Complementarity in atomic (finite-level quantum) systems: an information-theoretic approach

Abstract

We develop an information theoretic interpretation of the numberphase complementarity in atomic systems, where phase is treated as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as an upper bound on a sum of knowledge of these two observables for the case of two-level systems. A tighter bound characterizing the uncertainty relation is obtained numerically in terms of a weighted knowledge sum involving these variables. We point out that complementarity in these systems departs from mutual unbiasededness in two signalificant ways: first, the maximum knowledge of a POVM variable is less than log(dimension) bits; second, surprisingly, for higher dimensional systems, the unbiasedness may not be mutual but unidirectional in that phase remains unbiased with respect to number states, but not vice versa. Finally, we study the effect of non-dissipative and dissipative noise on these complementary variables for a single-qubit system.2 log 1 f (A,B) , where f (A, B) = max a,b | a|b |, and H(·) is the Shannon binary entropy. We note

Maximally informative ensembles for SIC-POVMs in dimension 3

“…where ρ is a pure state [18]. Additionally, the problems of finding the informational power of group- [27]. Indeed, the greaterH is, the more we know about the measurement outcomes.…”

Complementarity in atomic and oscillator systems

“…In Ref. [13], we numerically found this to be the case for the continued-valued phase obervable in a two-level system. However, we know of no (published) proof that this is true in general, which was the motivation behind adopting the concept of entropic knowledge in the uncertainty relation.…”

“…[13], where the problem that the Shannon entropy of a continuous random variable may be negative is circumvented by instead using relative entropy (also called Kullbäck-Leibler divergence, which is always positive) [14,15] with respect to a uniform distribution. This quantity is a measure of knowledge [13]. Note that recourse to entropic knowledge may not be always necessary, and other ways might exist to circumvent the problem.…”

“…A particular case of discrete-continuous conjugacy, considered in detail in Ref. [13], is the number and phase of an atomic system. This generalization of the entropic uncertainty principle to cover discrete-continuous systems still suffers from the restriction that the system must be finite dimensional, since in the case of an infinite-dimensional system, such as an oscillator, entropic knowledge of the number distribution can diverge, making it unsuitable for infinite-dimensional systems.…”

“…In this section, we briefly recapitulate, for convenience, some useful formulas of quantum phase distributions for oscillator systems. For the case of atomic systems, the basic formulas were presented in [13].…”