Nolan R. Wallach - University of California/San Diego

Nolan R. Wallach
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Nolan R. Wallach
University of California/San Diego
La Jolla
United States

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Pub Categories

Quantum Physics (12)
Mathematics - Representation Theory (9)
Mathematics - Mathematical Physics (8)
Mathematical Physics (8)
Mathematics - Differential Geometry (3)
Mathematics - Symplectic Geometry (2)
Mathematics - Analysis of PDEs (2)
Mathematics - Group Theory (2)
Mathematics - Combinatorics (2)
Mathematics - Algebraic Geometry (1)

Publications Authored By Nolan R. Wallach

Gleason's theorem shows the equivalence of von Neumann's density operator formalism of quantum mechanics and frame functions that arise from associating measurement probabilities to vectors on a sphere. Wallach's Unentangled Gleason's Theorem extends it to the regime of measurements by unentangled orthogonal bases (UOB) in a multi-partite system in which each subsystem is at least 3-dimensional. In this paper, we determine the structure of unentangled frame functions in general. Read More

The main result in this paper is the generalization of the Kostant-Rallis multiplicity formula to general {\theta}--groups (in the sense of Vinberg). The special cases of the two most interesting examples one for $E_6$ (three qubits) and one for $E_8$ are given explicit formulas. Read More

The stabilizer group of an n-qubit state \psi is the set of all matrices of the form g=g_1\otimes\cdots\otimes g_n, with g_1,... Read More

In his brilliant but sketchy paper on the strucure of quotient varieties of affine actions of reductive algebraic groups over C, Amnon Neeman introduced a gradiant flow with remarkable properties. The purpose of this paper is to study several applications of this flow. In particular we prove that the cone on a Zariski closed subset of n-1 dimensional real projective space is a deformation retract of n dimensional Euclidean space. Read More

An orthonormal basis consisting of unentangled (pure tensor) elements in a tensor product of Hilbert spaces is an Unentangled Orthogonal Basis (UOB). In general, for $n$ qubits, we prove that in its natural structure as a real variety, the space of UOB is a bouquet of products of Riemann spheres parametrized by a class of edge colorings of hypercubes. Its irreducible components of maximum dimension are products of $2^n-1$ two-spheres. Read More

This article has two objectives. The first is to give a guide to the proof of the (so-called) Casselman-Wallach theorem as it appears in Real Reductive Groups II. The emphasis will be on one aspect of the original proof that leads to the new result in this paper which is the second objective. Read More

We give an algorithm to solve the quantum hidden subgroup problem for maximal cyclic non-normal subgroups of the affine group of a finite field (if the field has order $q$ then the group has order $q(q-1)$) with probability $1-\varepsilon$ with (polylog) complexity $O(\log(q)^{R}\log(\varepsilon)^{2})$ where $R<\infty.$ Read More

We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such an SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Read More

We find the generating set of SL-invariant polynomials in four qubits that are also invariant under permutations of the qubits. The set consists of four polynomials of degrees 2,6,8, and 12, for which we find an elegant expression in the space of critical states. In addition, we show that the Hyperdeterminant in four qubits is the only SL-invariant polynomial (up to powers of itself) that is non-vanishing precisely on the set of generic states. Read More

The purpose of this note was to give a proof that Shor's algorithm for period search is polynomial using only the standard $2^{n}$ quantum Fourier thansform and some simple trigonometry. There is an error that was pointed out to the author by Pavel Wocjan. Read More

For homogeneous metrics on the spaces of the title it is shown that the Ricci flow can move a metric of stricly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature. Read More

The representation of the conformal group (PSU(2,2)) on the space of solutions to Maxwell's equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Fr\'echet representations of moderate growth. An explicit inner product is defined on each representation. The frequency spectrum of each of these representations is analyzed. Read More

Suppose several parties jointly possess a pure multipartite state, |\psi>. Using local operations on their respective systems and classical communication (i.e. Read More

A holomorphic continuation of Jacquet type integrals for parabolic subgroups with abelian nilradical is studied. Complete results are given for generic characters with compact stabilizer and arbitrary representations induced from admissible representations. A description of all of the pertinent examples is given. Read More

Let $G$ be a complex simple Lie group and let $\g = \hbox{\rm Lie}\,G$. Let $S(\g)$ be the $G$-module of polynomial functions on $\g$ and let $\hbox{\rm Sing}\,\g$ be the closed algebraic cone of singular elements in $\g$. Let ${\cal L}\s S(\g)$ be the (graded) ideal defining $\hbox{\rm Sing}\,\g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $\g$. Read More

We find an operational interpretation for the 4-tangle as a type of residual entanglement, somewhat similar to the interpretation of the 3-tangle. Using this remarkable interpretation, we are able to find the class of maximally entangled four-qubits states which is characterized by four real parameters. The states in the class are maximally entangled in the sense that their average bipartite entanglement with respect to all possible bi-partite cuts is maximal. Read More

We explore in this paper the spaces of common zeros of several deformations of Steenrod operators. Read More

This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single formal power series with a multifaced interpretation. Read More

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the $\kappa$-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor, which includes the rotationally symmetric ones. Read More

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional gradient shrinking solitons to dimension four. Read More

We introduce the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can play an important role in the study of quantum error correction codes. Read More

In this paper we give a classification of parabolic subalgebras of simple Lie algebras over $\mathbb{C}$ that satisfy two properties. The first property is Lynch's sufficient condition for the vanishing of certain Lie algebra cohomology spaces for generalized Whittaker modules associated with the parabolic subalgebra and the second is that the moment map of the cotangent bundle of the corresponding generalized flag variety be birational onto its image. We will call this condition the moment map condition. Read More

In this paper, Part II, of a two part paper we apply the results of [KW], Part I, to establish, with an explicit dual coordinate system, a commutative analogue of the Gelfand-Kirillov theorem for M(n), the algebra of $n\times n$ complex matrices. The function field F(n) of M(n) has a natural Poisson structure and an exact analogue would be to show that F(n) is isomorphic to the function field of a $n(n-1)$-dimensional phase space over a Poisson central rational function field in $n$ variables. Instead we show that this the case for a Galois extension, $F(n, {\frak e})$, of F(n). Read More

This paper gives a classification of parabolic subalgebras of simple Lie algebras over $\CC$ that are complexifications of parabolic subalgebras of real forms for which Lynch's vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split cases) and analyzes the important special case when the parabolic is defined by an even embedded TDS (three dimensional simple Lie algebra). Read More

A commutative Poisson subalgebra of the Poisson algebra of polynomials on the Lie algebra of n x n matrices over ${\Bbb C}$ is introduced which is the Poisson analogue of the Gelfand-Zeitlin subalgebra of the universal enveloping algebra. As a commutative algebra it is a polynomial ring in $n(n+1)/2$ generators, $n$ of which can be taken to be basic generators of the polynomial invariants. Any choice of the next $n(n-1)/2$ generators yields a Lie algebra of vector fields that generates a global holomorphic action of the additive group ${\Bbb C}^{n(n -1)/2}$. Read More

Affiliations: 1University of California/San Diego, 2University of California/San Diego
Category: Quantum Physics

We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin-1/2 particles. Read More

The purpose of this note is to give a generalization of Gleason's theorem inspired by recent work in quantum information theory on "nonlocality without entanglement." For multipartite quantum systems, each of dimension three or greater, the only nonnegative frame functions over the set of unentangled states are those given by the standard Born probability rule. However, if one system is of dimension 2 this is not necessarily the case. Read More