Symmetric informationally complete quantum measurements

Renes, Joseph M.; Blume-Kohout, Robin; Scott, A. J.; Caves, Carlton M.

doi:10.1063/1.1737053

Cited by 949 publications

(1,203 citation statements)

References 22 publications

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“…The natural choice is to let E i = (1/2)P i , where the one-dimensional projectors P i correspond to four tetrahedrally related points on the Bloch sphere. The operators E i in this case constitute what is called a symmetric informationally complete POVM, or SIC POVM [15].…”

Section: Symmetric Informationally Complete Positiveoperator-valued Mmentioning

confidence: 99%

Quantum Measurements and Finite Geometry

2006

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A complete set of mutually unbiased bases for a Hilbert space of dimension N is analogous in some respects to a certain finite geometric structure, namely, an affine plane. Another kind of quantum measurement, known as a symmetric informationally complete positive-operator-valued measure, is, remarkably, also analogous to an affine plane, but with the roles of points and lines interchanged. In this paper I present these analogies and ask whether they shed any light on the existence or non-existence of such symmetric quantum measurements for a general quantum system with a finite-dimensional state space.

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Section: Symmetric Informationally Complete Positiveoperator-valued Mmentioning

confidence: 99%

Quantum Measurements and Finite Geometry

2006

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“…Bases such that the angles between arbitrary pairs of elements of different bases are all equal are known as MU bases (MUBs) [32]; a set of d+1 MUBs of a system of d levels is informationally complete [33]. When the Hilbert-Schmidt inner products between every pair of different operators of an IC-POVM are all equal, this POVM is a SIC-POVM [34]. Despite the lack of a formal proof, it is widely believed that SIC-POVMs exist in any Hilbert space of finite dimension [35].…”

Section: Quantum Tomographymentioning

confidence: 99%

Single plane minimal tomography of double slit qubits

Sogamoso¹,

Angulo²,

Romero³

2017

Optics Communications

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The determination of the density matrix of an ensemble of identically prepared quantum systems by performing a series of measurements, known as quantum tomography, is minimal when the number of outcomes is minimal.The most accurate minimal quantum tomography of qubits, sometimes called a tetrahedron measurement, corresponds to projections over four states which can be represented on the Bloch sphere as the vertices of a regular tetrahedron. We investigate whether it is possible to implement the tetrahedron measurement of double slit qubits of light, using measurements performed on a single plane. Assuming Gaussian slits and free propagation, we demonstrate that a judicious choice of the detection plane and the double slit geometry allows the implementation of a tetrahedron measurement. Finally, we consider possible sets of values which could be used in actual experiments.

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“…Symmetric informationally-complete positive operator-valued measures (SICs) [1,2] represent a general form of measurement in quantum theory. As the more familiar projective measurements are associated to an orthonormal basis in Hilbert space, a SIC is associated to an over-complete set of N 2 unit vectors in H N such that the absolute value of the scalar product between any distinct two is always constant, i.e.…”

Section: Introductionmentioning

confidence: 99%

Linear dependencies in Weyl–Heisenberg orbits

Dang

Blanchfield

Bengtsson

et al. 2013

Quantum Inf Process

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Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl-Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.

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Symmetric informationally complete quantum measurements

Cited by 949 publications

References 22 publications

Quantum Measurements and Finite Geometry

Quantum Measurements and Finite Geometry

Single plane minimal tomography of double slit qubits

Linear dependencies in Weyl–Heisenberg orbits

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