Mutually unbiased bases and trinary operator sets for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>qutrits

Abstract

A compete orthonormal basis of $N$-qutrit unitary operators drawn from the Pauli group consists of the identity and ${9}^{N}\ensuremath{-}1$ traceless operators. The traceless ones partition into ${3}^{N}+1$ maximally commuting subsets (MCS's) of ${3}^{N}\ensuremath{-}1$ operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order ${3}^{N}$ are isomorphic to all MCS's and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like). For three qutrits, 728 operators partition into 28 MCS's with less rigid structure, allowing for the coexistence of separable, partially entangled, and totally entangled (GHZ-like) bases. However a minimum of $16\phantom{\rule{0.3em}{0ex}}\mathrm{GHZ}$-like bases must occur. Every basis state is described by an $N$-digit trinary number consisting of the eigenvalues of $N$ observables constructed from the corresponding MCS.