_{1,i},\theta_{2,i}$ and $\phi_{1,i},\phi_{2,i}$ such that $U_{i}=R_{2}(\theta_{2,i},\phi_{2,i})\cdot R_{1}(\theta_{1,i},\phi_{1,i})$. Observe that defining $\Pi_{1|i}:=U_{i}|0\rangle\langle 0|U_{i}^{\dagger}$ and $\Pi_{0|i}:=U_{i}(\mathbb{I}-|0\rangle\langle 0|)U_{i}^{\dagger}$ results in the KCBS configuration.

If experimental imperfections are neglected (which are indeed negligible in our experiment; see \SubsecrefnoiseComment), it is evident that LABEL:Assu:KCBSexclusivity is satisfied, i.e., $\Pi_{1|i}\cdot\Pi_{1|j}=0$ for $(i,j)\in E$ and $\Pi_{1|i}^{2}=\Pi_{1|i}$. Further, it is clear that

where $\Pr(ab|ij)$ denotes the probability of obtaining outcomes $(a,b)$ when measurements $(M_{i},M_{j})$ are performed and $\Pr(a|i)$ denotes the probability of obtaining outcome $a$ given $M_{i}$ was measured. Finally, we note that $\Pr(10|ij)=\Pr(01|ji)$.

As illustrated in LABEL:Fig:experiment_result, $\sum_{k}p_{k}$ is estimated by randomly choosing (using a quantum random number generator; see \SubsecrefnoiseComment) $(i,j)\in E$, measuring $M_{i}$ followed by $M_{j}$ and counting $N(10|ij)$, the number of times the output was $(1,0)$, i.e.,

The Calcium ions, in which two dark states and one bright state are chosen to encode the states {$\left|1\right\rangle$,$\left|2\right\rangle$} and $\left|0\right\rangle$, respectively. Every experimental sequence starts with 1 ${\rm ms}$ Doppler cooling using a 397 ${\rm nm}$ laser and a resonant 866 nm laser. The 397 nm laser is red detuned approximately half a natural linewidth from resonating with the cycling transition between $S_{1/2}$ and $P_{1/2}$ manifolds. This is followed by 300 $\mu s$ electromagnetically-induced transparency (EIT) cooling sequence which cools the ion down to near the motional ground state. After that, 10 $\mu s$ optical pumping laser is applied to initialize the qutrit to the $\left|0\right\rangle$ state with 99.5% fidelity. When needed, 729 ${\rm nm}$ pulse sequence is applied to prepare a certain initial state. Measurements of the observables $\{A_{i}\}$ are performed through coherent rotations $R_{2}(\theta_{2},\phi_{2})R_{1}(\theta_{1},\phi_{1})$, a fluorescence detection followed by the operation $R_{1}^{\dagger}(\theta_{1},\phi_{1})R_{2}^{\dagger}(\theta_{2},\phi_{2})$ which undoes the previous rotation. The first fluorescence detection lasts for 220 $\mu s$ and uses laser settings close to those of Doppler cooling, to minimize the motional heating due to photon recoil (in case of a bright state). However, this brings the temperature of the bright state close to the Doppler limit. Therefore, a 150 $\mu s$ EIT cooling sequence followed by 10 $\mu s$ optical pumping pulses is applied to cool the bright state ion. The detection and cooling time are greatly suppressed to minimize the random phase accumulation between the sequential measurement. Neither fluorescence detection nor EIT cooling process affects the temperature of a dark state ion, which is slightly heated at a rate of 140 quanta per second due to electronic noise. The second fluorescence detection uses resonant laser pulse and 300 $\mu s$ integrating time, which lowers detection error. The result is acquired by counting the number of photons collected from photo multiplier tube (PMT) within the detection time. The statistical probability is calculated by repeating the same measurement 10000 times.

In our experiments the $2\pi$ pulse time for both transitions are adjusted to around 142 $\mu s$, that is, the Rabi frequency is as low as $\Omega_{1/2}=(2\pi)7$ KHz, making the AC-Stark shift below 100 Hz. The separation between $\left|1\right\rangle$ and $\left|2\right\rangle$ is $\omega_{2}-\omega_{1}=(2\pi)8.69$ MHz with corresponding magnetic field $B=5.18$ G. The maximum probability of off-resonant excitation $\Omega^{2}/(\omega_{2}-\omega_{1})^{2}$ is about 6.5 $\times$ $10^{-7}$, small enough to ensure the independence of each Rabi oscillation (48; 49).

To study the self-testing property of device, we perform experiments which deviate from the standard KCBS scenario, in two qualitatively different ways.

First, we consider the realisation $(\rho^{\prime}(p),\{M_{i}\})$ parametrised by $0\leq p\leq 1$. Here, $M_{i}$ correspond to projectors along $|u_{i}\rangle$ as before, but instead of initialising the state of the device to $\rho=|0\rangle\langle 0|$, we initialise it to a mixture of $\rho$ and the maximally mixed state, i.e.,

For $p=0$, we recover the ideal KCBS configuration for which $\mathcal{I}=0$. However, for $p>0$, LABEL:Assu:KCBSexclusivity still holds (neglecting the imperfections in the measurements) but $\mathcal{I}=\epsilon>0$. The experiment was performed for $p=0,0.1$ and $0.2$.

Second, we consider the realisation $(|u_{0}^{\prime}\rangle\langle u_{0}^{\prime}|,\{M_{i}^{\prime}(\theta)\})$ parametrised by the vector $|u_{0}^{\prime}\rangle$ and the angle $\theta$. Here, unlike the previous case, the state is pure. Further, the measurements $M_{i}^{\prime}=(\Pi_{1|i}^{\prime},\Pi_{0|i}^{\prime})$ now correspond to projections on¹⁸¹⁸18Defined exactly as $M_{i}$ was defined using $|u_{i}\rangle$; $\Pi_{1|i}^{\prime}=U_{i}^{\prime}|0\rangle\langle 0|U_{i}^{\prime\dagger}$ and $\Pi_{0|i}^{\prime}=U_{i}(\mathbb{I}-|0\rangle\langle 0|)U_{i}^{\prime\dagger}$ with $U^{\prime}_{i}$ encoding a rotation from $|u^{\prime}_{i}\rangle$ to $|0\rangle$. $|u^{\prime}_{i}\rangle$ for $i\in\{1,2\dots 5\}$. These vectors $|u^{\prime}_{i}\rangle$ are chosen to be

which ensures that for all angles $\theta$, LABEL:Assu:KCBSexclusivity holds. We performed the experiment for two values of these parameters:

For the normal ordered case, the results are summarised in LABEL:Fig:experiment_result. See Figure D.1 in Appendix D for the summary of reverse ordered experiments. Note that we have two robustness curves in both of the aforementioned figures. To see why we have two robustness curves, notice that the observed statistic constraint in the SDP corresponding to LABEL:Alg:The-algorithm can be tightened by specifying the value of each $\left\langle\Pi_{i}\right\rangle$ term. Obviously, if one specifies the value of $\left\langle\Pi_{i}\right\rangle$, the value for their sum automatically gets fixed. Based on the experimental data, we have access to the value of each of the $\left\langle\Pi_{i}\right\rangle$ and thus we can always obtain a tighter bound compared with the bound obtained via the sum constraint in the SDP for LABEL:Alg:The-algorithm. We illustrate this by plotting the robustness curve obtained in the special case $p_{1}=p_{2}=p_{3}=p_{4}=p_{5}$. One could have just as easily taken a different choice for the values of $p_{i}$ such that their sum remains equal to the appropriate KCBS value and obtain a different robustness curve. All such “curves” obtained by providing the value for $p_{i}$s will be lower bounded by the curve obtained via the sum constraint (the blue curve). For our experimental data point with mean $\mu$ and standard deviation $\sigma$, we base our analysis on $\mu-1.96\sigma$, which is the smallest possible value with $95$ percent confidence for a normally distributed data. The experimental data has been summarised in Table 2 and in Appendix D. The complete dataset can be accessed on GitHub (47). We now present the detailed analysis.