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.     

Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality

We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so–called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information.     

Inequalities for nonnegative numbers and information properties of qudit tomograms

We discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The inequalities characterize the degree of quantum correlations in addition to noncontextuality and quantum discord. We use the subadditivity and strong subadditivity conditions for qudit tomographic-probability distributions depending on the unitary-group parameters in order to derive new inequalities for Shannon, Rényi, and Tsallis entropies of spin states.     

Subadditivity Condition for Spin Tomograms and Density Matrices of Arbitrary Composite and Noncomposite Qudit Systems

We obtain a new quantum entropic inequality for the states of a system of n ≥ 1 qudits. The inequality has the form of the quantum subadditivity condition of a bipartite qudit system and coincides with the subadditivity condition for the system of two qudits. We formulate a general statement on the existence of the subadditivity condition for an arbitrary probability distribution and an arbitrary qudit-system tomogram. We discuss the nonlinear quantum channels creating the entangled states from separable states.     

Hidden Correlations and Entanglement in Single-Qudit States<Superscript>†</Superscript>

We discuss the notion of hidden correlations in classical and quantum indivisible systems along with such characteristics of the correlations as the mutual information and conditional information corresponding to the entropic subadditivity condition and the entropic strong subadditivity condition. We present an analog of the Bayes formula for systems without subsystems, study entropic inequality for von Neumann entropy and Tsallis entropy of the single-qudit state, and discuss the inequalities for qubit and qutrit states as an example.     

Deformed Subadditivity Condition for Qudit States and Hybrid Positive Maps

We extend the subadditivity condition for q-deformed entropy of a bipartite quantum system to the case of an arbitrary quantum system including the single qudit state. We present the subadditivity condition for the density matrix of the single qutrit state in an explicit form. We obtain the inequality for the purity parameters of a bipartite quantum system and its subsystems. We propose a positive map construction using the fiducial density matrix.     

A New Inequality for the von Neumann Entropy

Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states.     

Subadditivity of states on quantum logics

We give various definitions of subadditivity of states on quantum logics and present several results stating when a quantum logic with sufficiently enough properly subadditive states has to be (almost) a Boolean algebra.     

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.     

The Saturation of Several Universal Inequalities in Information-Processing

In this paper, we characterize the saturation of four universal inequalities in quantum information theory, including a variant version of strong subadditivity inequality for von Neumann entropy, the coherent information inequality, the Holevo quantity, and average entropy inequalities. These results shed new light on quantum information inequalities.     

Tomographic and Improved Subadditivity Conditions for Two Qubits and a Qudit with j = 3/2

We obtain a new entropic inequality for quantum and tomographic Shannon information for systems of two qubits. We derive the inequality relating quantum information and spin-tomographic information for particles with spin j = 3/2. We recommend the method for obtaining new entropic and information inequalities for composite systems of qudits, as well as for one qudit.     

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincaré or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher (C^k, k ≥ 1) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities), which yields, by a new “multiscale” Poincaré inequality, quantitative estimates on the sublinearity of the corrector.     

Information Processing Using Three-Qubit and Qubit–Qutrit Encodings of Noncomposite Quantum Systems

We study quantum information properties of a seven-level system realized by a particle in a onedimensional square-well trap and discuss the features of encodings of seven-level systems in a form of three-qubit or qubit–qutrit systems. We use the three-qubit encoding of the system in order to investigate the subadditivity and strong subadditivity conditions for the particle’s thermal state. We employ the qubit–qutrit encoding to suggest a single qudit algorithm for calculating the parity of a bit string. The results obtained indicate on the potential resource of multilevel systems for realization of quantum information processing.     

On the Relation Between Strong Subadditivity and Entanglement

Strong subadditivity is used to improve the triangular inequality for the entropy of tensorproducts by the amount of entanglement.     

On the failure of subadditivity of the Wigner–Yanase entropy

It was recently shown by Hansen that the Wigner–Yanase entropy is, for general states of quantum systems, not subadditive with respect to decomposition into two subsystems, although this property is known to hold for pure states. We investigate the question whether the weaker property of subadditivity for pure states with respect to decomposition into more than two subsystems holds. This property would have interesting applications in quantum chemistry. We show, however, that it does not hold in general, and provide a counterexample. Work partially supported by U.S. National Science Foundation grant PHY-0353181 and by an Alfred P. Sloan Fellowship. This paper may be reproduced, in its entirety, for non-commerical purposes.     

中间态引入量子干涉的三光子共振非简并六波混频

理论研究了五能级系统中三光子共振非简并六波混频(NSWM) 由于中间能级加入耦合光场而产生的量子干涉效应. 分析了耦合光场对三光子共振NSWM信号以及频谱的影响. 研究发现, 在强耦合场作用下, NSWM的频谱出现了Autler-Townes分裂, 它反映的是两个缀饰态的能级, 量子干涉可以使NSWM信号被抑制或增强. 提出利用量子干涉对NSWM信号产生增强作用, 可以由耦合场产生的缀饰态代替原子固有能级, 成为三光子共振的中间态, 从而控制耦合场来选择三光子共振的中间态的位置.     

Control of Spontaneous Emission from a Microwave Field Coupled Five-Level M-type Atom in Photonic Crystals

We investigate the spontaneous emission spectrum of a five-level M-type atom driven by a microwave field, in which one lower level is coupled by the same modified reservoir to two upper levels. The results show that a few interesting phenomena in spontaneous emission spectra, such as spectral-line shift, spectral-line enhancement and spectral-line suppression, which can be controlled by adjusting the proper parameters of the system. These phenomena can originate from quantum interference of the strong coupling system.     

Modification of spontaneous emission from a five-level M-type atom in coupled cavity waveguide

We study the spontaneous emission spectrum of a five-level M-type atom driven by a microwave field, in which two upper levels are coupled by the same-coupled cavity waveguide reservoir to a lower level. The spectrum behavior presents a strong non-Lorentzian shape that originates from effective quantum interference in Markovian reservoir, in which the spectral line can be significantly enhanced and eliminated by adjusting the proper parameters of the system. However, for non-Markovian reservoir, it seems that the shape of emission spectrum is quite dependent on the geometry behavior of a coupled cavity waveguide.     

Quantum Mutual Entropy for a Multilevel Atom Interacting with a Cavity Field

We derive an explicit formula for the quantum mutual entropy as a measure of the total correlations in a multi-level atom interacting with a cavity field. We describe its theoretical basis and discuss its practical relevance. The effect of the number of levels involved on the quantum mutual entropy is demonstrated via examples of three-, four- and five-level atom. Numerical calculations under current experimental conditions are performed and it is found that the number of levels present changes the general features of the correlations dramatically. PACS numbers: 32.80.−t, 42.50.Ct, 03.65.Ud, 03.65.Yz.     

On Convexity of Generalized Wigner–Yanase–Dyson Information

The convexity of the Wigner–Yanase–Dyson information, as first proved by Lieb, is a deep and fundamental result because it leads to the strong subadditivity of quantum entropy. The Wigner–Yanase–Dyson information is a particular kind of quantum Fisher information with important applications in quantum estimation theory. But unlike the quantum entropy, which is the unique natural quantum extension of the classical Shannon entropy, there are many different variants of quantum Fisher information, and it is desirable to investigate their convexity. This article is devoted to studying the convexity of a direct generalization of the Wigner–Yanase–Dyson information. Some sufficient conditions are obtained, and some necessary conditions are illustrated. In a particular case, a surprising necessary and sufficient condition is obtained. Our results reveal the intricacy and subtlety of the convexity issue for general quantum Fisher information.