The certainty of quantum randomness

Carlton M. Caves, Christopher A. Fuchs, Rüdiger Schack

Aleksandra Dimić, Borivoje Dakić

We study the weak-convergence properties of random variables generated by unsharp quantum measurements. More precisely, for a sequence of random variables generated by repeated unsharp quantum measurements, we study the limit distribution of relative frequency. We provide a representation theorem for all separable states, showing that the distribution can be well approximated by a mixture of normal distributions. Furthermore, we investigate the convergence rates and show that the relative frequency can stabilize to some constant at best at the rate of order for all separable inputs. On the other hand, we provide an example of a strictly unsharp quantum measurement where the better rates are achieved by using entangled inputs. This means that in certain cases the noise generated by the measurement process can be suppressed by using entanglement. We deliver our result in the form of quantum information task where the player achieves the goal with certainty in the limiting case by using entangled inputs or fails with certainty by using separable inputs.

Scott Aaronson

A. Diebra, S. Llorens, D. González-Lociga +4 more

We establish a sharp quantum advantage in determining the parity (even-odd) of an unknown permutation applied to any number n≥3 of particles. Classically, this is impossible with fewer than n labels, being that the success is limited to random guessing. Quantum mechanics does it with certainty with as few as ⌈sqrt[n] ⌉ distinguishable states per particle, thanks to entanglement. Below this threshold, not even quantum mechanics helps: both classical and quantum success are limited to random guessing. For small n, we provide explicit expressions for states that ensure perfect parity identification. We also assess the minimum entanglement these states need to carry, finding it to be close to maximal, and even maximal in some cases. The task requires no oracles or contrived setups and provides a simple, rigorous example of genuine quantum advantage.

Hongxing Li, Wei Guo Zhou, Honghai Mi

The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors where every random vector is defined on one of these probability spaces, such that the sequence of conditional mathematical expectations formed by the random vectors uniformly converges to the continuous function. This is a random vector representation of continuous functions, which is regarded as a bridge to be set up between real function theory and probability theory. By means of this conclusion, an interesting result about function approximation theory can be obtained. The random vector representation of continuous functions is of important applications in physics. Based on the conclusion, if a large proportion of certainty phenomena can be described by continuous functions and random phenomena can also be described by random variables or vectors, then any certainty phenomenon must be the limit state of a sequence of random phenomena. Then, in the approximation from a sequence of random vectors to a continuous function, the base functions are appropriately selected by us, and an important conclusion for quantum mechanics is deduced: classical mechanics and quantum mechanics are unified. Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.