Grasshopper jumping on Bloch sphere finds new quantum insights

Date:
August 10, 2020
Source:
University of Warwick
Summary:
New research has (pardon the pun) put a new spin on a mathematical
analogy involving a jumping grasshopper and its ideal lawn
shape. This work could help us understand the spin states of
quantum-entangled particles.



FULL STORY ==========================================================================
New research at the University of Warwick has (pardon the pun) put a new
spin on a mathematical analogy involving a jumping grasshopper and its
ideal lawn shape. This work could help us understand the spin states of quantum-entangled particles.


==========================================================================
The grasshopper problem was devised by physicists Olga Goulko (then at
UMass Amherst), Adrian Kent and Damia'n Pitalu'a-Garci'a (Cambridge). They asked for the ideal lawn shape that would maximize the chance that a grasshopper, starting from a random position on the lawn and jumping a
fixed distance in a random direction, lands back on the lawn. Intuitively
one might expect the answer to be a circular lawn, at least for small
jumps. But Goulko and Kent actually proved otherwise: various shapes
from a cogwheel pattern to some disconnected patches of lawn performed
better for different jump sizes (link to the technical paper).

Beyond surprises about lawn shapes and grasshoppers, the research provided useful insight into Bell-type inequalities relating probabilities of
the spin states of two separated quantum-entangled particles. The
Bell inequality, proved by physicist John Stewart Bell in 1964 and
later generalised in many ways, demonstrated that no combination
of classical theories with Einstein's special relativity is able to
explain the predictions (and later actual experimental observations)
of quantum theory.

The next step was to test the grasshopper problem on a sphere. The
Bloch sphere is a geometrical representation of the state space of
a single quantum bit. A great circle on the Bloch sphere defines
linear polarization measurements, which are easily implemented and
commonly used in Bell and other cryptographic tests. Because of the
antipodal symmetry for the Bloch sphere, a lawn covers half the total
surface area, and the natural hypothesis would be that the ideal lawn
is hemispherical. Researchers in the Department of Computer Science
at the University of Warwick, in collaboration with Goulko and Kent, investigated this problem and found that it too requires non-intuitive
lawn patterns. The main result is that the hemisphere is never optimal,
except in the special case when the grasshopper needs exactly an even
number of jumps to go around the equator.

This research shows that there are previously unknown types of Bell inequalities.

One of the paper's authors -- Dmitry Chistikov from the Centre for
Discrete Mathematics and its Applications (DIMAP) and the Department
of Computer Science, at the University of Warwick, commented: "Geometry
on the sphere is fascinating. The sine rule, for instance, looks nicer
for the sphere than the plane, but this didn't make our job easy."
The other author from Warwick, Professor Mike Paterson FRS, said:
"Spherical geometry makes the analysis of the grasshopper problem more complicated. Dmitry, being from the younger generation, used a 1948
textbook and pen-and-paper calculations, whereas I resorted to my good old Mathematica methods." The paper, entitled 'Globe-hopping', is published
in the Proceedings of the Royal Society A. It is interdisciplinary work involving mathematics and theoretical physics, with applications to
quantum information theory.

The research team: Dmitry Chistikov and Mike Paterson (both from the
University of Warwick), Olga Goulko (Boise State University, USA),
and Adrian Kent (Cambridge), say that the next steps to give even more
insight into quantum spin state probabilities are looking for the most grasshopper-friendly lawns on the sphere or even letting the grasshopper
boldly go jumping in three or more dimensions.


========================================================================== Story Source: Materials provided by University_of_Warwick. Note: Content
may be edited for style and length.


========================================================================== Journal Reference:
1. Dmitry Chistikov, Olga Goulko, Adrian Kent, Mike
Paterson. Globe-hopping.

Proceedings of the Royal Society A: Mathematical, Physical and
Engineering Sciences, 2020; 476 (2238): 20200038 DOI: 10.1098/
rspa.2020.0038 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2020/08/200810103235.htm

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