Abstract
Bell nonlocality as a resource for deviceindependent certification schemes has been studied extensively in recent years. The strongest form of deviceindependent certification is referred to as selftesting, which given a device, certifies the promised quantum state as well as quantum measurements performed on it without any knowledge of the internal workings of the device. In spite of various results on selftesting protocols, it remains a highly nontrivial problem to propose a certification scheme of qudit–qudit entangled states based on violation of a single doutcome Bell inequality. Here we address this problem and propose a selftesting protocol for the maximally entangled state of any local dimension using the minimum number of measurements possible, i.e., two per subsystem. Our selftesting result can be used to establish unbounded randomness expansion, \({{{\mathrm{log}}}\,}_{2}d\) perfect random bits, while it requires only one random bit to encode the measurement choice.
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Introduction
The advent of quantum theory has not just changed the understanding of physics but has also given rise to new phenomena that would have never been possible in the classical world. Arguably, one of the most interesting features of quantum theory is the existence of quantum correlations, which cannot be explained by any local hiddenvariable model, a phenomenon commonly referred to as Bell nonlocality^{1,2}. It has been understood that apart from its fundamental significance, nonlocality is a powerful resource for certain deviceindependent applications such as quantum cryptography^{3}, randomness generation^{4,5}, or, more recently, for deviceindependent certification methods^{3,4,5,6}.
The strength of the deviceindependent (DI) certification methods lies in the fact that given an unknown quantum system, one can make nontrivial statements on some of its key features based solely on the nonlocal correlations obtained from it. They are advantageous over standard certification methods such as those based on quantum tomography (cf. Ref. ^{7}) because they do not require making assumptions on the system under study, apart from that, it is governed by quantum mechanics. An example of such a DIcertification scheme would be verifying whether a quantum device produces entanglement^{8} or certification of the dimension of a quantum system^{9}, both based on a violation of some Bell inequality by the corresponding quantum system.
The strongest form of DI certification is selftesting. First introduced in^{6}, it allows one to provide a full description, up to certain wellunderstood equivalences, of the considered quantum system and also the measurements performed on it based on observing the maximal violation of some Bell inequality. Such a form of certification is particularly interesting from the application point of view as it provides a way of verifying that a given quantum device functions properly without the need of knowing its internal working.
A lot of attention has thus been devoted to proposing selftesting schemes for entangled quantum states^{10,11,12,13,14}. However, most of the obtained results focus on states that are locally qubits, such as, for instance, the selftesting statement for any twoqubit entangled state^{11,12} based on a violation of the tilted version of the famous Clauser–Horne–Shimony–Holt (CHSH) Bell inequality^{15,16}. At the same time, quantum systems of higher local dimension remain mostly unexplored and, in fact, few results are devoted to them^{11,17,18,19}. In particular, in Ref. ^{17}, the twoqubit results^{11,12} were combined to design a selftesting protocol for any entangled state of arbitrary local dimension. Still, these results are based on a violation of many two–outcome Bell inequalities and the question whether one can design a self–testing statement for qudit quantum systems relying on violation of a single and truly doutcome Bell inequality remains open. Moreover, the self–testing statement of Refs. ^{11,17} is not optimal in terms of the number of measurements that the observers need to perform (three and four, respectively) to certify the state and the corresponding measurements. Taking into account the possibility of experimental implementations of self–testing, it is a highly relevant question whether alternative protocols can be derived, which rely on the minimal number of two measurements per observer.
The main aim of this work is to address the above questions. We provide a self–testing statement for a maximally entangled state of local dimension d:
from a violation of a truly doutcome Bell inequality exploiting the minimal possible number of measurements per party, which is two. To this end, we use the Bell inequality introduced in Ref. ^{20} as a generalization of the well–known CHSH Bell inequality to doutcome Bell scenarios. A straightforward implication of our self–testing statement is a novel and simpler, as compared with the previous approaches^{21,22}, scheme for parallel self–testing of N copies of the two–qubit maximally entangled state \(\left{\phi }_{2}^{+}\right\rangle\). Another implication is that the outcomes of local measurements maximally violating the respective Bell inequality are perfectly random, which allows us to propose a quantum protocol for unbounded expansion of quantum randomness.
Results
Preliminaries
Before presenting our results, let us set up the scenario and introduce the relevant notions.
Bell scenario
The simplest experimental setup exhibiting quantum nonlocality, namely, the bipartite Bell setup, comprises one preparation and two measurement devices. The latter are possessed by distant and noncommunicating parties, Alice and Bob, and can in general perform one of m measurements, denoted by M_{x} and N_{y} with x, y = 1, …, m. In each run of the experiment, a bipartite quantum system ρ_{AB} is prepared by the preparation device and subsequently each measurement device performs a measurement on one of its subsystems, yielding an outcome a and b, respectively, with a, b ∈ {0, …, d − 1}.
Correlations obtained in this experiment are captured by a vector of probability distributions
Here, p(a, b∣x, y) is the probability of obtaining outcomes a and b by Alice and Bob after performing measurements M_{x} and N_{y}, respectively, and it is given by the well–known formula
where \({M}_{x}^{(a)}\) and \({N}_{y}^{(b)}\) are the measurement operators defining the measurements M_{x} and N_{y}, respectively. In what follows, we also refer to the vector p as correlations.
In a fixed Bell scenario with mdoutcome measurements, such quantum correlations, that is, those obtained by performing local measurements on composite quantum systems, form a convex set \({{{\mathcal{Q}}}}\). For further purposes, it is important to recall that any point \({{{\bf{p}}}}\in {{{\mathcal{Q}}}}\) can be achieved with a pure state \(\left{\psi }_{{{{\rm{AB}}}}}\right\rangle\), whose local dimensions might in general be higher than those of ρ_{AB}, and projective measurements \({M}_{x}=\{{P}_{x}^{(a)}\}\) and \({N}_{y}=\{{Q}_{y}^{(b)}\}\).
It turns out that the quantum set \({{{\mathcal{Q}}}}\) contains correlations that even if obtained from a quantum state, can be simulated by Alice and Bob in a purely classical way. Such correlations are said to admit a local–realistic description, and for brevity, we refer to them as local or classical. More formally, local correlations are those for which p(a, b∣x, y) can be represented as a convex combination of product–deterministic correlations
where λ is a random variable distributed according to a distribution μ(λ) and p(a∣x, λ), p(b∣y, λ) ∈ {0, 1} for every x, y, a, b. For any m and d, such local correlations form a convex polytope that we denote \({{{\mathcal{L}}}}\).
Bell inequalities
As proven by Bell, the local set \({{{\mathcal{L}}}}\) is a proper subset of \({{{\mathcal{Q}}}}\)^{1} and those quantum points that are outside \({{{\mathcal{L}}}}\) are termed nonlocal. The most natural way to show that a given point \({{{\bf{p}}}}\in {{{\mathcal{Q}}}}\) is not an element of \({{{\mathcal{L}}}}\), is to use Bell inequalities. Recall their generic form to be
where t = {t_{abxy}} is a collection of some real coefficients t_{abxy} and \({\beta }_{{{{\rm{L}}}}}=\mathop{\max }\limits_{{{{\bf{p}}}}\in {{{\mathcal{L}}}}}{{{\mathcal{I}}}}\) is the local bound of inequality (5). Analogously, the quantum or the Tsirelson bound of (5) is defined as \({\beta }_{{{{\rm{Q}}}}}={\sup }_{{{{\bf{p}}}}\in {{{\mathcal{Q}}}}}{{{\mathcal{I}}}}\).
Clearly, violation of a Bell inequality by correlation p implies that it is nonlocal. Moreover, any point p violating maximally some Bell inequality (or, equivalently, achieving its Tsirelson bound) necessarily belongs to the boundary of the quantum set \({{{\mathcal{Q}}}}\).
Correlation picture
Let us finally notice that it is often beneficial to describe correlations obtained in the Bell experiment by expectation values instead of probability distributions. A convenient way to do so in Bell scenarios involving doutcome measurements is to consider the two–dimensional Fourier transform of the conditional probabilities p(a, b∣x, y)
where ω is the dth root of unity \(\omega =\exp (2\pi {\mathbb{i}}/d)\) and k, l = 0, …, d − 1. Importantly, for any quantum point p, the expectation values (6) can be represented as \(\langle {A}_{x}^{k}{B}_{y}^{l}\rangle =\langle {\psi }_{{{{\rm{AB}}}}} {A}_{x}^{(k)}\otimes {B}_{y}^{(l)} {\psi }_{{{{\rm{AB}}}}}\rangle\) with \(\{{A}_{x}^{(k)}\}\) and \(\{{B}_{y}^{(l)}\}\) being collections of unitary operators with eigenvalues ω^{i}(i = 0, …, d − 1) defined as the Fourier transforms of the corresponding projective measurements
It is not difficult to see that \({A}_{x}^{(k)}\) is simply the kth power of A_{x} (and similarly for Bob’s operators), and thus, in what follows, we simply write \({A}_{x}^{k}\). In this correlator picture, the unitary operators A_{x} and B_{y} are doutcome observables measured in the Bell setup.
Self–testing
We are now ready to present our main result: the self–testing statement for the two–qudit maximally entangled state and certain doutcome measurements usually referred to as Collins–Gisin–Linden–Massar–Popescu (CGLMP) measurements^{23,24,25} (see Supplementary Methods for their explicit form).
To recall the task of self–testing, or more generally, DI certification, let us consider a quantum device performing a Bell experiment on some state \(\left{\psi }_{{{{\rm{AB}}}}}\right\rangle \in {{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{B}}}}}\) with some quantum doutcome observables _{Ax} and B_{y}, where the dimensions of the \({{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}\) and \({{{{\mathcal{H}}}}}_{{{{\rm{B}}}}}\) are unknown but finite. The only information accessible to the user about how this device functions is the observed correlation p. The aim of DI certification is to reveal the form of the state and observables from violation of some Bell inequality by p (given that the observed correlations violate some Bell inequality). However, there are two degrees of freedom, which can never be detected from the observed statistics. One is the set of local unitaries U_{A}, U_{B} that act on \({{{{\mathcal{H}}}}}_{{{{\rm{A}}}}},{{{{\mathcal{H}}}}}_{{{{\rm{B}}}}}\), that is, the state \({U}_{{{{\rm{A}}}}}\otimes {U}_{{{{\rm{B}}}}}\left{\psi }_{{{{\rm{AB}}}}}\right\rangle\) together with \(\{{U}_{{{{\rm{A}}}}}{A}_{x}{U}_{{{{\rm{A}}}}}^{{\dagger} }\},\{{U}_{{{{\rm{B}}}}}{B}_{y}{U}_{{{{\rm{B}}}}}^{{\dagger} }\}\) will generate the same p. Another one is the presence of an auxiliary system on which the measurements act trivially. A particular instance of such DI certification, termed self–testing, infers a unique state and two sets of unique measurements up to these equivalences.
Clearly, a necessary condition to derive an exact self–testing statement for a state and measurements is that the obtained correlation p lies on the boundary of \({{{\mathcal{Q}}}}\), and thus violates some Bell inequality maximally. Consequently, in order to derive a self–testing statement for \(\left{\phi }_{d}^{+}\right\rangle\) for any d, the first task is to identify a class of Bell inequalities in the bipartite scenario with the minimum number of measurements for which β_{Q} is achieved by it for any d. The only known Bell inequality meeting these requirements is the Salavrakos–Augusiak–Tura–Wittek–Acín–Pironio (SATWAP) Bell inequality^{20}, which in the correlator picture reads
where \({a}_{k}=(1{\mathbb{i}}){\omega }^{k/4}/2\) and the classical value is given by \({\beta }_{{{{\rm{L}}}}}=(1/2)[3\cot (\pi /4d)\cot (3\pi /4d)]2\).
As proven in Ref. ^{20}, the maximal quantum value of this inequality is β_{Q} = 2(d − 1) and it is achieved by the maximally entangled state of two qudits \(\left{\phi }_{d}^{+}\right\rangle\) and the aforementioned optimal CGLMP measurements (cf. Supplementary Methods). In what follows, we show that this is in fact the only quantum system, up to additional freedoms as discussed above, realizing the maximal quantum violation of inequality (8) (Fig. 1).
Self–testing of maximally entangled state of arbitrary local dimension
Let us first introduce Z_{d} = diag[1, ω, …, ω^{d−1}] to be the ddimensional generalization of the σ_{z}Pauli matrix in the standard basis and T_{d} to be the following matrix:
where δ_{i,j} is the Kronecker delta: δ_{i,j} = 1 for i = j and δ_{i,j} = 0 otherwise. It is not difficult to see that both matrices are unitary and have eigenvalues ω^{i}(i = 0, …, d − 1) (see Supplementary Methods) and thus represent doutcome quantum observables.
Now, we can state our main theorem.
Theorem 1
Assume that the SATWAP Bell inequality (8) is maximally violated by a state \(\left{\psi }_{{{{\rm{AB}}}}}\right\rangle \in {{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{B}}}}}\) and unitary observables A_{i}, B_{i}(i ∈ {1, 2}). Then, for any d, \({{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}={{\mathbb{C}}}^{d}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{A}}}}^{\prime} }\) as well as \({{{{\mathcal{H}}}}}_{{{{\rm{B}}}}}={{\mathbb{C}}}^{d}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{B}}}}^{\prime} }\) with some auxiliary Hilbert spaces \({{{{\mathcal{H}}}}}_{{{{\rm{A}}}}^{\prime} }\) and \({{{{\mathcal{H}}}}}_{{{{\rm{B}}}}^{\prime} }\) of unknown but finite dimensions. Moreover, there exist local unitary transformations \({U}_{{{{\rm{A}}}}}:{{\mathbb{C}}}^{d}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{A}}}}^{\prime} }\to {{\mathbb{C}}}^{d}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{A}}}}^{\prime} }\) and \({U}_{{{{\rm{B}}}}}:{{\mathbb{C}}}^{d}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{B}}}}^{\prime} }\to {{\mathbb{C}}}^{d}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{B}}}}^{\prime} }\), such that
and
where \({a}_{1}=(1+{\mathbb{i}}){\omega }^{1/4}/2\), Z_{d} and T_{d} are defined above, and \({{\mathbb{1}}}_{{{{\rm{A}}}}^{\prime} },{{\mathbb{1}}}_{{{{\rm{B}}}}^{\prime} }\) are identity matrices acting on \({{{{\mathcal{H}}}}}_{{{{\rm{A}}}}^{\prime} }\) and \({{{{\mathcal{H}}}}}_{{{{\rm{B}}}}^{\prime} }\). Finally, the state \(\left{\psi }_{{{{\rm{AB}}}}}\right\rangle\) is equivalent to \(\left{\phi }_{d}^{+}\right\rangle\) in the following sense:
where \(\left{{{{\rm{aux}}}}}_{A^{\prime} B^{\prime} }\right\rangle\) is some auxiliary state from \({{{{\mathcal{H}}}}}_{{{{\rm{A}}}}^{\prime} }\otimes {{{{\mathcal{H}}}}}_{{{{\rm{B}}}}^{\prime} }\).
Before proving this theorem, a few comments are in order. First, as proven in Supplementary Methods, the observables Z_{d} and T_{d} are unitarily equivalent to the CGLMP measurements. Second, our self–testing statement exploits a single Bell inequality involving the minimal number of two measurements per observer for which one can observe nonlocality in quantum systems. The number of local observables one needs to measure in order to self–test a state is of key importance from the application point of view. Finally, in the case of d = 2^{N}, the theorem implies a parallel selftesting protocol for \({\left{\phi }_{2}^{+}\right\rangle }^{\otimes N}\), i.e., N copies of maximally twoqubit entangled state (see Refs. ^{21,22} for the previous approaches).
Proof
The proof is quite technical and thus is deferred to the Supplemental Information (cf. Supplementary Methods). Here we only sketch its main ingredients. Let a state \(\left{\psi }_{{{{\rm{AB}}}}}\right\rangle\) and observables A_{x} and B_{y} maximally violate the SATWAP Bell inequality. Without loss of generality, we can assume that the reduced states of \(\left{\psi }_{{{{\rm{AB}}}}}\right\rangle\) are full rank. Exploiting the sumofsquares decomposition of the corresponding Bell operator, we can establish that Alice’s and Bob’s observables satisfy
for x, y = 1, 2 and any n < d, which is a divisor of the number of outcomes d. These conditions imply that the multiplicities of the eigenvalues of all the observables A_{x} and B_{y} are equal, which has two consequences. First, their matrix dimensions are a multiple of d, meaning that they act on, respectively, \({{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}={{\mathbb{C}}}^{d}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{A}}}}^{\prime} }\) and \({{{{\mathcal{H}}}}}_{{{{\rm{B}}}}}={{\mathbb{C}}}^{d}\otimes {{{{\mathcal{H}}}}}_{{{{\rm{B}}}}^{\prime} }\) with \({{{{\mathcal{H}}}}}_{{{{\rm{A}}}}^{\prime} }\) and \({{{{\mathcal{H}}}}}_{{{{\rm{B}}}}^{\prime} }\) being some auxiliary Hilbert spaces of unknown but finite dimensions. Second, there exist unitary operations \({U}_{{{{\rm{A}}}}}:{{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}\to {{{{\mathcal{H}}}}}_{{{{\rm{A}}}}}\) and \({U}_{{{{\rm{B}}}}}:{{{{\mathcal{H}}}}}_{{{{\rm{B}}}}}\to {{{{\mathcal{H}}}}}_{{{{\rm{B}}}}}\) which allow us to bring Alice’s and Bob’s observables to the form given in Eqs. (10) and (11).
Finally, to obtain an analogous statement for the state we again employ the sumofsquares decomposition, which in virtue of relations (10) and (11), imposes the following conditions on the state \({\left\psi \right\rangle }_{{{{\rm{AB}}}}}\):
where we have denoted \(\left{\hat{\psi }}_{{{{\rm{AB}}}}}\right\rangle ={U}_{{{{\rm{A}}}}}\otimes {U}_{{{{\rm{B}}}}}\left{\psi }_{{{{\rm{AB}}}}}\right\rangle\). These conditions can easily be solved, leading to Eq. (12), which completes the proof.□
Applications to randomness expansion
An interesting application of our selftesting statement is DI randomness certification^{5}. Let us consider the distrustful scenario where one of the observers performing the Bell test, say Bob, wants to generate random bits from the outcomes of his measurements, and Eve who supplies the measurement devices wants to access that outcome.
We show with the aid of the above selftesting result that the maximal violation of the SATWAP Bell inequality certifies \({{{\mathrm{log}}}\,}_{2}d\) bits of perfect randomness in the outcomes of Alice’s and Bob’s measurements. Without loss of generality, we can focus on Bob’s measurements. One quantifies the randomness obtained in a Bell experiment as \({{{\mathrm{log}}}\,}_{2}G(x,{{{\bf{p}}}})\), where G(x, p) is the local guessing probability defined through the following formula:
where \(\left{\psi }_{{{{\rm{ABE}}}}}\right\rangle\) is a threepartite state shared by Alice, Bob, and Eve, \(\{{Q}_{y}^{(b)}\}\) is a projective measurement performed by Bob, and E = {E^{(c)}} is a doutcome measurement on \({{{{\mathcal{H}}}}}_{{{{\rm{E}}}}}\) whose result is Eve’s best guess of Bob’s outcome. Finally, S_{p} is the set of all possible strategies that Eve implements to guess the outcome of Bob’s measurements, consisting of the state \(\left{\psi }_{{{{\rm{ABE}}}}}\right\rangle\) and the measurement E, which reproduce the correlation p observed by Alice and Bob, i.e.,
Assume now that inequality (8) is maximally violated by p. This means that, up to local unitary operations, \(\left{\psi }_{{{{\rm{ABE}}}}}\right\rangle =\left{\phi }_{d}^{+}\right\rangle \left{{{{\rm{aux}}}}}_{{{{\rm{A}}}}^{\prime} {{{\rm{B}}}}^{\prime} {{{\rm{E}}}}}\right\rangle\) as well as \({P}_{x}^{(a)}={\bar{P}}_{x}^{(a)}\otimes {{\mathbb{1}}}_{{{{\rm{A}}}}^{\prime} }\) and \({Q}_{y}^{(b)}={\bar{Q}}_{y}^{(b)}\otimes {{\mathbb{1}}}_{{{{\rm{B}}}}^{\prime} }\), where \({\bar{P}}_{x}^{(a)}\) and \({\bar{Q}}_{y}^{(b)}\) are eigenprojectors of the observables in Eqs. (10) and (11), respectively. Taking all this into account along with the fact that Bob’s observables are traceless, one finally finds that G(y, p) = 1/d.
Thus, \({{{\mathrm{log}}}\,}_{2}G(x,{{{\bf{p}}}})={{{\mathrm{log}}}\,}_{2}d\) bits of randomness can be certified using our selftesting statement. This is the maximum randomness that could be extracted from a system of local dimension d, whereas it requires one bit of randomness to encode the inputs. This gives rise to unbounded randomness expansion as d can be arbitrarily large.
Discussions
We propose a selftesting statement for quantum system of arbitrary local dimension that unlike the previous approaches^{11,17} does not rely on selftesting results for qubit states and exploits a truly doutcome Bell inequality. Moreover, our selftesting result exploits only two measurements per party, which is the minimal number allowing the parties to observe nonlocal correlations. This makes our results interesting from the experimental point of view. In fact, violation of the SATWAP Bell inequality has already been experimentally tested in Ref. ^{26}.
Several followup questions arise from our work. The robustness of the above selftesting statement has been previously studied numerically in^{20} for local dimension three. However, there does not exist any analytical method for deriving robustness bounds for local dimension greater than two. Finding a general methodology for deriving analytical robustness bounds for any arbitrary local dimension and then comparing it to that of the scheme of^{11,17} could be an interesting followup question. Another interesting direction is to explore whether the doutcome SATWAP Bell inequality can be modified to be maximally violated by a partially entangled state and whether the resulting modifications can be used to make selftesting statements for those states, again with the minimal number of measurements per observer. Finally, it is also interesting to investigate whether, analogously to Refs. ^{27,28}, our selftesting statement can be used to establish certification of the optimal amount of \(2{{{\mathrm{log}}}\,}_{2}d\) bits of local or global randomness from the maximally entangled state of two qudits with the aid of a local POVM or two projective measurements, respectively. From a more general perspective, our selftesting result might provide nontrivial insights into the structure of the set of quantum correlations along the lines of^{29}, or might find applications in delegated quantum computation^{30}.
Note added—While working on this project, we became aware of the work^{31} exploring parallel selftesting with minimal number of measurements.
Data availability
No data set was generated or analyzed in the current study.
Code availability
No codes were generated or used in the current study.
References
Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964).
Bell, J. S. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966).
Acín, A. et al. Deviceindependent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007).
Colbeck, R. & Renner, R. No extension of quantum theory can have improved predictive power. Nat. Commun. 2, 411 (2011).
Pironio, S. et al. Random numbers certified by Bell’s theorem. Nature 464, 1021–1024 (2010).
Mayers, D. & Yao, A. Quantum cryptography with imperfect apparatus, Proceedings 39th Annual Symposium on Foundations of Computer Science (FOCS), 503 (IEEE, 1998) https://ieeexplore.ieee.org/document/743501.
Nielsen, M. & Chuang, I. Quantum Computation and Quantum Information (Cambridge University Press, 2000) http://mmrc.amss.cas.cn/tlb/201702/W020170224608149940643.pdf.
Bancal, J.D. et al. Deviceindependent witnesses of genuine multipartite entanglement. Phys. Rev. Lett. 106, 250404 (2011).
Brunner, N. et al. Testing the dimension of Hilbert spaces. Phys. Rev. Lett. 100, 210503 (2008).
McKague, M., Yang, T. H. & Scarani, V. Robust selftesting of the singlet. J. Phys. A Math. Theor. 45, 455304 (2012).
Yang, T. H. & Navascués, M. Robust selftesting of unknown quantum systems into any entangled twoqubit states. Phys. Rev. A 87, 050102(R) (2013).
Bamps, C. & Pironio, S. Sumofsquares decompositions for a family of ClauserHorneShimonyHoltlike inequalities and their application to selftesting. Phys. Rev. A 91, 052111 (2015).
Wang, Y., Wu, X. & Scarani, V. All the selftestings of the singlet for two binary measurements. New J. Phys. 18, 025021 (2016).
Šupić, I., Augusiak, R., Salavrakos, A. & Acín, A. Selftesting protocols based on the chained Bell inequalities. New J. Phys. 18, 035013 (2016).
Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test local hiddenvariable theories. Phys. Rev. Lett. 23, 880 (1969).
Acín, A., Massar, S. & Pironio, S. Randomness versus nonlocality and entanglement. Phys. Rev. Lett. 108, 100402 (2012).
Coladangelo, A., Goh, K. T. & Scarani, V. All pure bipartite entangled states can be selftested. Nat. Commun. 8, 15485 (2017).
Coladangelo, A. Generalization of the ClauserHorneShimonyHolt inequality selftesting maximally entangled states of any local dimension. Phys. Rev. A 98, 052115 (2018).
Kaniewski, J. et al. Maximal nonlocality from maximal entanglement and mutually unbiased bases, and selftesting of twoqutrit quantum systems. Quantum 3, 198 (2019).
Salavrakos, A. et al. Bell inequalities tailored to maximally entangled states. Phys. Rev. Lett. 119, 040402 (2017).
Wu, X., Bancal, J.D., McKague, M. & Scarani, V. Deviceindependent parallel selftesting of two singlets. Phys. Rev. A 93, 062121 (2016).
McKague, M. Selftesting in parallel. New J. Phys. 18, 045013 (2016).
Collins, D., Gisin, N., Linden, N., Massar, S. & Popescu, S. Bell inequalities for arbitrarily highdimensional systems. Phys. Rev. Lett. 88, 040404 (2002).
Barrett, J., Kent, A. & Pironio, S. Maximally nonlocal and monogamous quantum correlations. Phys. Rev. Lett. 97, 170409 (2006).
Żukowski, M., Zeilinger, A. & Horne, M. A. Realizable higherdimensional twoparticle entanglements via multiport beam splitters. Phys. Rev. A 55, 2564 (1997).
Wang, J. et al. Multidimensional quantum entanglement with largescale integrated optics. Science 360, 285–291 (2018).
Acín, A., Pironio, S., Vértesi, T. & Wittek, P. Optimal randomness certification from one entangled bit. Phys. Rev. A 93, 040102(R) (2016).
Woodhead, E. et al. Maximal randomness from partially entangled states. Phys. Rev. Res. 2, 042028(R) (2020).
Coladangelo, A. A twoplayer dimension witness based on embezzlement, and an elementary proof of the nonclosure of the set of quantum correlations. Quantum 4, 282 (2020).
Reichardt, B. W., Unger, F. & Vazirani, U. Classical command of quantum systems. Nature 496, 456–460 (2013).
Šupić, I., Cavalcanti, D. & Bowles, J. Deviceindependent certification of tensor products of quantum states using singlecopy selftesting protocols. Quantum 5, 418 (2021).
Acknowledgements
We would like to thank A. Acín, P. Nurowski, and A. Sawicki for useful discussions. This work is supported by Foundation for Polish Science through the First Team project (No First TEAM/20174/31) cofinanced by the European Union under the European Regional Development Fund. JK acknowledges support from the Homing programme of the Foundation for Polish Science (grant agreement no. POIR.04.04.00005F4F/1800) cofinanced by the European Union under the European Regional Development Fund.
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Sarkar, S., Saha, D., Kaniewski, J. et al. Selftesting quantum systems of arbitrary local dimension with minimal number of measurements. npj Quantum Inf 7, 151 (2021). https://doi.org/10.1038/s41534021004903
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DOI: https://doi.org/10.1038/s41534021004903
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