Geometry Of Hilbert Space Research Articles

Gleason’s theorem for composite systems

Gleason’s theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.

Assembling RKHS with Pick kernels and assembling polyhedra in $\mathbb {CH} ^{n}$

AbstractWe study the geometry of Hilbert spaces with complete Pick kernels and the geometry of sets in complex hyperbolic space, taking advantage of the correspondence between the two topics. We focus on questions of assembling Hilbert spaces into larger spaces and of assembling sets into larger sets. Model questions include describing the possible three-dimensional subspaces of four-dimensional Hilbert spaces with Pick kernels and describing the possible triangular faces of a tetrahedron in $\mathbb {CH}^{n}$ . A novel technical tool is a complex analog of the cosine of a vertex angle.

Robust portfolio choice under the interest rate uncertainty

ABSTRACT In this paper, we investigate a portfolio selection which is robust with respect to the interest rate uncertainty. For the portfolios with random assets, liabilities and interest rates, we bound from above Value-at-Risk of the change in the portfolio value, due to the interest rate model violation. Next, we find the robust portfolio applicable for a wide range of model perturbations. The robust optimization problem is treated in general and under additional restrictions, for example, on the expected duration of the portfolio. The results are based on the Hilbert space geometry methods applied to incomplete markets.

Hunting for the non-Hermitian exceptional points with fidelity susceptibility

The fidelity susceptibility has been used to detect quantum phase transitions in the Hermitian quantum many-body systems over a decade, where the fidelity susceptibility density approaches $+\infty$ in the thermodynamic limits. Here the fidelity susceptibility $\chi$ is generalized to non-Hermitian quantum systems by taking the geometric structure of the Hilbert space into consideration. Instead of solving the metric equation of motion from scratch, we chose a gauge where the fidelities are composed of biorthogonal eigenstates and can be worked out algebraically or numerically when not on the exceptional point (EP). Due to the properties of the Hilbert space geometry at EP, we found that EP can be found when $\chi$ approaches $-\infty$. As examples, we investigate the simplest $\mathcal{PT}$ symmetric $2\times2$ Hamiltonian with a single tuning parameter and the non-Hermitian Su-Schriffer-Heeger model.

Geometry of radial Hilbert spaces with unconditional bases of reproducing kernels

Geometry of radial Hilbert spaces with unconditional bases of reproducing kernels

Non-Hermitian Hamiltonians and no-go theorems in quantum information

Recently, apparent nonphysical implications of non-Hermitian quantum mechanics (NHQM) have been discussed in the literature. In particular, the apparent violation of the no-signaling theorem, discrimination of nonorthogonal states, and the increase of quantum entanglement by local operations were reported, and therefore NHQM was not considered as a fundamental theory. Here we show that these and other no-go principles (including the no-cloning and no-deleting theorems) of conventional quantum mechanics still hold in finite-dimensional non-Hermitian quantum systems, including parity-time symmetric and pseudo-Hermitian cases, if its formalism is properly applied. We have developed a modified formulation of NHQM based on the geometry of Hilbert spaces which is consistent with the conventional quantum mechanics for Hermitian systems. Using this formulation the validity of these principles can be shown in a simple and uniform approach.

Quantum approaches to music cognition

Quantum cognition emerged as an important discipline of mathematical psychology during the last two decades. Using abstract analogies between mental phenomena and the formal framework of physical quantum theory, quantum cognition demonstrated its ability to resolve several puzzles from cognitive psychology. Until now, quantum cognition essentially exploited ideas from projective (Hilbert space) geometry, such as quantum probability or quantum similarity. However, many powerful tools provided by physical quantum theory, e.g., symmetry groups have not been utilized in the field of quantum cognition research so far. Inspired by seminal work by Guerino Mazzola on the symmetries of tonal music, our study aims at elucidating and reconciling static and dynamic tonal attraction phenomena in music psychology within the quantum cognition framework. Based on the fundamental principles of octave equivalence, fifth similarity and transposition symmetry of tonal music that are reflected by the structure of the circle of fifths, we develop different wave function descriptions over this underlying tonal space. We present quantum models for static and dynamic tonal attraction and compare them with traditional computational models in musicology. Our approach replicates and also improves predictions based on symbolic models of music perception.

Canonical Agler decompositions and transfer function realizations

A seminal result of Agler proves that the natural de Branges-Rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift-invariant pieces. Agler’s decomposition is non-constructive—a problem remedied by work of Ball-Sadosky-Vinnikov, which uses scattering systems to produce Agler decompositions through concrete Hilbert space geometry. This method, while constructive, so far has not revealed the rich structure shown to be present for special classes of functions—inner and rational inner functions. In this paper, we show that most of the important structure present in these special cases extends to general bounded analytic functions. We give characterizations of all Agler decompositions, we prove the existence of coisometric transfer function realizations with natural state spaces, and we characterize when Schur functions on the bidisk possess analytic extensions past the boundary in terms of associated Hilbert spaces.

Foundations of Signal Processing [Book Reviews

The landscape of signal processing books is quite populated, and it is a tall order to produce a new book that can constructively add to this landscape. This book not only accomplishes this but also stands out for its distinct, consistent, and strong personality. The book’s identity starts from a presentation that is faithful to its title since the topics covered in this book are thorough in explaining the foundations of signal processing. However, foundations can be built in many ways and with different materials. The distinct element in this book is that these foundations are built based on the use of Hilbert space geometry, which allows extending Euclidean geometric insights to signals. As such, the geometry of Hilbert spaces forms the common thread across the multiple topics explained in the book. This allows for a presentation where the topics, such as Fourier representations, sampling, interpolation, approximation, compression, and filter design, can be seamlessly unified across finite dimensions, discrete time, and continuous time.

Time-limited optimal dynamics beyond the quantum speed limit

The quantum speed limit sets the minimum time required to transfer a quantum system completely into a given target state. At shorter times the higher operation speed has to be paid with a loss of fidelity. Here we quantify the trade-off between the fidelity and the duration in a system driven by a time-varying control. The problem is addressed in the framework of Hilbert space geometry offering an intuitive interpretation of optimal control algorithms. This approach is applied to non-uniform time variations which leads to a necessary criterion for control optimality applicable as a measure of algorithm convergence. The time fidelity trade-off expressed in terms of the direct Hilbert velocity provides a robust prediction of the quantum speed limit and allows to adapt the control optimization such that it yields a predefined fidelity. The results are verified numerically in a multilevel system with a constrained Hamiltonian, and a classification scheme for the control sequences is proposed based on their optimizability.

Mini-maximizing two qubit quantum computations

Two qubit quantum computations are viewed as two player, strictly competitive games and a game-theoretic measure of optimality of these computations is developed. To this end, the geometry of Hilbert space of quantum computations is used to establish the equivalence of game-theoretic solution concepts of Nash equilibrium and mini-max outcomes in games of this type, and quantum mechanisms are designed for realizing these mini-max outcomes.

Effect of Entanglement on Geometric Phase for Multi-Qubit States

When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state projective Hilbert space. A correction term to this geometric phase, in addition to the local subsystem phases, may appear from correlations between the subsystems. We find that this correction term can be characterized completely either by the entanglement or by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest that the cause of the entanglement correction is a recently introduced gauge field-like SL(2,ℂ)-invariant called twist.

Manifolds of Hilbert space projections

The Hardy space H2 (ℝ) for the upper half-plane together with a multiplicative group of unimodular functions u(λ) = exp(i(λ1ψ1 + … +λnψn)), with λ ∈ ℝn, gives rise to a manifold ℳ of orthogonal projections for the subspaces u(λ) H2 (ℝ) of L2 (ℝ). For classes of admissible functions ψi the strong operator topology closures of ℳ and ℳ ∪ ℳ⊥ are determined explicitly as various n-balls and n-spheres. The arguments used are direct and rely on the analysis of oscillatory integrals (E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43 (Princeton University Press, Princeton, NJ, 1993)) and Hilbert space geometry. Some classes of these closed projection manifolds are classified up to unitary equivalence. In particular, the Fourier–Plancherel 2-sphere and the hyperbolic 3-sphere of Katavolos and Power (A. Katavolos and S. C. Power, Translation and dilation invariant subspaces of L2(ℝ), J. reine angew. Math. 552 (2002) 101–129) appear as distinguished special cases admitting non-trivial unitary automorphism groups, which are explicitly described.

MONSTERS, BLACK HOLES AND THE STATISTICAL MECHANICS OF GRAVITY

We review the construction of monsters in classical general relativity. Monsters have finite ADM mass and surface area, but potentially unbounded entropy. From the curved space perspective, they are objects with large proper volume that can be glued on to an asymptotically flat space. At no point is the curvature or energy density required to be large in Planck units, and quantum gravitational effects are, in the conventional effective field theory framework, small everywhere. Since they can have more entropy than a black hole of equal mass, monsters are problematic for certain interpretations of black hole entropy and the AdS/CFT duality. In the second part of the paper we review recent developments in the foundations of statistical mechanics which make use of properties of high-dimensional (Hilbert) spaces. These results primarily depend on kinematics — essentially, the geometry of Hilbert space — and are relatively insensitive to dynamics. We discuss how this approach might be adopted as a basis for the statistical mechanics of gravity. Interestingly, monsters and other highly entropic configurations play an important role.

Entanglement and Hilbert space geometry for systems with a few qubits

This paper reviews recent attempts to describe the two-and three-qubit Hilbert space geometries. In the first part, this is done with the help of Hopf fibrations of hyperspheres. It is shown that the associated Hopf map is strongly sensitive to states’ entanglement content. In the two-qubit case, a generalisation of the celebrated one-qubit Bloch sphere representation is described. In the second part, we present Hilbert space discrete versions, which are comparable to polyhedral approximations of spheres in standard geometry.

Quantic Superpositions and the Geometry of Complex Hilbert Spaces

The concept of a superposition is a revolutionary novelty introduced by Quantum Mechanics. If a system may be in any one of two pure states x and y, we must consider that it may also be in any one of many superpositions of x and y. An in-depth analysis of superpositions is proposed, in which states are represented by one-dimensional subspaces, not by unit vectors as in Dirac’s notation. Superpositions must be considered when one cannot distinguish between possible paths, i.e., histories, leading to the current state of the system. In such a case the resulting state is some compound of the states that result from each of the possible paths. States can be compounded, i.e., superposed in such a way only if they are not orthogonal. Since different classical states are orthogonal, the claim implies no non-trivial superpositions can be observed in classical systems. The parameter that defines such compounds is a proportion defining the mix of the different states entering the compound. Two quantities, p and θ, both geometrical in nature, relate one-dimensional subspaces in complex Hilbert spaces: the first one is a measure of proximity relating two rays, the second one is an angle relating three rays. The properties of superpositions with respect to those two quantities are studied. The algebraic properties of the operation of superposition are very different from those that govern linear combination of vectors.

Quantum computaton from a quantum logical perspective

It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the information-processing in the three algorithms are presented from a different perspective, in terms of the way in which the algorithms exploit the non-Boolean quantum logic represented by the projective geometry of Hilbert space. From this quantum logical perspective, the XOR algorithm appears directly as a special case of Simon's algorithm, and all three algorithms can be seen as exploiting the non-Boolean logic represented by the subspace structure of Hilbert space in a similar way. Essentially, a global property of a function (such as a period, or a disjunctive property) is encoded as a subspace in Hilbert space representing a quantum proposition, which can then be efficiently distinguished from alternative propositions, corresponding to alternative global properties, by a measurement (or sequence of measurements) that identifies the target proposition as the proposition represented by the subspace containing the final state produced by the algorithm.

A course in the modern mathematical physics: groups, Hilbert space and differential geometry

Preface 1. Sets and structures 2. Groups 3. Vector spaces 4. Linear operators and matrices 5. Inner product spaces 6. Algebras 7. Tensors 8. Exterior algebra 9. Special relativity 10. Topology 11. Measure theory and integration 12. Distributions 13. Hilbert space 14. Quantum theory 15. Differential geometry 16. Differentiable forms 17. Integration on manifolds 18. Connections and curvature 19. Lie groups and lie algebras.

The Hilbert space geometry of the Rihaczek distribution for stochastic analytic signals

The Rihaczek distribution for stochastic signals is a time- and frequency-shift covariant bilinear time-frequency distribution (TFD) based on the Crame/spl acute/r-Loe/spl grave/ve spectral representation for a harmonizable process. It is a complex Hilbert space inner product (or cross correlation) between the time series and its infinitesimal stochastic Fourier generator. To this inner product, we may attach an illuminating geometry, wherein the cosine squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. The Rihaczek distribution also determines a time-varying Wiener filter for estimating a time series from its infinitesimal stochastic Fourier generator and measures the resulting error covariance. We propose a factored kernel to construct estimators of the Rihaczek distribution that are contained in Cohen's class of bilinear TFDs.

Are There Dynamical Laws

The nature of a physical law is examined, and it is suggested that there may not be any fundamental dynamical laws. This explains the intrinsic indeterminism of quantum theory. The probabilities for transition from a given initial state to a final state then depends on the quantum geometry that is determined by symmetries, which may exist as relations between states in the absence of dynamical laws. This enables the experimentally well-confirmed quantum probabilities to be derived from the geometry of Hilbert space and gives rise to effective probabilistic laws. An arrow of time which is consistent with the one given by the second law of thermodynamics, regarded as an effective law, is obtained. Symmetries are used as the basis for a new proposed paradigm of physics. This naturally gives rise to the gravitational and gauge fields from the symmetry group of the standard model and a general procedure for obtaining interactions from any symmetry group.