MUBs and some other quantum designs

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

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History of combinatorial designs
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History of combinatorial designs

• 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.

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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
(4n2)X(4n2) case.
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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
  (4n2)X(4n2) 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.

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Combinatorial designs

• Kirkmans schoolgirl problem (1850) and Steiner
  triples (solved)
• Finite geometries (projective, affine,...)
• Difference sets
• Hadamard matrices
• Modern combinatorial design theory started with
  R. Fishers work on design of statistical
  experiments in 1930s.

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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?

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Combinatorial designs

• Example
• v7,
• b7,
• k3,
• r3,
• ?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
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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
• ?? ?00? ?11? ... ?n-1n-1?.
• When measured in this basis, one of the basis
  states i? is obtained with probability ?i2,
  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.

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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?

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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
  n1 mutually unbiased bases (MUBs) if such
  exists.

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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.

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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
• ?xy?
• There can be no more than n1 such bases in Cn.

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MUBs

• An example 3-MUB in C2.

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MUBs

• I.D. Ivanovic (1981),
• W.K.Wootters, B.D.Fields (1989)
• (n1)-MUB exists for any dimension npm, 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).

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MUBs

• Does an (n1)-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 n6. At the moment
  only a 3-MUB is known in 6 dimensions.
• If an (n1)-MUB does not exist, then what is the
  maximal number of MUB that exist in any given
  dimension?

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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
• ?xy? .

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SIC POVMs

• An example SIC POVM in C2.

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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 lt 45 by
  finding approximate SIC POVMs numerically.

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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 ?
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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.

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Quantum designs

• MUBs as quantum designs
• If we consider MUB as consisting not of vectors,
  but of projections on their lines, then an
  (n1)-MUB in Cn is a quantum design with
  parameters
• v n, b n(n1), k 1, r n1,
• the degree s 2, and ?1 0, ?2 1/n.

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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/(n1).

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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).

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Quantum designs

• Welch inequalities
• For any set X of unit vectors in Cn and any
  natural number k holds
• (L.R.Welch, 1974)

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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.

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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.

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MUBs

• The known (n1)-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
  (n1)-MUB iff this function is perfect non-linear.

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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(xa) ? 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 n2m.

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Link with combinatorial designs

• Difference sets
• A set Dd1,...,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.

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Link with combinatorial designs

• Relative difference sets
• If G is an Abelian group, and N its subgroup,
  then a subset Dd1,...,dk of G is called an
  (m,n,k,?)-relative difference set iff Nn,
  G/Nm, 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.

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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.

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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.

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Link with combinatorial designs

• Collineation of a projective plane
• a transformation of the plane that maps collinear
  points into collinear points.

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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.

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What further?

• The formula
• gives an (n1)-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.

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Thank you for the attention!Questions?