Hilbert-Schmidt Norm Research Articles

Characterizing nonclassical correlation using affinity

Geometric discord, a measure of quantumness of bipartite system, captures minimal nonlocal effects of a quantum state due to locally invariant von Neumann projective measurements. Original version of this measure is suffered by the local ancilla problem. In this article, we propose a new version of geometric discord using affinity. This quantity satisfies all criteria of a good measure of quantum correlation of the bipartite system and resolves local ancilla problem of Hilbert-Schmidt norm based discord. We evaluate analytically the proposed quantity for both pure and mixed states. For an arbitrary pure state, it is shown that affinity based geometric discord equal to geometric measure of entanglement. Further, we obtain a lower bound of this measure for m \times n dimensional arbitrary mixed state and a closed formula of proposed version of geometric discord for 2 \times n dimensional mixed state is obtained. Finally, as an illustration, we have studied the nonlocality of Bell diagonal state, isotropic and Werner states.

Inequalities for Contraction Matrices

Let and Y be n× n matrices such that A and B are positive definite contractions. It is shown that if and thenMoreover, if then for with where and denote the Hilbert-Schmidt norm, the spectral matrix norm, the j th singular value, and the condition number of the n×n matrix T, respectively.

Semiclassical Limit to the Vlasov Equation with Inverse Power Law Potentials

We consider mixed quasi-free states describing $N$ fermions in the mean-field limit. In this regime, the time evolution is governed by the nonlinear Hartree equation. In the large $N$ limit, we study the convergence towards the classical Vlasov equation. Under integrability and regularity assumptions on the initial state, we prove strong convergence in trace and Hilbert-Schmidt norm and provide explicit bounds on the convergence rate for a class of singular potentials of the form $V(x)=|x|^{-\alpha}$, for $\alpha\in(0,1/2)$.

Upper and lower bounds for the Hilbert–Schmidt norm of a potential operator

We give upper and lower bounds for the Hilbert–Schmidt norm of logarithmic potential on a planar domain in terms of its area and inradius.

On a Refined Operator Version of Young's Inequality and Its Reverse

In this note, some refinements of Young's inequality and its reverse for positive numbers are proved, and using these inequalities, some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.

In-sphere property and reverse inequalities for matrix means

The in-sphere property for matrix means is studied. It is proved that the matrix power mean satisfies in-sphere property with respect to the Hilbert-Schmidt norm. A new characterization of the matrix arithmetic mean is provided. Some reverse AGM inequalities involving unitarily invariant norms and operator monotone functions are also obtained.

Non-commutative measure of quantum correlations under local operations

We study some desirable properties of recently introduced measures of quantum correlations based on the amount of non-commutativity quantified by the Hilbert-Schmidt norm (Sci Rep 6:25241, 2016, and Quantum Inf. Process. 16:226, 2017). Specifically, we show that: 1) for any bipartite ($A+B$) state, the measures of quantum correlations with respect to subsystem $A$ are non-increasing under any Local Commutative Preserving Operation on subsystem $A$, and 2) for Bell diagonal states, the measures are non-increasing under arbitrary local operations on $B$. Our results accentuate the potentialities of such measures, and exhibit them as valid monotones in a resource theory of quantum correlations with free operations restricted to the appropriate local channels.

Reverses of Young Type Inequalities for Matrices Using the Classical Kantorovich Constant

In this article, we give some reverses of Young type inequalities which were established by Burqan and Khandaqji (J Math Inequal 9:113–120, 2015) applying the Kantorovich constant. As an application of these numerical versions, we study some matrix inequalities for the Hilbert–Schmidt norm and the trace norm.

Norm estimations, continuity, and compactness for Khatri-Rao products of Hilbert Space operators

We provide estimations for the operator norm, the trace norm, and the Hilbert-Schmidt norm for Khatri-Rao products of Hilbert space operators. It follows that the Khatri-Rao product is continuous on norm ideals of compact operators equipped with the topologies induced by such norms. Moreover, if two operators are represented by block matrices in which each block is nonzero, then their Khatri-Rao product is compact if and only if both operators are compact. The Khatri-Rao product of two operators are trace-class (Hilbert-Schmidt class) if and only if each factor is trace-class (Hilbert-Schmidt class, respectively).

Revisiting Connes’ finite spectral distance on noncommutative spaces: Moyal plane and fuzzy sphere

Beginning with a review of the existing literature on the computation of spectral distances on noncommutative spaces like Moyal plane and fuzzy sphere, adaptable to Hilbert–Schmidt operatorial formulation, we carry out a correction, revision and extension of the algorithm provided in [1] i.e. [F. G. Scholtz and B. Chakraborty, J. Phys. A, Math. Theor. 46 (2013) 085204] to compute the finite Connes’ distance between normal states. The revised expression, which we provide here, involves the computation of the infimum of an expression which involves the “transverse” [Formula: see text] component of the algebra element in addition to the “longitudinal” component [Formula: see text] of [1], identified with the difference of density matrices representing the states, whereas the expression given in [1] involves only [Formula: see text] and corresponds to the lower bound of the distance. This renders the revised formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a nontrivial task, but under rather generic conditions it turns out that the Connes’ distance is proportional to the Hilbert-Schmidt norm of [Formula: see text], leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [P. Martinetti and L. Tomassini, Commun. Math. Phys. 323 (2013) 107–141]. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite-dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere, is never reached for any finite [Formula: see text]-representation of [Formula: see text]. Indeed, for the case of maximal noncommutativity ([Formula: see text]), the finite distance is shown to coincide exactly with the above-mentioned lower bound, with the transverse component playing no role. This, however, starts changing from [Formula: see text] onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our revised algorithm. The contrasting features of these types of noncommutative spaces becomes quite transparent through the analysis, carried out in the eigen-spinor bases of the respective Dirac operators.

A class of optimal estimators for the covariance operator in reproducing kernel Hilbert spaces

The covariance operator plays an important role in modern statistical methods and is critical for inference. It is most often estimated by the empirical covariance operator. In spite of its simple and appealing properties, however, this estimator can be improved by a class of shrinkage operators. In this paper, we study shrinkage estimation of the covariance operator in reproducing kernel Hilbert spaces. A data-driven shrinkage estimator enjoying desirable theoretical and computational properties is proposed. The procedure is easily implemented and its numerical performance is investigated through simulations. In finite samples, the estimator outperforms the empirical covariance operator, especially when the data dimension is much larger than the sample size. We also show that the rate of convergence in Hilbert–Schmidt norm is of the order n−1∕2. Furthermore, we establish the minimax optimal rate of convergence over suitable classes of probability measures and demonstrate that these shrinkage operators are all minimax rate-optimal.

Higher order [formula omitted]-differentiability and application to Koplienko trace formula

Let A be a selfadjoint operator in a separable Hilbert space, K a selfadjoint Hilbert–Schmidt operator, and f∈Cn(R). We establish that φ(t)=f(A+tK)−f(A) is n-times continuously differentiable on R in the Hilbert–Schmidt norm, provided either A is bounded or the derivatives f(i), i=1,…,n, are bounded. This substantially extends the results of [3] on higher order differentiability of φ in the Hilbert–Schmidt norm for f in a certain Wiener class. As an application of the second order S2-differentiability, we extend the Koplienko trace formula from the Besov class B∞12(R)[20] to functions f for which the divided difference f[2] admits a certain Hilbert space factorization.

A note on reverses of the young type inequalities via Kantorovich constant

In present paper, we give some new reverses of the Young type inequalities which were established by X. Hu and J. Xue [7] via Kantorovich constant. Then we apply these inequalities to establish corresponding inequalities for the Hilbert-Schmidt norm and the trace norm.

Magnetic shielding of quantum entanglement states

The measure of quantum entanglement is determined for any dimer, either ferromagnetic or antiferromagnetic, spin-1/2 Heisenberg systems in the presence of external magnetic field. The physical quantity proposed as a measure of thermal quantum entanglement is the distance between states defined through the Hilbert-Schmidt norm. It has been shown that for ferromagnetic systems there is no entanglement at all. However, although under applied magnetic field, antiferromagnetic spin-1/2 dimers exhibit entanglement for temperatures below the decoherence temperature -- the one above which the entanglement vanishes. In addition to that, the decoherence temperature shows to be proportional to the exchange coupling constant and independent on the applied magnetic field, consequently, the entanglement may not be destroyed by external magnetic fields -- the phenomenon of {\it magnetic shielding effect of quantum entanglement states}. This effect is discussed for the binuclear nitrosyl iron complex [Fe$_2$(SC$_3$H$_5$N$_2$)$_2$(NO)$_4$] and it is foreseen that the quantum entanglement survives even under high magnetic fields of Tesla orders of magnitude.

Quantifying Discriminating Strength for Gaussian States**Supported by Natural Science Foundation of China under Grant Nos. 11671006, 11671294, and Outstanding Youth Foundation of Shanxi Province (201701D211001)

The discriminating strength induced by local Gaussian unitary operators for any (n + m)-mode Gaussian state is introduced in [Phys. Rev. A 83 (2011) 042325]. In this paper, we further discuss the quantity by restricting to Hilbert-Schmidt norm. The analytic formulas of for two-mode squeezed thermal states and mixed thermal states are given. Then, the relationship between and for some special Gaussian channels Φ is discussed. In addition, is compared with Gaussian entanglement for symmetric squeezed thermal states.

Optimal rate for covariance operator estimators of functional autoregressive processes with random coefficients

We improve a result of Allam and Mourid (2014) by deriving the optimal n rate for the empirical covariance operators of a Hilbert-valued autoregressive process with random coefficients. Our approach is based on a suitable autoregressive representation of a sequence of covariance operators related to the model, which leads to a decomposition with Hilbert-valued martingale differences. Using large deviation inequalities for Hilbert-valued martingale differences, we then establish exponential bounds and derive the almost sure convergence of the empirical covariance operators in the Hilbert–Schmidt norm, achieving the parametric rate n up to a ln(n) factor in the bounded process case.

Rhaly Operators on Small Weighted Hardy Spaces

We study several properties of Rhaly operators on small weighted Hardy spaces of holomorphic functions in the unit disc \(\mathbb {D}\). In particular, we obtain criteria for boundedness, compactness, Hilbert-Schmidt norm, and p-Schatten class membership of such operators. A closedness of Rhaly operator is also studied.

Average of Uncertainty Product for Bounded Observables

The goal of this paper is to calculate exactly the average of uncertainty product of two bounded observables and to establish its typicality over the whole set of finite dimensional quantum pure states. Here we use the uniform ensembles of pure and isospectral states as well as the states distributed uniformly according to the measure induced by the Hilbert-Schmidt norm. Firstly, we investigate the average uncertainty of an observable over isospectral density matrices. By letting the isospectral density matrices be of rank-one, we get the average uncertainty of an observable restricted to pure quantum states. These results can help us check how large is the gap between the uncertainty product and any lower bounds obtained for the uncertainty product. Although our method in the present paper cannot give a tighter lower bound of uncertainty product for bounded observables, it can help us drop any one that is not substantially tighter than the known one.

Stability of group relations under small Hilbert–Schmidt perturbations

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here “almost” and “close” are in terms of the Hilbert–Schmidt norm. Using tracial 2-norm on II1-factors we similarly define II1-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are II1-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characters: each character must be a pointwise limit of traces of finite-dimensional representations. This allows us to prove matricial stability for the discrete Heisenberg group H3 and for all virtually abelian groups. For non-amenable groups the same approximation property is a necessary condition for being matricially stable. We study this approximation property and show that RF groups with character rigidity have it.

Norm Ratios Under a Weak Order Relation in M m ⊗ M n ${\mathbb {M}}_{m}\otimes {\mathbb {M}}_{n}$

In the real Hilbert space of self-adjoint elements of the tensor product ${\mathbb {M}}_{m}\otimes {\mathbb {M}}_{n}$ , there are two natural cones besides the cone ${\mathfrak {P}}_{0}$ of positive semi-definite elements. The one is and the other is the cone ${\mathfrak {P}}_{-}$ , dual to ${\mathfrak {P}}_{+}$ with respect to the inner product. Then, ${\mathfrak {P}}_{+} \subset {\mathfrak {P}}_{0} \subset {\mathfrak {P}}_{-}.$ A weak order relation ≽ is introduced by Our interest is in finding bounds for the ratio | | |T| | |/| | |S| | | for S ≽T ≽ 0, where | | |⋅| | | is one of the operator norm, the trace norm, and the Hilbert-Schmidt norm.