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Europhysics Letters (EPL)

A novel quantum Shannon entropy as a sensitivity of Anderson transitions: It simultaneously relates to periodic and antiperiodic boundary conditions

Longyan Gong, Jingjing Zhang, Kaixin Ma, Xingfei Zhou, Xuechao Zhai, W. W. Cheng, Shengmei Zhao

Published Jun 27, 2018• 4 citations
Publisher

Abstract

In general, extended states are more sensitive to boundary conditions (BCs) than localized ones. We expand an eigenstate of a Hamiltonian with periodic BC as the superposition of all eigenstates of the same Hamiltonian but with antiperiodic BC, i.e. , , where and N is the Hilbert space dimension. Thus, we propose a novel quantum Shannon entropy , which simultaneously relates to periodic and antiperiodic BCs. It measures the information of contained in all the N eigenstates . Based on it, we define a reduced quantum Shannon entropy , where . We apply it to several typical models. Results show that are relatively smaller for delocalized states and are less than and close to one for localized states as N increases. There are rapid changes in around Anderson transition points. Therefore, can be used as a sensitivity measure to characterize Anderson transitions.

DOI
10.1209/0295-5075/122/37002

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In general, extended states are more sensitive to boundary conditions (BCs) than localized ones. We expand an eigenstate of a Hamiltonian with periodic BC as the superposition of all eigenstates of the same Hamiltonian but with antiperiodic BC, i.e. , , where and N is the Hilbert space dimension. Thus, we propose a novel quantum Shannon entropy , which simultaneously relates to periodic and antiperiodic BCs. It measures the information of contained in all the N eigenstates . Based on it, we define a reduced quantum Shannon entropy , where . We apply it to several typical models. Results show that are relatively smaller for delocalized states and are less than and close to one for localized states as N increases. There are rapid changes in around Anderson transition points. Therefore, can be used as a sensitivity measure to characterize Anderson transitions.
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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.
Sensitivity of entanglement Hamiltonian spectrum to boundary conditions is considered as a phase detection parameter for delocalized-localized phase transition. By employing one-dimensional models that undergo delocalized-localized phase transition, we study the shift in the entanglement energies and the shift in the entanglement entropy when we change boundary conditions from periodic to antiperiodic. Specifically, we show that both these quantities show a change of several orders of magnitude at the transition point in the models considered. Therefore, this shift can be used to indicate the phase-transition points in the models. We also show that both these quantities can be used to determine mobility edges separating localized and delocalized states.
The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission. Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does. We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional C∗C^*C∗-algebras throughout, modeling finite quantum systems that may have superselection sectors.
CONFINED HARMONIC OSCILLATOR.The aim of this study was a didactic presentation of the Shannon entropy in the quantum theory context, followed by application to the case of a one-dimensional harmonic oscillator in its ground state, both in the free case and confined case.The study of these systems allows us to highlight notions such as location or delocation of a particle, a possible interpretation that the Shannon entropy can adopt.The Shannon entropy in position (S x ) and momentum (S p ) spaces were calculated for both systems, beyond the entropic sum (S t = S x + S p ).With this procedure it was possible to identify trends in the behavior of the Shannon entropy (S x and S p ) and test the compliance of the entropic uncertainty relation (S t = S x + S p ≥ n(1 + ln(p))).
The notion of Shannon entropy is crucial for the theory of classical information. In quantum information theory, an analogous key role is played by the von Neumann entropy: quantum information processing is closely related to entropy dynamics. This reveals a direct link with the theory of quantum chaotic systems, which can be characterized by a positive entropy production. Furthermore, noise, which inevitably affects any quantum system, can be modeled by a random quantum operation or by coupling to an environment in a generic chaotic state. In this contribution, we emphasize the universality of quantum chaotic dynamics and discuss its implications for quantum information processing.