[seqfan] Help wanted: A counting problem concerning the generalized Pauli group

[Simone Severini]

Dear Seqfans,


I would like to propose a counting problem. It requires little
background definitions. I shall then include a link.

The problem is related to the generalized Pauli group. This is
sometimes called Pauli group on n-qubits.

Please see:

http://www.quantiki.org/wiki/index.php/Pauli_group

The Pauli group on n-qubits has order 4^n.

Disregarding the identity element, we partition the group into 2^n+1
sets of 2^n-1 elements each.

We care that each set forms an abelian subgroup.

Problem: In how many different ways can we partition the Pauli group
on n qubits, such that the obtained partitions satisfy the above
constraint?

Numerics have shown the following:

for n = 2, there are 6 ways;

for n = 3, there are 960 ways.

I hope that I stated the problem clearly. Please email me if you are
interested and you find my statements unclear.

Together with Ernesto Galvao and Francesco Russo, we have worked on
this problem for sometime without any evident success.

The problem has some relevance in the context of quantum information
theory. A solution is potentially useful to approach some further
problems or a more physical nature.

See, e.g., [J. Lawrence, Č. Brukner, A. Zeilinger, Mutually unbiased
binary observable sets on N qubits, Phys. Rev. A 65, 32320 (2002)].

If any of you has time to give a look, I would be very happy to
discuss the problem together.

Thanks a lot for your time and your usual kindness.


Best,
Simone
-- 
Simone Severini
http://www.iqc.ca/~sseverin