Finite-dimensional Hilbert Space Research Articles

Bounded below composition operators on the space of Bloch functions on the unit ball of a Hilbert space

Let B_E be the open unit ball of a complex finite or infinite dimensional Hilbert space E and consider the space mathcal {B}(B_E) of Bloch functions on B_E. Using Lipschitz continuity of the dilation map on B_E given by x mapsto (1-Vert xVert ^2) mathcal {R}f(x) for x in B_E, where mathcal {R}f denotes the radial derivative of f in mathcal {B}(B_E), we study when a composition operator on mathcal {B}(B_E) is bounded below.

Operator-difference schemes on non-uniform grids for second-order evolutionary equations

Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.

Equivalent quantum systems

We have studied quantum systems on finite-dimensional Hilbert spaces and found that all these systems are connected through local transformations. Actually, we have shown that these transformations give rise to a gauge group that connects the Hamiltonian operators associated with each quantum system. This bridge allows us to connect different quantum systems, in such a way that studying one of them allows to understand the other through a gauge transformation. Furthermore, we included the case where the Hamiltonian operator can be time-dependent. An application for this construction will be achieved in the theory of control quantum systems.

Passive Quantum Measurement: Arrival Time, Quantum Zeno Effect and Gambler's Fallacy

AbstractClassical measurements are passive, in the sense that they do not affect the physical properties of the measured system. Normally, quantum measurements are not passive in that sense. In the infinite dimensional Hilbert space, however, it is found that quantum projective measurement can be passive in a way which is impossible in finite dimensional Hilbert spaces. Specifically, it is found that expectation value of a hermitian Hamiltonian can have an imaginary part in the infinite dimensional Hilbert space and that such an imaginary part implies a possibility to avoid quantum Zeno effect, which can physically be realized in quantum arrival experiments. The avoidance of quantum Zeno effect can also be understood as avoidance of a quantum version of gambler's fallacy, leading to the notion of passive quantum measurement that updates information about the physical system without affecting its physical properties. The arrival time probability distribution of a particle is found to be given by the flux of the probability current. Possible negative fluxes correspond to regimes at which there is no arrival at all, physically understood as regimes at which the particle departs rather than arrives.

Effective operators and their variational principles for discrete electrical network problems

Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we study a general class of problems (called Z-problems) that arise in effective media theory, especially within the theory of composites, for defining the effective operator. A new and unified approach is developed, based on block operator methods, for obtaining solutions of the Z-problem, formulas for the effective operator in terms of the Schur complement, and associated variational principles (e.g., the Dirichlet and Thomson minimization principles) that lead to upper and lower bounds on the effective operator. In the case of finite-dimensional Hilbert spaces, this allows for a relaxation of the standard hypotheses on positivity and invertibility for the classes of operators usually considered in such problems by replacing inverses with the Moore–Penrose pseudoinverse. As we develop the theory, we show how it applies to the classical example from the theory of composites on the effective conductivity in the periodic conductivity problem in the continuum (2d and 3d) under the standard hypotheses. After that, we consider the following three important and diverse examples (increasing in complexity) of discrete electrical network problems in which our theory applies under the relaxed hypotheses. First, an operator-theoretic reformulation of the discrete Dirichlet-to-Neumann (DtN) map for an electrical network on a finite linear graph is given and used to relate the DtN map to the effective operator of an associated Z-problem. Second, we show how the classical effective conductivity of an electrical network on a finite linear graph is essentially the effective operator of an associated Z-problem. Finally, we consider electrical networks on periodic linear graphs and develop a discrete analog to the classical example of the periodic conductivity equation and effective conductivity in the continuum.

Path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold

If H is a Hilbert space, the non-compact Stiefel manifold St(n, H) consists of independent n-tuples in H. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that ∩j∈J (U(j) + St(n, H)) is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating J-family. Then, in the second part, we show that the topological closure of St(n, H)∩S contains all polynomial paths contained in S and passing through a point in St(n, H). As a consequence, we prove that St(n, H) is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since St(n, L2(X, μ; F)) is isometric to, FF(X, μ), n, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.

Mixed discrete-variable Gaussian states

The quantum systems with finite-dimensional Hilbert space have several applications and are intensively explored theoretically and experimentally. The mathematical description of these systems follows the analogy with the usual infinite-dimensional case. There exist finite versions for most of the elements used in the continuous case, but (to our knowledge) there does not exist a finite version corresponding to the mixed Gaussian states. Our aim is to fill this gap. The definition we propose for the mixed discrete Gaussian states is based on the explicit formulas we have obtained in the case of pure discrete variable Gaussian states.

Selective and Efficient Quantum Process Tomography for Non-Trace-Preserving Maps: Implementation with a Superconducting Quantum Processor

Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace-preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show with the aid of a priori information on the losses structure of the quantum channel that the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension $d$. The method is experimentally verified on a superconducting quantum processor provided by IBM Quantum services, by estimating several non-trace-preserving quantum processes in dimensions up to $d=6$. Our results show that it is possible to efficiently reconstruct non-trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace preserving.

Schwinger model on an interval: Analytic results and DMRG

Quantum electrodynamics in $1+1$ dimensions (Schwinger model) on an interval admits lattice discretization with a finite-dimensional Hilbert space, and is often used as a testbed for quantum and tensor network simulations. In this work we clarify the precise mapping between the boundary conditions in the continuum and lattice theories. In particular we show that the conventional Gauss law constraint commonly used in simulations induces a strong boundary effect on the charge density, reflecting the appearance of fractionalized charges. Further, we obtain by bosonization a number of exact analytic results for local observables in the massless Schwinger model. We compare these analytic results with the simulation results obtained by the density matrix renormalization group (DMRG) method and find excellent agreements.

Biomolecular and quantum algorithms for the dominating set problem in arbitrary networks

A dominating set of a graph G = (V, E) is a subset U of its vertices V, such that any vertex of G is either in U, or has a neighbor in U. The dominating-set problem is to find a minimum dominating set in G. Dominating sets are of critical importance for various types of networks/graphs, and find therefore potential applications in many fields. Particularly, in the area of communication, dominating sets are prominently used in the efficient organization of large-scale wireless ad hoc and sensor networks. However, the dominating set problem is also a hard optimization problem and thus currently is not efficiently solvable on classical computers. Here, we propose a biomolecular and a quantum algorithm for this problem, where the quantum algorithm provides a quadratic speedup over any classical algorithm. We show that the dominating set problem can be solved in O(2^{n/2}) queries by our proposed quantum algorithm, where n is the number of vertices in G. We also demonstrate that our quantum algorithm is the best known procedure to date for this problem. We confirm the correctness of our algorithm by executing it on IBM Quantum’s qasm simulator and the Brooklyn superconducting quantum device. And lastly, we show that molecular solutions obtained from solving the dominating set problem are represented in terms of a unit vector in a finite-dimensional Hilbert space.

Invariant Theory and wheeled PROPs

We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra V=T(V)⊗T(V⋆), where T(V) is the tensor algebra on an n-dimensional vector space over a field K of characteristic 0. First we classify all the ideals of the initial object Z in the category of wheeled PROPs. We show that non-degenerate sub-wheeled PROPs of V are exactly subalgebras of the form VG where G is a closed, reductive subgroup of the general linear group GL(V). When V is a finite dimensional Hilbert space, a similar description of invariant tensors for an action of a compact group was given by Schrijver. We also generalize the theorem of Procesi that says that trace rings satisfying the n-th Cayley-Hamilton identity can be embedded in an n×n matrix ring over a commutative algebra R. Namely, we prove that a wheeled PROP can be embedded in R⊗V for a commutative K-algebra R if and only if it satisfies certain relations.

Fractional-order quantum kicked top map and its discrete dynamic behaviors.

A kind of top with a fractional operator is discussed in this paper. The top has a periodic nonlinear pulse kick sequence in the magnetic field and constant precessing around the magnetic field. Then, a fractional quantum kicked top map based on the Caputo derivative is proposed. The numerical solutions of the fractional difference equation are obtained, and the chaotic behavior is observed numerically in three aspects. Fractional quantum dynamics behaviors take place in a finite dimensional Hilbert space where the squared angular momentum is free precession. Finally, the dynamic behaviors of the fractional quantum kicked top map are systematically analyzed by using the bifurcation diagram, the phase diagram, and the maximum Lyapunov exponent.

Commutant hypercyclicity of Hilbert space operators

An operator T on a Hilbert space H is commutant hypercyclic if there is a vector x in H such that the set {Sx : TS = ST} is dense in H. We prove that operators on finite dimensional Hilbert space, a rich class of weighted shift operators, isometries, exponentially isometries and idempotents are all commutant hypercyclic. Then we discuss on commutant hypercyclicity of 2 ? 2 operator matrices. Moreover, for each integer number n ? 2, we give a commutant hypercyclic nilpotent operator of order n on an infinite dimensional Hilbert space. Finally, we study commutant transitivity of operators and give necessary and sufficient conditions for a vector to be a commutant hypercyclic vector.

Supporting vectors vs. principal components

<abstract><p>Let $ T:X\to Y $ be a bounded linear operator between Banach spaces $ X, Y $. A vector $ x_0\in {\mathsf{S}}_X $ in the unit sphere $ {\mathsf{S}}_X $ of $ X $ is called a supporting vector of $ T $ provided that $ \|T(x_0)\| = \sup\{\|T(x)\|:\|x\| = 1\} = \|T\| $. Since matrices induce linear operators between finite-dimensional Hilbert spaces, we can consider their supporting vectors. In this manuscript, we unveil the relationship between the principal components of a matrix and its supporting vectors. Applications of our results to real-life problems are provided.</p></abstract>

Quantifying causal influence in quantum mechanics

We extend Pearl's definition of causal influence to the quantum domain, where two quantum systems $A, B$ with finite-dimensional Hilbert space are embedded in a common environment $C$ and propagated with a joint unitary $U$. For a finite-dimensional Hilbert space $C$, we find the necessary and sufficient condition on $U$ for a causal influence of $A$ on $B$ and vice versa. We introduce an easily computable measure of the causal influence and use it to study the causal influence of different quantum gates, its mutuality, and quantum superpositions of different causal orders. For two two-level atoms dipole-interacting with a thermal bath of electromagnetic waves, the space-time dependence of causal influence almost perfectly reproduces the one of reservoir-induced entanglement.

Toolkit for scalar fields in universes with finite-dimensional Hilbert space

The holographic principle suggests that the Hilbert space of quantum gravity is locally finite-dimensional. Motivated by this point-of-view, and its application to the observable Universe, we introduce a set of numerical and conceptual tools to describe scalar fields with finite-dimensional Hilbert spaces, and to study their behaviour in expanding cosmological backgrounds. These tools include accurate approximations to compute the vacuum energy of a field mode k as a function of the dimension d k of the mode Hilbert space, as well as a parametric model for how that dimension varies with |k|. We show that the maximum entropy of our construction momentarily scales like the boundary area of the observable Universe for some values of the parameters of that model. And we find that the maximum entropy generally follows a sub-volume scaling as long as d k decreases with |k|. We also demonstrate that the vacuum energy density of the finite-dimensional field is dynamical, and decays between two constant epochs in our fiducial construction. These results rely on a number of non-trivial modelling choices, but our general framework may serve as a starting point for future investigations of the impact of finite-dimensionality of Hilbert space on cosmological physics.

Toward the continuum limit of a (1+1)D quantum link Schwinger model

The solution of gauge theories is one of the most promising applications of quantum technologies. Here, we discuss the approach to the continuum limit for $U(1)$ gauge theories regularized via finite-dimensional Hilbert spaces of quantum spin-$S$ operators, known as quantum link models. For quantum electrodynamics (QED) in one spatial dimension, we numerically demonstrate the continuum limit by extrapolating the ground state energy, the scalar, and the vector meson masses to large spin lengths $S$, large volume $N$, and vanishing lattice spacing $a$. By exactly solving Gauss' law for arbitrary $S$, we obtain a generalized PXP spin model and count the physical Hilbert space dimension analytically. This allows us to quantify the required resources for reliable extrapolations to the continuum limit on quantum devices. We use a functional integral approach to relate the model with large values of half-integer spins to the physics at topological angle $\Theta=\pi$. Our findings indicate that quantum devices will in the foreseeable future be able to quantitatively probe the QED regime with quantum link models.

Norm retrieval from few spatio-temporal samples

Vectors in a real finite-dimensional Hilbert space that accomplish phase and norm retrieval can be characterized. In this article, we consider the cases where an operator A is self adjoint, normal or an operator in Jordan canonical form in a real finite-dimensional Hilbert space. In each case, we give a structure to construct norm-retrievable frame sets that have a dynamical sampling structure. Given that phase retrieval always implies norm retrieval, we particularly concentrate on finding conditions for norm retrieval where phase retrieval is not possible.

Approximate solution of the Cauchy problem for a first-order integrodifferential equation with solution derivative memory

We consider the Cauchy problem for a first-order evolution equation with memory in a finite-dimensional Hilbert space when the integral term is related to the time derivative of the solution. The main problems of the approximate solution of such nonlocal problems are due to the necessity to work with the approximate solution for all previous time moments. We propose a transformation of the first-order integrodifferential equation to a system of local evolutionary equations. We use the approach known in the theory of Volterra integral equations with an approximation of the difference kernel by the sum of exponents. We formulate a local problem for a weakly coupled system of equations with additional ordinary differential equations. We have given estimates of the stability of the solution by initial data and the right-hand side for the solution of the corresponding Cauchy problem. The primary attention is paid to constructing and investigating the stability of two-level difference schemes, which are convenient for computational implementation. The numerical solution of a two-dimensional model problem for the evolution equation of the first order, when the Laplace operator conditions the dependence on spatial variables, is presented.

A note on phase (norm) retrievable real Hilbert space fusion frames

In this paper, we will present several new results in finite and countable dimensional separable real Hilbert space phase retrieval and norm retrieval by fusion frames. We will characterize of norm retrieval for fusion frames similar norm retrieval for vectors and we will show that only one direction holds for fusion frames. In similar vector case, we will show that every tight fusion frame can do norm retrieval. Also we will show that the unitary operators preserve phase (norm) retrievability of fusion frames. We will make a detailed study of when hyperplanes do norm retrieval and show a general result about it. We will provide numerous examples to show that our results are best possible.