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Mutually Unbiased Bases and Symmetric Informationally Complete Measurements in Bell Experiments

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Mutually Unbiased Bases and Symmetric Informationally Complete Measurements in Bell Experiments Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments Armin Tavakoli,1 Máté Farkas,2, 3 Denis Rosset,4 Jean-Daniel Bancal,1 and J˛edrzejKaniewski5 1Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland 2Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland 3International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdansk, Poland 4Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada 5Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland (Dated: October 21, 2020) Mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) are crucial to many conceptual and practical aspects of quantum theory. Here, we develop their role in quantum nonlocality by: i) introducing families of Bell inequalities that are maximally violated by d-dimensional MUBs and SICs respectively, ii) proving device-independent certification of natural operational notions of MUBs and SICs, and iii) using MUBs and SICs to develop optimal-rate and nearly optimal-rate protocols for device independent quantum key distribution and device-independent quantum random number generation respectively. Moreover, we also present the first example of an extremal point of the quantum set of correlations which admits physically inequivalent quantum realisations. Our results elaborately demonstrate the foundational and practical relevance of the two most important discrete Hilbert space structures to the field of quantum nonlocality. One-sentence summary — Quantum nonlocality is de- a single measurement. Since MUBs and SICs are both con- veloped based on the two most celebrated discrete struc- ceptually natural, elegant and (as it turns out) practically im- tures in quantum theory. portant classes of measurements, they are often studied in the same context [9–14]. Let us briefly review their importance to foundational and applied aspects of quantum theory. I. INTRODUCTION MUBs are central to the concept of complementarity in quantum theory i.e. how the knowledge of one quantity Measurements are crucial and compelling processes at the limits (or erases) the knowledge of another quantity (see heart of quantum physics. Quantum measurements, in their e.g. Ref. [15] for a review of MUBs). This is often high- diverse shapes and forms, constitute the bridge between the lighted through variants of the famous Stern–Gerlach exper- abstract formulation of quantum theory and concrete data pro- iment in which different Pauli observables are applied to a duced in laboratories. Crucially, the quantum formalism of qubit. For instance, after first measuring (say) σx, we know measurement processes gives rise to experimental statistics whether our system points up or down the x-axis. If we then that elude classical models. Therefore, appropriate measure- measure σz, our knowledge of the outcome of yet another σx ments are indispensible for harvesting and revealing quantum measurement is entirely erased since σz and σx are MUBs. phenomena. Sophisticated manipulation of quantum measure- This phenomenon leads to a inherent uncertainty for the out- ments is both at the heart of the most well-known features of comes of MUB measurements on all quantum states, which quantum theory such as contextuality [1] and the violation of can be formalised in terms of entropic quantities, leading to Bell inequalities [2] as well as its most groundbreaking appli- so-called entropic uncertainty relations. It is then natural that cations such as quantum cryptography [3] and quantum com- MUBs give rise to the strongest entropic uncertainties in quan- putation [4]. In the broad landscape of quantum measure- tum theory [16]. Moreover, MUBs play a prominent role in ments [5], certain classes of measurements are outstanding quantum cryptography, where they are employed in many of due to their breadth of relevance in foundations of quantum the most well-known quantum key distribution protocols [17– arXiv:1912.03225v2 [quant-ph] 20 Oct 2020 theory and applications in quantum information processing. 21] as well as in secret sharing protocols [22–24]. Their ap- Two widely celebrated, intensively studied and broadly use- peal to cryptography stems from the idea that eavesdroppers ful classes of measurements are known as mutually unbiased who measure an eigenstate of one basis in another basis unbi- bases (MUBs) and symmetric informationally complete mea- ased to it obtain no useful information, while they also induce surements (SICs). Two measurements are said to be mutually a large disturbance in the state which allows their presence to unbiased if by preparing any eigenstate of the first measure- be detected. Furthermore, complete (i.e. largest possible in ment and then performing the second measurement, one finds a given dimension) sets of MUBs are tomographically com- that all outcomes are equally likely [6]. A typical example of plete and their symmetric properties make them pivotal for MUBs corresponds to measuring two perpendicular compo- quantum state tomography [25, 26]. In addition, MUBs are nents of the polarisation of a photon. A SIC is a quantum mea- useful for a range of other problems such as quantum random surement with the largest number of possible outcomes such access coding [27–31], quantum error correction [32, 33] and that all measurement operators have equal magnitude overlaps entanglement detection [34]. This broad scope of relevance [7,8]. Thus, the former is a relationship between two differ- has motivated much efforts towards determining the largest ent measurements whereas the latter is a relationship within number of MUBs that exist in general Hilbert space dimen- 2 sions [15]. two results of this type have been proven: (a) a triple of MUBs The motivations behind the study of SICs are quite simi- in dimension three [59] and (b) the measurements conjectured lar to the ones discussed for MUBs. It has been shown that to be optimal for the Collins–Gisin–Linden–Massar–Popescu SICs are natural measurements for quantum state tomogra- Bell inequality [60] (the former is a single result, while the lat- phy [35], which has also prompted several experimental real- ter is a family parameterised by the dimension d ≥ 2). None isations of SICs [36–38]. Also, some protocols for quantum of these results can be used to certify MUBs in dimension key distribution derive their success directly from the defin- d ≥ 4. ing properties of SICs [39, 40], which have also been exper- imentally demonstrated [41]. Furthermore, a key property of SICs is that they have the largest number of outcomes possi- Since mutual unbiasedness and symmetric informational ble while still being extremal measurements i.e. they cannot completeness are natural and broadly important concepts in be simulated by stochastically implementing other measure- quantum theory, they are prime candidates of interest for such ments. This gives SICs a central role in a range of applica- certification in general Hilbert space dimensions. This chal- tions which include random number generation from entan- lenge is increasingly relevant due to the broader experimental gled qubits [42], certification of non-projective measurements advances towards high-dimensional systems along the fron- [43–46], semi-device-independent self-testing [45] and entan- tier of quantum information theory. This is also reflected in glement detection [47, 48]. Moreover, SICs have a key role in the fact that recent experimental implementations of MUBs quantum Bayesianism [49] and they exhibit interesting con- and SICs can go well beyond the few lowest Hilbert space nections to several areas of mathematics, for instance Lie and dimensions [38, 41, 61]. Jordan algebras [50] and algebraic number theory [51]. Due to their broad interest, much research effort has been directed towards proving the existence of SICs in all Hilbert space di- Focusing on mutual unbiasedness and symmetric informa- mensions (presently known, at least, up to dimension 121) tional completeness, we solve the above challenges. To this [7,8, 52, 53]. See e.g. Ref. [54] for a recent review of SICs. end, we first construct Bell inequalities that are maximally In this work, we broadly investigate MUBs and SICs in violated using a maximally entangled state of local dimen- the context of Bell nonlocality experiments. In these experi- sion d and, respectively, a pair of d-dimensional MUBs and ments, two separated observers perform measurements on en- a d-dimensional SIC. In the case of MUBs, we show that the tangled quantum systems which can produce nonlocal corre- maximal quantum violation of the proposed Bell inequality lations that elude any local hidden variable model [55]. In device-independently certifies that the measurements satisfy recent years, Bell inequalities have played a key role in the an operational definition of mutual unbiasedness, and also that rise of device-independent quantum information processing the shared state is essentially a maximally entangled state of where they are used to certify properties of quantum systems. local dimension d. Similarly, in the case of SICs, we find Naturally, certification of a physical property can be achieved that the maximal quantum violation device-independently cer- under different assumptions of varying strength. Device- tifies that the measurements satisfy an analogous
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