1. MUBs and some other quantum designs Aleksandrs Belovs and Juris Smotrovs

2. 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

3. History of combinatorial designs

4. 36 officer problem (1782): INVESTIGATIONS ON A NEW TYPE OF MAGIC SQUARE LEONHARD EULER A very curious question that has taxed the brains of many inspired me to undertake the following research that has seemed to open a new path in Analysis and in particular in the area of combinatorics. This question concerns a group of thirty-six officers of six different ranks, taken from six different regiments, and arranged in a square in a way such that in each row and column there are six officers, each of a different rank and regiment. But after spending much effort to resolve this problem, we must acknowledge that such an arrangement is absolutely impossible, though we cannot give a rigorous proof.

5. Combinatorial designs 36 officer problem (L.Euler, 1782) An example with a simpler case with 9 officers: 123 231 312 123 312 231 Euler conjectured that there is no solution for the 6X6 case, and, in general, for the (4n+2)X(4n+2) case.

6. 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.

7. 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.

8. 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?

9. Combinatorial designs Example v=7, b=7, k=3, r=3, =1 B1B1 B2B2 B3B3 B4B4 B5B5 B6B6 B7B7 1234560 2345601 4560123

10. Optimal quantum measurement A pure quantum state is a vector (denoted something like |  ) of unit length in the vector space C n. 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:  = p 1 |  1  1 | + p 2 |  2  2 | +... + p k |  k  k |.

11. Optimal quantum measurement Problem Suppose we have many instances of the same state  in C n. 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?

12. Optimal quantum measurement Case 1: we are allowed measurements only within the given space C n ; 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.

13. Optimal quantum measurement Case 2: we are allowed to measure in a larger space C m which contains the given space C n Such measurement from the viewpoint of the given space C n 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.

14. MUBs A number of orthonormal bases in C n 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 C n.

15. MUBs An example: 3-MUB in C 2.

16. MUBs I.D. Ivanovic (1981), W.K.Wootters, B.D.Fields (1989): (n+1) -MUB exists for any dimension n=p m, where p is prime: r is base index, k is vector index, l is component index; r, k, l  GF(p m ), Tr is the trace GF(p m )  GF(p).

17. 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?

18. SIC POVMs A set of n 2 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  | =.

19. SIC POVMs An example: SIC POVM in C 2.

20. 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.

21. Quantum designs G.Zauner (1999): Block designQuantum design v elements orthonormal basis in C v b blocks b orthogonal projections k elements in each blockeach projection is to a k -dimensional subspace each element in r blockseach basis vector is in r projection subspaces each 2 blocks have common elements each 2 proj. subspaces intersection dim =

22. Quantum designs G.Zauner (1999): Quantum design is a set {P 1,..., P b } of projection operators in C v. It is called regular iff there is such k that Tr(P i ) = k for all i. It is called coherent iff there is such r that P 1 +... + P b = rE. Its degree s is the number of elements in the set  = {Tr(P i P j ) | i  j} = { 1,..., s }.

23. 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 C n 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.

24. Quantum designs SIC POVMs as quantum designs SIC POVM in C n is a quantum design with parameters: v = n, b = n 2, k = 1, r = n, the degree s = 1, and 1 = 1/(n+1).

25. Quantum designs Complex projective t -design: A set X of unit vectors in C n such that for any polynomial f of degree t on the complex projective sphere CS n-1 (formed by equivalence classes of unit vectors in C n where collinear vectors are considered equivalent).

26. Quantum designs Welch inequalities For any set X of unit vectors in C n and any natural number k holds: (L.R.Welch, 1974)

27. 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.

28. Quantum designs A.Belovs, J.Smotrovs (2008): Let X be a set of unit vectors in C n. Let B be a matrix formed by vectors from X as columns. Let w 1,..., w n 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.

29. 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 /n 1 Z ...  Z /n m Z of size n= n 1... n m ; 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.

30. Link with combinatorial designs Perfect non-linear functions A function f: G  G 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)=x 2 in Z /p Z, where p is prime, is perfect non-linear. These functions are much studied in cryptography, but mostly in the binary case n=2 m.

31. Link with combinatorial designs Difference sets A set D={d 1,...,d k } of k elements from an Abelian group G of size v is said to form a ( v, k, )-difference set iff the differences d i  d j with i  j contain each non-zero element of G exactly times. A long-known special case of balanced incomplete block designs.

32. Link with combinatorial designs Relative difference sets If G is an Abelian group, and N its subgroup, then a subset D={d 1,...,d k } of G is called an ( m, n, k, )-relative difference set iff |N|=n, |G|/|N|=m, and the differences d i  d j with i  j contain no element from N, and each of the other non-zero elements of G exactly times.

33. 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 G 2 and its subgroup N={(x,0) | x  G}.

34. 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.

35. Link with combinatorial designs Collineation of a projective plane: a transformation of the plane that maps collinear points into collinear points.

36. Link with combinatorial designs A.Blokhuis, D.Jungnickel, B.Schmidt (2001): If G is an Abelian collineation group of order n 2 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 C n where n is a prime power.

37. What further? The formula gives an (n+1) -MUB in C n also when f is a function of a more general kind: Z /n 1 Z ...  Z /n m Z  R /n 1 R ...  R /n m R with properties similar to those of perfect non- linear functions. The existence of such functions for arbitrary dimension is still an open question.

38. Thank you for the attention! Questions?