Quantum Marginal Inequalities and the Conjectured Entropic Inequalities

A conjecture – the modified super-additivity inequality of relative entropy – was proposed in Zhang et al. (Phys. Lett. A 377:1794–1796, 2013): There exist three unitary operators \(U_{A}\in \mathrm {U}(\mathcal {H}_{A}), U_{B}\in \mathrm {U}(\mathcal {H}_{B})\) , and \(U_{AB}\in \mathrm {U}(\mathcal {H}_{A}\otimes \mathcal {H}_{B})\) such that $$\mathrm{S}\left(U_{AB}\rho_{AB}U^{\dagger}_{AB}||\sigma_{AB}\right)\geqslant \mathrm{S}\left(U_{A}\rho_{A}U^{\dagger}_{A}||\sigma_{A}\right) + \mathrm{S}\left(U_{B}\rho_{B}U^{\dagger}_{B}||\sigma_{B}\right), $$ where the reference state σ is required to be full-ranked. A numerical study on the conjectured inequality is conducted in this note. The results obtained indicate that the modified super-additivity inequality of relative entropy seems to hold for all qubit pairs.     

Monogamy Relations of Measurement-Induced Disturbance

The standard monogamy imposes severe limitations to sharing quantum correlations in multipartite quantum systems, which is a star topology and is established by Coffman, Kundu and Wootters. In this work, we discuss some monogamy relations beyond it, and focus on the measurement-induced disturbance (MID) which quantifies the multipartite quantum correlation. We prove exactly that MID obeys the property of discarding quantum systems never increases in an arbitrary quantum state. Moreover, we define a new kind of sharper monogamy relation which shows that the sum of all bipartite MID can not exceed the amount of total MID. This restriction is similarly called a mesh monogamy. We numerically study how MID is distributed in a 4-qubit mixed state, and which relation exists between the mesh monogamy of MID and the level of obeying the standard monogamy.     

Monogamy Relations for Squared Entanglement Negativity

This paper contains two main contents. In the first part, we provide two counterexamples of monogamy inequalities for the squared entanglement negativity: one three-qutrit pure state which violates of the He-Vidal monogamy conjecture, and one four-qubit pure state which disproves the squared-negativity-based Regula-Martino-Lee-Adesso-class strong monogamy conjecture. In the second part, we investigate the sharing of the entanglement negativity in a composite cavity-reservoir system using the corresponding multipartite entanglement scores, and then we find that there is no simple dominating relation between multipartite entanglement scores and the entanglement negativity in composite cavity-reservoir systems. As a by-product, we further validate that the entanglement of two cavity photons is a decreasing function of the evolution time, and the entanglement will suddenly disappear interacting with independent reservoirs.     

Entropic Inequalities for Two Coupled Superconducting Circuits

We discuss the known construction of two interacting superconducting circuits based on Josephson junctions, which can be precisely engineered and easily controlled. In particular, we use the parametric excitation of two circuits realized by an instant change of the qubit coupling to study entropic and information properties of the density matrix of a composite system. We obtain the density matrix from the initial thermal state and perform its analysis in the approximation of small perturbation parameter and sufficiently low temperature. We also check the subadditivity condition for this system both for the von Neumann entropy and deformed entropies and check the dependence of mutual information on the system temperature. Finally, we discuss the applicability of this approach to describe the two coupled superconducting qubits as harmonic oscillators with limited Hilbert space.     

Entropic Inequalities and Properties of Some Special Functions

Using known entropic and information inequalities, we obtain new inequalities for some classical polynomials. We consider examples of Jacobi and Legendre polynomials.     

Weighted Information and Weighted Entropic Inequalities for Qutrit States

We review the notion of weighted quantum entropy and consider the weighted quantum entropy for bipartite and noncomposite quantum systems. We extend the subadditivity condition, the inequality known for the weighted entropy information, to the case of indivisible qudit system, such as a qutrit. We discuss the new inequality for the qutrit density matrix for different weights and states, as well as the role of weighted entropy with respect to nonlinear quantum channels.     

Generalized Entanglement Monogamy and Polygamy Relations for N-Qubit Systems

We investigate the generalized monogamy and polygamy relations N-qubit systems. We give a general upper bound of the αth (0 ≤ α ≤ 2) power of concurrence for N-qubit states. The monogamy relations satisfied by the αth (0 ≤ α ≤ 2) power of concurrence are presented for N-qubit pure states under the partition AB and C₁...C_N− 2, as well as under the partition ABC₁ and C₂⋯C_N− 2. These inequalities give rise to the restrictions on entanglement distribution and the trade off of entanglement among the subsystems. Similar results are also derived for negativity.

     

Monogamy Relations in Terms of Tsallis-Q Entropy in any Dimensional System

Monogamy of entanglement is a fundamental property of multipartite entangled states. In this article, due to the convexity of Trρ^q with respect to q when q ≥ 1, we give a monogamy-like relation in terms of Tsallis-q entanglement entropy of assistance (TqEEA) for pure states over an n- partite any dimensional system and monogamy-like relations in terms of Tsallis-q entanglement entropy (TqEE) for mixed states for any dimensional system, we also give a lower bound for the TqEE of a four-partite pure state. At last, we show that the generalized W-class states satisfy the polygamy relation in terms of TqEE when q = 2.     

Tighter Monogamy and Polygamy Relations of Quantum Entanglement in Multi-qubit Systems

International Journal of Theoretical Physics - We investigate the monogamy relations related to the concurrence, the entanglement of formation, convex-roof extended negativity, Tsallis-q...     

Quantum‐Memory‐Assisted Entropic Uncertainty Relations

Uncertainty relations take a crucial and fundamental part in the frame of quantum theory, and are bringing on many marvelous applications in the emerging field of quantum information sciences. Especially, as entropy is imposed into the uncertainty principle, entropy‐based uncertainty relations lead to a number of applications including quantum key distribution, entanglement witness, quantum steering, quantum metrology, and quantum teleportation. Herein, the history of the development of the uncertainty relations is discussed, especially focusing on the recent progress with regard to quantum‐memory‐assisted entropic uncertainty relations and dynamical characteristics of the measured uncertainty in some explicit physical systems. The aims are to help deepen the understanding of entropic uncertainty relations and prompt further explorations for versatile applications of the relations on achieving practical quantum tasks.     

Entropic and Information Inequalities for Indivisible Qudit Systems<Superscript>*</Superscript>

We present the idea that in both classical and quantum systems all correlations available for composite multipartite systems, e.g., bipartite systems, exist as “hidden correlations” in indivisible (noncomposite) systems. The presence of correlations is expressed by entropic-information inequalities known for composite systems like the subadditivity condition. We show that the mathematically identical subadditivity condition and the mutual information nonnegativity are available as well for noncomposite systems like a single-qudit state. We demonstrate an explicit form of the subadditivity condition for a qudit with j = 2 or the five-level atom. We consider the possibility to check the subadditivity condition (entropic inequality) in experiments where such a system is realized by the superconducting circuit based on Josephson-junction devices.     

Entropic pressure between fluctuating membranes in multilayer systems

Gaining insights into the fluctuation-induced entropic pressure between membranes that mediates cell adhesion and signal transduction is of great significance for understanding numerous physiological processes driven by intercellular communication. Although much effort has been directed toward investigating this entropic pressure, there still exists tremendous controversy regarding its quantitative nature, which is of primary interest in biophysics, since Freund challenged the Helfrich’s well-accepted results on the distance dependence. In this paper, we have investigated the entropic pressure between fluctuating membranes in multilayer systems under pressure and tension through theoretical analysis and Monte Carlo simulations. We find that the scaling relations associated with entropic pressure depend strongly on the magnitude of the external pressures in both bending rigidityand surface tension-dominated regimes. In particular, both theoretical and computational results consistently demonstrate that, in agreement with Helfrich, the entropic pressure p decays with inter-membrane separations c as p~c^–3 for the tensionless multilayer systems confined by small external pressures. However, our results suggest that the entropic pressure law follows to be p~c^–1 and p~c^–3, respectively, in the limit of large and small thermal wavelengths for bending fluctuations of the membranes in a tension-independent manner for the case of large external pressures.     

熵测不准关系与光场的熵压缩

用熵作为光场量子涨落的量度,根据熵测不准关系,建立了熵压缩的概念,具体研究了光场与原子相互作用时的熵压缩,结果显示,熵压缩实现了对光场压缩效应的高灵敏量度。     

Inequalities between different classical spin models

We apply correlation inequalities to show that in a multicomponent spin model some quantities (e.g. magnetisation) decrease from a one-component model to a two-component, or from two to n > 2.     

On Entropic Uncertainty Relations for Measurements of Energy and Its “Complement”

Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays an important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non‐commuting observables. Since quantum theory was originated, many approaches to energy–time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, uncertainty relations of the “energy–time” type are obtained in terms of Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state‐dependent and state‐independent formulations are of interest. Some of the derived state‐independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.     

Entropic characteristics of photon tomograms

We study the electromagnetic-field tomograms for classical and quantum states. We use the violation of the positivity of entropy for the photon-probability distributions for distinguishing the classical and quantum domains. We show that the photon-probability distribution expressed in terms of optical or symplectic tomograms of the photon quantum state must be a nonnegative function, which yields the nonnegative Shannon entropy. We also show that the optical tomogram of the photon classical state provides the expression for the Shannon entropy, which can be nonpositive.     

Entropic formulation of uncertainty relations

We review the recent investigations on the improved formulation of uncertainty relations which employ the information-theoretic entropy rather than variance as a measure of uncertainty. We show that this formulation also brings out clearly the relation between the overall uncertainty and the quantum mechanical interference due to measurements. Lecture delivered at the International Symposium on Theoretical Physics, Bangalore, November 1984.