Let us unpack all these technical terms. First, unitary operators: these are linear transformations (or matrices) which are rotation-like, meaning that they are invertible and they do not modify the magnitude of the vectors they act on. Here is a counterclockwise 45° (or ) rotation:
and below is a unitary matrix realizing the symmetry with respect to the first coordinate axis in a plane
In the general case, unitary matrices satisfy the condition , where the star operation corresponds to taking the (Hermitian) adjoint (complex conjugation, followed by transposition). The set of unitary matrices form a compact group under multiplication, and this group admits a unique probability measure which is invariant under left/right translations by group elements: the Haar measure. As it turns out, in the case of the unitary group (and even more so for its relatives, the orthogonal and the symplectic groups), the Haar measure is complicated, and computing integrals with respect to it is a daunting task.
But before we get into computing integrals with respect to the Haar measure, let me address an even more fundamental question regarding it: sampling. Producing a Haar-distributed random unitary matrix on a computer is quite easy: construct a random matrix with enough symmetry (say, a Ginibre or a GUE matrix), compute its singular value/eigenvalue decomposition, and just take one of the unitary matrices appearing in that decomposition. The initial symmetry of your random matrix will translate into the invariance property that uniquely identifies the Haar measure (although there are some minor subtleties, see here or here). Numerous software implementations of this procedure exist, here is one example in a MATLAB library popular with the Quantum Information Theory crowd.
Once that we have properly defined the Haar measure and that we know how to produce samples from it, the next question that comes to mind is how to compute integrals with respect to it. This problem was first considered in the physics literature by Weingarten in the asymptotic limit of large dimension. The mathematical theory was developed, rigorously and in full generality, by Benoit Collins (here, and later with Piotr Śniady), who also coined the name Weingarten formula for the following result.
If then
Here we denoted by the unitary group acting on an -dimensional Hilbert space, and by the symmetric group on elements. The integrals are taken with respect to the normalized Haar measure on . The function is called the unitary Weingarten function.
The above formula allows one to compute the expectation of any monomial (and thus, by linearity, any polynomial) in the entries of a Haar-distributed random unitary matrix. But there is one caveat: the Weingarten function appearing in the formula above is an object of combinatorial nature, which has no explicit formula for general . Nonetheless, for fixed monomial order , there exist algorithmic procedures to compute the values for all permutations and thus one can compute averages of monomials such as
As it turns out, in practical applications, one does not encounter often polynomials in the entries of a random unitary matrix . It is more likely to be faced with expressions which involve (traces of) polynomials in the matrix itself, together with some other given matrices. For instance, the following type of expression is quite common (in, e.g., Quantum Information Theory): , where are two given, fixed matrices. One could, of course, write the trace in coordinates, and compute the expectation as a sum where the individual terms are calculated with the help of the Weingarten formula. Here's where RTNI comes to the rescue, saving you all that trouble (we use here the Mathematica implementation of RTNI):
In[1]:= MultinomialexpectationvalueHaar[d, {1, 2}, {X, Y}, True]
Out[1]= (Tr[X] Tr[Y])/d
Above, the MultinomialexpectationvalueHaar function computes , where (meaning we simply take ) and (meaning we take ). Using this function, one can compute (the trace of) any linear expression involving Haar-distributed random unitary matrices and deterministic matrices.
But RTNI can do much more than that! It can compute expectation values of (non-linear) graphs of matrices (i.e. tensors) involving random unitary matrices which can act on tensor products of vector spaces. Such situations are very common in quantum information theory and in theoretical physics in general, where networks of tensors are becoming very popular. One starts from the description of a tensor in Penrose graphical notation; is a graph where the vertices correspond to tensors, and edges correspond to tensor contraction. Importantly, the tensor product operation corresponds to the disjoint union of graphs. Here is an example coming from the theory of random quantum channels. Consider a random quantum channel given by a Haar-distributed random isometry ( is just the truncation to the first columns of a random unitary matrix )
Such random quantum channels were first considered by Hayden and Winter in relation to the additivity conjecture in quantum information theory, solved in the negative by Hastings. In such considerations, one bounds the minimum output entropy of the tensor product channel by using the overlap
between the output of the channels acting on a maximally entangled state with another maximally entangled state (on the output spaces) . Recall that the maximally entangled state is defined, in general, by , where the vector reads
with being an orthonormal basis of . Here is the complex conjugate channel, obtained by taking the entry-wise conjugate of . The diagram for the scalar is given below
In[1]:= e1 = {{"U*", 2, "out", 1}, {"U", 1, "in", 1}};
In[2]:= e2 = {{"U*", 1, "out", 1}, {"U", 2, "in", 1}};
In[3]:= e3 = {{"U", 1, "out", 1}, {"U*", 1, "in", 1}};
In[4]:= e4 = {{"U", 2, "out", 1}, {"U*", 2, "in", 1}};
In[5]:= e5 = {{"U", 1, "out", 2}, {"U*", 2, "in", 2}};
In[6]:= e6 = {{"U", 2, "out", 2}, {"U*", 1, "in", 2}};
In[7]:= g = {e1, e2, e3, e4, e5, e6};
In[8]:= listg = {{g, 1/(d k)}};
Note that the final command assigns a weight to the graph ; this corresponds to the normalization of the two maximally entangled states . The edge numbering in the code matches the numbering in the picture, as do the numberings of the different boxed and of the tensor factors. One has to call then the integrateHaarUnitary function of the RTNI package to compute the expected value of the scalar defined above with respect to the Haar measure on . The routine integrateHaarUnitary implements the graphical Weingarten calculus which I developed together with Benoit in this paper. It is basically the Weingarten formula, read in the Penrose graphical calculus. RTNI gives the following output, which we further manipulate in order to obtain the asymptotic overlap in the regime where :
In[9]:= Eg = integrateHaarUnitary[listg, "U", {d}, {n, k}, n k]
In[10]:= overlap = Eg[[1, 2]];
In[11]:= overlapt = overlap /. {d -> t n k};
In[12]:= Assuming[t > 0 && k > 1, Limit[overlapt, n -> Infinity]] // Simplify
Out[1]:= {{{}, -((d^2 k^2 n)/(d k - d k^3 n^2)) - (d k n^2)/(
d k - d k^3 n^2) + (d k^2 n)/(d k^2 n - d k^4 n^3) + (d^2 k n^2)/(
d k^2 n - d k^4 n^3)}}
Out[2]:= (1 - t)/k^2 + t
This study case emphasizes the strengths of the RTNI package: computing the expected overlap using the algebraic Weingarten formula would have required tedious summations and case analysis for index collisions. The RTNI package provides a simple, conceptual way of computing such expectations. RTNI can also display the different tensor networks one inputs, here are some examples:
If you are interested in using RTNI for your research, we would be happy to hear about it. Also, if you want to implement the killer feature of RTNI which is missing in the current release, please contact us!
Quantum measurements (mathematically modeled by POVMs) are a bunch of positive semidefinite operators which sum up to the identity: , where and . It is always useful to think of quantum theory as a non-commutative generalization of classical probability theory. In this situation, one can understand POVMs as matricial versions of probability vectors: both are objects having some positivity property and a normalization constraint (elements sum up to ).
The question we start from is the following:
How to define a natural probability measure on the set of POVMs?
Let us try and get some inspiration from the classical case, that is probability vectors. There exist distinguished measures on the probability simplex, the Dirichlet distributions.
To sample from a symmetrical Dirichlet distribution, one can start from independent Gamma-distributed random variables of parameter , and then set
To start, let us recall how one samples from the Wishart ensemble. Consider a Ginibre random matrix, i.e. a matrix with independent, identically distributed standard complex Gaussian entries; one produces such a matrix in MATLAB with the following command:
G = (randn(d, s) + 1i*randn(d, s))/sqrt(2)
The is there because the real and imaginary parts of a standard complex Gaussian random variable are independent real centered Gaussians, of variance . From such a random Ginibre matrix , one produces a Wishart matrix by setting . The result is a random positive semidefinite matrix, of typical rank .
Now, back to random POVMs. Produce independent Wisharts of parameters , call them . Being positive semidefinite, these almost look like random effect operators (the elements of a random POVM); they lack however the normalization property. We achieve this by "dividing" each by their sum. Careful here: when dividing operators, one has to do this in the proper way, by multiplying from the left and from the right with the half-inverse of the denominator (which needs to be positive semidefinite):
The MATLAB code which produces such a random quantum measurement is as follows (you can find this, and much more, in the ancillary files of our arXiv paper)
function ret = RandomHaarPOVM(d,k,s) % outputs a sample from the Haar-random POVM ensemble % INPUT % d = Hilbert space dimension of the POVM effect % k = number of outcomes (POVM elements) % s = environment parameter (must satisfy d lt;= ks) % OUTPUT % A = a kxdxd matrix, where A_i = A(i,:,:) is the i-th POVM element % METHOD % sample according to the Wishart ensemble: each effect is a Wishart % random matrix, normalized by the sum. ret = zeros(k, d, d); S = zeros(d); for i = 1:k Gi = RandomGinibre(d,s); Wi = Gi*Gi'; ret(i,:,:) = Wi; S = S + Wi; end S = mpower(S, -1/2); for i = 1:k Wi = squeeze(ret(i,:,:)); ret(i,:,:) = S*Wi*S; end end
In the paper, we actually use a different method to produce such random measurements, which is based on Haar-distributed random unitary matrices, and which more pleasant to work with analytically. Proving that these two methods are equivalent is one of our main results.
Having defined random POVMs, we then study their properties. We start with the spectrum of individual effect operators . The behavior of their eigenvalues, in the large limit, is given by Voiculescu's free probability theory.
We also study the probability that two random POVMs and are compatible: is there another POVM with elements such that the marginals of are and ? This question can be decided for concrete examples using semidefinite programming. However, in practice, when the size of the matrices becomes large, this technique can be computationally expensive. We then resort to compatibility and incompatibility criteria, which are simpler conditions that are only sufficient or only necessary for compatibility to hold (the situation is similar to the one in entanglement theory). We compare such compatibility criteria for generic POVMs, and we declare a winner! I will let you read the paper to find out which one that is. For incompatibility, the situation is more complicated: the known incompatibility criteria are very weak for random POVMs of large dimension with few outcomes. It is an open problem to invent new such criteria which can be insightful in this asymptotic regime.
]]>Start with copies of a finite dimensional complex Hilbert space, say . Consider a random complex Gaussian tensor, that is a vector of size with i.i.d. standard complex Gaussian entries. for a subset of cardinality , we define the marginal
obtained by partial tracing the tensor legs belonging to , the complementary set of . Moreover, we allow for twisted marginals: for a permutation , let
where permutes the tensor legs as follows:
We prove the following result.
Theorem 1 The set of twisted marginals are asymptotically free as . In other words, they jointly converge in distribution to a family of free standard free Poisson elements.
Recall that a free Poisson (or a Marchenko-Pastur) random variable has distribution
where is a positive parameter (in our result, ). We display below the density (red curve) and numerical simulations for
and for
The message of our main theorem is that although the (overlapping) marginals are correlated random variables, their distributions become asymptotically free as the size of the individual Hilbert spaces grows to infinity.
The freeness result proven in the balanced asymptotical regime (the tensor factors scale in the same way) fails in the unbalanced asymptotical regime (some of the tensor factors have fixed dimensions, whereas others grow to infinity). We emphasize this fact in a particular situation, for (4-partite tensors). Write , where
and denote, respectively, the four tensor factors above. It is natural to consider the joint distribution of the marginals , which are not asymptotically free anymore:
Theorem 2 As and are fixed, the marginals have the following asymptotical free cumulants:
where is an arbitrary word in the letters , and is the number of different consecutive values of , counted cyclically:
where .
The difference with the previous situation is that the amount of ''fresh'' randomness in the marginals is relatively small (fixed vs. ), and not enough to ensure the asymptotical freeness. The natural framework to tackle such situations is that of freeness with amalgamation, and we shall study these cases in some future work.
]]>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.
]]>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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