#### Document Type

Article

#### Publication Date

7-2-2004

#### Publication Title

Physical Review A - Atomic, Molecular, and Optical Physics

#### Abstract

A complete orthonormal basis of N-qutrit unitary operators drawn from the Pauli Group consists of the identity and 9^N-1 traceless operators. The traceless ones partition into 3^N+1 maximally commuting subsets (MCS's) of 3^N-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 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.

#### DOI

10.1103/PhysRevA.70.012302

#### Recommended Citation

Lawrence, Jay, "Mutually Unbiased Bases and Trinary Operator Sets for N Qutrits" (2004). *Open Dartmouth: Faculty Open Access Articles*. 2961.

https://digitalcommons.dartmouth.edu/facoa/2961