Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments

Tavakoli, Armin; Farkas, Máté; Rosset, Denis; Bancal, Jean-Daniel; Kaniewski, Jędrzej

doi:10.48550/arxiv.1912.03225

Cited by 5 publications

(6 citation statements)

References 73 publications

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“…The aim of this definition is to convey the most important property of the original MUBs, namely, that localization with respect to one set implies even spreading from the perspective of all other sets. We note in passing that this definition has recently been utilized for device independent applications [34].…”

Section: Mutual Unbiasednessmentioning

confidence: 99%

Periodic discretized continuous observables are neither continuous nor discrete

Silva,

Rudnicki,

Tasca

et al. 2020

Preprint

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Continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number "d" of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of mutual unbiasedness to show that such coarse-grained observables are of a different kind, being in a sense neither continuous nor discrete. In particular, we study periodic binning of continuous observables and prove that the maximum number of mutually unbiased measurements of this type equals three for even d, which is analogous to the continuous case. On the other hand, for odd d we can find more mutually unbiased measurements, though the known results for discrete systems are not entirely reproduced. However, we can show that there are never more than d + 1 in general, which is a known feature of discrete systems. To illustrate and explore these results, we present an example for construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with d = 3 outcomes in a continuous variable system.

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Section: Mutual Unbiasednessmentioning

confidence: 99%

Periodic discretized continuous observables are neither continuous nor discrete

Silva,

Rudnicki,

Tasca

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…These interesting and highly symmetric structures have broad applications in quantum information science. Examples include quantum tomography [26,27], quantum key distribution [28,29], Bell inequalities [30], entropic uncertainty relations [31] and entanglement detection [32,33]. They also accommodate (as special cases) some of the most intensely researched and celebrated discrete Hilbert space structures such as rank-one generalised measurements, complete sets of mutually unbiased bases [34] and symmetric informationally complete sets of states [35], as well as the Platonic solids [36].…”

mentioning

confidence: 99%

Semi-device-independent certification of independent quantum state and measurement devices

Tavakoli

2020

Preprint

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Certifying that quantum devices behave as intended is crucial for quantum information science. Here, methods are developed for certification of both state preparation devices and measurement devices based on prepareand-measure experiments with independent devices. The experimenter is only assumed to know the Hilbert space dimension and thus no precise characterisation of any part of the experiment is required. The certification is based on a randomised version of unambiguous state discrimination and targets the broad class of state ensembles corresponding to quantum t-designs of any size and any dimension. These quantum designs are sets of states over which the average of any t-degree polynomial equals its average over all pure states -and they accommodate many of the most useful discrete structures in quantum information processing. Furthermore, it is shown that the same experiments also certify the detection efficiency of the measurement devices, as well as their non-projective nature. The presented methods can readily be implemented in experiments.

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“…SIC-POVMs are key tools for quantum state tomography [11][12][13], which has motivated their experimental realisation in highdimensional Hilbert spaces [14][15][16]. Generally, SICs and SIC-POVMs are used in a range of protocols: quantum key distribution (QKD) [17][18][19], entanglement detection [20][21][22], device-independent random number generation [23,24], dimension witnessing [25] and characterisation of quantum de-vices [26][27][28][29][30]. Moreover, SICs have been studied in the context of quantum nonlocality [24,[31][32][33] and they have an interesting foundational role in QBism [34].…”

mentioning

confidence: 99%

“…Generally, SICs and SIC-POVMs are used in a range of protocols: quantum key distribution (QKD) [17][18][19], entanglement detection [20][21][22], device-independent random number generation [23,24], dimension witnessing [25] and characterisation of quantum de-vices [26][27][28][29][30]. Moreover, SICs have been studied in the context of quantum nonlocality [24,[31][32][33] and they have an interesting foundational role in QBism [34]. All this has triggered much interest in addressing the existence of SICs in general Hilbert space dimensions.…”

mentioning

confidence: 99%

Compounds of symmetric informationally complete measurements and their application in quantum key distribution

Tavakoli,

Bengtsson,

Gisin

et al. 2020

Preprint

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Symmetric informationally complete measurements (SICs) are elegant, celebrated and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC-compound is defined to be a collection of d 3 vectors in d-dimensional Hilbert space that can be partitioned in two different ways: into d SICs and into d 2 orthonormal bases. While a priori their existence may appear unlikely when d > 2, we surprisingly answer it in the positive through an explicit construction for d = 4. Remarkably this SIC-compound admits a close relation to mutually unbiased bases, as is revealed through quantum state discrimination. Going beyond fundamental considerations, we leverage these exotic properties to construct a protocol for quantum key distribution and analyze its security under general eavesdropping attacks. We show that SIC-compounds enable secure key generation in the presence of errors that are large enough to prevent the success of the generalisation of the six-state protocol.

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Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments

Cited by 5 publications

References 73 publications

Periodic discretized continuous observables are neither continuous nor discrete

Periodic discretized continuous observables are neither continuous nor discrete

Semi-device-independent certification of independent quantum state and measurement devices

Compounds of symmetric informationally complete measurements and their application in quantum key distribution

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