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Self-Testing of Quantum Systems: a Review

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Self-Testing of Quantum Systems: a Review Self-testing of quantum systems: a review Ivan Šupić1 and Joseph Bowles2 1Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland 2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain September 21, 2020 Self-testing is a method to infer the underlying physics of a quantum experiment in a black box scenario. As such it represents the strongest form of certification for quantum systems. In recent years a considerable self-testing literature has developed, leading to progress in related device-independent quantum information protocols and deepening our understanding of quantum correlations. In this work we give a thorough and self-contained introduction and review of self-testing and its application to other areas of quantum information. 1 Introduction works, and includes the first usage of the term ‘self-testing’ in this context. A similar idea was In contrast to classical theories, states in quan- already present in [MY98] in a cryptographic tum physics can be entangled and sets of mea- context, using the term ‘self-checking’ instead surements can be incompatible. As shown by of ‘self-testing’. These early works also intro- Bell in 1964 [Bel64], these features imply striking duced the paradigm of device-independence, to observable phenomena. In particular, the out- which self-testing is intimately related. In par- comes of incompatible measurements made on ticular, a self-testing protocol can be seen as a the local subsystems of an entangled quantum device-independent—or black box—certification state can exhibit correlations that are provably of a quantum system, assuming that the sys- stronger than any resulting from a classical the- tem can be prepared many times in an inde- ory, a phenomenon known as Bell nonlocality. pendent, identically distributed manner. Self- The field of Bell nonlocality has since grown con- testing is consequently relevant to many device- siderably (see [BCP+14] for a recent review ar- independent quantum information protocols and ticle), and the existence of Bell nonlocal corre- has led to related progress in this area. More re- lations in nature is now a well established fact cently, self-testing has become synonymous with [HBD+15, GVW+15, SMSC+15]. any protocol for certifying any type of quantum As more was understood about Bell nonlocal- system under a small set of assumptions. ity, a number of works [SW87, PR92, BMR92, In this work we give a up-to-date review of the Tsi93] eventually pointed out that there exist Bell field of self-testing. We hope that it will be of nonlocal correlations that—as well as requiring use to people both unfamiliar with the field, as entanglement and incompatibility—can only be well as serving as a reference for those within arXiv:1904.10042v4 [quant-ph] 18 Sep 2020 produced by making particular sets of incompati- it. The review is organised as follows. In sec- ble measurements on particular entangled states. tion2 we give a gentle introduction to device- These works have since given birth to the field independence and its connection to self-testing. of self-testing, which broadly speaking aims to We then formally introduce self-testing, giving understand the structure of the set of quantum the mathematical definitions in section3 and a correlations and identify those correlations that simple example in section4 that illustrates many admit a unique physical realisation. important concepts. Sections5 to8 are a thor- An important milestone in the development of ough literature review of state and measurement self-testing was the 2004 work of Mayers and self-testing, explaining the tools and techniques Yao [MY04]. This work set the terminology that are commonly used along the way. In sec- and formalism that was to be adopted by later tion9 we review extensions of self-testing to other Accepted in Quantum 2020-23-08, click title to verify 1 scenarios, and in section 10 the application of 6 Self-testing of multipartite states 21 self-testing to other fields in quantum informa- 6.1 Self-testing of graph states from tion theory. In section 11 we cover experimental stabilizer operators . 21 realisations of self-testing protocols. Finally, in 6.2 Tailoring Bell inequalities . 22 section 12 we discuss some possible future direc- 6.3 Reductions to bipartite methods . 23 tions for the field and a number of open problems. 6.4 Parallel self-testing of multipartite We point the reader to the related review ar- states . 23 ticles [MdW16] and [Sca12] where discussions 6.5 Self-testing using only marginal in- about self-testing can also be found. [MdW16] formation . 23 deals with the classical and quantum certifi- cation of both classical and quantum proper- 7 Robust self-testing of states 23 ties of an object, with self-testing being identi- 7.1 Robust self-testing methods . 24 fied as classical certification of quantum prop- 7.2 Robust certification of large entan- erties. [Sca12] provides a pedagogical review glement . 28 of the device-independent approach to quan- tum physics. Self-testing is discussed as one of 8 Self-testing of measurements 29 device-independent protocols. We also recom- 8.1 Measurement self-testing results . 29 mend [McK10, Kan17, Kan16] as valuable texts 8.2 Methods in measurement self-testing 30 for first time readers. 8.3 Robust measurement self-testing . 32 9 Extensions of self-testing to other Contents scenarios 33 9.1 Self-testing of quantum gates and 1 Introduction1 circuits . 33 9.2 Semi-device-independent scenarios 34 2 Self-testing as a device-independent protocol3 10 Applications of self-testing 37 10.1 Device-independent randomness 3 Definitions5 generation . 38 3.1 Notation . 5 10.2 Device-independent quantum 3.2 The self-testing scenario . 6 cryptography . 39 3.3 Self-testing of states . 7 10.3 Entanglement detection . 40 3.4 Self-testing of measurements . 8 10.4 Delegated quantum computing . 41 3.5 Self-testing via a Bell inequality 10.5 Structure of the set of quantum and the geometry of the set of correlations . 41 quantum correlations . 8 3.6 Robust self-testing . 9 11 Experiments 42 3.7 Generalisations and alternative definitions . 10 12 Concluding remarks and open ques- tions 43 4 A first example 11 4.1 Geometrical proof of anticommu- Acknowledgements 45 tativity . 13 4.2 Algebraic proof of anticommuta- A Appendix 45 tivity . 13 A.1 Self-testing complex measurements 45 4.3 Swap gate . 13 A.2 Regularisation trick . 45 4.4 Self-testing of measurements . 16 A.3 Swap isometries . 46 A.4 Localising matrices in the Swap 5 Self-testing of bipartite states 16 method . 46 5.1 Self-testing of two-qubit states . 16 5.2 Self-testing of qudit states . 18 B State and measurement assumptions 47 5.3 Self-testing n maximally entangled B.1 State . 47 pairs of qubits . 19 B.2 Measurements . 47 Accepted in Quantum 2020-23-08, click title to verify 2 References 49 derstand the experimental setup, so they do not know what the different settings do. Moreover, even if they were told what the settings do, they 2 Self-testing as a device-independent do not have a good understanding of quantum protocol optics. As a result, they will not be able to re- construct the state of the source in order to check The treatment of complex systems as black boxes if it is entangled, as was the case for Carmela and is a powerful tool in many scientific domains, pro- Deng. viding a minimalist level of abstraction that al- Alice, however, proposes the following: even lows one to focus on what a device or system does though they do not understand what the set- without the need to model precisely how this is tings do, they can still change them and observe achieved. In quantum information theory, this something. That is, they can simply model their approach is known as the device-independent (DI) laboratories as black boxes. Each laboratory is approach. treated as a device (a black box) that takes an In order to explain the idea of the device- input (the settings) and returns and output (the independent approach we imagine the following result), but the physical mechanism behind how scenario. Consider two laboratories, run by two this occurs is unknown (see figure1, centre). Sim- experimenters called Carmela and Deng. In their ilarly, they do not assume anything about the laboratories (let’s imagine they are quantum op- source; all they know is that it is distributing tics laboratories) both Carmela and Deng have some physical systems that may or may not be access to some equipment (e.g. lasers, beamsplit- entangled. Alice denotes each of her possible set- ters, waveplates, photon detectors,...) which they tings as x = 0, 1,... and Bob denotes each of his can use to perform different experiments. A given possible settings as y = 0, 1,... Similarly Al- experiment consists of a choice of settings (e.g. ice and Bob denote the possible results of their laser intensity, angle of the waveplates, type of experiments by a = 0, 1,... and b = 0, 1,... beamsplitter,...) that after a run of the exper- After trying the different settings sufficiently iment provides a result (e.g. photon detection many times and collecting statistics, Alice and location, time of detection,...). Furthermore, a Bob can estimate the probabilities (also called the source is positioned between the laboratories and correlations) emits physical systems (e.g. photons) that are sent to Carmela’s and Deng’s laboratories; see p(a, b|x, y), (1) figure1, left. that is, the probabilities to see the results a and Suppose that Carmela and Deng would like to b given that the settings x and y are used.
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