Noncommuting Variables Research Articles

De Finetti theorems for easy quantum groups

We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups S_n, O_n, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of K\"ostler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.

Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

Let V be a complex vector space with basis {x 1, x 2, . . . , x n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x 1, x 2, . . . , x n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words.

The tracial moment problem and trace-optimization of polynomials

The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace \({f(\underline {A})}\) can attain for a tuple of matrices \({\underline {A}}\)? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satisfies a certain condition called flatness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the Artin-Wedderburn theorem for matrix *-algebras due to Murota, Kanno, Kojima, Kojima, and Maehara. Trace-optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side—two topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics.

Noncommutative plurisubharmonic polynomials part I: Global assumptions

We consider symmetric polynomials, p, in the noncommutative (nc) free variables { x 1 , x 2 , … , x g } . We define the nc complex hessian of p as the second directional derivative (replacing x T by y) q ( x , x T ) [ h , h T ] : = ∂ 2 p ∂ s ∂ t ( x + t h , y + s k ) | t , s = 0 | y = x T , k = h T . We call an nc symmetric polynomial nc plurisubharmonic (nc plush) if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n × n matrices for every size n; i.e., q ( X , X T ) [ H , H T ] ≽ 0 for all X , H ∈ ( R n × n ) g for every n ⩾ 1 . In this paper, we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables; i.e., an nc symmetric polynomial p is nc plush if and only if it has the form (0.1) p = ∑ f j T f j + ∑ k j k j T + F + F T where the sums are finite and f j , k j , F are all nc analytic. In this paper, we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper (J.M. Greene, preprint [6]), proves that if the nc complex hessian, q, of p takes positive semidefinite values on an “nc open set” then q takes positive semidefinite values on all tuples X, H. Thus, p has the form in Eq. (0.1). The proof, in J.M. Greene (preprint) [6], draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions.

Multilinear polynomials and cocentralizing conditions in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x 1,..., x n ) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x 1, …, xn) on R. If F and G are generalized derivations of R such that [[F(x), x], [G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds: (1) there exists α ∈ C such that F(x) = αx for all x ∈ R (2) there exists β ∈ C such that G(x) = βx for all x ∈ R (3) f(x 1, …, x n )2 is central valued on R and either there exist a ∈ U and α ∈ C such that F(x) = ax + xa + αx for all x ∈ R or there exist c ∈ U and β ∈ C such that G(x) = cx + xc + βx for all x ∈ R (4) R satisfies the standard identity s 4(x 1,…, x 4) and either there exist a ∈ U and α ∈ C such that F(x) = ax + xa + αx for all x ∈ R or there exist c ∈ U and β ∈ C such that G(x) = cx + xc + βx for all x ∈ R.

The cyclicity problem for the images of Q-rational series

We show that it is decidable whether or not a given Q-rational series in several noncommutative variables has a cyclic image. By definition, a series r has a cyclic image if there is a rational number q such that all nonzero coefficients of r are integer powers of q .

Freeness of linear and quadratic forms in von Neumann algebras

We characterize the semicircular distribution by freeness of linear and quadratic forms in noncommutative random variables from tracial W ⁎ -probability spaces with relaxed moment conditions.

Quantum mechanics onSO(3)via noncommutative dual variables

We formulate quantum mechanics on SO(3) using a non-commutative dual space representation for the quantum states, inspired by recent work in quantum gravity. The new non-commutative variables have a clear connection to the corresponding classical variables, and our analysis confirms them as the natural phase space variables, both mathematically and physically. In particular, we derive the first order (Hamiltonian) path integral in terms of the non-commutative variables, as a formulation of the transition amplitudes alternative to that based on harmonic analysis. We find that the non-trivial phase space structure gives naturally rise to quantum corrections to the action for which we find a closed expression. We then study both the semi-classical approximation of the first order path integral and the example of a free particle on SO(3). On the basis of these results, we comment on the relevance of similar structures and methods for more complicated theories with group-based configuration spaces, such as Loop Quantum Gravity and Spin Foam models.

The images of non-commutative polynomials evaluated on 2×2 matrices

Let p p be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field K K of any characteristic. It has been conjectured that for any n n , the image of p p evaluated on the set M n ( K ) M_n(K) of n n by n n matrices is either zero, or the set of scalar matrices, or the set s l n ( K ) sl_n(K) of matrices of trace 0, or all of M n ( K ) M_n(K) . We prove the conjecture for n = 2 n=2 , and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of M n ( K ) M_n(K) .

NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials

NCSOStools is a Matlab toolbox for • symbolic computation with polynomials in noncommuting (NC) variables; • constructing and solving sum of Hermitian squares (with commutators) programs for polynomials in NC variables. It can be used in combination with semidefinite programming software, such as SeDuMi, SDPA or SDPT3, to solve these constructed programs. This paper provides an overview of the theoretical underpinning of the sum of Hermitian squares (with commutators) programs and provides a gentle introduction to the primary features of NCSOStools.

Non-commutative integrability, paths and quasi-determinants

In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.

Time-symmetric quantum mechanics questioned and defended

The article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is stimulating and raises some interesting and profound issues regarding the foundations of quantum mechanics and quantum measurements. One point the authors make is that a measurement of spin component √2‾/2 of a spin-1⁄2 particle is unphysical and can be attributed only to errors in weak measurements performed on a collection of N spins. I offer a more mundane interpretation that does not require the introduction of any error or postselection concepts. The physical, textbook picture of spin-1⁄2 is a vector of length √S(S + 1) = √3‾/2 with some distribution of orientations (that is, polar coordinates θ and φ). A conventional single measurement of Sz can yield only one of the eigenvalues ±1⁄2. This may be interpreted in the classical vector model by envisioning that the spin lies on a cone with θ defined by cosθ = √3‾/3. The expectation value of Sx then vanishes due to the uniform distribution of φ. Having some control over φ can yield a finite value of Sx . Thus I would argue that only an expectation value greater than √3‾/2 is unphysical; a value of √2‾/2 is quite physical and can even be interpreted classically in terms of some distribution of φ and θ.I would phrase the important observation of Aharonov and coauthors in a different way. While spin-1⁄2 has a magnitude of √3‾/2, a fundamental limitation of conventional single-point quantum measurements is that they can yield only the values +1⁄2 and −1⁄2 and any expectation value must therefore lie between those two extremes. Classically, therefore, the spin may not be fully aligned along the z-axis with θ = 0. Such alignment is impossible since it would yield well-defined values of the three noncommuting variables Sz = √3‾/2 and Sx = Sy = 0. However, once multiple time measurements are performed, one can define a reduced distribution of some of the measurements that is conditional on the outcome in the other measurements; such a distribution can be interpreted in terms of Sz greater than 1⁄2 but less than or equal to √3‾/2. That interpretation does not violate any of quantum mechanics’ fundamental rules, which do not usually consider such quantities.Rather than a new, time-symmetric formulation of quantum mechanics that involves pre- and postselection, one can simply treat the scheme of the authors’ figure 1b as a three-point correlation function, whereas figures 2a and 2b show a four-point correlation function, each panel having its own set of conditional probabilities. Error, time reversal, and postselection need not be invoked. Instead, one may think in multiple dimensions and develop the right language for the interpretation of the observables.The combination of pre- and postselection is an attempt to create an artificial ensemble that reproduces the results of multiple-point measurements in a single one-point measurement. As shown by Aharonov and coauthors, that is possible by introducing errors. An alternative physical picture is obtained by retaining the multipoint analysis. Multidimensional thinking is well developed in coherent nonlinear spectroscopy, where a system of spins or optical chromophores are subjected to sequences of short impulsive pulses.11. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . , 22. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . Similar ideas may be applied for the interpretation of multiple measurements. One can think of various types of n-point observables obtained by combining n − m perturbations and m measurements. The nonlinear n-point response functions in spectroscopy represent n − 1 impulsive perturbations followed by a single measurement. The objects Aharonov and colleagues considered correspond to n = m. Proper multiple-distribution functions could then be naturally used for the interpretation of such generalized measurements.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . Google Scholar2. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Google ScholarCrossref, ISI Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; , Google ScholarCrossref, ISI Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . , Google Scholar© 2011 American Institute of Physics.

Quantum invariant families of matrices in free probability

We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued R-cyclicity condition. This is a surprising contrast with the Aldous–Hoover characterization of jointly exchangeable arrays.

On a New Family of Generalized Stirling and Bell Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

Trace-positive polynomials

In this paper positivity of polynomials in free noncommuting variables in a dimension-dependent setting is considered. That is, the images of a polynomial under finite-dimensional representations of a fixed dimension are investigated. It is shown that unlike in the dimension-free case, every trace-positive polynomial is (after multiplication with a suitable denominator—a a Hermitian square of a central polynomial) a sum of a positive semidefinite polynomial and commutators. Together with our previous results this yields the following Positivstellensatz: every trace-positive polynomial is modulo sums of commutators and polynomial identities a sum of Hermitian squares with weights and denominators. Understanding trace-positive polynomials is one of the approaches to Connes' embedding conjecture.

Joint similarity to operators in noncommutative varieties

In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in $[B(\cH)^n]_1$ associated with positive regular free holomorphic functions in $n$ noncommuting variables and with sets of noncommutative polynomials in $n$ indeterminates, where $B(\cH)$ is the algebra of all bounded linear operators on a Hilbert space $\cH$. In particular, if $f=X_1+...+X_n$ and $\cP=\{0\}$, the elements of the corresponding variety can be seen as noncommutative multivariable analogues of Agler's $m$-hypercontractions.

The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables

We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the antipode.

Partitions, Rooks, and Symmetric Functions in Noncommuting Variables

Let $\Pi_n$ denote the set of all set partitions of $\{1,2,\ldots,n\}$. We consider two subsets of $\Pi_n$, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let ${\cal E}_n\subseteq\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, ${\cal T}_{n-1}$. Given $\pi\in\Pi_m$ and $\sigma\in\Pi_n$, define their slash product to be $\pi|\sigma=\pi\cup(\sigma+m)\in\Pi_{m+n}$ where $\sigma+m$ is the partition obtained by adding $m$ to every element of every block of $\sigma$. Call $\tau$ atomic if it can not be written as a nontrivial slash product and let ${\cal A}_n\subseteq\Pi_n$ denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of $NCSym$, the symmetric functions in noncommuting variables. We show that, despite their very different definitions, ${\cal E}_n={\cal A}_n$ for all $n\ge0$. Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to $NCSym$. We end with some remarks.

Skew quasisymmetric Schur functions and noncommutative Schur functions

Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset LC that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ:NSym→Sym. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ.As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.

The f-Vector of the Descent Polytope

For a positive integer n and a subset S⊆[n−1], the descent polytope DP S is the set of points (x 1,…,x n ) in the n-dimensional unit cube [0,1] n such that x i ≥x i+1 if i∈S and x i ≤x i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F S (t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.