                    Title    Mutually unbiased bases (MUBs)   Author    Christoph Spengler
 Organization: University of Innsbruck
 Department: Theoretical Physics   Revision date    2013-10-04   Description    Given the coefficients c={c(0),...,c(n-1)} of a (monic) irreducible polynomial of degree n over Z_p, where p is a prime number, this file automatically generates a complete set of 'Mutually Unbiased Bases' (MUBs). This is done using the construction presented in the paper  C. Spengler and B. Kraus, 'A graph state formalism for mutually unbiased bases' (http://arxiv.org/abs/1309.6557).

The file contains 5 functions: -QMatrix[c,p] is the implementation of the algorithm from  which symmetrizes the companion matrix C corresponding to the polynomial f(x) with coefficients c={c(0),...,c(n-1)}, i.e. yields Q=P C P^(-1) where Q=Q^T. -AdjMatrix[a,Q,p] yields a linear combination of the powers of the symmetric matrix Q with the coefficients a={a(0),...,a(n-1)} modulo p. Here, each resulting matrix constitutes an adjacency matrix of a generalized multi-graph. -Graphstatebasis[A,p] yields a graph state basis for the adjacency matrix A over Z_p, as defined in . For the p^n different settings a={a(0),...,a(n-1)} of A=AdjMatrix[a,Q,p], these bases together with the computational basis give rise to a complete set of p^n+1 MUBs for d=p^n. -MUBCheck[Bx,By] can be used to check if a pair of bases, Bx any By, are mutually unbiased. -AdjCheck[Ax,Ay,p] can be used to check if a pair of adjacency matrices, Ax and Ay, satisfy the MUB condition [det(Ax-Ay) mod p] unequal 0.   Subject     Science > Physics > Quantum Physics   Keywords    mutually unbiased bases, complementarity, quantum information, quantum mechanics, quantum physics, finite fields, graph states, quantum state tomography, entanglement, 2-design   Downloads     MUBS.nb (320.9 KB) - Mathematica Notebook       