# A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

@article{Planat2004ASO, title={A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements}, author={M. Planat and H. Rosu and Serge Perrine}, journal={Foundations of Physics}, year={2004}, volume={36}, pages={1662-1680} }

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite, and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.

#### 42 Citations

Galois algebras of squeezed quantum phase states

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Coding, transmission and recovery of quantum states with high security and efficiency, and with as low fluctuations as possible, is the main goal of quantum information protocols and their proper… Expand

MUBs: From Finite Projective Geometry to Quantum Phase Enciphering

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This short note highlights the most prominent mathematical problems and physical questions associated with the existence of the maximum sets of mutually unbiased bases (MUBs) in the Hilbert space of… Expand

Weighted complex projective 2-designs from bases : Optimal state determination by orthogonal measurements

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We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then… Expand

On group theory for quantum gates and quantum coherence

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Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates… Expand

Viewing sets of mutually unbiased bases as arcs in finite projective planes

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Abstract This note is a short conceptual elaboration of the conjecture of Saniga et al. [J. Opt. B: Quantum Semiclass 6 (2004) L19–L20] by regarding a set of mutually unbiased bases (MUBs) in a d… Expand

Constructing Mutually Unbiased Bases in Dimension Six

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The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist… Expand

Mutually Unbiased Product Bases

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A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are… Expand

Test of mutually unbiased bases for six-dimensional photonic quantum systems

- Physics, Medicine
- Scientific reports
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This work implements and test different sets of three MUBs for a single photon six-dimensional quantum state (a “qusix”) encoded exploiting polarization and orbital angular momentum of photons by exploiting a newly developed holographic technique. Expand

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A comprehensive theory of phase for finite-dimensional quantum systems is developed. The only physical requirement imposed is that phase is complementary to amplitude. This complementarity is… Expand

Quantifying Measurement Incompatibility of Mutually Unbiased Bases.

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This work quantifies precisely the degree of incompatibility of mutually unbiased bases (MUB) using the notion of noise robustness using the standard construction for d being a prime power, and provides upper and lower bounds on this quantity for sets of k MUB in dimension d. Expand

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