Download Quantum design

MUBs and some other
quantum designs
Aleksandrs Belovs
and
Juris Smotrovs
Outline of the talk
• Combinatorial designs
• Optimal quantum measurement problem
(MUBs, SIC POVMs)
• Quantum designs
• MUBs and SIC POVMs as quantum
designs
• Links with problems in combinatorics
• Conclusion
Combinatorial designs
• 36 officer problem (L.Euler, 1782)
An example with a simpler case with 9 officers:
1
2
3
1
2
3
2
3
1
3
1
2
3
1
2
2
3
1
Euler conjectured that there is no solution for the 6X6 case,
and, in general, for the (4n+2)X(4n+2) case.
Combinatorial designs
• 36 officer problem:
– Modern name of the general problem: Mutually
orthogonal latin squares (MOLS)
– Euler conjectured that there is no solution for the 6X6
case, and, in general, for the (4n+2)X(4n+2) case.
– G. Tarry, 1900: proved by exhaustive search of 6X6
latin squares that no two of them are orthogonal
– Bose, Shrikhande, and Parker, 1960: found with
computer search orthogonal 10X10 latin squares,
then proved that they do not exist only for dimensions
2X2 and 6X6.
Combinatorial designs
• Kirkman’s schoolgirl problem (1850) and
Steiner triples (solved)
• Finite geometries (projective, affine,...)
• Difference sets
• Hadamard matrices
Modern combinatorial design theory started
with R. Fisher’s work on design of
statistical experiments in 1930s.
Combinatorial designs
• Balanced incomplete block designs (BIBD)
v elements
must be arranged into b blocks (sets) so that
each block contains k elements,
each element is in r blocks, and
each two elements are both contained in 
blocks.
For which parameter quintuples (v,b,k,r,)
such design can be constructed and how?
Combinatorial designs
• Example
v=7,
b=7,
k=3,
r=3,
=1
B1 B2 B3 B4 B5 B6 B7
1
2
3
4
5
6
0
2
3
4
5
6
0
1
4
5
6
0
1
2
3
Optimal quantum measurement
• A pure quantum state is a vector (denoted something like
| ) of unit length in the vector space Cn.
• In an orthonormal basis |0, |1, ..., |n-1 it can be
represented as
| = 0|0 + 1|1 + ... + n-1|n-1.
• When measured in this basis, one of the basis states |i
is obtained with probability |i|2, and the state |
collapses to |i. This is called von Neumann
measurement.
• A mixed quantum state is a probabilistic composition of
pure states:  = p1|11| + p2|22| + ... + pk|kk|.
Optimal quantum measurement
• Problem
Suppose we have many instances of the
same state  in Cn. Then we can perform
many measurements of this state using
different bases. How should we choose
the bases so that we learn the state with
maximum precision?
Optimal quantum measurement
• Case 1: we are allowed measurements
only within the given space Cn; we use
each base for the same number of
measurements
Then the optimum would be obtained with a
set of n+1 mutually unbiased bases
(MUBs) – if such exists.
Optimal quantum measurement
• Case 2: we are allowed to measure in a
larger space Cm which contains the given
space Cn
Such measurement from the viewpoint of
the given space Cn is called positive
operator valued measurement or POVM.
Solution to the problem would then be
provided by a symmetric informationally
complete POVM (SIC POVM) – if it exists.
MUBs
A number of orthonormal bases in Cn is said to be
mutually unbiased iff any two basis vectors |x, |y
from different bases have the same scalar
product by absolute value:
1
| x|y | =
n
There can be no more than n+1 such bases in Cn.
MUBs
An example: 3-MUB in C2.
1
 
 0
 0
 ,
1
1

1

2


2
 1 2 

,
 1 
2

i

1

2


2
1

i




2
2
MUBs
I.D. Ivanovic (1981),
W.K.Wootters, B.D.Fields (1989):
(n+1)-MUB exists for any dimension n=pm, where p is
prime:
(v ) 
(r )
k
l
1
p
m
e
( 2i / p )Tr ( rl 2  kl )
r is base index, k is vector index, l is component index;
r,k,l  GF(pm), Tr is the trace GF(pm)  GF(p).
MUBs
• Does an (n+1)-MUB exist for a dimension
n not being a prime power?
Up to now the answer has not been found
for any of these dimensions, even for n=6.
At the moment only a 3-MUB is known in 6
dimensions.
• If an (n+1)-MUB does not exist, then what
is the maximal number of MUB that exist
in any given dimension?
SIC POVMs
A set of n2 unit vectors form a symmetric
informationally complete POVM (SIC POVM) iff
any two of these vectors |x, |y have the same
scalar product by absolute value:
| x|y | =
1
.
n 1
SIC POVMs
An example: SIC POVM in C2.


3

1




2 3

,
3 1 
 ei / 4


2
3




3

1




2 3

,
3 1 
  e i / 4


2
3




3

1
i

/
4
e


2 3 

,
3 1 



2
3




3

1
i

/
4
 e


2 3 


3 1 



2
3


SIC POVMs
• Does there exist a SIC POVM for any
dimension?
It has been conjectured that the answer is
positive, however it has been proven only
for a finite amount of dimensions: for small
n by finding SIC POVMs analytically, and
for n < 45 by finding approximate SIC
POVMs numerically.
Quantum designs
G.Zauner (1999):
Block design
Quantum design
orthonormal basis in Cv
v elements
b blocks
b orthogonal projections
k elements in each block each projection is to a k
-dimensional subspace
each element in r blocks each basis vector is in r
projection subspaces
each 2 blocks have 
each 2 proj. subspaces
common elements
intersection dim = 
Quantum designs
G.Zauner (1999):
Quantum design is a set {P1, ..., Pb} of projection
operators in Cv.
It is called regular iff there is such k that Tr(Pi) = k
for all i.
It is called coherent iff there is such r that
P1 + ... + Pb = rE.
Its degree s is the number of elements in the set
 = {Tr(PiPj) | i  j} = {1, ..., s}.
Quantum designs
• MUBs as quantum designs
If we consider MUB as consisting not of
vectors, but of projections on their lines,
then an (n+1)-MUB in Cn is a quantum
design with parameters:
v = n, b = n(n+1), k = 1, r = n+1,
the degree s = 2, and 1 = 0, 2 = 1/n.
Quantum designs
• SIC POVMs as quantum designs
SIC POVM in Cn is a quantum design with
parameters:
v = n, b = n2, k = 1, r = n,
the degree s = 1, and 1 = 1/(n+1).
Quantum designs
• Complex projective t-design:
A set X of unit vectors in Cn such that
 f ( x) 

xX
CS
X
n1
f ( x)d ( x)
 (CS
n 1
)
for any polynomial f of degree t on the complex
projective sphere CSn-1 (formed by equivalence
classes of unit vectors in Cn where collinear
vectors are considered equivalent).
Quantum designs
• Welch inequalities
For any set X of unit vectors in Cn and any natural
number k holds:

x y
x , yX
X
(L.R.Welch, 1974)
2
2k
1

 n  k  1


 k 
Quantum designs
A.Klappenecker, M.Rötteler (2005):
A set X is a complex projective t-design iff
with its vectors the Welch inequality turns
into an equality for all k between 0 and t.
MUBs and SIC POVMs are complex
projective 2-designs.
Quantum designs
A.Belovs, J.Smotrovs (2008):
Let X be a set of unit vectors in Cn. Let B be a
matrix formed by vectors from X as columns.
Let w1, ..., wn be the rows of matrix B. The
Welch inequality turns into an equality for X
and natural number k iff all vectors from


k
 ( k1 )
 

( kn )
 w1    wn k1    kn  k 
W   
k , , k n 


  1

are of equal length and pairwise orthogonal.
MUBs
The known (n+1)-MUBs can be expressed in form:
1
(v ) 
 r ( f (l ))  k (l )
n
(r )
k
l
where base index r, vector index k, component index l are
elements of an Abelian group G = Z/n1Z  ...  Z/nmZ of
size n= n1...nm;
 m 2i

 a (b)  exp  
a jb j 

n
j

1
j


is a character of this group, and f is some function in this
group. It follows from the result of the previous slide that
we have (n+1)-MUB iff this function is perfect non-linear.
Link with combinatorial designs
• Perfect non-linear functions
A function f: GG is said to be perfect nonlinear iff for any a  0 and b there is exactly
one x such that f(x+a)  f(x) = b.
Example: f(x)=x2 in Z/pZ, where p is prime, is
perfect non-linear.
These functions are much studied in
cryptography, but mostly in the binary case
n=2m.
Link with combinatorial designs
• Difference sets
A set D={d1,...,dk} of k elements from an
Abelian group G of size v is said to form a
(v,k,)-difference set iff the differences
di  dj with i  j contain each non-zero
element of G exactly  times.
A long-known special case of balanced
incomplete block designs.
Link with combinatorial designs
• Relative difference sets
If G is an Abelian group, and N its subgroup,
then a subset D={d1,...,dk} of G is called an
(m,n,k,)-relative difference set iff |N|=n,
|G|/|N|=m, and the differences di  dj with
i  j contain no element from N, and each
of the other non-zero elements of G
exactly  times.
Link with combinatorial designs
A function f: GG is perfect non-linear iff the
set D={(x,f(x)) | x  G} is a relative
difference set with respect to group G2 and
its subgroup N={(x,0) | x  G}.
Link with combinatorial designs
• Finite projective plane:
a finite set P of points together with a
collection of subsets of P called lines, such
that
– for any two points there is exactly one line
containing both of them;
– the intersection of any two lines contains
exactly one point;
– there are 4 points such that no 3 of them
belong to the same line.
Link with combinatorial designs
• Collineation of a projective plane:
a transformation of the plane that maps
collinear points into collinear points.
Link with combinatorial designs
A.Blokhuis, D.Jungnickel, B.Schmidt (2001):
If G is an Abelian collineation group of order n2 of a
projective plane, then n is a prime power.
Proof essentially is a proof about relative
difference sets.
It follows from this result that perfect non-linear
functions can exist only in groups whose order is
power of a prime.
Thus MUBs of the form described above can exist
only in spaces Cn where n is a prime power.
What further?
The formula
1
(v ) 
 r ( f (l ))  k (l )
n
(r )
k
l
gives an (n+1)-MUB in Cn also when f is a function
of a more general kind:
Z/n1Z  ...  Z/nmZ  R/n1R  ...  R/nmR
with properties similar to those of perfect nonlinear functions. The existence of such functions
for arbitrary dimension is still an open question.
Thank you for the attention!
Questions?