that include the properties that cannot
They proved this did not apply to one
class of measurements: those referring
to continuous properties such as momentum. They thought, but could not
prove, the answer would also be ‘no’ for
discrete situations, such as energy states.
ASOLUTION TO a problem in mathematics that lingered unsolved for more than 50 years could help deliver faster computer algorithms to many problems
in physics and signal processing.
However, it may take years for mathematicians to fully digest the result,
which was first published online
three years ago.
The roots of the problem defined
by Richard Kadison and Isadore Singer in the late 1950s lie in attempts to
give the physics of quantum mechanics a footing in abstract mathematics.
The concept it deals with traces back
to Werner Heisenberg’s initial work
on quantum mechanics. Heisenberg
used matrix mathematics to develop
his model of the quantum world that
says it is not possible to accurately
measure simultaneously different
properties of a physical system at the
A decade later, John von Neumann
applied graph theory to the problem of
reconciling what physicists had postu-
lated about quantum mechanics with
mathematics. One result of this work
was the development of a specialized
algebra known as C* (pronounced
“see-star”) that could model quantum
states. In this algebra, Kadison and
Singer formulated a question of ab-
stract mathematics that mirrored one
of the key issues facing quantum me-
chanics: do measurements of quan-
tum properties in the observable world
map into uniquely identifiable states
Physics with Math
Mathematicians explore the root of many problems
in developing a proof for the Kadison-Singer problem.
Science | DOI: 10.1145/2967975 Chris Edwards
From left, Nikhil Srivastava, Adam Marcus, and Daniel Spielman shortly after completing the
proof of the Kadison-Singer problem.