Download Some Problems in Quantum Information Theory

Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Some Problems in Quantum Information Theory
William J. Martin
Department of Mathematical Sciences
and
Department of Computer Science
Worcester Polytechnic Institute
BIRS Workshop on Mathematics of Communications
Banff, Alberta
January 29, 2015
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Thanks
Thanks to the organizers!
Yesterday, along the Bow River
In preparing this talk, I benefited from conversations with Bill
Kantor.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
The Whole Talk in One Slide
You were born to solve these problems :
I
Find, as many as you can, equiangular lines in Cd (unit
vectors whose inner products have constant modulus).
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
The Whole Talk in One Slide
You were born to solve these problems :
I
Find, as many as you can, equiangular lines in Cd (unit
vectors whose inner products have constant modulus).
Find d 2 , if possible.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
The Whole Talk in One Slide
You were born to solve these problems :
I
I
Find, as many as you can, equiangular lines
vectors whose inner products have constant
Find d 2 , if possible.
Find, as many as you can, equiangular lines
vectors whose inner products have constant
William J. Martin
in Cd (unit
modulus).
in Rd (unit
abs value).
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
The Whole Talk in One Slide
You were born to solve these problems :
I
I
Find, as many as you can, equiangular lines
vectors whose inner products have constant
Find d 2 , if possible.
Find, as many as you can, equiangular lines
vectors whose
inner products have constant
d+1
Find 2 , if possible.
William J. Martin
in Cd (unit
modulus).
in Rd (unit
abs value).
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
The Whole Talk in One Slide
You were born to solve these problems :
I
I
I
Find, as many as you can, equiangular lines in Cd (unit
vectors whose inner products have constant modulus).
Find d 2 , if possible.
Find, as many as you can, equiangular lines in Rd (unit
vectors whose
inner products have constant abs value).
d+1
Find 2 , if possible.
Find, as many as you can, orthonormal bases in Cd where unit
vectors from distinct bases have inner products with constant
modulus. Find d + 1, if possible.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
The Whole Talk in One Slide
You were born to solve these problems :
I
I
I
I
Find, as many as you can, equiangular lines in Cd (unit
vectors whose inner products have constant modulus).
Find d 2 , if possible.
Find, as many as you can, equiangular lines in Rd (unit
vectors whose
inner products have constant abs value).
d+1
Find 2 , if possible.
Find, as many as you can, orthonormal bases in Cd where unit
vectors from distinct bases have inner products with constant
modulus. Find d + 1, if possible.
Find, as many as you can, orthonormal bases in Rd where unit
vectors from distinct bases have inner products with constant
absolute value. Find d2 + 1, if possible.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Uncertainty Principle
Exact knowledge of one diminishes knowledge of the other
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
A Cartoon Version of Quantum Physics
Kraus: “Two observables A and B of an n-level system
(i.e., a quantum system with n-dimensional state space)
are called complementary, if knowledge of the measured value of A
implies maximal uncertainty of the measured value of B, and vice
versa.”
Also called “maximally incompatible”.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
A Cartoon Version of Quantum Physics
Complementary Observables: If we measure one exactly, a
subsequent measurement of the other produces all outcomes with
equal probability.
I have more to say about measurements in an
William J. Martin
appendix
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
States and Measurements
I
basis states 1, 2, . . . , d
I
pure state (superposition) v ∈ Cd
I
A mixed state is a probability distribution on pure states
state v1 v2 · · · vk
probability p1 p2 · · · pk
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
States and Measurements
I
basis states 1, 2, . . . , d
I
pure state (superposition) v ∈ Cd
I
A mixed state is a probability distribution on pure states
state v1 v2 · · · vk
probability p1 p2 · · · pk
A mixed state is encoded as a positive semidefinite Hermitian
matrix with trace one.
ρ=
k
X
pj vj vj∗
j=1
(specified by d 2 − 1 real parameters)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
States and Measurements
In the simplest model, a measurement is an orthogonal
decomposition of Cn . We identify the measurement with an
orthonormal basis {v1 , . . . , vn }.
Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }:
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
States and Measurements
In the simplest model, a measurement is an orthogonal
decomposition of Cn . We identify the measurement with an
orthonormal basis {v1 , . . . , vn }.
Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }:
(1) the state is expressed as a linear combination
ψ = p1 v1 + p2 v2 + · · · + pn vn
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
States and Measurements
In the simplest model, a measurement is an orthogonal
decomposition of Cn . We identify the measurement with an
orthonormal basis {v1 , . . . , vn }.
Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }:
(1) the state is expressed as a linear combination
ψ = p1 v1 + p2 v2 + · · · + pn vn
P
we assume ψ is a unit vector so that j |pj |2 = 1.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
States and Measurements
In the simplest model, a measurement is an orthogonal
decomposition of Cn . We identify the measurement with an
orthonormal basis {v1 , . . . , vn }.
Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }:
(1) the state is expressed as a linear combination
ψ = p1 v1 + p2 v2 + · · · + pn vn
P
we assume ψ is a unit vector so that j |pj |2 = 1.
(2) an index j is chosen at random with probability |pj |2
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
States and Measurements
In the simplest model, a measurement is an orthogonal
decomposition of Cn . We identify the measurement with an
orthonormal basis {v1 , . . . , vn }.
Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }:
(1) the state is expressed as a linear combination
ψ = p1 v1 + p2 v2 + · · · + pn vn
P
we assume ψ is a unit vector so that j |pj |2 = 1.
(2) an index j is chosen at random with probability |pj |2
(3) our measurement returns only the index j
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
States and Measurements
In the simplest model, a measurement is an orthogonal
decomposition of Cn . We identify the measurement with an
orthonormal basis {v1 , . . . , vn }.
Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }:
(1) the state is expressed as a linear combination
ψ = p1 v1 + p2 v2 + · · · + pn vn
P
we assume ψ is a unit vector so that j |pj |2 = 1.
(2) an index j is chosen at random with probability |pj |2
(3) our measurement returns only the index j
(4) and the state collapses to vj .
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complementary Observables — I repeat myself:
Given measurements (“observables”) A and B on an n-dimensional
quantum system,
we say A and B are complementary if whenever we measure one
exactly, a subsequent measurement of the other produces all
outcomes with equal probability.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
The cost of clarity
”Truth and clarity are complementary.”
- Niels Bohr
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complex MUBs
Two orthonormal bases B and B 0 for Cd are unbiased if
|hu, vi| = α
for some constant α, for every u ∈ B and every v ∈ B 0 .
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complex MUBs
Two orthonormal bases B and B 0 for Cd are unbiased if
|hu, vi| = α
0
for some constant α, for every
√ u ∈ B and every v ∈ B .
Easy to check that α = 1/ d.
Schwinger (1960), Ivanovic (1981) - measurements
Wootters & Fields (1989) - “MUB”
Alltop (1980) - autocorrelation
Seidel, et al. (1970s) - line systems
various physicists (1980s)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complete sets of MUBs
A collection {B1 , . . . , BM } of orthonormal bases for Cd is a
collection/set of mutually unbiased bases if each pair is unbiased.
We say we have a set of “k MUBs”.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complete sets of MUBs
A collection {B1 , . . . , BM } of orthonormal bases for Cd is a
collection/set of mutually unbiased bases if each pair is unbiased.
We say we have a set of “k MUBs”.
Let M(d) denote the maximum size of a set of MUBs in Cd .
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complete sets of MUBs
A collection {B1 , . . . , BM } of orthonormal bases for Cd is a
collection/set of mutually unbiased bases if each pair is unbiased.
We say we have a set of “k MUBs”.
Let M(d) denote the maximum size of a set of MUBs in Cd .
Theorem[Delsarte, Goethals, Seidel (1978)]
(proven in terms of line systems)
M(d) ≤ d + 1
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complex Hadamard matrices
Say a matrix is flat if all entries have the same modulus.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complex Hadamard matrices
Say a matrix is flat if all entries have the same modulus.
I
We represent each basis Bj as the set of columns of some
unitary matrix Hj
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complex Hadamard matrices
Say a matrix is flat if all entries have the same modulus.
I
We represent each basis Bj as the set of columns of some
unitary matrix Hj
I
applying a unitary transformation, we may assume the first
matrix is the identity
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complex Hadamard matrices
Say a matrix is flat if all entries have the same modulus.
I
We represent each basis Bj as the set of columns of some
unitary matrix Hj
I
applying a unitary transformation, we may assume the first
matrix is the identity
I
in this case, all the other √
matrices (bases) in a set of MUBs
are flat, with modulus 1/ d
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complex Hadamard matrices
Say a matrix is flat if all entries have the same modulus.
I
We represent each basis Bj as the set of columns of some
unitary matrix Hj
I
applying a unitary transformation, we may assume the first
matrix is the identity
I
in this case, all the other √
matrices (bases) in a set of MUBs
are flat, with modulus 1/ d
√
Since these are unitary, Hj∗ Hj = I and each dHj is a
complex Hadamard matrix
I
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complex Hadamard matrices
Say a matrix is flat if all entries have the same modulus.
I
We represent each basis Bj as the set of columns of some
unitary matrix Hj
I
applying a unitary transformation, we may assume the first
matrix is the identity
I
in this case, all the other √
matrices (bases) in a set of MUBs
are flat, with modulus 1/ d
√
Since these are unitary, Hj∗ Hj = I and each dHj is a
complex Hadamard matrix
I
I
so we seek mutually unbiased Hadamard matrices (“MUCH”,
Best and Kharaghani, 2010): Hj∗ Hk flat ∀ j 6= k
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Retrieving Complete Information about a State
Note that each basis corresponds to a measurement which,
statistically, reveals d − 1 pieces of information (probabilities sum
to one). So d + 1 measurements represent d 2 − 1 real parameters,
which is the amount of information needed to represent a general
mixed state
k
X
ρ=
pj vj vj∗
j=1
we have d real values along the diagonal summing to one and 2 × d(d − 1)/2 real parameters to specify the
complex entries above the diagonal
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Planar Functions
Let F be an additive group (e.g., F = Fq ). A mapping f : F → F
is a planar function if, for every non-zero a ∈ F the map
x 7→ f (x + a) − f (x)
is a bijection. (E.g., for q odd f : Fq → Fq via f (x) = x 2 .)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Planar Functions
Hypothesis: We suppose that S is a set of functions Fq → Fq ,
including the zero function, such that the difference of any two is
planar. For q = p r , Fq is a vector space over Zp (p prime); let ζ
P
j
be a primitive complex p th root of unity so that p−1
j=0 ζ = 0.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Complete Sets of MUBs in Prime Power Dimensions
Roy-Scott (2007), Godsil-Roy (2009)
Write the standard basis for Cd (d prime) as {eb | b ∈ Fq }.
Let B∞ = {eb | b ∈ F }.
For f ∈ S, let
)
(
1 X a·b+f (b)
ζ
eb | a ∈ F .
Bf = √
d b∈F
Then the collection {Bf : f ∈ S} ∪ {B∞ } is a set of |S| + 1
MUBs in Cd .
Here d can also be a prime power and we can find S of size q
using symplectic spreads (CCKS, K).
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Bent Functions
Another flavour of construction arises from a bent function on a
vector space over Fp .
These two techniques, using bent functions f : V → F and planar
functions f : F → F cover all the constructions I know of complete
sets of complex MUBs.
Can we do something in between? f : V → W for a non-trivial
subspace W of V ? (Or modules over rings?)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
MUBs and planes
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
MUBs and planes
I
All known constructions of complete sets are somehow related
to affine planes
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
MUBs and planes
I
All known constructions of complete sets are somehow related
to affine planes
I
Kantor (2012): symplectic spreads yield complete sets of
MUBs and inequivalent spreads give rise to inequivalent
MUBs, so there are a lot of them
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
MUBs and planes
I
All known constructions of complete sets are somehow related
to affine planes
I
Kantor (2012): symplectic spreads yield complete sets of
MUBs and inequivalent spreads give rise to inequivalent
MUBs, so there are a lot of them
I
Yet this does not mean that all complete sets of MUBs must
come from planes.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
What is needed?
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
What is needed?
I
Ideas!
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
What is needed?
I
Ideas!
I
New constructions — particularly when d is not a prime power
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
What is needed?
I
Ideas!
I
New constructions — particularly when d is not a prime power
I
large, but not complete, sets for non-prime power orders
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
What is needed?
I
Ideas!
I
New constructions — particularly when d is not a prime power
I
large, but not complete, sets for non-prime power orders
I
M(d) = Ω(d) (or even a lower bound that grows with d)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
What is needed?
I
Ideas!
I
New constructions — particularly when d is not a prime power
I
large, but not complete, sets for non-prime power orders
I
M(d) = Ω(d) (or even a lower bound that grows with d)
I
Is there any combinatorics in the inner products? A coherent
configuration?
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Examples of non-complete sets
I
Analogue of MacNeish construction: if d has prime
factorization d = p1r1 p2r2 · · · pkrk , then
M(d) ≥ min{M(p1r1 ), M(p2r2 ), . . . , M(pkrk )}.
via taking tensor products of this many MUBs in each
prime-power dimension
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Examples of non-complete sets
I
Analogue of MacNeish construction: if d has prime
factorization d = p1r1 p2r2 · · · pkrk , then
M(d) ≥ min{M(p1r1 ), M(p2r2 ), . . . , M(pkrk )}.
via taking tensor products of this many MUBs in each
prime-power dimension
I
Beth-Wocjan construction (2004): If d = s 2 and there exist k
MOLS(s) then M(d) ≥ k + 2. So, for square dimensions d,
we obtain M(d) ≥ d 1/29.6
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Examples of non-complete sets
I
Analogue of MacNeish construction: if d has prime
factorization d = p1r1 p2r2 · · · pkrk , then
M(d) ≥ min{M(p1r1 ), M(p2r2 ), . . . , M(pkrk )}.
via taking tensor products of this many MUBs in each
prime-power dimension
I
Beth-Wocjan construction (2004): If d = s 2 and there exist k
MOLS(s) then M(d) ≥ k + 2. So, for square dimensions d,
we obtain M(d) ≥ d 1/29.6
(Beats MacNeish lower bound infinitely often)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Just a bit of physics
Basics of MUBs
MUB Constructions
MUB Challenges
Examples of non-complete sets
I
Analogue of MacNeish construction: if d has prime
factorization d = p1r1 p2r2 · · · pkrk , then
M(d) ≥ min{M(p1r1 ), M(p2r2 ), . . . , M(pkrk )}.
via taking tensor products of this many MUBs in each
prime-power dimension
I
Beth-Wocjan construction (2004): If d = s 2 and there exist k
MOLS(s) then M(d) ≥ k + 2. So, for square dimensions d,
we obtain M(d) ≥ d 1/29.6
(Beats MacNeish lower bound infinitely often)
I
Kharaghani and co-authors have found mutually unbiased
Hadamard matrices in various small dimensions
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
SIC-POVMs
A set of d + 1 MUBs give us d + 1 measurements that together
estimate d 2 − 1 real parameters. Can we do this with one
measurement?
Whereas MUBs give physicists a way to perform quantum state
tomography on a single mixed state (prepared repeatedly) by
performing a series of measurements (one for each basis),
SIC-POVMs achieve the same thing with a single measurement
(repeatedly applied).
But this is not our usual sort of measurement: the outcome states
are not pairwise orthogonal.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Complex Equiangular Lines
In each complex vector space Cd , we seek the maximum size of a
set L of equiangular lines.
These were studied by Seidel and co-authors in the 1960s and
1970s.
Example: In C2 , the lines spanned by
(τ, i),
(τ, −i),
are equiangular for τ =
√
1+ 3
2 (1
William J. Martin
(1, iτ ),
(1, −iτ )
+ i).
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
SIC-POVMs
Physicists are interested in equiangular line sets of size exactly d 2 :
these are called symmetric informationally complete positive
operator-valued measures (“SIC-POVMs”)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Why are these measurements?
Why are physicists interested?
In the simplest model, a measurement is an orthogonal
decomposition of Cn . We identify the measurement with an
orthonormal basis {v1 , . . . , vn }.
As we said before, the state is expressed as a linear combination
ψ = p1 v1 + p2 v2 + · · · + pn vn
P
where j |pj |2 = 1. When we perform this measurement, an index
j is chosen at random with probability |pj |2 and the state collapses
to vj .
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Von Neumann Measurements
This P
measurement is represented by the Hermitian matrix
ψ = j pj vj vj∗ , a sum of n rank-one projections, or by the
ensemble {vj vj∗ } of n pairwise orthogonal rank-one Hermitian
matrices summing to I .
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Measure the whole system to determine state of a
subsystem
Now suppose we are interested in a quantum subsystem of some
system. Let matrix P denote orthogonal projection onto this
subspace. A measurement on the overall system is an ensemble of
rank-one projections. Restricted to this subspace, these correspond
to projections of rank one (or zero) Pvj vj∗ P. These are not
necessarily pairwise orthogonal, but they still sum to the identity.
A POVM is an ensemble {Ej } of positive semidefinite Hermitian
matrices which sum to I .
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Special POVMs
Symmetric: Constant inner product Tr(Eh Ej∗ ) over all h 6= j.
But Eh = vh vh∗ , Ej = vj vj∗ gives
Tr(Eh Ej∗ ) = Tr(vh vh∗ vj vj∗ ) = Tr(vj∗ vh vh∗ vj ) = |hvh , vj i|2 .
So, for rank one projectors, “symmetry” means “equiangular”.
By expanding I 2 inside Tr(I 2 ) = d, we find the common modulus
of inner products must be 1/(d + 1).
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Special POVMs
Informationally Complete: Now we can get more than d − 1 real
values from the same measurement. So we consider repeatedly
applying the same POVM. As before, the general mixed state is
described by d 2 − 1 real parameters. Our POVM has probabilities
summing to one, so we need d 2 outcomes in order to be
“informationally complete”.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
SIC-POVMs
Physicists are interested in these for both practical and theoretical
connections:
I
quantum state tomography
I
quantum cryptography
I
evaluating “hidden variable” theories (Kochen-Specker
Theorem)
(Some classical applications are mentioned as well.)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Some Nice Constructions
Let N(d) denote the max size of a set of equiangular lines in Cd .
Theorem[Delsarte, Goethals, Seidel (1978)]
N(d) ≤ d 2
Here are the general techniques to obtain exactly d 2 lines in Cd :
I
I
I
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Three Constructions of Exactly d 2 Lines
I
S. Hoggar (1981,1998): 64 lines in C8 from a 4-dimensional
quaternionic polytope
I
Jedwab & Wiebe (2014): Elegant solutions in dimensions
d = 2, 3, 8 from Hadamard matrices
I
Various authors: “Zauner method”
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Some Nice Constructions of Less Than d 2 Lines
I
König (1995,1999): Characters of abelian groups restricted to
(v , k, 1)-difference sets (rediscovered several times)
I
Godsil & Roy (2012): line systems from relative difference sets
I
Greaves, et al. (2014)/Jedwab & Wiebe (2014): equiangular
lines from MUBs
I
equiangular tight frame constructions
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Sets of Quadratic Size
Recall that the upper bound is N(d) ≤ d 2
I
The König construction gives
N(d) ≥ d 2 − d + 1
for d = p r + 1 (p prime)
I
Converting MUBs to equiangular lines (Greaves, et al.,
Jedwab & Wiebe) gives
N(d) = Θ(d 2 )
for many other dimensions d.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Equiangular Lines from Difference Sets
Let G be a finite abelian group. A character of G is a group
homomorphism
χ : G → C∗
Example: For G = Z7 we have seven distinct characters, χa
(0 ≤ a < 7) given by
χa (b) = ω ab
where ω is any primitive complex 7th root of unity.
The quadratic residues in Z7 form a difference set D = {1, 2, 4}:
1−2 = 6,
1−4 = 4,
2−1 = 1,
William J. Martin
2−4−5,
4−1 = 3,
Quantum Information Theory
4−2 = 2.
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Lines from Difference sets
The quadratic residues in Z7 form a difference set D = {1, 2, 4}
with parameters (7, 3, 1).
The corresponding characters χ1 , χ2 , χ4 give us 7 vectors by
arranging them in a 3 × 7 array and transposing:
With ω = e 2πi/7
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Lines from Difference sets
The quadratic residues in Z7 form a difference set D = {1, 2, 4}
with parameters (7, 3, 1).
The corresponding characters χ1 , χ2 , χ4 give us 7 vectors by
arranging them in a 3 × 7 array and transposing:
With ω = e 2πi/7
χ1
χ2
χ4
=
=
=
0
[1
[1
[1
William J. Martin
1
ω
ω2
ω4
2
ω2
ω4
ω
3
ω3
ω6
ω5
4
ω4
ω
ω2
Quantum Information Theory
5
ω5
ω3
ω6
6
ω6]
ω5]
ω3]
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Lines from Difference sets
The quadratic residues in Z7 form a difference set D = {1, 2, 4}
with parameters (7, 3, 1).
The corresponding characters χ1 , χ2 , χ4 give us 7 vectors by
arranging them in a 3 × 7 array and transposing:
χ1 = [
With ω = e 2πi/7
χ2 = [
χ4 = [
This gives us a configuration
0
1
1
1
of
1
2
3
ω
ω2 ω3
ω2 ω4 ω6
ω4 ω
ω5
7 vectors in C3
4
ω4
ω
ω2
5
ω5
ω3
ω6
X = (1, 1, 1), (ω, ω 2 , ω 4 ), (ω 2 , ω 4 , ω), (ω 3 , ω 6 , ω 5 ),
(ω 4 , ω, ω 2 ), (ω 5 , ω 3 , ω 6 ), (ω 6 , ω 5 , ω 3 )
which span 7 equiangular lines.
William J. Martin
Quantum Information Theory
6
ω6]
ω5]
ω3]
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
Conjecture (G. Zauner, 1999) SIC-POVMs exist in all dimensions
d. Moreover, in each dimension d, there is a “fiducial” vector (a
common eigenvector of all operators in some specific abelian
subgroup of the Heisenberg-Weyl group) whose orbit under the
Heisenberg-Weyl group has size d 2 and forms a SIC-POVM.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
From a computational viewpoint, the conjecture seems to work!
“Numerical” examples of SIC-POVMs have been found to high
precision in all dimensions d ≤ 67.
Exact analytic solutions have been found (mostly by Gröbner basis
techniques) in the following dimensions
d = 2, 3, . . . , 16, 19, 24, 28, 35, 48
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
From a computational viewpoint, the conjecture seems to work!
“Numerical” examples of SIC-POVMs have been found to high
precision in all dimensions d ≤ 67.
Exact analytic solutions have been found (mostly by Gröbner basis
techniques) in the following dimensions
d = 2, 3, . . . , 16, 19, 24, 28, 35, 48
So what’s holding us up?
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
Grassl and Scott summarize these solutions in a 16-page paper
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
Grassl and Scott summarize these solutions in a 16-page paper
(with a 206-page supplement).
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
Grassl and Scott summarize these solutions in a 16-page paper
(with a 206-page supplement).
Fiducial vector for numerical example in C10 .
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
Second part of Grassl-Scott supplement gives exact solutions.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
Second part of Grassl-Scott supplement gives exact solutions.
Fiducial vector for analytic example in C10 requires some
preliminary definitions of constants.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
Second part of Grassl-Scott supplement gives exact solutions.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Zauner’s Conjecture
Second part of Grassl-Scott supplement gives exact solutions.
Here is the first entry of the fiducial vector for the example in C10 .
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Nine Lines in C3
Jedwab & Wiebe (2014):
With ω = e 2πi/3 , consider the complex Hadamard matrix


1 1
1
H =  1 ω ω2  .
ω 1 ω2
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Nine Lines in C3
Jedwab & Wiebe (2014):
With ω = e 2πi/3 , consider the complex Hadamard matrix


1 1
1
H =  1 ω ω2  .
ω 1 ω2
Then the 9 rows of the following matrices form a SIC-POVM in C3 .






−2
1
1
1 −2
1
1 1 −2
 −2
 1 −2ω ω 2  ,
 1 ω −2ω 2 
ω ω2  ,
−2ω 1 ω 2
ω −2
ω2
ω 1 −2ω 2
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Nine Lines in C3
Jedwab & Wiebe (2014):
With ω = e 2πi/3 , consider the complex Hadamard matrix


1 1
1
H =  1 ω ω2  .
ω 1 ω2
Then the 9 rows of the following matrices form a SIC-POVM in C3 .






−2
1
1
1 −2
1
1 1 −2
 −2
 1 −2ω ω 2  ,
 1 ω −2ω 2 
ω ω2  ,
−2ω 1 ω 2
ω −2
ω2
ω 1 −2ω 2
Good News! This works again in dimensions 2 and 8.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Nine Lines in C3
Jedwab & Wiebe (2014):
With ω = e 2πi/3 , consider the complex Hadamard matrix


1 1
1
H =  1 ω ω2  .
ω 1 ω2
Then the 9 rows of the following matrices form a SIC-POVM in C3 .






−2
1
1
1 −2
1
1 1 −2
 −2
 1 −2ω ω 2  ,
 1 ω −2ω 2 
ω ω2  ,
−2ω 1 ω 2
ω −2
ω2
ω 1 −2ω 2
Good News! This works again in dimensions 2 and 8.
Bad News. That’s it for this technique, in this form at least.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
We Need More Ideas Like This
Even if Zauner’s conjecture is true, it will be helpful to find more
constructions — efficiently computable, reasonably verifiable
examples.
Clearly there is better chance for progress if we are happy with less
than d 2 lines. But even for d 2 lines, we must be careful to avoid
assumptions that apparent connections to finite design theory are
in fact constraints.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Some nonsense on Wikipedia
“This seems similar in nature to the symmetric property of
SIC-POVMs. In fact, the problem of finding a SIC-POVM is
precisely the problem of finding equiangular lines in Cd ; whereas
mutually unbiased bases are analogous to affine spaces. In fact
it can be shown that the geometric analogy of finding a ‘complete
set of N+1 mutually unbiased bases is identical to the geometric
structure analogous to a SIC-POVM’. It is important to note that
the equivalence of these problems is in the strict sense of an
abstract geometry, and since the space on which each of these
geometric analogues differs, there’s no guarantee that a solution
on one space will directly correlate with the other. An example of
where this analogous relation has yet to necessarily produce results
is the case of 6-dimensional Hilbert space, in which a SIC-POVM
has been analytically computed using mathematical software, but
no complete mutually unbiased bases has yet been discovered.”
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
The Welch Bound
In telecommunications, it is beneficial to spread n unit vectors as
far apart as possible on the unit sphere.
If we have a set X of N unit vectors in Cd with maximum norm on all pairwise inner products, then
s
N −d
≥
.
d(N − 1)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
The Welch Bound
In telecommunications, it is beneficial to spread n unit vectors as
far apart as possible on the unit sphere.
If we have a set X of N unit vectors in Cd with maximum norm on all pairwise inner products, then
s
N −d
≥
.
d(N − 1)
If equality holds, then the Gram matrix G = [x · y]x,y has only two
eigenvalues and all off-diagonal entries equal to ±ε. (“ETF”)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
The Welch Bound
In telecommunications, it is beneficial to spread n unit vectors as
far apart as possible on the unit sphere.
If we have a set X of N unit vectors in Cd with maximum norm on all pairwise inner products, then
s
N −d
≥
.
d(N − 1)
If equality holds, then the Gram matrix G = [x · y]x,y has only two
eigenvalues and all off-diagonal entries equal to ±ε. (“ETF”)
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Equiangular Tight Frames (ETFs)
Only four non-trivial general constructions known:
I
N = 2d where ∃ conference matrix
(d = 2s+1 or d = p s + 1, s ≥ 1, p odd prime)
I
characters from difference sets
I
Incidence matrices of Steiner systems 2-(v , k, 1)
I
eigenspace projections of strongly regular graphs with special
parameters
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Definition and physics
Constructions
Equiangular Tight Frames
Equiangular Tight Frames (ETFs)
Only four non-trivial general constructions known:
I
N = 2d where ∃ conference matrix
(d = 2s+1 or d = p s + 1, s ≥ 1, p odd prime)
I
characters from difference sets
I
Incidence matrices of Steiner systems 2-(v , k, 1)
I
eigenspace projections of strongly regular graphs with special
parameters
Note also:
• orthonormal basis has N = d
• regular simplex has N = d + 1
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Real MUBs and Equiangular Lines
A few brief remarks about the real case:
I
Upper bounds are roughly half the corresponding LP bounds
for MUBs and equiangular lines
I
more combinatorics and number-theoretic restrictions (|x| = α
for only two values of x)
I
known extremal examples of d2 + 1 MUBs in Rd (d = 4t )
come from Kerdock sets
examples of d+1
equiangular lines in Rd only known for
2
d = 2, 3, 7, 23
I
I
many examples yield or come from association schemes
(regular two-graphs, all sets of real MUBs, de Caen’s set of
2 2
9 d lines)
And we can ask both questions over the quaternions, as well!
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
The End. Thank You!
You are the optimal people to solve these challenges! :
I
I
I
I
Find, as many as you can, equiangular lines in Cd (unit
vectors whose inner products have constant modulus).
Find d 2 , if possible.
Find, as many as you can, equiangular lines in Rd (unit
vectors whose
inner products have constant abs value).
d+1
Find 2 , if possible.
Find, as many as you can, orthonormal bases in Cd where unit
vectors from distinct bases have inner products with constant
modulus. Find d + 1, if possible.
Find, as many as you can, orthonormal bases in Rd where unit
vectors from distinct bases have inner products with constant
absolute value. Find d2 + 1, if possible.
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Quantum States
pure state (superposition)
ψ=
√
0.40 +
√
1
0.61 = √
5
√ √2
3
with amplitudes a1 , a2 satisfying |a1 |2 + |a2 |2 = 1.
If we measure state ψ in the standard basis, we will see 0 with
probability 0.4 and 1 with probability 0.6.
Back to
the introduction
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Measurements
A quantum measurement of a d-dimensional quantum system is
specified by an orthonormal basis
{w1 , w2 , . . . , wd }
of the space.
What happens:
I the current state is expressed as a linear combination
ψ = a1 v1 + a2 v2 + · · · + ad vd
with |a1 |2 + |a2 |2 + · · · + |ad |2 = 1.
I an oracle chooses j, 1 ≤ j ≤ d with probability |aj |2
I only the integer j is returned by the measurement
I the state ψ collapses to vj
Back to the introduction
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
“Collapse of the wave function”
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
States as Rank One Trace One Hermitian Matrices
We may represent this as a rank one Hermitian matrix
√
2/5
6/5
∗
√
Ψ = ψψ =
6/5
3/5
with trace one.
Then, the probability of a measurement yielding basis state b is
pb = b∗ Ψb.
Back to
the introduction
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Entanglement of Qubits
entanglement (superposition of basis states in a multiparticle
system)
Suppose the basis states are
 
 
 
 
1
0
0
0
 0 
 1 
 0 
 0 

 
 
 
00 = 
 0  , 01 =  0  , 10 =  1  , 11 =  0  .
0
0
0
1
Then the collection of pure states includes, for example,
1
1
ψ = √ 00 − √ 11
2
2
in which the two qubits are “entangled”.
(ψ is not expressible as a tensor product of two single-qubit state
vectors).
Back to the introduction
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Mixed States
Physicists also allow probability distributions on pure states.
Suppose the system is in state φ1 with probability p1 , in state φ2
with probability p2 , etc. Such a mixed state is prescribed by a pair
of tuples
[φ1 , φ2 , . . . , φk ] ,
[p1 , p2 , . . . , pk ].
Here, each pj is non-negative and p1 + p2 + · · · + pk = 1.
A general mixed state is represented by its corresponding density
matrix
k
X
ρ=
pj φj φ∗j
j=1
a positive semidefinite Hermitian matrix of trace one.
Back to
the introduction
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Measuring Mixed States
A measurement with respect to an orthonormal basis containing
basis vector b returns b with probability
pb = b∗ ρb =
k
X
pj b∗ φj φ∗j b =
j=1
k
X
pj |hb, φj i|2
j=1
which, intuitively, can be thought of as a two-stage probabilistic
event.
Back to
the introduction
William J. Martin
Quantum Information Theory
Complex MUBs
SIC-POVMs
Problems over the Real Numbers
Two notes about mixed states
I
The mixed state described by
[φ1 , φ2 , . . . , φk ] ,
[p1 , p2 , . . . , pk ]
P
is distinct from the pure state j pj φj . For example, the
√
√
density matrix for pure state 0.40 + 0.61 is
√
6/5
√2/5
6/5
3/5
while the mixed state which is equal to 0 with probability 0.4
and equal to 1 with probability 0.6 has a density matrix with
zeros off the diagonal.
I Two mixed states are indistinguishable if and only if they have
the same density matrix. So we may as well identify (mixed)
quantum states with positive semidefinite Hermitian matrices
with trace one.
Quantum Information Theory
Back to the introduction William J. Martin