A construction of the Wehrl entropy is proposed for an arbitrary locally compact abelian group (G). It is proved that the Wehrl entropy is not less than a certain nonnegative integer, which is an invariant of the group (G). The minimum of the Wehrl entropy is attained on coherent states.
We consider a metric (D) describing the difference between real (noisy) and ideal processes that is based on the operator norm of the maximum deviation between the final real and ideal states of a quantum system. We discuss the properties as well as geometric and experimental interpretations of the metric.
We study the properties of the fractional derivative (D_alpha l(t,x)) of order (alpha<1/2) of the Brownian local time (l(t,x)) with respect to the variable (x). This derivative is understood as the convolution of the local time with the generalized function (|x|^{-1-alpha}). We show that (D_alpha l(t,x)) appears naturally in Itô’s formula for the process (|w(t)|^{1-alpha}). Using the martingale technique, we also study the limit behavior of (D_alpha l(t,x)) as (ttoinfty).
Let a von Neumann algebra (mathcal M) of operators act on a Hilbert space (mathcal{H}), and let (tau) be a faithful normal semifinite trace on (mathcal M). Let (t_{tautext{l}}) be the topology of (tau)-local convergence in measure on the *-algebra (S(mathcal M,tau)) of all (tau)-measurable operators. We prove the (t_{tautext{l}})-continuity of the involution on the set of all normal operators in (S(mathcal M,tau)), investigate the (t_{tautext{l}})-continuity of operator functions on (S(mathcal M,tau)), and show that the map (Amapsto |A|) is (t_{tautext{l}})-continuous on the set of all partial isometries in (mathcal M).
We formulate and prove Bell’s inequalities in the realm of JB(^*) triples and JB(^*) algebras. We show that the maximal violation of Bell’s inequalities occurs in any JBW(^*) triple containing a nonassociative (2)-Peirce subspace. Moreover, we show that the violation of Bell’s inequalities in a nonmodular JBW(^*) algebra and in an essentially nonmodular JBW(^*) triple is generic. We describe the structure of maximal violators and its relation to the spin factor. In addition, we present a synthesis of available results based on a unified geometric approach.
We obtain a representation of the set of quantum states in terms of barycenters of nonnegative normalized finitely additive measures on the unit sphere (S_1(mathcal H)) of a Hilbert space (mathcal H). For a measure on (S_1(mathcal H)), we find conditions in terms of its properties under which the barycenter of this measure belongs to the set of extreme points of the family of quantum states and to the set of normal states. The unitary elements of a unital (mathrm C^*)-algebra are characterized in terms of extreme points. We also study extreme points (mathrm{extr}(mathcal E^1)) of the unit ball (mathcal E^1) of a normed ideal operator space (langlemathcal E,|kern1pt{cdot}kern1pt|_{mathcal E}rangle) on (mathcal H). If (Uinmathrm{extr}(mathcal E^1)) for some unitary operator (Uinmathcal{B}(mathcal H)), then (Vinmathrm{extr}(mathcal E^1)) for all unitary operators (Vinmathcal{B}(mathcal H)). In addition, we construct quantum correlations corresponding to singular states on the algebra of all bounded operators in a Hilbert space.
We study tightness-type properties such as tightness, minitightness, and local density of the space of weakly additive functionals with finite support. We also investigate some generalizations of continuous functions. Furthermore, we present an extension of the functor of weakly additive functionals with finite support to the class of strictly (tau)-continuous mappings. We introduce two extensions of the categories (mathrm{Comp}) and (mathrm{Tych}) (of compact and Tychonoff spaces, respectively). One of the main results of the paper is that the functor (O_n) of weakly additive functionals with finite support preserves the tightness character of infinite compact spaces. In addition, we show that the local densities of the spaces (X) and (O_n(X)) coincide for any infinite compact space (X).
We study the problem of maximizing the Jensen gap with respect to the probability distribution in a fairly general case, and prove a theorem on the optimal distribution. Using the results obtained, we calculate the one-shot capacity of a certain family of non-unital quantum channels. We show that in sufficiently large dimensions the channel admits one of two modes of an optimal input ensemble depending on the parameters. We also prove that both the fulfillment and the violation of the entanglement-breaking property are possible in any dimension depending on the parameters of the channel.
We study the Chernoff averages for random generalized shift operators in the case of noncanonical commutation relations between creation and annihilation operators. We introduce the concepts of shift-dual ladder operators and generalized shift operators. As an example, we consider a one-parameter family of commutation relations for which generalized shift operators are unitary and satisfy the semigroup property on straight lines passing through the origin. For this family, we prove that the sequence of expectations of Feynman–Chernoff iterations of random shift operators converges to a limit strongly continuous semigroup.
Inadequate management and control of hyperglycemia predisposes diabetic patients to a wide range of complications. Thus, this opens new windows for exploring and scrutinizing novel candidate biomarkers. This study was designed to scrutinize the relationship between HbA1c, osteocalcin, calcium, phosphorus, and expression levels of miR-143 and miR-145 in individuals with T1DM and explore their correlations and diagnostic potential for T1DM. 120 unrelated participants were included (i.e., 90 participants with type 1 diabetes mellitus and 30 healthy controls) and were allocated into two groups. Participants with T1DM were allocated into three subgroups (i.e., below 1 year, 1-8 years, and over 8 years) based on diabetic duration. Participants with T1DM experienced noticeable HbA1c elevation. However, osteocalcin, phosphorus, and calcium profiles notably declined in participants with diabetes compared with those in healthy controls. Moreover, the expression levels of miR-143 and miR-145 decreased in participants with diabetes with a significant difference between participants with diabetes and healthy controls. Additionally, significant alterations in HbA1c, osteocalcin, phosphorus, and calcium profiles and expression levels of miR-143 and miR-145 were observed with increasing diabetic duration (T1DM > 8 years compared with those with a diabetes duration of less than 1 year). This study suggests that miR-143 and miR-145 are prospective biomarkers of diabetes mellitus, which may help predict the progression of complications.
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