Preliminaries Lines Colouring Quantum Walks Quantum Problems Chris Godsil University of Waterloo Plzeň, 4 October, 2016 Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Outline Preliminaries 1 Lines 2 Equiangular Lines Mutually unbiased bases Colouring 3 Quantum Walks 4 Basics Which vertices are involved in PST? Near Enough Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Cosmology Quote Hydrogen is a colorless, odorless gas which given sufficient time, turns into people. (Henry Hiebert) Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Axioms Quote “The axioms of quantum physics are not as strict as those of mathematics” Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Outline Preliminaries 1 Lines 2 Equiangular Lines Mutually unbiased bases Colouring 3 Quantum Walks 4 Basics Which vertices are involved in PST? Near Enough Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Complex lines A line in C d can be represented by any vector that spans it. If x spans a line then P = ( x ∗ x ) − 1 xx ∗ represents orthogonal projection onto the line spanned by x (and is independent of the choice of spanning vector). Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Angles between complex lines Definition The angle between the lines spanned by unit vectors x and y is determined by |� x, y �| = | x ∗ y | . If P and Q are the projections xx ∗ and yy ∗ , then tr( PQ ) = tr( xx ∗ yy ∗ ) = tr( y ∗ xx ∗ y ) = |� x, y �| 2 . Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Equiangular sets of lines Definition A set of complex lines in C d given by projections P 1 , . . . , P n is equiangular if there is a real scalar α such that tr( P r P s ) = α whenever r � = s . Theorem The maximum size of an equiangular set of lines in C d is d 2 . Physicists refer to an equiangular set of d 2 lines in C d as a SIC-POVM. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Equiangular sets of lines Definition A set of complex lines in C d given by projections P 1 , . . . , P n is equiangular if there is a real scalar α such that tr( P r P s ) = α whenever r � = s . Theorem The maximum size of an equiangular set of lines in C d is d 2 . Physicists refer to an equiangular set of d 2 lines in C d as a SIC-POVM. [It’s better not to ask.] Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases A conjecture Conjecture For each d ≥ 2 , there is a set of d 2 equiangular lines in C d . A more refined version of this asserts that the set of lines is an orbit under the action of the Weyl-Heisenberg group. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases A question about chromatic number Let X ( d ) be the graph on lines in C d , where lines given by projections P and Q are adjacent if tr( PQ ) = ( d + 1) − 1 . Then ω ( X ( d )) ≤ d 2 . Problem What is the chromatic number of X ( d ) ? If there is an equiangular set of d 2 lines in C d then tr ( PQ ) = ( d + 1) − 1 . Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Outline Preliminaries 1 Lines 2 Equiangular Lines Mutually unbiased bases Colouring 3 Quantum Walks 4 Basics Which vertices are involved in PST? Near Enough Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Flat matrices We now concern ourselves with orthonormal bases in C d . Any such basis can be presented as the columns of a d × d unitary matrix. Definition A square (real or complex) matrix is flat if all its entries have the same absolute value. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Flat matrices We now concern ourselves with orthonormal bases in C d . Any such basis can be presented as the columns of a d × d unitary matrix. Definition A square (real or complex) matrix is flat if all its entries have the same absolute value. A flat real orthogonal matrix is a scalar multiple of a Hadamard matrix. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Flat matrices We now concern ourselves with orthonormal bases in C d . Any such basis can be presented as the columns of a d × d unitary matrix. Definition A square (real or complex) matrix is flat if all its entries have the same absolute value. A flat real orthogonal matrix is a scalar multiple of a Hadamard matrix. A flat unitary matrix is a scalar multiple of a complex Hadamard matrix. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Mutually unbiased bases Definition Let β 1 and β 2 be two orthonormal bases in C d , and let U 1 and U 2 respectively be unitary matrices whose columns are β 1 and β 2 . The two bases are mutually unbiased if the product U ∗ 1 U 2 is flat. Equivalently the two bases are mutually unbiased if the change-of-basis matrix is flat. (It is necessarily unitary.) Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Sets of mutually unbiased bases Theorem The maximum size of a set of mutually unbiased bases in C d is d + 1 . Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Sets of mutually unbiased bases Theorem The maximum size of a set of mutually unbiased bases in C d is d + 1 . A set of d + 1 mutually unbiased bases in C d are only known to exist if d is a prime power. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Sets of mutually unbiased bases Theorem The maximum size of a set of mutually unbiased bases in C d is d + 1 . A set of d + 1 mutually unbiased bases in C d are only known to exist if d is a prime power. All known examples can be constructed using commutative semifields and translation planes. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Sets of mutually unbiased bases Theorem The maximum size of a set of mutually unbiased bases in C d is d + 1 . A set of d + 1 mutually unbiased bases in C d are only known to exist if d is a prime power. All known examples can be constructed using commutative semifields and translation planes. If d is not a prime power then, in most cases, the best lower bound is three. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases Cayley graphs Definition ∈ C and g − 1 ∈ C for Suppose G is a group and C ⊆ G such that 1 / each g in C . The Cayley graph X ( G, C ) with connection set C is the graph with vertex set G , where group elements g and h are adjacent if hg − 1 ∈ C . Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases MUBs and cliques Let F denote the set of flat unitary matrices of order d × d and let U ( d ) be the unitary group. Then the maximum size of a set of mutually unbiased bases in C d is the maximum size of a clique in the Cayley graph X ( U ( d ) , F ) . Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks A finite subgraph of the unit sphere in R d Definition Let Φ( d ) denote the graph with the ± 1 -vectors of length d as its vertices, where two vectors are adjacent if and only if they are orthogonal. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks A finite subgraph of the unit sphere in R d Definition Let Φ( d ) denote the graph with the ± 1 -vectors of length d as its vertices, where two vectors are adjacent if and only if they are orthogonal. 1 If d is odd, Φ( d ) has no edges. Chris Godsil University of Waterloo Quantum Problems
Preliminaries Lines Colouring Quantum Walks A finite subgraph of the unit sphere in R d Definition Let Φ( d ) denote the graph with the ± 1 -vectors of length d as its vertices, where two vectors are adjacent if and only if they are orthogonal. 1 If d is odd, Φ( d ) has no edges. 2 If d ≡ 2 modulo four, Φ( d ) is bipartite. Chris Godsil University of Waterloo Quantum Problems
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