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Nature Physics

Going random

Federico Levi

Published Oct 1, 2017• 1 citation
Publisher
DOI
10.1038/nphys4291

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Motivated by the fact that random quantum states are generic, i. e., that they occur as eigenvectors or during time evolution of sufficiently complex quantum system, we are going to address three questions: (i) what is the entanglement content of random quantum states, (ii) how can we efficiently generate them, and (iii) how come that there is apparently no entanglement in macroscopic world. To explain the later we will use properties of random quantum states. To answer the first question we are going to calculate average values of all Schmidt coefficient and to address the second one we are going to explicitly calculate convergence times of various random protocols.
We discusss the role of information entropy on the behaviour of random processes, and how this might take effect in the dynamics of financial market prices. We then go on to show how the Open Quantum Systems approach can be used as a more flexible alternative to classical methods in terms of modelling the entropy gain of a random process. We start by describing an open quantum system that can be used to model the state of a financial market. We then go on to show how to represent an essentially classical diffusion in this framework. Finally, we show how by relaxing certain assumptions, one can generate interesting and essentially non-classical results, which are highlighted through numerical simulations.
To discuss secure key generation from imperfect random numbers, we address the secure key generation length. There are several studies for its asymptotic expansion up to the order √n or log n. However, these expansions have errors of the order o(√n) or o(log n), which does not go to zero asymptotically. To resolve this problem, we derive the asymptotic expansion up to the constant order for upper and lower bounds of these optimal values. While the expansions of upper and lower bonds do not match, they clarify the ranges of these optimal values, whose errors go to zero asymptotically.