# On SIC-POVMs and MUBs in Dimension 6

@article{Grassl2004OnSA, title={On SIC-POVMs and MUBs in Dimension 6}, author={M. Grassl}, journal={arXiv: Quantum Physics}, year={2004} }

We provide a partial solution to the problem of constructing mutually unbiased bases (MUBs) and symmetric informationally complete POVMs (SIC-POVMs) in non-prime-power dimensions.
An algebraic description of a SIC-POVM in dimension six is given. Furthermore it is shown that several sets of three mutually unbiased bases in dimension six are maximal, i.e., cannot be extended.

#### Figures from this paper

#### 142 Citations

On PSI-complete and PSIR-complete measurements

- Physics
- 2004

I construct a POVM which has 2d rank-one elements and which is informationally complete for generic pure states in d dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves… Expand

An operational link between MUBs and SICs

- Physics
- 2013

The existence of symmetric informationally complete positive operator valued measures (SIC POVMs, or simply SICs) in arbitrary dimensions and the existence of complete systems of d + 1 mutually… Expand

Two new constructions of approximately SIC-POVMs from multiplicative characters

- Mathematics, Computer Science
- Quantum Inf. Process.
- 2017

This paper proposes two new constructions of ASIC-POVMs for prime power dimensions only by using multiplicative characters over finite fields. Expand

Maximal sets of mutually unbiased quantum states in dimension 6

- Physics
- 2008

We study sets of pure states in a Hilbert space of dimension $d$ which are mutually unbiased (MU), that is, the moduli of their scalar products are equal to zero, one, or $1∕\sqrt{d}$. Each of these… Expand

On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states

- Mathematics, Physics
- 2005

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform. This is… Expand

SIC-POVMS AND THE STARK CONJECTURES

- 2018

The existence of a set of d pairwise equiangular complex lines (a SIC-POVM) in ddimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if there exists a… Expand

Constructing Mutually Unbiased Bases in Dimension Six

- Physics
- 2009

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

Pure-state informationally complete and "really" complete measurements (3 pages)

- Physics
- 2004

I construct a positive-operator-valued measure (POVM) which has 2d rank-1 elements and which is informationally complete for generic pure states in d dimensions, thus confirming a conjecture made by… Expand

SIC-POVMs and the Stark conjectures

- Mathematics, Physics
- 2018

The existence of a set of d^2 pairwise equiangular complex lines (equivalently, a SIC-POVM) in d-dimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if… Expand

Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6

- Physics, Mathematics
- 2013

We study the problem of constructing mutually unbiased bases in dimension 6. This approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction… Expand

#### References

SHOWING 1-10 OF 11 REFERENCES

Symmetric informationally complete quantum measurements

- Mathematics, Physics
- 2003

It is conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim. Expand

A New Proof for the Existence of Mutually Unbiased Bases

- Mathematics, Computer Science
- Algorithmica
- 2002

A constructive proof of the existence of mutually biased bases for dimensions that are powers of primes is presented and it is proved that in any dimension d the number of mutually unbiased bases is at most d+1. Expand

Quantum Measurements and Finite Geometry

- Physics, Mathematics
- 2004

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… Expand

Constructions of Mutually Unbiased Bases

- Mathematics, Computer Science
- International Conference on Finite Fields and Applications
- 2003

This work gives a simplified proof of this fact based on the estimation of exponential sums that extremal sets containing d+1 mutually unbiased bases are known to exist. Expand

Discrete phase space based on finite fields

- Physics
- 2004

The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete… Expand

The Magma Algebra System I: The User Language

- Computer Science, Mathematics
- J. Symb. Comput.
- 1997

The MAGMA language is presented, the design principles and theoretical background are outlined, and the constructors for structures, maps, and sets are outlined. Expand

Elements of the Representation Theory of the Jacobi Group

- Mathematics
- 1998

The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group. It is an important example for a non-reductive group and sets the frame within which to treat theta functions… Expand

Leprobì eme de 36 officiers, Comptes Rendues de l'Assoc

- Français Avanc. Sci. Naturel
- 1900