Single Qubit Private Quantum Channels and 3-Dimensional Regular Polyhedra

Jeong, Kabgyun; Lee, Jun Seo; Choi, Jin; Hong, Seok Min; Jung, M.; Kim, Gyeong Beom; Kim, Jae Kwon; Kim, Suntaek

doi:10.3938/npsm.68.232

Cited by 4 publications

(6 citation statements)

References 8 publications

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“…In Ref. [13], different types of single qubit private quantum channels have been constructed and linked to threedimensional Platonic solids. In a more recent work [14] four-dimensional Platonic solids were connected to qutrit-based private quantum channels.…”

Section: Platonic Solids In Quantum Physicsmentioning

confidence: 99%

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Platonic Bell inequalities for all dimensions

Pál

Vértesi

2022

Quantum

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In this paper we study the Platonic Bell inequalities for all possible dimensions. There are five Platonic solids in three dimensions, but there are also solids with Platonic properties (also known as regular polyhedra) in four and higher dimensions. The concept of Platonic Bell inequalities in the three-dimensional Euclidean space was introduced by Tavakoli and Gisin [Quantum 4, 293 (2020)]. For any three-dimensional Platonic solid, an arrangement of projective measurements is associated where the measurement directions point toward the vertices of the solids. For the higher dimensional regular polyhedra, we use the correspondence of the vertices to the measurements in the abstract Tsirelson space. We give a remarkably simple formula for the quantum violation of all the Platonic Bell inequalities, which we prove to attain the maximum possible quantum violation of the Bell inequalities, i.e. the Tsirelson bound. To construct Bell inequalities with a large number of settings, it is crucial to compute the local bound efficiently. In general, the computation time required to compute the local bound grows exponentially with the number of measurement settings. We find a method to compute the local bound exactly for any bipartite two-outcome Bell inequality, where the dependence becomes polynomial whose degree is the rank of the Bell matrix. To show that this algorithm can be used in practice, we compute the local bound of a 300-setting Platonic Bell inequality based on the halved dodecaplex. In addition, we use a diagonal modification of the original Platonic Bell matrix to increase the ratio of quantum to local bound. In this way, we obtain a four-dimensional 60-setting Platonic Bell inequality based on the halved tetraplex for which the quantum violation exceeds the 2 ratio.

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Section: Platonic Solids In Quantum Physicsmentioning

confidence: 99%

“…In this case, one can show that the local bound L above is obtained with a deterministic strategy, where a m A +1 b m B +1 = 1. Then the local bound L in (12) can be obtained by a simple subtraction from the local bound L in (13):…”

Section: The Local Boundmentioning

confidence: 99%

Platonic Bell inequalities for all dimensions

Pál

Vértesi

2022

Quantum

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show abstract

“…In Ref. [13], different types of single qubit private quantum channels have been constructed and linked to three-dimensional Platonic solids. In a more recent work [14] four-dimensional Platonic solids were connected to qutrit-based private quantum channels.…”

Section: Platonic Solids In Quantum Physicsmentioning

confidence: 99%

Platonic Bell inequalities for all dimensions

Pál,

Vértesi

2021

Preprint

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“…To this end, we need to exploit the notion of an isotropic (or unitarily invariant) measure on the unitary group, and two modifications of the Gell-Mann matrix [19] and quantum Fourier transform [20]. In this study, we highlight a new connection between qutrit-based PQCs and the regular 4-polytope by generalizing our previous research on qubit-based PQCs and regular polyhedra [21] equipped with an isotropic measure.…”

Section: Introductionmentioning

confidence: 99%

High-dimensional Private Quantum Channels and Regular Polytopes

Lee

Jeong

2021

Comm. in Phys.

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As the quantum analog of the classical one-time pad, the private quantum channel (PQC) plays a fundamental role in the construction of the maximally mixed state (from any input quantum state), which is very useful for studying secure quantum communications and quantum channel capacity problems. However, the undoubted existence of a relation between the geometric shape of regular polytopes and private quantum channels in the higher dimension has not yet been reported. Recently, it was shown that a one-to-one correspondence exists between single-qubit PQCs and three-dimensional regular polytopes (i.e., regular polyhedra). In this paper, we highlight these connections by exploiting two strategies known as a generalized Gell-Mann matrix and modified quantum Fourier transform. More precisely, we explore the explicit relationship between PQCs over a qutrit system (i.e., a three-level quantum state) and regular 4-polytopes. Finally, we attempt to devise a formula for such connections on higher dimensional cases.

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Single Qubit Private Quantum Channels and 3-Dimensional Regular Polyhedra

Cited by 4 publications

References 8 publications

Platonic Bell inequalities for all dimensions

Platonic Bell inequalities for all dimensions

Platonic Bell inequalities for all dimensions

High-dimensional Private Quantum Channels and Regular Polytopes

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