In this paper we present the quantity, which is an entanglement parameter. Its origin is very intriguing, because its construction is motivated by separability criteria based on uncertainty relation. We show that this quantity is asymptotically continuous. We also find the lower and upper bounds for it. Our entanglement parameter has the same feature as the coherent information: both can be negative.
There are also some classes of states for which these quantities coincide with each other.
We discuss the relationship between entropic uncertainty relations and entanglement. We present two methods for deriving separability criteria in terms of entropic uncertainty relations. Especially we show how any entropic uncertainty relation on one part of the system results in a separability condition on the composite system. We investigate the resulting criteria using the Tsallis entropy for two and three qubits.
The Wehrl entropy is an entropy associated to the Husimi quasi-probability distribution. We discuss how it can be used to formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. We show that the Wehrl-Lieb inequality is closer to equality than the usual Białynicki-Birula and Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how a Wehrl mutual information can be used to obtain a measurable perfect witness for pure state bipartite entanglement, which additionally provides a lower bound on the entanglement entropy.
Entropic uncertainty and entanglement are two distinct aspects of quantum mechanical procedures. To estimate entropic uncertainty relations, entropies are used: the greater the entropy bound, the less effective the quantum operations and entanglement are. In this regard, we analyze the entropic uncertainty, entropic uncertainty lower bound, and concurrence dynamics in two non-interacting qubits. The exposure of two qubits is studied in two different qubit-noise configurations, namely, common qubit-noise and independent qubit-noise interactions. To include the noisy effects of the local externa…
The quantum entropic uncertainty relation and entanglement witness in the two-atom system coupling with the non-Markovian environments are studied by the time-convolutionless master-equation approach. The influence of non-Markovian effect and detuning on the lower bound of the quantum entropic uncertainty relation and entanglement witness is discussed in detail. The results show that, only if the two non-Markovian reservoirs are identical, increasing detuning and non-Markovian effect can reduce the lower bound of the entropic uncertainty relation, lengthen the time region during which the enta…
Entanglement criteria for general (pure or mixed) states of systems consisting of two identical fermions are introduced. These criteria are based on appropriate inequalities involving the entropy of the global density matrix describing the total system, on the one hand, and the entropy of the one particle reduced density matrix, on the other one. A majorization-related relation between these two density matrices is obtained, leading to a family of entanglement criteria based on Rényi's entropic measure. These criteria are applied to various illustrative examples of parametrized families of mix…
We relate the principle of quantum-memory-assisted entropic uncertainty to quantum teleportation and show geometrically that any two-qubit state which lowers the upper bound of this uncertainty relation is useful for teleportation. We also explore the efficiency of this entropic uncertainty principle on witnessing entanglement in a general class of bosonic structured reservoirs. The entanglement regions witnessed by different estimates are determined, which may have no relation with the explicit form of the spectral density of the reservoir for certain special chosen sets of the initial states…
We apply the Walborn [Walborn S P, Taketani B G, Salles A, Toscano F and de Matos Filho R L 2009 Entropic Entanglement Criteria for Continuous Variables Phys. Rev. Lett. 103 160505] Shannon Entropic Entanglement Criterion (SEEC), to simple harmonic oscillators, a system typically discussed in undergraduate quantum mechanics courses. In particular, we investigate the entanglement of a system of coupled harmonic oscillators. A simple form of entanglement criterion in terms of their interaction is found. It is shown that a pair of interacting ground state coupled oscillators are more likely to be…
We derive entropic inseparability criteria for the phase space representation of quantum states. In contrast to criteria involving differential entropies of marginal phase space distributions, our criteria are based on a joint distribution known as the Husimi Q-distribution. This distribution is experimentally accessible in cold atoms, circuit QED architectures and photonic systems and bears practical advantages compared to the detection of marginals. We exemplify the strengths of our entropic approach by considering several classes of non-Gaussian states where second-order criteria fail. We s…
We derive several entanglement criteria for bipartite continuous variable quantum systems based on the Shannon entropy. These criteria are more sensitive than those involving only second-order moments, and are equivalent to well-known variance product tests in the case of Gaussian states. Furthermore, they involve only a pair of quadrature measurements, and will thus should prove extremely useful the experimental identification of entanglement.
We show how to quantify tri-partite entanglement using entropies derived from experimental correlations. We use a multi-partite generalization of the entanglement of formation that is greater than zero if and only if the state is genuinely multi-partite entangled. We develop an entropic witness for tripartite entanglement, and show that the degree of violation of this witness places a lower limit on the tripartite entanglement of formation. We test our results in the three-qubit regime using the GHZ-Werner state and the W-Werner state, and in the high-dimensional pure-state regime using the tr…
The uncertainty principle, which bounds the uncertainties involved in obtaining precise outcomes for two complementary variables defining a quantum particle, is a crucial aspect in quantum mechanics. Recently, the uncertainty principle in terms of entropy has been extended to the case involving quantum entanglement. With previously obtained quantum information for the particle of interest, the outcomes of both non-commuting observables can be predicted precisely, which greatly generalises the uncertainty relation. Here, we experimentally investigated the entanglement-assisted entropic uncertai…
Quantum correlations have fundamental and technological interest, and hence many measures have been introduced to quantify them. Some hierarchical orderings of these measures have been established, e.g., discord is bigger than entanglement, and we present a class of bipartite states, called premeasurement states, for which several of these hierarchies collapse to a single value. Because premeasurement states are the kind of states produced when a system interacts with a measurement device, the hierarchy collapse implies that the uncertainty of an observable is quantitatively connected to the q…
The effects of mixedness and entanglement on the lower bound and tightness of the entropic uncertainty in the Heisenberg model with Dzyaloshinski-Moriya (DM) interaction have been investigated. It is found that the mixedness can reflect the essence of the entropic uncertainty better than the entanglement. Meanwhile, the uncertainty of measurement results will be reduced by the entanglement and improved by the mixedness.The entanglement can destroy the tightness of the uncertainty, while the tightness will be improved with the increasing of the mixedness. In addition, the tightness of the uncer…
We study the dynamics of quantum-memory-assisted entropic uncertainty for a hybrid qutrit-qubit system interacting with fluctuating quantum scalar field in the background of expanding de Sitter space. We firstly derive the master equation that the system evolution obeys. As evolution time goes by, for different initial states, entropic uncertainty develops to different fixed values for different parameter values, whereas entanglement always decays to zero, and there exist monotonous relations between entropic uncertainty, entanglement and various parameters for a fixed initial state, but mixed…
In this paper, we discuss some general connections between the notions of positive map, weak majorization and entropic inequalities in the context of detection of entanglement among bipartite quantum systems. First, basing on the fact that any positive map $Λ:M_{d}(\mathbb{C})\to M_{d}(\mathbb{C})$ can be written as the difference between two completely positive maps $Λ=Λ_{1}-Λ_{2}$, we propose a possible way to generalize the Nielsen--Kempe majorization criterion. Then we present two methods of derivation of some general classes of entropic inequalities useful for the detection of entanglemen…
Quantum correlation often refers to correlations exhibited by two or more local subsystems under a suitable measurement. These correlations are beyond the framework of classical statistics and the associated classical probability distribution. Quantum entanglement is the most well known of such correlations and plays an important role in quantum information theory. However, there exist non-entangled states that still possess quantum correlations which cannot be described by classical statistics. One such measure that captures these nonclassical correlations is discord. Here we introduce a new …
In this paper, we study the interaction between a moving $Λ$-type three-level atom and a single-mode cavity field in the presence of intensity-dependent atom-field coupling. After obtaining the state vector of the entire system explicitly, we study the nonclassical features of the system such as quantum entanglement, position-momentum entropic squeezing, quadrature squeezing and sub-Poissonian statistics. According to the obtained numerical results we illustrate that the squeezed period, the duration of entropy squeezing and the maximal squeezing can be controlled by choosing the appropriate n…
In this paper, the interaction between a $Λ$-type three-level atom and two-mode cavity field is discussed. The detuning parameters and cross-Kerr nonlinearity are taken into account and it is assumed that atom-field coupling and Kerr medium to be $f$-deformed. Even though the system seems to be complicated, the analytical form of the state vector of the entire system for considered model is exactly obtained. The time evolution of nonclassical properties such as quantum entanglement and position-momentum entropic uncertainty relation (entropy squeezing) of the field are investigated. In each ca…
Recently it was shown that if a given state fulfils the reduction criterion it must also satisfy the known entropic inequalities. Now the questions arises whether on the assumption that stronger criteria based on positive but not completely positive maps are satisfied, it is possible to derive some scalar inequalities stronger than the entropic ones. In this paper we show that under some assumptions the extended reduction criterion [H.-P. Breuer, Phys. Rev. Lett 97, 080501 (2006); W. Hall, J. Phys. A 40, 6183 (2007)] leads to some entropic--like inequalities which are much stronger than their …
We investigate some properties of the entanglement of hypergraph states in purely hypergraph theoretical terms. We first introduce an approach for computing local entropic measure on qubit t of a hypergraph state by using the Hamming weight of the so-called t-adjacent subhypergraph. Then we quantify and characterize the entanglement of hypergraph states in terms of local entropic measures obtained by using the above approach. Our results show that a class of n-qubit hypergraph states can not be converted into any graph state under local unitary transformations.
The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find entanglement-dependent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem~\ref{th:bound}. Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on t…
Distinguishability and predictability are part of complementarity relations which apply to two different kinds of interference experiments, with and without a path-detector, respectively. In [Opt. Comm. 179, 337 (2000)], Englert and Bergou pointed out the possible connection between distinguishability, predictability, and entanglement. They even conjectured that an entanglement measure was hidden between the measures of distinguishability and predictability. Here, we push forward this conjecture. We start defining a new entropic distinguishability measure and suggesting an entanglement measure…
Entanglement is not only the resource that fuels many quantum technologies but also plays a key role for some of the most profound open questions of fundamental physics. Experiments controlling quantum systems at the single quantum level may shed light on these puzzles. However, measuring, or even bounding, entanglement experimentally has proven to be an outstanding challenge, especially when the prepared quantum states are mixed. We use entropic uncertainty relations for bipartite systems to derive measurable lower bounds on distillable entanglement. We showcase these bounds by applying them …
We examine the inference of quantum density operators from incomplete information by means of the maximization of general non-additive entropic forms. Extended thermodynamic relations are given. When applied to a bipartite spin 1/2 system, the formalism allows to avoid fake entanglement for data based on the Bell-CHSH observable, and, in general, on any set of Bell constraints. Particular results obtained with the Tsallis entropy and with an introduced exponential entropic form are also discussed.
In this paper, we study the interaction between a $Λ$-type three-level atom and two quantized electromagnetic fields which are simultaneously injected in a bichromatic cavity surrounded by a Kerr medium in the presence of the field-field interaction (parametric down conversion) and detuning parameters. By applying a canonical transformation, the introduced model is reduced to a well-known form of the generalized Jaynes-Cummings model. Under particular initial conditions which may be prepared for the atom and the field, the time evolution of state vector of the entire system is analytically eva…
The uncertainty principle can be expressed in entropic terms, also taking into account the role of entanglement in reducing uncertainty. The information exclusion principle bounds instead the correlations that can exist between the outcomes of incompatible measurements on one physical system, and a second reference system. We provide a more stringent formulation of both the uncertainty principle and the information exclusion principle, with direct applications for, e.g., the security analysis of quantum key distribution, entanglement estimation, and quantum communication. We also highlight a f…
The qudit state for j = 3=2 with density matrix of the form corresponding to X-state of two-qubits is studied from the point of view of entanglement and separability properties. The method of qubit portrait of qudit states is used to get the entropic inequalities for the entangled state of the single qudit. The tomographic probability representation of the qudit X-state under consideration and its Shannon and q-entropic characteristics are presented in explicit form.
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.
We discuss conditional Renyi and Tsallis entropies for bipartite quantum systems of finite dimension. We investigate the relation between the positivity of conditional entropies and entanglement properties. It is in particular shown that any state having a negative conditional entropy with respect to any value of the entropic parameter is distillable since it violates the reduction criterion. Moreover we show that the entanglement of Werner states in odd dimensions can neither be detected by entropic criteria nor by any other spectral criterion.
The measurement range problem, where one cannot determine the data outside the range of the detector, limits the characterization of entanglement in high-dimensional quantum systems when employing, among other tools from information theory, the entropic uncertainty relations. Practically, the measurement range problem weakens the security of entanglement-based large-alphabet quantum key distribution (QKD) employing degrees of freedom including time-frequency or electric field quadrature. We present a modified entropic uncertainty relation that circumvents the measurement range problem under ce…
We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of en…
We study experimentally accessible lower bounds on entanglement measures based on entropic uncertainty relations. Experimentally quantifying entanglement is highly desired for applications of quantum simulation experiments to fundamental questions, e.g., in quantum statistical mechanics and condensed-matter physics. At the same time it poses a significant challenge because the evaluation of entanglement measures typically requires the full reconstruction of the quantum state, which is extremely costly in terms of measurement statistics. We derive an improved entanglement bound for bipartite sy…
We investigate bipartite entanglement in spin-1/2 systems on a generic lattice. For states that are an equal superposition of elements of a group $G$ of spin flips acting on the fully polarized state $\ket{0}^{\otimes n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems $A$ and $B$. These states are stabilized by the group $G$. A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus $\mathfrak{g}$. For a square lattice, we find that the entropy of entanglement is bounded from ab…
We qualify the entanglement of arbitrary mixed states of bipartite quantum systems by comparing global and marginal mixednesses quantified by different entropic measures. For systems of two qubits we discriminate the class of maximally entangled states with fixed marginal mixednesses, and determine an analytical upper bound relating the entanglement of formation to the marginal linear entropies. This result partially generalizes to mixed states the quantification of entaglement with marginal mixednesses holding for pure states. We identify a class of entangled states that, for fixed marginals,…
The thermal entanglement of the Hubbard dimer (two-site Hubbard model) has been studied with the nonextensive statistics. We have calculated the auto-correlation ($O_q$), pair correlation ($L_q$), concurrence ($Γ_q$) and conditional entropy ($R_q$) as functions of entropic index $q$ and the temperature $T$. The thermal entanglement is shown to considerably depend on the entropic index. For $q < 1.0$, the threshold temperature where $Γ_q$ vanishes or $R_q$ changes its sign is more increased and the entanglement may survive at higher temperatures than for $q=1.0$. Relations among $L_q$, $Γ_q$ an…
In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between majorization and the trace distance between quantum states. Namely, the majorization pre-order attains a minimum over $ε$-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance $ε$, we obtain the path of "majorization flow". This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which ex…
We evaluate the entanglement entropy of exactly solvable Hamiltonians corresponding to general families of three-dimensional topological models. We show that the modification to the entropic area law due to three-dimensional topological properties is richer than the two-dimensional case. In addition to the reduction of the entropy caused by non-zero vacuum expectation value of contractible loop operators a new topological invariant appears that increases the entropy if the model consists of non-trivially braiding anyons. As a result the three-dimensional topological entanglement entropy provid…
We investigate the scaling of the entanglement entropy in an infinite translational invariant Fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. Thus, in contrast to analogous Bosonic systems, the entropic area law is violated for Fermions. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven, that…
We study the entanglement entropy, $S_C$, of a massless free scalar field on the outside region $C$ of two black holes $A$ and $B$ whose radii are $R_1$ and $R_2$ and how it depends on the distance, $r(\gg R_1,R_2)$, between two black holes. If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force acting on the two black holes from the $r$ dependence of $S_C$. We develop the computational method based on that of Bombelli et al to obtain the $r$ dependence of $S_C$ of scalar fields whose Lagrangian is quadratic with respect to the scalar fields. First …
The entropic uncertainty principle in the form proven by Maassen and Uffink yields a fundamental inequality that is prominently used in many places all over the field of quantum information theory. In this work, we provide a family of versatile generalizations of this relation. Our proof methods build on a deep connection between entropic uncertainties and interpolation inequalities for the doubly stochastic map that links probability distributions in two measurements bases. In contrast to the original relation, our generalization also incorporates the von Neumann entropy of the underlying qua…
Equi-Entropic Maps for Four-Partite Quantum States
Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）；Center for Quantum-Enabled Computing, Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心量子使能计算中心）；Department for Quantum Technology, University of Münster（明斯特大学量子技术系）
Squashed entanglement and entanglement of purification are quantum mechanical correlation measures and defined as certain minimisations of entropic quantities. We present the first non-trivial calculations of both quantities. Our results lead to the conclusion that both measures can drop by an arbitrary amount when only a single qubit of a local system is lost. This property is known as "locking" and has previously been observed for other correlation measures, such as the accessible information, entanglement cost and the logarithmic negativity. In the case of squashed entanglement, the results…
We present a new paradigm for capturing the complementarity of two observables. It is based on the entanglement created by the interaction between the system observed and the two measurement devices used to measure the observables sequentially. Our main result is a lower bound on this entanglement and resembles well-known entropic uncertainty relations. Besides its fundamental interest, this result directly bounds the effectiveness of sequential bipartite operations---corresponding to the measurement interactions---for entanglement generation. We further discuss the intimate connection of our …
Spin-outcome correlations and memory-assisted entropic uncertainty in Bhabha scattering with an entangled spectator
State Key Laboratory of Opto-Electronic Information Acquisition and Protection Technology, School of Physics, Anhui University（安徽大学光电信息获取与保护技术国家重点实验室物理学院）
Quantum steering, loosely speaking the distribution of entanglement from an untrusted party, is a form of quantum nonlocality which is intermediate between entanglement and Bell nonlocality. Determining which states can be steered is important from a conceptual point of view, but also for applications, e.g. in quantum cryptography. Here we show that bound entanglement, although it represents the weakest form of entanglement, can nevertheless lead to quantum steering. This is done by noticing that steering inequalities can be derived from entropic uncertainty relations. Our result has implicati…
The advent of structured, high-dimensional entangled states brings new possibilities for quantum imaging, information processing and quantum key distribution. We experimentally generate and characterize a spatially entangled state stored in a quantum memory system, using entropic EPR-steering inequality, yielding genuine violation of $1.06\pm0.15$ bits and certifying entanglement of formation of at least $0.70\pm0.15$ ebits. The supremacy of the entropic witness is demonstrated for a wide class of experimentally available states, giving prospects for EPR-steering applications in noisy systems …
The entanglement properties of some novel quantum systems are studied that are inspired by recent developments in cold-atom technology. A triangular optical lattice of two atomic species can be employed to generate a variety of spin-1/2 Hamiltonians including effective three-spin interactions. A variety of one or two dimensional systems can thus be realized that possess multi-degenerate ground states or non-vanishing chirality. The properties of these ground states and their phase transitions are probed with appropriate measures such as the entropic entanglement and the spin chirality.
Some bounds on the entropic informational quantities related to a quantum continual measurement are obtained and the time dependencies of these quantities are studied.
We investigate the irreversibility of entanglement distillation for a symmetric d-1 parameter family of mixed bipartite quantum states acting on Hilbert spaces of arbitrary dimension d x d. We prove that in this family the entanglement cost is generically strictly larger than the distillable entanglement, such that the set of states for which the distillation process is asymptotically reversible is of measure zero. This remains true even if the distillation process is catalytically assisted by pure state entanglement and every operation is allowed, which preserves the positivity of the partial…
The present Thesis covers the subject of the characterization of entangled states by recourse to entropic measures, as well as the description of entanglement related to several issues in quantum mechanics, such as the speed of a quantum evolution or the connections existing between quantum entanglement and quantum phase transitions.
Entropic entanglement measures of a two-dimensional system of two Coulombically interacting particles confined in an anisotropic harmonic potential are discussed in dependence on the anisotropy and the interaction strength. The harmonic approximation appears exact in the strong interaction limit, allowing determination of the asymptotic expression for the linear entropy. Entanglement properties are dramatically influenced by the anisotropy of the confining potential in the strong-correlation regime.
