Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six

Abstract

We study the problem of constructing mutually unbiased bases in dimension six. This approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions. Our technique suggests the existence of previously unknown symmetries in Karlsson's non-affine family $K_6^{(2)}$ which we confirm analytically. Also, we obtain strong evidence that no more than three mutually unbiased bases can be constructed from pairs which contain members of some non-affine families of complex Hadamard matrices.

arXiv commentImproved version, published in JPA. 22 pages, 6 figures

References in corpus (19)

• Security of quantum key distribution using d-level systems
• Optimal eavesdropping in quantum cryptography with six states
• On mutually unbiased bases
• A new proof for the existence of mutually unbiased bases
• Discrete phase space based on finite fields
• Entanglement detection via mutually unbiased bases
• Mubs and Hadamards of Order Six
• Constructing Mutually Unbiased Bases in Dimension Six
• Three ways to look at mutually unbiased bases
• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
• Numerical evidence for the maximum number of mutually unbiased bases in dimension six
• The mean king's problem: Spin 1
• On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six
• Quantum mechanics in finite dimensional Hilbert space
• Mutually Unbiased Bases and Semi-definite Programming
• Two-parameter complex Hadamard matrices for N=6
• MUBs, polytopes, and finite geometries
• State determination: an iterative algorithm

Cited by in corpus (13)

• Test of mutually unbiased bases for six-dimensional photonic quantum systems
• Product states and Schmidt rank of mutually unbiased bases in dimension six
• Small sets of complementary observables
• States that "look the same" with respect to every basis in a mutually unbiased set
• Mutually unbiased bases with free parameters
• Quantum measurements with prescribed symmetry
• A family of quantum graph vertex couplings interpolating between different symmetries
• Bloch sphere analog of qudits using Heisenberg-Weyl Operators
• Quantum tomography meets dynamical systems and bifurcations theory
• Comment on 'Product states and Schmidt rank of mutually unbiased bases in dimension six'
• A characterization of complex Hadamard matrices appearing in families of MUB triplets
• Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three
• Generalized Numerical Construction of MUBs: A Group Theoretical Investigation