Probability Distributions for Random Quantum Operations

Mark Kon

Kevin Schultz

Motivated by the engineering applications of uncertainty quantification, in this work we draw connections between the notions of random quantum states and operations in quantum information with probability distributions commonly encountered in the field of orientation statistics. This approach identifies natural probability distributions that can be used in the analysis, simulation, and inference of quantum information systems. The theory of exponential families on Stiefel manifolds provides the appropriate generalization to the classical case, and fortunately there are many existing techniques for inference and sampling that exist for these distributions. Furthermore, this viewpoint motivates a number of additional questions into the convex geometry of quantum operations relative to both the differential geometry of Stiefel manifolds as well as the information geometry of exponential families defined upon them. In particular, we draw on results from convex geometry to characterize which quantum operations can be represented as the average of a random quantum operation.

Tian Chen, Xiangdong Zhang

We study the localization of probability distribution in a discrete quantum random walk on an infinite chain. With a phase defect introduced in any position of the quantum random walk (QRW), we have found that the localization of the probability distribution in the QRW emerges. Different localized behaviors of the probability distribution in the QRW are presented when the defect occupies different positions. Given that the coefficients of the localized stationary eigenstates relies on the coin operator, we reveal that when the defect occupies different positions, the amplitude of localized probability distribution in the QRW exhibits a non-trivial dependence on the coin operator.

Stanley Gudder

E. B. Davies, John T. Lewis

Arie Bar-Haim

A new model maps a quantum random walk described by a Hadamard operator to a particular case of a birth and death process. The model is represented by a 2D Markov chain with a stochastic matrix, i.e., all the transition rates are positive, although the Hadamard operator contains negative entries (this is possible by increasing the dimensionality of the system). The probability distribution of the walker population is preserved using the Markovian property. By applying a proper transformation to the population distribution of the random walk, the probability distributions of the quantum states |0>, 1> are revealed. Thus, the new model has two unique properties: it reveals the probability distribution of the quantum states as a unitary system and preserves the population distribution of the random walker as a Markovian system.