Quantum guessing games with posterior information

Abstract

Quantum guessing games form a versatile framework for studying different tasks of information processing. A quantum guessing game with posterior information uses quantum systems to encode messages and classical communication to give partial information after quantum measurement has been performed. We present a general framework for quantum guessing games with posterior information and derive structure and reduction theorems that enable to analyze any such game. We formalize symmetry of guessing games and characterize the optimal measurements in cases where the symmetry is related to an irreducible representation. The application of guessing games to incompatibility detection is reviewed and clarified. All the presented main concepts and results are demonstrated with examples.

Experimentally determining the incompatibility of two qubit measurements

“…In this case, by the discussion of Sec. II D, the incompatibility generalized robustness η g (A γ a , B γ b ) coincides with χ(E; A γ a , B γ b ) for all values of the noise parameter γ within the range (12). The latter values are exacty those that render the two measurements A γ a and B γ b incompatible.…”

“…Then, we use the well-known characterization of compatibility for dichotomic qubit measurements in order to conclude the calculation of η g (A γ a , B γ b ). In this way, we provide the proof of ( 11), (21) for γ assuming the values (12).…”

“…Therefore, by comparing this formula with the experimental data, we empirically verify the value of η g (A γ a , B γ b ) predicted by the theory when A γ a and B γ b are incompatible. For the values of γ below the interval (12), the two measurements are compatible, hence η g (A γ a , B γ b ) = 1 and accordingly we must find χ(E;…”

“…which motivates the constraint on γ given in (12). Summarizing, the success probability in the partitioned state ensemble ( 5)-( 6) is the quantity we are going to estimate through our experiments, as it allows to evaluate the incompatibility of the measurements A α a and B β b via ( 9) and ( 11), or at least upper bound it via (13), depending on the values of the noise parameters α and β and on the directions a and b.…”

Experimentally determining the incompatibility of two qubit measurements

“…Such a task does not involve multipartite quantum systems, and for this reason it is easier to be performed in the laboratory. The task is actually a variation of the usual state discrimination, from which it differs only in the following extra step: before guessing the unknown state, the experimenter receives some classical information, and, based on it, he performs a measurement chosen from some predetermined possible alternatives [9][10][11][12]. It then turns out that the probability of guessing the correct state is connected to the incompatibility of the measurements used in the experiment.…”