Euclidean Jordan Algebras Research Articles

Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

The commutation principle of Ramírez et al. (SIAM J Optim 23:687–694, 2013) proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function h and a spectral function $$\Phi $$ is minimized/maximized over a spectral set E, any local optimizer a at which h is Fréchet differentiable operator commutes with the derivative $$h^{\prime }(a)$$ . In this note, we describe some analogs of the above result by assuming the existence of a subgradient in place of the derivative (of h) and obtaining strong operator commutativity relations. We show, for example: if a solves the problem $$\underset{E}{\max }\,(h+\Phi )$$ , then a strongly operator commutes with every element in the subdifferential of h at a; If E and h are convex and a solves the problem $$\underset{E}{\min }\,h$$ , then a strongly operator commutes with the negative of some element in the subdifferential of h at a. These results improve known operator commutativity relations for linear h and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in a broader setting.

Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems

This article deals with necessary and sufficient conditions for a family of elements in a Euclidean Jordan algebra to have simultaneous (order) spectral decomposition. Motivated by a well-known matrix theory result that any family of pairwise commuting complex Hermitian matrices is simultaneously (unitarily) diagonalizable, we show that in the setting of a general Euclidean Jordan algebra, any family of pairwise operator commuting elements has a simultaneous spectral decomposition, i.e. there exists a common Jordan frame relative to which every element in the given family has the eigenvalue decomposition of the form . The simultaneous order spectral decomposition further demands the ordering of eigenvalues . We characterize this by a pairwise strong operator commutativity condition or, equivalently, , where denotes the vector of eigenvalues of x written in the decreasing order. Going beyond Euclidean Jordan algebras, we formulate commutativity conditions in the setting of the so-called Fan–Theobald–von Neumann system that includes normal decomposition systems (Eaton triples) and certain systems induced by hyperbolic polynomials.

Standard Subspaces of Hilbert Spaces of Holomorphic Functions on Tube Domains

In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains E + i C^0, where C \subeq E is a pointed generating cone invariant under e^{R h} for some endomorphism h \in \End(E), diagonalizable with the eigenvalues 1,0,-1 (generalizing a Lorentz boost). This data specifies a wedge domain W(E,C,h) \subeq E and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple (E,C,e^{\pi i h}) and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces H. For the imaginary part of these distributions, we find similarities to the well known Huygens' principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.

How dynamics constrains probabilities in general probabilistic theories

We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence with the spherical representations of the dynamical group. We apply our methods to classify all probabilistic structures when the dynamical structure is that of complex Grassmann manifolds acted on by the unitary group. This is a generalisation of quantum theory where the pure states, instead of being represented by one-dimensional subspaces of a complex vector space, are represented by subspaces of a fixed dimension larger than one. We also show that systems with compact two-point homogeneous dynamical structures (i.e. every pair of pure states with a given distance can be reversibly transformed to any other pair of pure states with the same distance), which include systems corresponding to Euclidean Jordan Algebras, all have rigid probabilistic structures.

SPECIAL VINBERG CONES

The paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; \U0001d524) and {a}_{ji}={a}_{ij}^{ast }=mathfrak{g}left({a}_{ij},cdot right)in {V}_{ij}^{ast } belongs to the dual space {V}_{ij}^{ast }. The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric \U0001d524S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone \U0001d4b1 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone \U0001d4b12 ⊂ ℋ2(V ) to the special rank 3 case.

An analog of Thompson's triangle inequality in Euclidean Jordan algebras

In a recent paper [Linear Algebra Appl., 461:92--122, 2014], Tao et al. proved an analog of Thompson's triangle inequality for a simple Euclidean Jordan algebra by using a case-by-case analysis. In this short note, we provide a direct proof that is valid on any Euclidean Jordan algebras.

The Araki-Lieb-Thirring inequality and the Golden-Thompson inequality in Euclidean Jordan algebras

ABSTRACT Motivated by the Araki-Lieb-Thirring inequality for ( for ) for Hermitian positive semidefinite matrices and the Golden-Thompson inequality for Hermitian matrices, in this paper, we extend these inequalities to the setting of Euclidean Jordan algebras in the form for ( for ) for and for all a and b, where and denote, respectively, the quadratic representation and Jordan product of x and y in Euclidean Jordan algebras.

A pointwise weak-majorization inequality for linear maps over Euclidean Jordan algebras

Given a linear map T on a Euclidean Jordan algebra of rank n, we consider the set of all nonnegative vectors q in with decreasing components that satisfy the pointwise weak-majorization inequality , where λ is the eigenvalue map and * denotes the componentwise product in . With respect to the weak-majorization ordering, we show the existence of the least vector in this set. When T is a positive map, the least vector is shown to be the join (in the weak-majorization order) of eigenvalue vectors of T(e) and , where e is the unit element of the algebra. These results are analogous to the results of Bapat [Majorization and singular values. III. Linear Algebra Appl. 1991;145:59–70] on singular values. We also extend two recent results of Tao et al. [Some log and weak majorization inequalities in Euclidean Jordan algebras. 2020. arXiv:2003.12377v2] proved for quadratic representations and Schur product induced transformations. As an application, we provide an estimate on the norm of a general linear map relative to spectral norms.

Refinements of Ky Fan's eigenvalue inequality for simple Euclidean Jordan algebras by using gradients of K-increasing functions

Refinements of Ky Fan's eigenvalue inequality for simple Euclidean Jordan algebras by using gradients of K-increasing functions

Composites and Categories of Euclidean Jordan Algebras

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities.These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit.Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. {In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents.}

Some log and weak majorization inequalities in Euclidean Jordan algebras

Motivated by Horn's log-majorization (singular value) inequality and the related weak-majorization inequality for square complex matrices, we consider their Hermitian analogs for positive semidefinite matrices and for general (Hermitian) matrices, where denotes the Jordan product of A and B and denotes the componentwise product in . In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form for and for all a and b, where and denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form , where A is a real symmetric positive semidefinite matrix and is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality , where denotes the spectral p-norm of x and with . We also give precise values of the norms of the Lyapunov transformation and relative to two spectral p-norms.

Superconnection in the spin factor approach to particle physics

The notion of superconnection devised by Quillen in 1985 and used in gauge-Higgs field theory in the 1990's is applied to the spin factors (finite-dimensional euclidean Jordan algebras) recently considered as representing the finite quantum geometry of one generation of fermions in the Standard Model of particle physics.

Euclidean Jordan Algebras and Inequalities over the Spectrum of a Strongly Regular Graph

Let G be a primitive strongly regular graph of order n with three distinct eigenvalues and A its adjacency matrix. In this paper we associate to A the 3—dimensional real Euclidean Jordan algebra A spanned by In and the natural powers of A equipped with the Jordan product of matrices and with inner product of two matrices being the trace of their Jordan product and next, by the spectral analysis of some elements of A we establish some inequalities over the spectrum and the parameters of a strongly regular graph.

Non-representable hyperbolic matroids

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the first author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non- representable hyperbolic matroid. The Va ́mos matroid and a generalization of it are to this day the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection, due to Jordan, between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non-representable. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.

Some majorization inequalities induced by Schur products in Euclidean Jordan algebras

In a Euclidean Jordan algebra V of rank n, an element x is said to be majorized by an element y, symbolically x≺y, if the corresponding eigenvalue vector λ(x) is majorized by λ(y) in Rn. In this article, we describe pointwise majorization inequalities of the form T(x)≺S(x), where T and S are linear transformations induced by Schur products. Specializing, we recover analogs of majorization inequalities of Schur, Hadamard, and Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as majorization inequalities connecting quadratic and Lyapunov transformations on V. We also show how Schur products induced by certain scalar means (such as arithmetic, geometric, harmonic, and logarithmic means) naturally lead to majorization inequalities.

Weak majorization, doubly substochastic maps, and some related inequalities in Euclidean Jordan algebras

In this paper, we extend the notion of weak majorization and doubly substochastic maps, and the Hardy-Littlewood-Pólya theorem on majorization to Euclidean Jordan algebras. We also provide a characterization of doubly substochastic maps on Euclidean Jordan algebras in terms of Jordan algebra automorphisms and quadratic representations. In conjunction with that, various weak majorization inequalities of Aujla-Silva are generalized to Euclidean Jordan algebras.

Self-concordance and matrix monotonicity with applications to quantum entanglement problems

Let V be a Euclidean Jordan algebra and Ω be a cone of invertible squares in V. Suppose that g:R+→R is a matrix monotone function on the positive semiaxis which naturally induces a function g˜:Ω→V. We show that −g˜ is compatible (in the sense of Nesterov–Nemirovski) with the standard self-concordant barrier B(x)=−lndet(x) on Ω. As a consequence, we show that for any c ∈ Ω, the functions of the form −tr(cg˜(x))+B(x) are self-concordant on Ω. In particular, the function x↦−tr(clnx) is a self-concordant barrier function on Ω. Using these results, we apply a long-step path-following algorithm developed in [L. Faybusovich and C. Zhou Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions. Comput Optim Appl, 72(3):769-795, 2019] to a number of important optimization problems arising in quantum information theory. Results of numerical experiments and comparisons with existing methods are presented.

On the algebraic Riccati inequality arising in cone-preserving time-delay systems

This paper studies the problem of Riccati stability of a pair of matrices. For a matrix pair (A,B), it was recently shown that if its corresponding time-delay system is internally positive, meaning that A is Metzler and B is nonnegative, then the pair (A,B) is diagonally Riccati stable if and only if A+B is Hurwitz. We extend this to the case when the pair (A,B) corresponds to a time-delay system with a more general cone-preserving property. We show that if the time-delay system relating to the pair (A,B) is invariant on a symmetric cone, the corresponding algebraic Riccati inequality admits positive definite solutions, which can be constructed via the scaling transformation on the Euclidean Jordan algebra associated with the symmetric cone. For the special case when the symmetric cone is the positive semi-definite cone, an application to a class of stochastic systems is discussed.

Generalized Subdifferentials of Spectral Functions over Euclidean Jordan Algebras

This paper is devoted to the study of generalized subdifferentials of spectral functions over Euclidean Jordan algebras. Spectral functions appear often in optimization problems playing the role of "regularizer", "barrier", "penalty function" and many others. We provide formulae for the regular, approximate and horizon subdifferentials of spectral functions. In addition, under local lower semicontinuity, we also furnish a formula for the Clarke subdifferential, thus extending an earlier result by Baes. As application, we compute the generalized subdifferentials of the function that maps an element to its k-th largest eigenvalue. Furthermore, in connection with recent approaches for nonsmooth optimization, we present a study of the Kurdyka-Lojasiewicz (KL) property for spectral functions and prove a transfer principle for the KL-exponent. In our proofs, we make extensive use of recent tools such as the commutation principle of Ram\'irez, Seeger and Sossa and majorization principles developed by Gowda.

An effect-theoretic reconstruction of quantum theory

An often used model for quantum theory is to associate to every physical system a C∗-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we find a set of physically meaningful assumptions such that any physical theory satisfying these assumptions must embed into the category of finite-dimensional C∗-algebras. These assumptions were originally introduced in the setting of effectus theory, a categorical logical framework generalizing classical and quantum logic. As these assumptions have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory.In contrast to other reconstructions of quantum theory, we do not start with the framework of generalized probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to `forgetting' and `measuring' the validity of an effect.We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We show that any convex finite-dimensional PET must embed into the category of Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure, then we show that it must embed into either the category of real or complex C∗-algebras, which completes our reconstruction.