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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 [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(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 [1]. 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
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MUBS.nb (320.9 KB) - Mathematica Notebook