Subadditivity and Strong Subadditivity Conditions for the Density Matrix of the Five-Level Atom

We obtain new quantum inequalities for von Neumann entropy of the five-level atom, which are analogs of the subadditivity condition known for bipartite quantum systems and the strong subadditivity condition known for tripartite quantum systems. We discuss the possibility to check the inequalities for the single qudit with j = 2, which can be realized as a five-level atom in the experiments with superconducting circuits. We present the strong subadditivity conditions for the finite-level atomic populations.     

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.     

Maps of Matrices and Portrait Maps of the Density Operators of Composite and Noncomposite Systems

We obtain a new inequality for arbitrary Hermitian matrices. We describe particular linear maps called the matrix portrait of arbitrary N × N matrices. The maps are obtained as analogs of partial tracing of density matrices of multipartite qudit systems. The structure of the maps is inspired by “portrait” map of the probability vectors corresponding to the action on the vectors by stochastic matrices containing either unity or zero matrix elements. We obtain new entropic inequalities for arbitrary qudit states including a single qudit and discuss entangled single qudit state. We consider in detail the examples of N = 3 and 4. Also we point out the possible use of entangled states of systems without subsystems (e.g., a single qudit) as a resource for quantum computations.     

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.     

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.     

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.     

Qubit portrait of qudit states and Bell inequalities

A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. In view of the qubit-portrait method, the Bell inequalities for two qubits and two qutrits are discussed within the framework of the probability-representation of quantum mechanics. A semigroup of stochastic matrices is associated with tomographic-probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as an ansatz to provide a necessary condition for the separability of quantum states.     

Generalized Qubit Portrait of the Qutrit-State Density Matrix

We obtain new inequalities for tomographic probability distributions and density matrices of qutrit states by generalization of the qubit-portrait method. We propose an approach based on the quditportrait method of obtaining new entropic inequalities. Our approach can be applied to the case of arbitrary nonnegative hermitian matrices, including the density matrices of multipartite qudit states.     

Separability and Entanglement of the Qudit X-State with j = 3/2

We study the qudit state with spin j = 3/2 and the density matrix of the form corresponding to the X state of two qubits and consider the entanglement and separability properties. We use the qubit portrait of qudit states to obtain the entropic inequalities for the entangled state of a single qudit. We present the tomographic-probability representation of the qudit X-state and obtain the Shannon and q entropic characteristics in explicit forms.     

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.     

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.     

New Inequality for Density Matrices of Single Qudit States

Using the monotonicity of relative entropy of composite quantum systems, we obtain new entropic inequalities for arbitrary density matrices of single qudit states. Examples of qutrit state inequalities and the “qubit portrait” bound for the distance between the qutrit states are considered in explicit form.     

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.     

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.     

Semigroup of positive maps for qudit states and entanglement in tomographic probability representation

Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups IGL(n,R) and GL(n,R) respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities are given in the framework of the tomographic probability representation using the stochastic semigroup properties.     

Inequalities for Purity Parameters of Multiqudit and Single-Qudit States

We analyze the recently found inequality for eigenvalues of the density matrix and purity parameters describing either a bipartite-system state or a single-qudit state. We rewrite the Minkowski-type trace inequality for the density matrices of the qudit states in terms of the purity parameters and discuss the properties of the inequality obtained, paying special attention to the X-states of two qubits and a single qudit. Also we study the relation of the purity inequalities obtained with the entanglement.     

Atomic Wehrl entropy and negativity as entanglement measures for qudit pure states in a trapped ion

We analyze the atomic Wehrl entropy and negativity as compared with concurrence for qudit pure states in a trapped ion. We use the density matrix in calculating the three measures of quantum correlations. We find that a long surviving entangled qudit can be established between the three atomic levels and vibrational modes. We observe three distinct entanglements in response to an increasing Lamb–Dicke parameter.     

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.     

弱测量对四个量子比特量子态的保护

廖湘萍等(Chin.Phys.B 23 020304,2014)指出弱测量和弱测量反转操作可以保护三个量子比特的纠缠,提高保真度.本文将弱测量方法推广至四个量子比特的情况,研究了几种典型四个量子比特量子态的演化.结果表明:在振幅阻尼通道中,弱测量方法能够有效地提高系统量子态的保真度.分析了影响量子态保真度的各种因素,对比了不同量子态的演化特征,划分了量子态保真度提高的敏感区域.最后,对弱测量方法抑制量子态衰减的内在机制做了合理的物理解释.     

Distribution of Mutual Information in Multipartite States

Using the relative entropy of total correlation, we derive an expression relating the mutual information of n-partite pure states to the sum of the mutual informations and entropies of its marginals and analyze some of its implications. Besides, by utilizing the extended strong subadditivity of von Neumann entropy, we obtain generalized monogamy relations for the total correlation in three-partite mixed states. These inequalities lead to a tight lower bound for this correlation in terms of the sum of the bipartite mutual informations. We use this bound to propose a measure for residual three-partite total correlation and discuss the non-applicability of this kind of quantifier to measure genuine multiparty correlations.