Finite-dimensional Hilbert Space Research Articles

Balanced Frames: A Useful Tool in Signal Processing with Good Properties

So far there has not been paid attention to frames that are balanced, i.e. those frames which sum is zero. In this paper we consider balanced frames, and in particular balanced unit norm tight frames, in finite dimensional Hilbert spaces. Here we discover various advantages of balanced unit norm tight frames in signal processing. They give an exact reconstruction in the presence of systematic errors in the transmitted coefficients, and are optimal when these coefficients are corrupted with noises that can have non-zero mean. Moreover, using balanced frames we can know that the transmitted coefficients were perturbed, and we also have an indication of the source of the error. We analyze several properties of these types of frames. We define an equivalence relation in the set of the dual frames of a balanced frame, and use it to show that we can obtain all the duals from the balanced ones. We study the problem of finding the nearest balanced frame to a given frame, characterizing completely its existence and giving its expression. We introduce and study a concept of complement for balanced frames. Finally, we present many examples and methods for constructing balanced unit norm tight frames.

Holographic map for cosmological horizons

We propose a holographic map between Einstein gravity coupled to matter in a de Sitter background and large N quantum mechanics of a system of spins. Holography maps a spin model with a finite-dimensional Hilbert space defined on a version of the stretched horizon into bulk gravitational dynamics. The full Hamiltonian of the spin model contains a nonlocal piece which generates chaotic dynamics, widely conjectured to be a necessary part of quantum gravity, and a local piece which recovers the perturbative spectrum in the bulk.

A SCHEMATIC DEFINITION OF QUANTUM POLYNOMIAL TIME COMPUTABILITY

AbstractIn the past four decades, the notion of quantum polynomial-time computability has been mathematically modeled by quantum Turing machines as well as quantum circuits. This paper seeks the third model, which is a quantum analogue of the schematic (inductive or constructive) definition of (primitive) recursive functions. For quantum functions mapping finite-dimensional Hilbert spaces to themselves, we present such a schematic definition, composed of a small set of initial quantum functions and a few construction rules that dictate how to build a new quantum function from the existing ones. We prove that our schematic definition precisely characterizes all functions that can be computable with high success probabilities on well-formed quantum Turing machines in polynomial time, or equivalently uniform families of polynomial-size quantum circuits. Our new, schematic definition is quite simple and intuitive and, more importantly, it avoids the cumbersome introduction of the well-formedness condition imposed on a quantum Turing machine model as well as of the uniformity condition necessary for a quantum circuit model. Our new approach can further open a door to the descriptional complexity of quantum functions, to the theory of higher-type quantum functionals, to the development of new first-order theories for quantum computing, and to the designing of programming languages for real-life quantum computer

A new Halpern-type averaging algorithm with inertial and error terms for fixed points of asymptotically nonexpansive maps

We introduce and study a Halpern-type averaging algorithm with both inertial and error terms for the approximation of fixed points of asymptotically nonexpansive maps in real Hilbert spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional real Hilbert spaces. Our results extend recent results of Yekini, Iyiola and Ogbuisi, Numer Algor (2019), https://doi.org/10.1007/s11075-019-00727-5 from the important class of nonexpansive maps to the much more general class of asymptotically nonexpansive maps. Furthermore, our preliminary lemma is of independent interest.

Quantum walks: The mean first detected transition time

We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate $1/\tau$. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state $|\psi_{\rm in}\rangle$ of the walker is orthogonal to the detected state $|\psi_{\rm d}\rangle$. We focus on diverging mean transition times, where the total detection probability exhibits a discontinuous drop of its value, by mapping the problem onto a theory of fields of classical charges located on the unit disk. Close to the critical parameter of the model, which exhibits a blow-up of the mean transition time, we get simple expressions for the mean transition time. Using previous results on the fluctuations of the return time, corresponding to $|\psi_{\rm in}\rangle = |\psi_{\rm d}\rangle$, we find close to these critical parameters that the mean transition time is proportional to the fluctuations of the return time, an expression reminiscent of the Einstein relation.

Skew compressions of positive definite operators and matrices

The paper is devoted to results connecting the eigenvalues and singular values of operators composed by $P^\ast G P$ with those composed in the same way by $QG^{−1}Q^\ast$. Here $P +Q = I$ are skew complementary projections on a finite-dimensional Hilbert space and $G$ is a positive definite linear operator on this space. Also discussed are graph theoretic interpretations of one of the results.

On a relation related to strong Birkhoff–James orthogonality

ABSTRACT In this paper, we discuss a version of Birkhoff–James orthogonality in the -algebra of all bounded linear operators on a finite-dimensional Hilbert space. We obtain a characterization of this relation and use it to describe some classes of operators. We also characterize linear preservers of this type of orthogonality.

From the Jordan Product to Riemannian Geometries on Classical and Quantum States.

The Jordan product on the self-adjoint part of a finite-dimensional -algebra is shown to give rise to Riemannian metric tensors on suitable manifolds of states on , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra of linear operators on a finite-dimensional Hilbert space , and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on , this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor.

Effective methods for constructing extreme quantum observables

We study extreme points of the set of finite-outcome positive-operator-valued measures (POVMs) on finite-dimensional Hilbert spaces and particularly the possible ranks of the effects of an extreme POVM. We give results discussing ways of deducing new rank combinations of extreme POVMs from rank combinations of known extreme POVMs and, using these results, show ways to characterize rank combinations of extreme POVMs in low dimensions. We show that, when a rank combination together with a given dimension of the Hilbert space solve a particular packing problem, there is an extreme POVM on the Hilbert space with the given ranks. This geometric method is particularly effective for constructing extreme POVMs with desired rank combinations.

Properties of operator systems, corresponding to channels

It was shown in arXiv:0906.2527, that in finite-dimensional Hilbert spaces each operator system corresponds to some channel, for which this operator system will be an operator graph. This work is devoted to finding necessary and sufficient conditions for this property to hold in infinite-dimensional case.

Why FHilb is Not an Interesting (Co)Differential Category

Differential categories provide an axiomatization of the basics of differentiation and categorical models of differential linear logic. As differentiation is an important tool throughout quantum mechanics and quantum information, it makes sense to study applications of the theory of differential categories to categorical quantum foundations. In categorical quantum foundations, compact closed categories (and therefore traced symmetric monoidal categories) are one of the main objects of study, in particular the category of finite-dimensional Hilbert spaces FHilb. In this paper, we will explain why the only differential category structure on FHilb is the trivial one. This follows from a sort of in-compatibility between the trace of FHilb and possible differential category structure. That said, there are interesting non-trivial examples of traced/compact closed differential categories, which we also discuss. The goal of this paper is to introduce differential categories to the broader categorical quantum foundation community and hopefully open the door to further work in combining these two fields. While the main result of this paper may seem somewhat "negative" in achieving this goal, we discuss interesting potential applications of differential categories to categorical quantum foundations.

Recoverability from direct quantum correlations

We address the problem of compressing density operators defined on a finite dimensional Hilbert space which assumes a tensor product decomposition. In particular, we look for an efficient procedure for learning the most likely density operator, according to ‘Jaynes’ principle, given a chosen set of partial information obtained from the unknown quantum system we wish to describe. For complexity reasons, we restrict our analysis to tree-structured sets of bipartite marginals. We focus on the tripartite scenario, where we solve the problem for the couples of measured marginals which are compatible with a quantum Markov chain, providing then an algebraic necessary and sufficient condition for the compatibility to be verified. We introduce the generalization of the procedure to the n-partite scenario, giving some preliminary results. In particular, we prove that if the pairwise Markov condition holds between the subparts then the choice of the best set of tree-structured bipartite marginals can be performed efficiently. Moreover, we provide a new characterization of quantum Markov chains in terms of quantum Bayesian updating processes.

Image Set Classification Using a Distance-Based Kernel Over Affine Grassmann Manifold.

Modeling image sets or videos as linear subspaces is quite popular for classification problems in machine learning. However, affine subspace modeling has not been explored much. In this article, we address the image sets classification problem by modeling them as affine subspaces. Affine subspaces are linear subspaces shifted from origin by an offset. The collection of the same dimensional affine subspaces of [Formula: see text] is known as affine Grassmann manifold (AGM) or affine Grassmannian that is a smooth and noncompact manifold. The non-Euclidean geometry of AGM and the nonunique representation of an affine subspace in AGM make the classification task in AGM difficult. In this article, we propose a novel affine subspace-based kernel that maps the points in AGM to a finite-dimensional Hilbert space. For this, we embed the AGM in a higher dimensional Grassmann manifold (GM) by embedding the offset vector in the Stiefel coordinates. The projection distance between two points in AGM is the measure of similarity obtained by the kernel function. The obtained kernel-gram matrix is further diagonalized to generate low-dimensional features in the Euclidean space corresponding to the points in AGM. Distance-preserving constraint along with sparsity constraint is used for minimum residual error classification by keeping the locally Euclidean structure of AGM in mind. Experimentation performed over four data sets for gait, object, hand, and body gesture recognition shows promising results compared with state-of-the-art techniques.

Uncertainty relations in terms of the Gini index for finite quantum systems

Lorenz values and the Gini index are popular quantities in Mathematical Economics, and are used here in the context of quantum systems with finite-dimensional Hilbert space. They quantify the uncertainty in the probability distribution related to an orthonormal basis. It is shown that Lorenz values are superadditive functions and the Gini indices are subadditive functions. The supremum over all density matrices of the sum of the two Gini indices with respect to position and momentum states is used to define an uncertainty coefficient which quantifies the uncertainty in the quantum system. It is shown that the uncertainty coefficient is positive, and an upper bound for it is given. Various examples demonstrate these ideas.

Processing of Sparse Signals and Mutual Coherence of ‘‘Measurable’’ Vectors

We define frames for a finite dimensional Hilbert space $$\mathbb{H}^{M}$$ as the complete systems in $$\mathbb{H}^{M}.$$ The basic frame families are classified such as tight and Parseval frames,equal norm frames and equiangular frames. The well-known theorems proved by Naimark, Maltsev and Kashin have received a new interpretation in the frame theory. The explicit construction of the equal norm Parseval frames in $$\mathbb{R}^{M}$$ for any $$N\geq M$$ is given in Theorem 3. Theorem 5 provides a simple proof of the Welch bound with the use of eigenvalues of the Gram matrix frame. There are examples of equiangular tight frames in the low-dimensional spaces. Sections 3 presents the definition of spark and the full spark systems and their usage in Compressed Sensing. There are examples of the full spark equiangular tight frames. In section 4 we use frames in Phase Reconstruction problem. The minimal number of measurable vectors in $$\mathbb{R}^{M}$$ is found with the help of Kashin’s stochastic matrix.

Optimal measures, nonzero projections, and irreducible representations

This brief note considers the problem of maximizing the probability of detection for a finite number of mixed quantum states. It is shown that for linearly independent density matrices, an optimal measure that is simple and unique contains nonzero operators that are orthogonal projections on subspaces associated to irreducible representations (irreps) of a finite-dimensional Hilbert space.

Fock representations of ZF algebras and R-matrices

A variation of the Zamolodchikov–Faddeev algebra over a finite-dimensional Hilbert space {mathcal {H}} and an involutive unitary R-Matrix S is studied. This algebra carries a natural vacuum state, and the corresponding Fock representation spaces {mathcal {F}}_S({mathcal {H}}) are shown to satisfy {mathcal {F}}_{Sboxplus R}({{mathcal {H}}}oplus {{mathcal {K}}}) cong {mathcal {F}}_S({{mathcal {H}}})otimes {mathcal {F}}_R({{mathcal {K}}}), where Sboxplus R is the box-sum of S (on {{mathcal {H}}}otimes {{mathcal {H}}}) and R (on {{mathcal {K}}}otimes {{mathcal {K}}}). This analysis generalises the well-known structure of Bose/Fermi Fock spaces and a recent result of Pennig. These representations are motivated from quantum field theory (short-distance scaling limits of integrable models).

Some Examples of m-Isometries

We obtain the admissible sets on the unit circle to be the spectrum of a strict $m$-isometry on an $n$-finite dimensional Hilbert space. This property gives a better picture of the correct spectrum of an $m$-isometry. We determine that the only $m$-isometries on $\mathbb{R}^2$ are $3$-isometries and isometries giving by $\pm I+Q$, where $Q$ is a nilpotent operator. Moreover, on real Hilbert space, we obtain that $m$-isometries preserve volumes. Also we present a way to construct a strict $(m+1)$-isometry with an $m$-isometry given, using ideas of Aleman and Suciu \cite[Proposition 5.2]{AS} on infinite dimensional Hilbert space.

Joint Schmidt-type decomposition for two bipartite pure quantum states

It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulae for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.

A new Halpern-type averaging algorithm with inertial and error terms for fixed points of asymptotically nonexpansive maps

We introduce and study a Halpern-type averaging algorithm with both inertial and error terms for the approximation of fixed points of asymptotically nonexpansive maps in real Hilbert spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional real Hilbert spaces. Our results extend recent results of Yekini, Iyiola and Ogbuisi, Numer Algor (2019), https://doi.org/10.1007/s11075-019-00727-5 from the important class of nonexpansive maps to the much more general class of asymptotically nonexpansive maps. Furthermore, our preliminary lemma is of independent interest.