# Rigidity and a common framework for mutually unbiased bases and k ‐nets

@article{Nietert2019RigidityAA, title={Rigidity and a common framework for mutually unbiased bases and k ‐nets}, author={Sloan Nietert and Zsombor Szil'agyi and Mih{\'a}ly Weiner}, journal={Journal of Combinatorial Designs}, year={2019}, volume={28}, pages={869 - 892} }

Many deep connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called k ‐nets (and in particular, between collections of MUBs and finite affine planes). Here we introduce the notion of a k ‐net over a C * ‐algebra, providing a common framework for both objects. In the commutative case, we recover (classical) k ‐nets, while the choice of M d ( C ) leads to collections of MUBs. In this framework, we derive a rigidity property which hence…

## One Citation

### On a conjecture regarding quantum hypothesis testing.

- Mathematics
- 2020

In this MSc thesis I consider the asymptotic behaviour of the symmetric error in composite hypothesis testing. In the classical case, when the null and alternative hypothesis are finite sets of…

## References

SHOWING 1-10 OF 25 REFERENCES

### A gap for the maximum number of mutually unbiased bases

- Mathematics
- 2009

A collection of (pairwise) mutually unbiased bases (in short: MUB) in d > 1 dimensions may consist of at most d + 1 bases. Such “complete” collections are known to exists in C when d is a power of a…

### New construction of mutually unbiased bases in square dimensions

- Mathematics, Computer ScienceQuantum Inf. Comput.
- 2005

The construction combines the design-theoretic objects (s, k)-nets and generalized Hadamard matrices of size s to show that k = w + 2 mutually unbiased bases can be constructed in any square dimension d = s2 provided that there are w mutually orthogonal Latin squares of order s.

### The limitations of nice mutually unbiased bases

- Mathematics, Computer Science
- 2004

It is shown that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches.

### A New Proof for the Existence of Mutually Unbiased Bases

- Mathematics, Computer ScienceAlgorithmica
- 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.

### Quantum Measurements and Finite Geometry

- Mathematics, Physics
- 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…

### On the monomiality of nice error bases

- Computer ScienceIEEE Transactions on Information Theory
- 2005

It is found that nice error bases have more structure than one can anticipate from their definition and can be written in a form in which at least half of the matrix entries are 0.

### On discrete structures in finite Hilbert spaces

- Mathematics
- 2017

We present a brief review of discrete structures in a finite Hilbert space, relevant for the theory of quantum information. Unitary operator bases, mutually unbiased bases, Clifford group and…