We consider a μ-deformation of the Segal-Bargmann transform, which is a unitary map from a μ-deformed quantum configuration space onto a μ-deformed quantum phase space (the μ-deformed Segal-Bargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases. We also consider two reverse log-Sobolev inequalities in the μ-deformed Segal-Bargmann space, which have been proved in a previous work, and show that a certain known coefficient in them is the best possible.
We consider a μ-deformation of the Segal-Bargmann transform, which is a unitary map from a μ-deformed quantum configuration space onto a μ-deformed quantum phase space (the μ-deformed Segal-Bargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases. We also consider two reverse log-Sobolev inequalities in the μ-deformed Segal-Bargmann space, which have been proved in a previous work, and show that a certain known coefficient in them is the best possible.
The Segal-Bargmann transform A is a unitary isomorphism of the quantum mechanical configuration space L2(Rn,ν) onto the Segal-Bargmann, or phase, space HL2(Cn,μ) of entire functions square integrable with respect to μ. Here μ and ν are Gaussian measures, and for z∈Cn and measurable f:Rn→CAf(z) :=∫Rn dν(t)f(t)exp(−z2/2+√z⋅t) if the integral converges. Af is defined for all f∈Lp(Rn,ν) if 1<p⩽∞, but A does not preserve Lp structure if p≠2. Specifically, it is proven here that A is a compact map from Lp(Rn,ν) to Lq(Cn,μ) if 1⩽q<2 and p>1+q/2 (which implies p>q) and in this case ‖A‖p→q⩽((1+q/2−p′q/2)(1−q/2))−n/2q where p′ is the dual of p. Moreover, a calculation with test functions shows that ‖A‖p→q=∞ if q>2 or p<1+q/2 (which is implied by p<q). Using an improved operator norm estimate (analogous to the Hausdorff-Young inequality) we prove Hirschman type entropy inequalities for f∈L2(Rn,ν), (p−1−2−1)S(f )⩽(q−1−2−1)S(Af )+nCp,q‖f‖22, where 1⩽q<2, p>1+q/2, Cp,q is given explicitly as a function of p and q and S(g) is the entropy of g. Examples given here show A is not entropy preserving. So these inequalities provide constraints on the change in entropy. Similarly, we get energy-entropy, or log-Sobolev, inequalities a1S(Af )+a2S(f )⩽a3‖f‖22+a4〈f,Nf〉, for f in a dense subspace of L2(Rn,ν). Here 〈f,Nf〉 is the Dirichlet form of the energy operator N in the configuration space and the ai can have various signs and are also given explicitly. The principal results of this article are generalized Heisenberg uncertainty relations, namely they are constraints on Af, the phase space representation, given information about f, the configuration space representation.
In this manuscript we study the Wehrl entropy of entangled oscillators. This semiclassical entropy associated with the phase-space description of quantum mechanics can be used for formulating uncertainty relations and for a quantification of entanglement. We focus on a system of two coupled oscillators described within its Segal-Bargmann space. This Hilbert space of holomorphic functions integrable with respect to a given Gaussian-like measure is particularly convenient to deal with harmonic oscillators. Indeed, the Stone-von Neumann theorem allows us to work in this space in a full correspondence with the ladder operators formalism. In addition, the Husimi pseudoprobability distribution is directly computed within the Segal-Bargmann formalism. Once we obtain the Husimi function, we analyze the Wehrl entropy and mutual information.
In this report, we present three generalized expressions of Shannon’s entropy using the electron densities of position and momentum spaces. Such expressions were obtained using the definition of deformed logarithm introduced by Tsallis. We show that only one expression fulfils the dimensionless criterion, it is strictly positive overall space and follows a growing behavior with respect to the electron number. We also found that by using some values of [Formula: see text], it is possible to reproduce the general trends of the radial distribution in position space and momentum space of the neon atom.
We shall state the following generalization of operator type Shannoninequal- ity and its reverse one as a simple corollary of parametric extensions of Shannon inequality in Hilbert space operators.Let {A1,A2,…,An} andn {B1,B2,…,Bn}n be two sequences of strictly positive operators on a Hilbert space H.If EAj=LBj=I, thenwhere Sq(A∣B)=A21(A2−1BA2−1)q(logA2−1BA2−1)Ai for A>0, B>0 and any real number q and S(A∣B)=S0(A∣B)=A21(logA2−1BA2−1)A21 which is the relative operator entropy of A>0 and B>0.Our results can be considered as parametric extensions of the following celebrated Shan- non inequality ( [3],[5] and [233 p,1] ) which is very useful and so famous in information theory.Let {a1,a2,…,an} and {b1,b2,…,bn} be two probability vectors.Then 0≥∑j=1najlogbj−∑j=1najlogaj (see inequalities (2.4) of Corollary 2.4).
Abstract. In the present paper, we consider the marginal entropy of the a pure state in the tensor product of two Hilbert spaces. The expectation of the marginal entropy of a random pure state with respect to the Haar measure of unitary transformation on the tensor product Hilbert space depends on group integrals of unitary group. For group integrals of ordinary representation matrix elements of U(n), we provide a elementary method to reobtain Weingarton’s result of asymptotical behavior of group integrals. For group integrals of matrix elements of irreducible representations of U(n),we generalize the Weyl-Schur duality theorem and get an algorithm. 1. Mathematical Preliminaries We adopt the following standard conventions of quantum mechanics: The states of an n-level quantum system (and we include the case infinity) are represented by an n-dimensional complex Hilbert space H. In this familiar representation, the states of a composite of two systems A and B say, the first one A n-level, and the second B m-level, is then represented by the tensor product of the respective Hilbert spaces HA and HB. As a result, the composite system (AB) is an (nm)-level system. This also makes sense in the infinite case where we then use standard geometry for Hilbert space, and suitable choices of orthonormal bases (ONBs). For a fixed system with Hilbert space H, the corresponding pure quantum states are vectors in H of norm one, or rather equivalence classes of such vectors: Equivalent vectors v and v ′ in H yield the same rank-one projection operator P, i.e., the projection of H onto Key words and phrases. random pure state, marginal entropy,group integral, Weyl-Schur duality
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.