The Pauli operators (tensor products of Pauli matrices) provide a complete basis of operators on the Hilbert space of N qubits. We prove that the set of 4 N −1 Pauli operators may be partitioned into 2 N + 1 distinct subsets, each consisting of 2 N − 1 internally commuting observables. Furthermore, each such partitioning defines a unique choice of 2 N + 1 mutually unbiased basis sets in the N -qubit Hilbert space. Examples for 2 and 3 qubit systems are discussed with emphasis on the nature and amount of entanglement that occurs within these basis sets.
We realize a Λ system in a superconducting circuit, with metastable states exhibiting lifetimes up to 8 ms. We exponentially suppress the tunneling matrix elements involved in spontaneous energy relaxation by creating a "heavy" fluxonium, realized by adding a capacitive shunt to the original circuit design. The device allows for both cavity-assisted and direct fluorescent readouts, as well as state preparation schemes akin to optical pumping. Since direct transitions between the metastable states are strongly suppressed, we utilize Raman transitions for coherent manipulation of the states.
A compete 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.
A few simply-stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets (MUBs), and constrain the combinations of such patterns that can coexist (ie, the stoichiometry) in full complements of (p N + 1) MUBs. We consider Hilbert spaces of prime power dimension (as realized by systems of N prime-state particles, or qupits), where full complements are known to exist, and we assume only that MUBs are eigenbases of generalized Pauli operators, without using a particular construction. The general rules include the following: 1) In any MUB, a particular qupit appears either in a pure state, or totally entangled, and 2) in any full MUB complement, each qupit is pure in (p+1) bases (not necessarily the same ones), and totally entangled in the remaining (p N − p). It follows that the maximum number of product bases is p + 1, and when this number is realized, all remaining (p N − p) bases in the complement are characterized by the total entanglement of every qupit. This "standard distribution" is inescapable for two qupits (of any p), where only product and generalized Bell bases are admissible MUB types. This and the following results generalize previous results for qubits [13,17] Such MUBs should play critical roles in filling complements.
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