Convex Roof Research Articles

Coherence measures based on coherence eigenvalue and their applications

The quantification of coherence is central in quantum information theory and far from being fully understood, particularly for multipartite quantum states. In this work, we first propose the coherence eigenvalue for multipartite pure quantum states and then define the coherence measure for a given multipartite pure or mixed state based on it. It is found that the convex roof coherence measure is equal to the geometric measure based on fidelity. This finding allows us to give an alternative way to prove that the geometric measure of coherence is monotone under selective measurements on average. The other two new coherence measures based on coherence eigenvalue are presented, and their properties are also investigated. As an application, the bounds of several well-known coherence measures are derived by using of coherence eigenvalue. In particular, by proposing a new coherence witness which is related to coherence eigenvalue, we obtain a bound to robustness of coherence and give it a corresponding geometric interpretation.

Monogamy of the entanglement of formation

We show that any measure of entanglement that on pure bipartite states is given by a strictly concave function of the reduced density matrix is monogamous on pure tripartite states. This includes the important class of bipartite measures of entanglement that reduce to the (von Neumann) entropy of entanglement. Moreover, we show that the convex roof extension of such measures (e.g., entanglement of formation) are monogamous also on \emph{mixed} tripartite states. To prove our results, we use the definition of monogamy without inequalities, recently put forward[Gour and Guo, Quantum \textbf{2}, 81 (2018)]. Our results promote the theme that monogamy of entanglement is a property of quantum entanglement and not an attribute of some particular measures of entanglement.

Strict entanglement monotonicity under local operations and classical communication

Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.

Coherence measure: Logarithmic coherence number

We introduce a measure of coherence, which is extended from the coherence rank via the standard convex roof construction, we call it the logarithmic coherence number. This approach is parallel to the Schmidt measure in entanglement theory, We study some interesting properties of the logarithmic coherence number, and show that this quantifier can be considered as a proper coherence measure. We also find that the logarithmic coherence number can be calculated exactly for a large class of states. We give the relationship between coherence and entanglement in bipartite system, and our results are generalized to multipartite setting. Finally, we find that the creation of entanglement with bipartite incoherent operations is bounded by the logarithmic coherence number of the initial system during the process.

Quantum coherence

Quantum coherence is not only a fundamental concept of quantum mechanics, but also an important physical resource for quantum information processing. Along with the formulation of the resource theoretic framework of quantum coherence, the quantification of coherence is still one of the recent research focuses. Quantum coherence is also very fragile, and the environmental noise usually induces a system to decohere. Hence it is also an important subject to make clear the dynamical behavior and to seek a flexible way of preserving quantum coherence of an open quantum system. Besides, there are many potential applications of quantum coherence in quantum many-body system, quantum thermodynamics, quantum biology and other related fields. We review in this paper the resource theoretic framework for quantifying coherence and the relevant quantum coherence measures defined within this framework which includes the relative entropy of coherence, the <i>l</i><sub>1</sub> norm of coherence, the entanglement-based measure of coherence, the convex roof measure of coherence, and the robustness of coherence. We also review the dynamical behaviors of quantum coherence for certain open quantum systems, the coherence generating and breaking power of typical quantum channels, and the freezing phenomenon of quantum coherence. Moreover, we exemplify applications of quantum coherence in Deutsch-Jozsa algorithm, Grover search algorithms, and the study of quantum phase transitions in multipartite systems. We hope that these results may provide not only an overview of the relevant field, but also an outlook of the future research direction of this exciting field.

Generalized W state of four qubits with exclusively the three-tangle

We single out a class of states possessing only threetangle but distributed all over four qubits. This is a three-site analogue of states from the $W$-class, which only possess globally distributed pairwise entanglement as measured by the concurrence. We perform an analysis for four qubits, showing that such a state indeed exists. To this end we analyze specific states of four qubits that are not convexly balanced as for $SL$ invariant families of entanglement, but only affinely balanced. For these states all possible $SL$-invariants vanish, hence they are part of the $SL$ null-cone. Instead, they will possess at least a certain unitary invariant. As an interesting byproduct it is demonstrated that the exact convex roof is reached in the rank-two case of a homogeneous polynomial $SL$-invariant measure of entanglement of degree $2m$, if there is a state which corresponds to a maximally $m$-fold degenerate solution in the zero-polytope that can be combined with the convexified minimal characteristic curve to give a decomposition of $\rho$. If more than one such state does exist in the zero polytope, a minimization must be performed. A better lower bound than the lowest convexified characteristic curve is obtained if no decomposition of $\rho$ is obtained in this way.

Family of coherence measures and duality between quantum coherence and path distinguishability

Coherence measures and their operational interpretations lay the cornerstone of coherence theory. In this paper, we introduce a class of coherence measures with $\alpha$-affinity, say $\alpha$-affinity of coherence for $\alpha \in (0, 1)$. Furthermore, we obtain the analytic formulae for these coherence measures and study their corresponding convex roof extension. We provide an operational interpretation for $1/2$-affinity of coherence by showing that it is equal to the error probability to discrimination a set of pure states with the least square measurement. Employing this relationship we regain the optimal measurement for equiprobable quantum state discrimination. Moreover, we compare these coherence quantifiers, and establish a complementarity relation between $1/2$-affinity of coherence and path distinguishability for some special cases.

Superadditivity of convex roof coherence measures

In this paper, we examine the superadditivity of convex roof coherence measures. We put forward a theorem on the superadditivity of convex roof coherence measures, which provides a sufficient condition to identify the convex roof coherence measures fulfilling the superadditivity. By applying the theorem to each of the known convex roof coherence measures, we prove that the coherence of formation and the coherence concurrence are superadditive, while the geometric measure of coherence, the convex roof coherence measure based on linear entropy, the convex roof coherence measure based on fidelity, and convex roof coherence measure based on -entropy are non-superadditive.

Monogamy of entanglement without inequalities

We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.

Thermal Comfort Modelling Based on House’s Architecture Using Ghost Point Quadratic Explicit Method

When a person builds a house, the different roof shapes become a factor of consideration to obtain the desired model. The shape of a roof is a very influential factor shaping a sense of comfort in the house. Henceforth, everyone will have to sort different forms in choosing the proper roof model that makes them feel comfortable. This research developed a mathematical modeling of the temperature distribution of a room that is influenced by the shape of the roof. This research uses different types of roof to be studied, namely, a rectangular house with a straight, triangular, convex and concave roof. Room temperature distribution is analogous to the concept of thermal fluid dynamics equations. Solution heat equation was solved numerically through explicit finite difference method. However, due to the random observation domain, this study uses ghost-point method. This method uses the quadratic extrapolation approach. The development of mathematical models in the program was compiled using python language, so that the program can represent the temperature of the room at any point well. This research shows that a house which with straight roof is the most efficient in terms of heat distribution

Useful States and Entanglement Distillation

We derive general upper bounds on the distillable entanglement of a mixed state under one-way and two-way LOCC. In both cases, the upper bound is based on a convex decomposition of the state into 'useful' and 'useless' quantum states. By 'useful', we mean a state whose distillable entanglement is non-negative and equal to its coherent information (and thus given by a single-letter, tractable formula). On the other hand, 'useless' states are undistillable, i.e., their distillable entanglement is zero. We prove that in both settings the distillable entanglement is convex on such decompositions. Hence, an upper bound on the distillable entanglement is obtained from the contributions of the useful states alone, being equal to the convex combination of their coherent informations. Optimizing over all such decompositions of the input state yields our upper bound. The useful and useless states are given by degradable and antidegradable states in the one-way LOCC setting, and by maximally correlated and PPT states in the two-way LOCC setting, respectively. We also illustrate how our method can be extended to quantum channels. Interpreting our upper bound as a convex roof extension, we show that it reduces to a particularly simple, non-convex optimization problem for the classes of isotropic states and Werner states. In the one-way LOCC setting, this non-convex optimization yields an upper bound on the quantum capacity of the qubit depolarizing channel that is strictly tighter than previously known bounds for large values of the depolarizing parameter. In the two-way LOCC setting, the non-convex optimization achieves the PPT-relative entropy of entanglement for both isotropic and Werner states.

Convex geometry of quantum resource quantification

We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that many of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, which provide a characterisation of the sets of resource witnesses.We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k, multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof–extended negativity, a measure of k-coherence which generalises the norm of coherence, and a hierarchy of norm-based quantifiers of k-partite entanglement generalising the greatest cross norm.

From Log-Determinant Inequalities to Gaussian Entanglement via Recoverability Theory

Many determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities, specialised to Gaussian distributed vectors with prescribed covariances. In particular, strong subadditivity (SSA) yields \begin{equation*} \ln\det V_{AC} + \ln\det V_{BC} - \ln\det V_{ABC} - \ln\det V_C \geq 0 \end{equation*} for all $3\times 3$-block matrices $V_{ABC}$, where subscripts identify principal submatrices. We shall refer to the above inequality as SSA of log-det entropy. In this paper we develop further insights on the properties of the above inequality and its applications to classical and quantum information theory. In the first part of the paper, we show how to find known and new necessary and sufficient conditions under which saturation with equality occurs. Subsequently, we discuss the role of the classical transpose channel (also known as Petz recovery map) in this problem and find its action explicitly. We then prove some extensions of the saturation theorem, by finding faithful lower bounds on a log-det conditional mutual information. In the second part, we focus on quantum Gaussian states, whose covariance matrices are not only positive but obey additional constraints due to the uncertainty relation. For Gaussian states, the log-det entropy is equivalent to the R\'enyi entropy of order $2$. We provide a strengthening of log-det SSA for quantum covariance matrices that involves the so-called Gaussian R\'enyi-$2$ entanglement of formation, a well-behaved entanglement measure defined via a Gaussian convex roof construction. We then employ this result to define a log-det entropy equivalent of the squashed entanglement, which is remarkably shown to coincide with the Gaussian R\'enyi-$2$ entanglement of formation. This allows us to establish useful properties of such measure(s), like monogamy, faithfulness, and additivity on Gaussian states.

Monogamy relations of quantum entanglement for partially coherently superposed states**Project partially supported by the National Key Research and Development Program of China (Grant No. 2016YFB1000902), the National Natural Science Foundation of China (Grant Nos. 61232015, 61472412, and 61621003), the Beijing Science and Technology Project (2016), Tsinghua-Tencent-AMSS-Joint Project (2016), and the Key Laboratory of Mathematics Mechanization Project: Quantum Computing and Quantum

Monogamy is a fundamental property of multi-partite entangled states. Recently, Kim J S [Phys. Rev. A 93 032331] showed that a partially coherent superposition (PCS) of a generalized W-class state and the vacuum saturates the strong monogamy inequality proposed by Regula B et al. [Phys. Rev. Lett. 113 110501] in terms of squared convex roof extended negativity; and this fact may present that this class of states are good candidates for studying the monogamy of entanglement. Hence in this paper, we will investigate the monogamy relations for the PCS states. We first present some properties of the PCS states that are useful for providing our main theorems. Then we present several monogamy inequalities for the PCS states in terms of some entanglement measures.

Operational one-to-one mapping between coherence and entanglement measures

We establish a general operational one-to-one mapping between coherence measures and entanglement measures: Any entanglement measure of bipartite pure states is the minimum of a suitable coherence measure over product bases. Any coherence measure of pure states, with extension to mixed states by convex roof, is the maximum entanglement generated by incoherent operations acting on the system and an incoherent ancilla. Remarkably, the generalized CNOT gate is the universal optimal incoherent operation. In this way, all convex-roof coherence measures, including the coherence of formation, are endowed with (additional) operational interpretations. By virtue of this connection, many results on entanglement can be translated to the coherence setting, and vice versa. As applications, we provide tight observable lower bounds for generalized entanglement concurrence and coherence concurrence, which enable experimentalists to quantify entanglement and coherence of the maximal dimension in real experiments.

Measuring coherence with entanglement concurrence

Quantum coherence is a fundamental manifestation of the quantum superposition principle. Recently, Baumgratz et al (2014 Phys. Rev. Lett. 113 140401) presented a rigorous framework to quantify coherence from the view of theory of physical resource. Here we propose a new valid quantum coherence measure which is a convex roof measure, for a quantum system of arbitrary dimension, essentially using the generalized Gell-Mann matrices. Rigorous proof shows that the proposed coherence measure, coherence concurrence, fulfills all the requirements dictated by the resource theory of quantum coherence measures. Moreover, strong links between the resource frameworks of coherence concurrence and entanglement concurrence is derived, which shows that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. Our work provides a clear quantitative and operational connection between coherence and entanglement based on two kinds of concurrence. This new coherence measure, coherence concurrence, may also be beneficial to the study of quantum coherence.

Entanglement monotones and transformations of symmetric bipartite states

The primary goal in the study of entanglement as a resource theory is to find conditions that determine when one quantum state can or cannot be transformed into another via local operations and classical communication operations. This is typically done through entanglement monotones or conversion witnesses. Such quantities cannot be computed for arbitrary quantum states in general, but it is useful to consider classes of symmetric states for which closed-form expressions can be found. In this paper, we show how to compute the convex roof of any entanglement monotone for all Werner states. The convex roofs of the well-known Vidal monotones are computed for all isotropic states, and we show how this method can generalize to other entanglement measures and other types of symmetries as well. We also present necessary and sufficient conditions that determine when a pure bipartite state can be deterministically converted into a Werner state or an isotropic state.

Exact zeros of entanglement for arbitrary rank-two mixtures derived from a geometric view of the zero polytope

Here I present a method how intersections of a certain density matrix of rank two with the zero-polytope can be calculated exactly. This is a purely geometrical procedure which thereby is applicable to obtaining the zeros of SL- and SU-invariant entanglement measures of arbitrary polynomial degree. I explain this method in detail for a recently unsolved problem. In particular, I show how a three-dimensional view, namely in terms of the Boch-sphere analogy, solves this problem immediately. To this end, I determine the zero-polytope of the three-tangle, which is an exact result up to computer accuracy, and calculate upper bounds to its convex roof which are below the linearized upper bound. The zeros of the three-tangle (in this case) induced by the zero-polytope (zero-simplex) are exact values. I apply this procedure to a superposition of the four qubit GHZand W-state. It can however be applied to every case one has under consideration, including an arbitrary polynomial convex-roof measure of entanglement and for arbitrary local dimension.

한국사원의 귀면장식과 김제 금산사 출토 「王」자명 귀면기와

Usually Monster face ornamentations in Korea Buddhist temple were divided into three categories such as exterior tile, interior tile and the rest which are mostly examples of Late Joseon Dynasty. After specified shape coding of ‘Wang’(王 : King) inscribed monster face tiles discovered in geumsansa temple, characteristics and various types of monster decoration tile have been brought into light thorugh in this study. Discovered ‘Wang’(王 : King) inscribed monster face tile in Bongchunwongu site of geumsansa temple is convex roof tile of Goryeo Dynasty. Through ‘Wang’(王 : King) inscribed monster face motifs like square rafter tiles discovered in Anapji pond, tiles of Wolseong castle of Gyeongju and Tang Dynasty silver and gold tea set from Famensi temple of China. Also this motif was discovered at the main Budhha altars of Jeondeungsa Temple in Gangwha, main hall of Jeonghyesa temple in Suncheon and Buddha’s stele of Tongdosa temple in Yangsan Main function of these tile motif is fabulous role to scare away evil forces. Fanciful Monster face has fabulous role to protect from evil forces. Simply docoration with chimerical frontal face stressed with big eyes and mouth, real reasons to crerate this monster face was to protect and to scare away evil forces. A motif of fish in monster mouth which is very unique and popular in Late Choseon Dynasty was prevent from fire disaster. Monster face and Monster tile is Korean traditional term not as Japanese term which are mostly recognized. Many historical records prove this. I also suggest that the terms of Beast face, Dragon face, Monster dragon instead of Monster Face must be deserve little consideration.

Witness for Non-Quasi Maximally Entangled States

Maximally entangled states, defined as those states that have the maximal entanglement of formation under some entanglement measure, are the ideal resource for many quantum missions. In this paper, we call a convex roof of maximally entangled pure states a quasi maximally entangled state. First, we present the concept of a witness for non-quasi maximally entangled states, which is an observable that can distinguish some non-quasi maximally entangled states from quasi maximally entangled ones. Then we prove that every non-quasi maximally entangled state can be witnessed by a witness and obtain some necessary and sufficient conditions for an observable to be a witness for non-quasi maximally entangled states. Lastly, we give some classes of Hermitian operators, which can become witnesses. Especially, we compute non-quasi maximally entangled states that can be detected by a specific product operator.