In combinatorics, a meander on points is a closed, self-avoiding plane curve, which intersects a given line times. Below, the meanders of a river (Rio Cauto, in Cuba) and a mathematical meander are represented.
The following more general objects are considered in the literature.
Definition 1 A meandric system on points with connected components is a collection of non-intersecting closed curves, which cross a given line times. A meander is a meandric system with component.
Below are represented two meandric systems on points, with , respectively connected components.
One easily recognizes the diagrams above and below the horizontal line: these are non-crossing pairings, a subclass of non-crossing partitions. Non-crossing pairings on points are known to be in bijection with (general) non-crossing partitions on points, through an operation called fattening; in the picture below, the same meanders as above are depicted, together with the corresponding non-crossing partitions on points; see how the meander (red) is a ``fattening'' of the pair of non-crossing partitions (black).
In our paper, we study meandric systems on points with connected components, for some fixed parameter . We obtain the general form of the generating function for the number of such meanders, with exact expressions for . Some trivial observations can be made right from the start. First, we have seen that meandric systems are in bijection with pairs of non-crossing partitions. Since non-crossing partitions are counted by Catalan numbers, we have
Note also that meandric systems with a maximal number of connected components (see the example on the right in the previous two pictures) correspond to the case where the two non-crossing partitions defining the meander are equal , and thus
To compute the numbers , it is not hard to see that the two non-crossing partitions and must be somehow ``close''. To make this precise, we consider another key bijection, the one between non-crossing partitions and the so-called geodesic permutations. In his seminal work Some properties of crossings and partitions, Biane realized that non-crossing partitions are in bijection with the following set of permutations
where is the full-cycle permutation and is the length function of permutations: is the minimal number of transpositions which multiply to . All these ideas are discussed in Lecture 23 of the excellent monograph of Nica and Speicher. In this framework, it has been known for some time that meanders on points with connected components are in bijection with the set
In particular, for , only pairs of geodesic permutations which differ by only one transposition contribute; one can easily show then
In our paper, we use the moment-cumulant formula from Free Probability Theory to show that, for fixed , the generating function for the number of meanders on points with connected components
after the change of variables , reads
where are polynomials of degree at most . With the help of a computer, we find these polynomials up to . We show that such meandric systems are made of simple building blocks, called irreducible meanders. Irreducible meanders were introduced by Lando and Zvonkin here, and they were recently featured in Andu Nica's recent paper.
Let me close this post by mentioning what is probably the most important problem related to the enumeration of meanders (on which we do not touch upon): the asymptotic growth of meanders. This problem and the one we discuss in our paper sit at opposite ends of the spectrum with respect to the number of components.
The sequence of meandric numbers is conjectured to grow like
for some constants . The exponential rate has been shown to satisfy . The polynomial speed is conjectured to be exactly .
The meandric numbers have a very simple geometric interpretation: they count diameters in the metric space :
The asymptotic behavior of the sequence is largely open. Old and recent results suggest that solving this many-faceted problem might require some new insight or techniques.
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In this post I would like to discuss a marginal problem discussed in the preprint, that of quantum Latin squares. Latin squares are combinatorial objects which have received a lot of attention: they are simple but deep mathematical objects, and have found many applications, mainly in statistics, experience design, and information theory.
Definition 1 A Latin square is a matrix with the property that each row and each column of are permutations of .
Here is an example of a Latin square of size :
We are interested in a ''vector space'' version of Latin squares, called quantum Latin squares, a notion introduced in the recent work of Musto and Vicary. The generalization is obtained via the classical information quantum information dictionary, where alphabets are replaced by vector spaces and letters by unit vectors in those spaces.
Definition 2 A quantum Latin square (QLS) is a -tensor with the property that the vectors in each row and each column of form orthonormal bases of .
Below is an example of a QLS, taken from Musto and Vicary's paper, where is a basis of .
One can make some trivial observations right off the bat. First, the elements of a QLS must have unit Euclidean norm . The condition in the definition can be stated using the following matrices, having the vectors of as columns
for some fixed orthonormal basis of . Then, is a QLS iff the matrices , are unitary operators. Moreover, using these matrices, we can define a natural ''distance'' function to the set of quantum Latin squares
Every classical latin square can be seen as a quantum Latin square by using the same basis for all rows and columns: . For , there exist moreover purely ''quantum'' examples, such as the one pictured above. The set of quantum Latin squares is a real algebraic variety, since the unitarity conditions for the matrices and can be expressed as polynomial equations in the variables , .
In our paper, QLS appear in connection to bipartite unitary operators with the property that for all ancilla space density matrices , the quantum channel
leaves the diagonal subalgebra invariant (see Theorem 4.4).
We raise the question of finding a natural probability measure of the set . We propose such a measure, based on a non-commutative analog of the Sinkhorn-Knopp algorithm (see here and here for the original papers). The following algorithm is discussed in our paper. Below, we make use of an operator , which returns the angular part of the polar decomposition: if is the polar decomposition of , then ; in case the polar decomposition is not unique, one of the valid decompositions is considered.
We conjecture that the algorithm above converges for almost any choice of the random points at step (2). We have some numerical evidence for this claim, as well as a convergence proof for a close variant of the problem, where we replace the vector entries of the matrix by positive definite matrices (see the appendix of our paper).
The algorithm above was also discussed in a paper with Teo Banica on the quantum permutation group ; there, we were interested in the probability measure induced on by the uniform measure in step (2).
Finally, let me mention that there exists a different non-commutative generalization of the Sinkhorn algorithm, introduced by Gurvits in his paper where he shows that the weak membership problem for the set of separable states is NP-hard.
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1. Anti-symmetric vectors and matrices
For order two tensors , there are only two symmetry classes, the symmetric and the antisymmetric vectors. The antisymmetric vectors form a vector space
where the vectors in the span can be shown to form an orthonormal basis of the -dimensional subspace , whenever form a basis of . Let be the orthogonal projection on the subspace . It is easy to see that has entries in and it looks as below ().
Via the usual isomorphism , one can see antisymmetric tensors as antisymmetric (or skew-symmetric) matrices: one simply has to rearrange the complex coordinates of the tensor in an matrix, respecting the ordering of bases. Note that in this way we obtain antisymmetric () and not anti-Hermitian () elements.
2. Singular values of vectors in the antisymmetric subspace
It is well known that antisymmetric (complex) matrices can be 2-block diagonalized using orthogonal rotations
The matrix in the middle has either diagonal blocks of size 2, as shown, or null diagonal elements. Hence, the non-zero eigenvalues of an antisymmetric matrix come in pairs . Since antisymmetric matrices are normal, their singular values are just the moduli of the eigenvalues, so non-zero singular values have multiplicity at least 2. We conclude that
Actually, it is easy to see we have equality, since the vector
Theorem 1. The set of all possible singular values of antisymmetric vectors inside is given by
References
[ghp] A. Grudka, M. Horodecki and L. Pankowski. Constructive counterexamples to the additivity of the minimum output Rényi entropy of quantum channels for all . J. Phys. A: Math. Theor. 43 425304.
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First, note that I am using the European notation for periodic decimal expansions (or repeating decimals). What you get is a periodic expansion, where all the two-digit numbers from 00 to 99, except 98, appear in the periodic part. Ok, there seems to be some magic going on here, let us try to understand what is happening.
First, note that , so probably there's nothing special about the 2 in there. Indeed, try
Good, let us go ahead and generalize this a little more. The 9 in there is the number of fingers you are left with when you lose one, so the same should be true for creatures anatomically different (by the way, there is ice on Mercury, and maybe more !):
Before stating and proving a general result, let us discuss the easiest case (for us humans), 1/81. The idea comes from [cgo] and the starting point is the formula
which is the power series expansion of around . For , we get
At this point, the expression above starts to look like a decimal expansion - the problem is that some of the numerators in the brackets are not digits! To get around this, write
which is the announced formula. The general result can be stated as follows.
Proposition 1. Let be a fixed basis. Then, when everything is written in basis , for any , one has
where each group containing in the expression above has digits and denotes the digit in basis .
Proof: We shall start from the RHS of the equation and work our way towards the LHS. Recall that a periodic decimal expansion in base can be written as a fraction as follows:
First, let us focus on the case and treat the general case at the end of the proof. In this particular case, the RHS of the equation in the statement is equal to
where the 1 in front of the sum stands for the fact that is replaced by as the last digit of the period. Using the formulas
we can write
from which the conclusion easily follows. For the general case, note that the following decimal expansion holds for groups of digits:
The proof follows then the same steps as above.
References
[cgo] A. Cheer and D. Goldston, Explaining the decimal expansion of 1/81 using calculus. Mathematics and Computer Education, 25-3 (1991).
]]>In what follows, all vector spaces shall be assumed to be complex and are fixed constants. For a vector , the symbol denotes its ordered version, i.e. and are the same up to permutation of coordinates and .
1. Singular values of vectors in a tensor product
Using the non-canonical isomorphism , one can see any vector
In QIT, the decomposition of above is called the Schmidt decomposition and the numbers are called the Schmidt coefficients of the pure state .
2. The singular value set of a vector subspace
Consider now a subspace of dimension and define the set
Below are some examples of sets , in the case , where the simplex is two-dimensional. In all the four cases, and . In the last two pictures, one of the vectors spanning the subspace has singular values .
3. Basic properties
Below is a list of very simple properties of the sets .
Proposition 1. The set is a compact subset of the ordered probability simplex having the following properties:
Proof: The first three statements are trivial. The last one is contained in [cmw], Proposition 6 and follows from a standard result in algebraic geometry about the dimension of the intersection of projective varieties.
4. So, what is the problem ?
The question one would like to answer is the following:
How does a typical look like ?
In order to address this, I will introduce random subspaces in the next post future. In the next post, I look at the special case of anti-symmetric tensors.
References
[bcn] S. Belinschi, B. Collins and I. Nechita, Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product, to appear in Invent. Math.
[cn2] B. Collins and I. Nechita, Random quantum channels II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems, Adv. in Math. 226 (2011), 1181--1201.
[cmw] T. Cubitt, A. Montanaro and A. Winter, On the dimension of subspaces with bounded Schmidt rank, J. Math. Phys. 49, 022107 (2008).
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