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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.

2014

Greenberger-Horne-Zeilinger (GHZ) states are characterized by their transformation properties under a continuous symmetry group, and N -body operators that transform covariantly exhibit a wealth of GHZ contradictions. We show that local or noncontextual hidden variables cannot duplicate the predicted measurement outcomes for covariant transformations, and we extract specific GHZ contradictions from discrete subgroups, with no restrictions on particle number N or dimension d except for the general requirement that N ≥ 3 for GHZ states. However, the specific contradictions fall into three regimes distinguished by increasing demands on the number of measurement operators required for the proofs. The first regime consists of proofs found recently by Ryu et. al. [33], the first operator-based theorems for all odd dimensions, d, covering many (but not all) particle numbers N for each d. We introduce new methods of proof that define second and third regimes and produce new theorems that fill all remaining gaps down to N = 3, for every d. The common origin of all such GHZ contradictions is that the GHZ states and measurement operators transform according to different representations of the symmetry group, which has an intuitive physical interpretation.

Apathy is a prominent neuropsychiatric symptom associated with human immunodeficiency virus (HIV). The increased frequency of apathy in this population may reflect the direct involvement of the virus on the central nervous system (CNS), but the severity of apathy has not been shown to consistently relate to markers of disease activity or other neuropsychiatric complications of the virus. We examined the relationship between ratings of apathy and performance on measures of cognitive function and immune system status in a sample of HIV-infected patients. Apathy was significantly elevated among HIV-infected individuals compared to healthy comparison subjects. Apathy was significantly related to performance on measures of learning efficiency and a measure of cognitive flexibility. Ratings of apathy did not relate to CD4 cell count, but they were associated with disease duration. In addition, ratings of depression were independent of ratings of apathy. These findings suggest that apathy does not co-vary with a proxy measure of active disease status, but apathy does relate to several measures of cognitive dysfunction in patients with HIV. As such, the increased prevalence of apathy among HIV-infected adults may reflect HIV-associated neurologic dysfunction.

The existence of GHZ contradictions in many-qutrit systems was a long-standing theoretical question until it's (affirmative) resolution in 2013. To enable experimental tests, we derive Mermin inequalities from concurrent observable sets identified in those proofs. These employ a weighted sum of observables, called M, in which every term has the chosen GHZ state as an eigenstate with eigenvalue unity. The quantum prediction for M is then just the number of concurrent observables, and this grows asymptotically as 2^N/3 as the number of qutrits (N) goes to infinity. The maximum classical value falls short for every N, so that the quantum to classical ratio (starting at 1.5 when N=3), diverges exponentially (~ 1.064^N) as N goes to infinity, where the system is in a Schroedinger cat-like superposition of three macroscopically distinct states.Comment: 12 pages, two figure

Quantum dynamics predicts that a metastable state should decay exponentially except at very early and very late times. We show through an exactly soluble model that if the decay products can interact weakly with their environment, then the exponential decay regime is prolonged to later times, while the exponential decay constant itself remains essentially unaffected. As the number of environmental degrees of freedom is increased, the asymptotic late-time decay follows higher powers of 1/t and the exponential decay regime is extended without limit.

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