The limitations of nice mutually unbiased bases

@article{Aschbacher2004TheLO,
  title={The limitations of nice mutually unbiased bases},
  author={Michael Aschbacher and Andrew M. Childs and Pawe l Wocjan},
  journal={Journal of Algebraic Combinatorics},
  year={2004},
  volume={25},
  pages={111-123}
}
Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show 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. We prove this by establishing a correspondence between nice mutually unbiased bases and abelian… 
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References

SHOWING 1-10 OF 29 REFERENCES
A New Proof for the Existence of Mutually Unbiased Bases
TLDR
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.
Constructions of Mutually Unbiased Bases
TLDR
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.
On the monomiality of nice error bases
TLDR
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.
There is no generalization of known formulas for mutually unbiased bases
In a quantum system having a finite number N of orthogonal states, two orthonormal bases {ai} and {bj} are called mutually unbiased if all inner products ⟨ai∣bj⟩ have the same modulus 1∕N. This
New construction of mutually unbiased bases in square dimensions
TLDR
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.
Beyond stabilizer codes I: Nice error bases
Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases
Optimal state-determination by mutually unbiased measurements
Non-binary unitary error bases and quantum codes
Error operator bases for systems of any dimension are defined and natural generalizations of the bit-flip/ sign-change error basis for qubits are given. These bases allow generalizing the
All teleportation and dense coding schemes
We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary
Discrete phase space based on finite fields
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
...
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