Title: Equiangular complex projective 2-designs
| Speaker: | Jon Yard |
| Affiliation: | IQC, C&O and Perimeter Institute |
| Location: | MC 6486 |
Abstract:
Tight
2-designs
in
complex
projective
space
correspond
to
maximal
sets
of
complex
equiangular
lines.
They
are
also
known
as
equiangular
tight
frames
in
signal
processing
and
as
SIC-POVMs
(Symmetric
Informationally
Complete
Positive
Operator-Valued
Measures)
in
quantum
information
theory.
While
they
are
believed
to
exist
in
every
finite-dimensional
complex
Hilbert
space
as
orbits
of
finite
Heisenberg
groups,
their
existence
has
so
far
only
been
mathematically
proven
for
finitely
many
dimensions.
The
proofs
in
these
cases
are
mainly
computational
and
require
computer-assisted
calculations
in
number
fields
of
degree
growing
roughly
quadratically
with
the
dimension.
In
this
talk,
I
will
illustrate
how
the
structure
of
these
number
fields
can
be
explained
using
class
field
theory.
The
resulting
uniformity
of
these
fields
constitutes
overwhelming
evidence
in
favor
of
their
existence
in
every
dimension
and
uncovers
a
surprising
connection
to
Hilbert's
still-unsolved
12th
problem
of
giving
explicit
generators
for
class
fields
of
real
quadratic
fields.
This talk is partially based on joint work (arxiv:1604.06098) with Appleby, Flammia and McConnell.