Maximal Sets of Mutually Unbiased Quantum States in Dimension Six

Abstract

We study sets of pure states in a Hilbert space of dimension d which are mutually unbiased (MU), that is, the squares of the moduli of their scalar products are equal to zero, one, or 1/d. These sets will be called a MU constellation, and if four MU bases were to exist for d=6, they would give rise to 35 different MU constellations. Using a numerical minimisation procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension six.

19 pages, 6 figures, 4 tables

References in corpus (4)

• On SIC-POVMs and MUBs in Dimension 6
• Mubs and Hadamards of Order Six
• Numerical evidence for the maximum number of mutually unbiased bases in dimension six
• A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

Cited by in corpus (53)

• On mutually unbiased bases
• Entanglement detection via mutually unbiased bases
• Constructing Mutually Unbiased Bases in Dimension Six
• Mutually unbiased measurements in finite dimensions
• Entanglement in mutually unbiased bases
• Quantifying measurement incompatibility of mutually unbiased bases
• A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
• Mutually unbiased bases in dimension six: The four most distant bases
• Entanglement detection using mutually unbiased measurements
• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Informationally complete joint measurements on finite quantum systems
• Unextendible maximally entangled bases and mutually unbiased bases
• Test of mutually unbiased bases for six-dimensional photonic quantum systems
• Mutually Unbiased Bases for Continuous Variables
• Linking Entanglement Detection and State Tomography via Quantum 2-Designs
• Quantum Conical Designs
• The Limited Role of Mutually Unbiased Product Bases in Dimension Six
• On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six
• Mutually Unbiased Bases and Semi-definite Programming
• Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
• Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six
• Entanglement Patterns in Mutually Unbiased Basis Sets for N Prime-state Particles
• Graph-state formalism for mutually unbiased bases
• Mutually unbiased bases in dimension six containing a product-vector basis
• Mutually Unbiased Product Bases for Multiple Qudits
• A Note on Mutually Unbiased Unextendible Maximally Entangled Bases in ${\mathbb{C}}^2⨂{\mathbb{C}}^3$
• Mutually unbiased measurement based entanglement witnesses
• Quantum Key Distribution Highly Sensitive to Eavesdropping
• Product states and Schmidt rank of mutually unbiased bases in dimension six
• Optimal reconstruction of states in qutrits system
• Weak mutually unbiased bases
• Orbits of mutually unbiased bases
• States that "look the same" with respect to every basis in a mutually unbiased set
• Communication capacity of mixed quantum t designs
• Structure of the sets of mutually unbiased bases with cyclic symmetry
• An improvement on the Delsarte-type LP-bound with application to MUBs
• Mutually unbiased bases with free parameters
• Affine Constellations Without Mutually Unbiased Counterparts
• Real and complex unbiased Hadamard matrices
• Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories
• Least-bias state estimation with incomplete unbiased measurements
• Practical implementation of mutually unbiased bases using quantum circuits
• Constructions of mutually unbiased entangled bases
• Choice of mutually unbiased bases and outcome labelling affects measurement outcome secrecy
• A rigidity property of complete systems of mutually unbiased bases
• Quantum state reconstruction from dynamical systems theory
• Real Entries of Complex Hadamard Matrices and Mutually Unbiased Bases in Dimension Six
• The problem of mutually unbiased bases in dimension 6
• How much complementarity?
• Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three
• Generalized Numerical Construction of MUBs: A Group Theoretical Investigation
• Some special complex Hadamard matrices of order six
• $H_2$-reducible matrices in six-dimensional mutually unbiased bases