MUBs and some other quantum designs

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

2 3 1

3 1 2

1 2 3

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.


• 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.


• 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.


• 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?


• 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|11| + p2|22| + ... + pk|kk|.


• 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?


• 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.


• 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:

| x|y | =

There can be no more than n+1 such bases in Cn.

n

1

MUBs

An example: 3-MUB in C2.

2

21

21

2

21

21

21

21

, ,1

0

0

1i

i

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:

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).

)()/2()( 21)( klrlTrpi

mlr

k ep

v

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

1

n

SIC POVMs

An example: SIC POVM in C2.

32

13

32

13

,

32

13

32

13

,

32

13

32

13

,

32

13

32

13 4/4/

4/4/

ii

ii

ee

ee

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

v elements orthonormal basis in Cv

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 common elements

each 2 proj. subspaces 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

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).

)(

)()()(

1

1

nCSXx

CS

xdxf

X

xfn

Quantum designs

• Welch inequalitiesFor any set X of unit vectors in Cn and any natural

number k holds:

(L.R.Welch, 1974)

k

knX

yxXyx

k

11

2

,

2

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

are of equal length and pairwise orthogonal.

kkkww

kk

kW n

kn

k

n

n 1

)()(1

1

1

,,

MUBs

The known (n+1)-MUBs can be expressed in form:

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;

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.

)())((1

)( )( llfn

v krlr

k

m

jjj

ja ba

n

ib

1

2exp)(

Link with combinatorial designs

• Perfect non-linear functionsA function f: GG is said to be perfect non-

linear 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.


• 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.


• 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.


A function f: GG 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}.


• 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.


• Collineation of a projective plane:

a transformation of the plane that maps collinear points into collinear points.


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

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 non-linear functions. The existence of such functions for arbitrary dimension is still an open question.

)())((1

)( )( llfn

v krlr

k

Thank you for the attention!

Questions?

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MUBs and some other quantum designs