Format dependent probabilities: An eye-tracking analysis of additivity neglect

When people are asked to estimate the probabilities of uncertain events, they often neglect the additivity principle, which requires that the probabilities assigned to an exhaustive set of outcomes should add up to 100%. Previous studies indicate that additivity neglect is dependent on response format, self-generated probability estimates being more coherent than estimates on rating scales. The present study made use of eye-tracking methodology, recording the movement, frequency and duration of fixations during the solution of ten additivity problems and two control tasks. Participants produced more non-additive estimates in the Scale format than in the Self-generated format. Self-generated estimates also led to longer decision time and a higher number of repeated inspections, suggesting a deliberate comparison process. In contrast, the Scale format seemed to encourage a case-based approach where each outcome is evaluated in isolation

LOW ENTROPY OUTPUT STATES FOR PRODUCTS OF RANDOM UNITARY CHANNELS

In this paper, we study the behavior of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [Superadditivity of communication capacity using entangled inputs, Nat. Phys.5 (2009) 255–257] and Hayden–Winter [Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys.284(1) (2008) 263–280] to the additivity problems. In particular, we study in depth the difference of behavior between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys.297(2) (2010) 345–370].