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Journal of Physics A Mathematical and Theoretical

Generation of bases with definite factorization for ann-qubit system and mutually unbiased sets construction

Andrés García, José Luis Romero, A. B. Klimov

Published Aug 4, 2010• 9 citations
Publisher

Abstract

We propose a systematic procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n -qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.

DOI
10.1088/1751-8113/43/38/385301

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We propose a systematic procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n -qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.
We define mutually complementary observable sets for N qubits via the operational requirement that a state with a definite outcome for one set of (commuting) binary observables must give completely random results in all other sets. The bases formed by the eigenvectors of such complementary sets are mutually unbiased. We prove that the full set of 4^N-1 Pauli operator products may be partitioned into 2^N+1 distinct sets, each set consisting of 2^N-1 internally commuting observables. Furthermore we prove that each such partitioning defines a unique choice of 2^N+1 mutually unbiased bases. Examples for 2 and 3 qubit systems are discussed with emphasis on the nature and amount of entanglement that occurs within these basis sets.
A compete orthonormal basis of NNN-qutrit unitary operators drawn from the Pauli group consists of the identity and ${9}^{N}\ensuremath{-}1$ traceless operators. The traceless ones partition into 3N+1{3}^{N}+13N+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 3N{3}^{N}3N 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 16GHZ16\phantom{\rule{0.3em}{0ex}}\mathrm{GHZ}16GHZ-like bases must occur. Every basis state is described by an NNN-digit trinary number consisting of the eigenvalues of NNN observables constructed from the corresponding MCS.
Mutually unbiased bases are an important tool in many applications of quantum information theory. We present a new algorithm for finding the mutually unbiased bases for two-qubit systems. We derive a system of four equations in the Galois field GF(4) and show that the solutions of this system are sufficient for obtaining the most general set of mutually unbiased bases. Further, our algorithm is applied to an example and we show that there are three possible solutions of the system of four equations, each solution leading to a different set of mutually unbiased bases.