| 1. |
- Andersson, Ole, et al.
(författare)
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Aligned SICs and embedded tight frames in even dimensions
- 2019
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Ingår i: Journal of Physics A. - : Institute of Physics (IOP). - 1751-8113 .- 1751-8121. ; 52:42
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Tidskriftsartikel (refereegranskat)abstract
- Alignment is a geometric relation between pairs of Weyl-Heisenberg SICs, one in dimension d and another in dimension d(d - 2), manifesting a well-founded conjecture about a number-theoretical connection between the SICs. In this paper, we prove that if d is even, the SIC in dimension d(d - 2) of an aligned pair can be partitioned into (d - 2)(2) tight d(2)-frames of rank d(d - 1)/2 and, alternatively, into d(2) tight (d - 2)(2) -frames of rank (d - 1) (d - 2)/2. The corresponding result for odd d is already known, but the proof for odd d relies on results which are not available for even d. We develop methods that allow us to overcome this issue. In addition, we provide a relatively detailed study of parity operators in the Clifford group, emphasizing differences in the theory of parity operators in even and odd dimensions and discussing consequences due to such differences. In a final section, we study implications of alignment for the symmetry of the SIC.
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| 2. |
- Andersson, Ole, et al.
(författare)
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Device-independent certification of two bits of randomness from one entangled bit and Gisin's elegant Bell inequality
- 2018
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Ingår i: Physical Review A. Atomic, Molecular, and Optical Physics. - : American Physical Society. - 1050-2947 .- 1094-1622. ; 97:1
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Tidskriftsartikel (refereegranskat)abstract
- We prove that as conjectured by Acín et al. [Phys. Rev. A 93, 040102(R) (2016)], two bits of randomness can be certified in a device-independent way from one bit of entanglement using the maximal quantum violation of Gisin's elegant Bell inequality. This suggests a surprising connection between maximal entanglement, complete sets of mutually unbiased bases, and elements of symmetric informationally complete positive operator-valued measures, on one side, and the optimal way of certifying maximal randomness, on the other.
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| 3. |
- Andersson, Ole, et al.
(författare)
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Self-testing properties of Gisin's elegant Bell inequality
- 2017
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Ingår i: Physical Review A: covering atomic, molecular, and optical physics and quantum information. - : American Physical Society. - 2469-9926 .- 2469-9934. ; 96:3
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Tidskriftsartikel (refereegranskat)abstract
- An experiment in which the Clauser-Horne-Shimony-Holt inequality is maximally violated is self-testing (i.e., it certifies in a device-independent way both the state and the measurements). We prove that an experiment maximally violating Gisin's elegant Bell inequality is not similarly self-testing. The reason can be traced back to the problem of distinguishing an operator from its complex conjugate. We provide a complete and explicit characterization of all scenarios in which the elegant Bell inequality is maximally violated. This enables us to see exactly how the problem plays out.
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| 4. |
- Appleby, Marcus, et al.
(författare)
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Dimension towers of SICs. I. Aligned SICs and embedded tight frames
- 2017
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 58:11
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Tidskriftsartikel (refereegranskat)abstract
- Algebraic number theory relates SIC-POVMsin dimension d > 3 to those in dimension d(d - 2). We define a SIC in dimension d(d - 2) to be aligned to a SIC in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d(d - 2) in a specifiedway. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension d, there exists an aligned SIC in dimension d(d - 2). In all our examples, the aligned SIC has lower dimensional equiangular tight frames embedded in it. If d is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If d - 2 is an odd prime number, we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.
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| 5. |
- Dumitru, Irina, 1989-
(författare)
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Exploring the Elegant Bell Inequality
- 2017
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In quantum information, device independent certification is a matter of both practical and fundamental interest. In this thesis, we explore the use of a particular Bell inequality, known as the elegant Bell inequality, in device independent certification. We first characterize all states and measurements that can lead to a maximal violation of the elegant Bell inequality. It turns out that, in all cases, the state involved in a maximal violation is a generalized singlet state, and the measurements of one of the parties are always maximally spread out on the Bloch sphere, forming a complete set of mutually unbiased bases. The measurements involved on the other party form two pairs of symmetric informationally complete vectors. The elegant Bell inequality, then, can be used to certify the presence of these elements.We also explore the usefulness of the elegant Bell inequality in randomness certification, in particular in a protocol for certification of maximal randomness from one entangled bit.The last part of this thesis is dedicated to a study of some of the special geometricstructures involved in the maximal violation of the elegant Bell inequality, namely the symmetric informationally complete vectors. The problem of the existence and of the construction of these sets of vectors in Hilbert spaces of any dimension is open, but there are solutions available in many dimensions. We look at these structures from both a geometric and an algebraic number theory perspective, and conjecture a relation between vectors in different dimensions. We introduce the relation of "alignment" between such sets of vectors.
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| 6. |
- Dumitru, Irina, 1989-
(författare)
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Studies in the Geometry of Quantum Measurements
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Quantum information studies quantum systems from the perspective of information theory: how much information can be stored in them, how much the information can be compressed, how it can be transmitted. Symmetric informationally-Complete POVMs are measurements that are well-suited for reading out the information in a system; they can be used to reconstruct the state of a quantum system without ambiguity and with minimum redundancy. It is not known whether such measurements can be constructed for systems of any finite dimension. Here, dimension refers to the dimension of the Hilbert space where the state of the system belongs.This thesis introduces the notion of alignment, a relation between a symmetric informationally-complete POVM in dimension d and one in dimension d(d-2), thus contributing towards the search for these measurements. Chapter 2 and the attached papers I and II also explore the geometric properties and symmetries of aligned symmetric informationally-complete POVMs.Chapter 3 and the attached papers III and IV look at an application of symmetric informationally-complete POVMs, the so-called Elegant Bell inequality. We use this inequality for device-independent quantum certification, the task of characterizing quantum scenarios without modelling the devices involved in these scenarios. Bell inequalities are functions that are bound in classical theories more tightly than in quantum theories, and can thus be used to probe whether a system is quantum. We characterize all scenarios in which the Elegant Bell inequality reaches its maximum quantum value. In addition, we show that this inequality can be used for randomness certification.
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