








Mutually unbiased bases (MUBs)






Organization:  University of Innsbruck 
Department:  Theoretical Physics 






20131004






Given the coefficients c={c(0),...,c(n1)} 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 [1] 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 [1] which symmetrizes the companion matrix C corresponding to the polynomial f(x) with coefficients c={c(0),...,c(n1)}, 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(n1)} modulo p. Here, each resulting matrix constitutes an adjacency matrix of a generalized multigraph. Graphstatebasis[A,p] yields a graph state basis for the adjacency matrix A over Z_p, as defined in [1]. For the p^n different settings a={a(0),...,a(n1)} 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(AxAy) mod p] unequal 0.












mutually unbiased bases, complementarity, quantum information, quantum mechanics, quantum physics, finite fields, graph states, quantum state tomography, entanglement, 2design






 MUBS.nb (320.9 KB)  Mathematica Notebook 







   
 
