We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular 2-qubit and 2-qutrit states -- and their generalizations to arbitrary dimensions -- where we calculate the optimal entanglement witnesses explicitly.
We characterize a convex subset of entanglement witnesses for two qutrits. Equivalently, we provide a characterization of the set of positive maps in the matrix algebra of 3 x 3 complex matrices. It turns out that boundary of this set displays elegant representation in terms of SO(2) rotations. We conjecture that maps parameterized by rotations are optimal, i.e. they provide the strongest tool for detecting quantum entanglement. As a byproduct we found a new class of decomposable entanglement witnesses parameterized by improper rotations from the orthogonal group O(2).
Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We investigate the inertias of bipartite EWs constructed by the partial transpose of NPT states. Furthermore, we find out most of the inertias of the patial transpose of the two-qutrit bipartite NPT states. As an application, we extend our results to high dimensional states.
We construct entanglement witnesses with regard to the geometric structure of the Hilbert--Schmidt space and investigate the geometry of entanglement. In particular, for a two--parameter family of two--qutrit states that are part of the magic simplex we calculate the Hilbert--Schmidt measure of entanglement. We present a method to detect bound entanglement which is illustrated for a three--parameter family of states. In this way we discover new regions of bound entangled states. Furthermore we outline how to use our method to distinguish entangled from separable states.
The Gell-Mann $λ$ matrices for Lie algebra su(3) are the natural basis for the Hilbert space of Hermitian operators acting on the states of a three-level system(qutrit). So the construction of EWs for two-qutrit states by using these matrices may be an interesting problem. In this paper, several two-qutrit EWs are constructed based on the Gell-Mann matrices by using the linear programming (LP) method exactly or approximately. The decomposability and non-decomposability of constructed EWs are also discussed and it is shown that the $λ$-diagonal EWs presented in this paper are all decomposable b…
In this paper, we use singularity theory to study the entanglement nature of pure three-qutrit systems. We first consider the algebraic variety $X$ of separable three-qutrit states within the projective Hilbert space $\mathbb{P}(\mathcal{H}) = \mathbb{P}^{26}$. Given a quantum pure state $|φ\rangle\in \mathbb{P}(\mathcal{H})$ we define the $X_φ$-hypersuface by cutting $X$ with a hyperplane $H_φ$ defined by the linear form $\langleφ|$ (the $X_φ$-hypersurface of $X$ is $X\cap H_φ\subset X$). We prove that when $|φ\rangle$ ranges over the SLOCC entanglement classes, the "worst" possible singular …
Based on the Liouville-von Neumann equation, we obtain closed system of equations for the description of a qutrit or coupled qutrits in arbitrary time-dependent external magnetic field. The dependence of the dynamics on the initial states and magnetic field modulation is studied analytically and numerically. We compare the relative entanglement measure's dynamics in the bi-qutrit system with permutation particle symmetry. We find the magnetic field modulation which retains the entanglement in the system of two coupled qutrits. Analytical formulas for entanglement measures in the chain from 2 t…
Teleportation witnesses are hermitian operators which can identify useful entanglement for quantum teleportation. Here we provide a systematic method to construct teleportation witnesses from entanglement witnesses corresponding to general qudit systems. The witnesses so constructed are shown to be optimal for qubit and qutrit systems, and therefore detect the largest set of states useful for teleportation within a given class. We demonstrate the action of the witness pertaining to different classes of states in qubits and qutrits. Decomposition of the witness in terms of spin operators facili…
We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3)xSU(3) transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator. These are identified up to quartic degree in the standard basis of Gell-Mann matrices, with the help of the calculus of f and d coefficients. Next, investigating local measurement operations, we study a SLOCC qutrit group, which plays the role of a `relativistic' transformation group analogous to…
We obtain formulas (bot. p. 12)--including $\frac{2}{121}$ and $\frac{4 \left(242 \sqrt{3} π-1311\right)}{9801}$--for the eight atoms (Fig. 11), summing to 1, which span a 256-dimensional three-set (P, S, PPT) entanglement-probability boolean algebra for the two-qutrit Hiesmayr-Loffler states. PPT denotes positive partial transpose, while P and S provide the Li-Qiao necessary and}sufficient conditions for entanglement. The constraints ensuring entanglement are $s> \frac{16}{9} \approx 1.7777$ and $p> \frac{2^{27}}{3^{18} \cdot 7^{15} \cdot13} \approx 5.61324 \cdot 10^{-15}$. Here, $s$ is the s…
Polarization features and entanglement of biphoton polarization qutrits are briefly outlined. Schmidt modes of qutrits are found analytically and in a general form by the method different from the standard one and based on the original approach of Erhard Schmidt (1906)
We examine the possibilities of non-trivial phenomena of time-invariant entanglement and freezing dynamics of entanglement for qutrit-qutrit quantum systems. We find no evidence for time-invariant entanglement, however, we do observe that quantum states freeze their entanglement after decaying for some time. It is interesting that quantum states are changing whereas their entanglement remains constant. We find that the combined action of decoherence free subspaces and subspaces where quantum states decay, facilitate this phenomenon. This study is an extension of similar phenomena observed for …
In the recent paper [Chruściński and Wudarski, arXiv:1105.4821], it was conjectured that the entanglement witnesses arising from some generalized Choi maps are optimal. We show that this conjecture is true. Furthermore, we show that they provide a one parameter family of indecomposable optimal entanglement witnesses.
We investigate the entanglement properties of a three--parameter family of states that are part of the magic simplex of two qutrits, which is a simplex of states that are mixtures of maximally entangled two--qutrit Bell states. Using entanglement witnesses we reveal large regions of bound entangled and separable states.
In this paper we consider a system consist of a qubit and a qutrit, and find a formula to evaluate the concurrence for it. We show that entanglement of formation for this system obeys the same relation as for two-qubits.
We analyze the recently introduced notion of quantumness witness and compare it to that of entanglement witness. We show that any entanglement witness is also a quantumness witness. We then consider some physically relevant examples and explicitly construct some witnesses.
The relationships between quantum entangled states and braid matrices have been well studied in recent years. However, most of the results are based on qubits. In this paper, We investigate the applications of 2-qutrit entanglement in the braiding associated with $\mathbb{Z}_3$ parafermion. The 2-qutrit entangled state $|Ψ(θ)\rangle$, generated by acting the localized unitary solution $\breve{R}(θ)$ of YBE on 2-qutrit natural basis, achieves its maximal $\ell_1$-norm and maximal von Neumann entropy simultaneously at $θ=π/3$. Meanwhile, at $θ=π/3$, the solutions of YBE reduces braid matrices, w…
Three-dimensional entanglement of orbital angular momentum states of an atomic qutrit and a single photon qutrit has been observed. Their full state was reconstructed using quantum state tomography. The fidelity to the maximally entangled state of Schmidt rank 3 exceeds the threshold 2/3. This result confirms that the density matrix cannot be decomposed into ensemble of pure states of Schmidt rank 1 or 2. That is, the Schmidt number of the density matrix must be equal to or greater than 3.
A new family of positive, trace-preserving maps is introduced. It is defined using the mutually unbiased measurements, which generalize the notion of mutual unbiasedness of orthonormal bases. This family allows one to define entanglement witnesses whose indecomposability depends on the characteristics of the associated measurement operators.
We provide a class of optimal nondecomposable entanglement witnesses for 4N x 4N composite quantum systems or, equivalently, a new construction of nondecomposable positive maps in the algebra of 4N x 4N complex matrices. This construction provides natural generalization of the Robertson map. It is shown that their structural physical approximations give rise to entanglement breaking channels.
We provide two 1-parameter families of indecomposable entanglement witnesses in C4 {\otimes} C4. Following recent paper by Ha and Kye [Phys. Rev. A 84, 024302 (2011)] we show that these EWs are optimal and hence provide the strongest tool in entanglement theory to discriminate between separable and entangled states. As a byproduct we show that these EWs detect quantum entanglement within a family of generalized Horodecki states.
Quantum entanglement, like othre resources, is now coonsidered to be a resource which can be produced, concentrated if required, transported and consumed. After its inception [1] in 1933, various schemes of quantum state teleportation have been proposed using different types of channels. Not restricting to qubit based systems, qutrit states and channels have also been of considerable interest. In the present paper we investigate the teleportation of an unknown single qutrit state as well as two qutrit state through a three qutrit quantum channel along with the required operations to recover th…
In this present paper we study the entanglement dynamics in qubit A-qutrit B pair under x component of Dzyaloshinshkii-Moriya $(D_{x})$ by taking an auxiliary qubit C. Here we consider an entangled qubit-qutrit pair initially prepared in two parameter qubit-qutrit states and one auxiliary qubit prepared in pure state interacts with the qutrit of the pair through DM interaction. We trace away the auxiliary qubit and calculate the reduced dynamics in qubit A-qutrit B pair to study the influence of the state of auxiliary qubit C and $D_{x}$ on entanglement. We find that the state (probability amp…
We study the thermal entanglement in a two-spin-qutrit system with anisotropy in the exchange coupling between two spins. We use the realignment criterion to distinguish the entangled states, and the negativity for measuring the entanglement in this system. We see that the anisotropy can provide an additional parameter for enhancing the entanglement.
We provide a generalization of the reduction and Robertson positive maps in matrix algebras. They give rise to a new class of optimal entanglement witnesses. Their structural physical approximation is analyzed. As a byproduct we provide a new examples of PPT (Positive Partial Transpose) entangled states.
We construct a new class of positive indecomposable maps in the algebra of 'd x d' complex matrices. Each map is uniquely characterized by a cyclic bistochastic matrix. This class generalizes a Choi map for d=3. It provides a new reach family of indecomposable entanglement witnesses which define important tool for investigating quantum entanglement.
We have given a novel formulation of the exact solutions for the lambda, vee and cascade three-level systems where the Hamiltonian of each configuration is expressed in the SU(3) basis. The solutions are discussed from the perspective of the Bloch equation and the atom-field entanglement scenario. For the semiclassical systems, the Bloch space structure of each configuration is studied by solving the corresponding Bloch equation and it is shown that at resonance, the eight-dimensional Bloch sphere is broken up into two distinct subspaces due to the existence of a pair of quadratic constants. B…
We characterize a set of positive maps in matrix algebra of 4x4 complex matrices. Equivalently, we provide a subset of entanglement witnesses parameterized by the rotation group SO(3). Interestingly, these maps/witnesses define two intersecting convex cones in the 3-dimensional parameter space. The existence of two cones is related to the topological structure of the underlying orthogonal group. We perform detailed analysis of the corresponding geometric structure.
Entangled states in high dimensional systems are of great interest due to the extended possibilities they provide in quantum information processing. Recently, Sun et al. [Phys. Rev. A 82, 052323 (2010)] and Kim et al. [Nat. Phys. 8, 117 (2012)] pointed out that weak measurement and quantum weak measurement reversal can actively combat decoherence. We generalize their studies from qubits to qutrits under amplitude damping decoherence. We find that the qutrit-qutrit entanglement can be partially retrieved for certain initial states when only weak measurement reversals are performed. However, we …
Entanglement plays a crucial role in quantum processes particularly those pertaining to quantum information and computation. An analytical expression for entanglement measure defined in terms of success rate of Grover's search algorithm has been obtained for a two-qutrit system and the calculated results agree well with the conventional entropy based measure. Qutrit system are of special interest because the Hilbert space dimensionality (for a given number of qudits-d-dimensional system)is optimal for d=3 and this may be of significance in the enhancement of computing power.
We consider the separability of various joint states for N qutrits. We derive two results: (i) the separability condition for a two-qutrit state that is a mixture of the maximally mixed state and a maximally entangled state (such a state is a generalization of the Werner state for two qubits); (ii) upper and lower bounds on the size of the neighborhood of separable states surrounding the maximally mixed state for N qutrits.
We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.
Based on maximally entangled states in the full- and sub-spaces of two qutrits, we present an alternative decomposition of two-qutrit pure states in a form $|Ψ>=\frac{p_{1}}{\sqrt{3}}(|00>+|11>+|22>) +\frac{p_{2}}{\sqrt{2}}(|01>+|12>)+ p_{3}e^{iθ}|02>$. Similar to the Schmidt decomposition, all two-qutrit pure states can be transformed into the alternative decomposition under local unitary transformations, and the parameter $p_1$ is shown to be an entanglement invariant.
We provide a recurrent construction of entanglement witnesses for a bipartite systems living in a Hilbert space corresponding to $2N$ qubits ($N$ qubits in each subsystem). Our construction provides a new method of generalization of the Robertson map that naturally meshes with $2N$ qubit systems, i.e., its structure respects the $2^{2N}$ growth of the state space. We prove that for $N>1$ these witnesses are indecomposable and optimal. As a byproduct we provide a new family of PPT (Positive Partial Transpose) entangled states.
Higher dimensional quantum systems are an important avenue for new explorations in quantum computing as well as quantum communications. One of the ubiquitous resources in quantum technologies is entanglement. However, so far, entanglement has been certified in higher dimensional systems through suitable bounds on known entanglement measures. In this work, we have, for the first time, quantified the amount of entanglement in bi-partite pure qutrit states by analytically relating statistical correlation measures and known measures of entanglement, and have determined the amount of entanglement i…
We investigate in a general form entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of collinear and, correspondingly, degenerate and non-degenerate Spontaneous Parametric Down-Conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and and polarization entanglement are fully taken into account. In the case of qutrits the most general 3-parametric families of maximally entangled and non-entangled states are found, and anti-correlation of the degree of entanglement and polarization is shown to occur and to be characterized b…
We demonstrate the existence of entanglement sudden death (ESD), the complete loss of entanglement in finite time, in qubit-qutrit systems. In particular, ESD is shown to occur in such systems initially prepared in a one-parameter class of entangled mixed states and then subjected to local dephasing noise. Together with previous results, this proves the existence of ESD for some states in all quantum systems for which rigorously defined mixed-state entanglement measures have been identified. We conjecture that ESD exists in all quantum systems prepared in appropriate bipartite states.
In this contribution, we investigate the entanglement behavior of a composite system consists of two different dimensional subsystems in non-inertial frames. In particular, we consider a composite system of qubit(two-dimensional) subsystem, and qutrit, (three-dimensional) subsystem. The degree of entanglement is quantified for different cases, where it is assumed that the two-subsystems are simultaneously or non- simultaneously accelerated. The entanglement decays as the acceleration of any subsystem increases.In general, the decay rate of entanglement increases as the dimensional of the accel…
The possibility of recovering and protecting the entanglement of accelerated 2-qubit and 2-qutrit systems is discussed using weak-reverse measurements. The accelerated partial entangled states are more responsive to be protected than the accelerated maximum entangled states. The accelerated coded local information in qutrit system is more robust than that encoded in a 2-qubit system, and it can be conserved even for larger values of accelerations. Meanwhile, the non-accelerated information in qubit system is not affected by the local operation compared with that depicted on qutrit system. For …
We study the thermal entanglement of a 2-qutrit spin chain with nonlinear coupling in the presence of nonuniform magnetic field. Thermal entanglement of an arbitrary (finite dimensional) m-partite system vanishes at some finite threshold temperature T_s. We investigate the dependence of T_s on the system's parameters, i.e. the nonlinear coupling and the magnetic field, for this 2-qutrit system. In addition, we compare two lower bounds of I-concurrence for this system and also study its dense coding capacity as a function of system's parameters.
We investigate the dynamics of entanglement for a two-parameter class of states in a hybrid qubit-qutrit system under the influence of various dissipative channels. Our results show that entanglement sudden death (ESD) is a general phenomenon and it usually takes place in a qubit-qutrit system interacting with various noisy channels, not only the case with dephasing and depolarizing channels observed by others. ESD can only be avoided for some initially entangled states under some particular noisy channels. Moreover, the environment affects the entanglement and the coherence of the system in v…
The thermal entanglement in a two-spin-qutrit system with two spins coupled by exchange interaction under a magnetic field in an arbitrary direction is investigated. Negativity, the measurement of entanglement, is calculated. We find that for any temperature the evolvement of negativity is symmetric with respect to magnetic field. The behavior of negativity is presented for four different cases. The results show that for different temperature, different magnetic field give maximum entanglement. Both the parallel and antiparallel magnetic field cases are investigated qualitatively (not quantita…
In this work, a mechanism for long time protection of entanglement of a two-qutrit V-type atomic system each of which interacts independently with a dissipative reservoir is investigated. It is shown that the protection process is related exclusively on the formation of bound states in the system-reservoir energy spectrum. It is demonstrated that the degree of boundedness for the system-reservoir bound states is well-controlled by entering other V-type qutrits into the reservoirs, and also through the manipulation of the spontaneously generated interference related to decay channels of the V-t…
The quantum entanglement is an important feature of many protocols in the field of quantum computing. In this paper we evaluate a level of entanglement in short qutrit chains. This evaluation is carried out with use of the CCNR criterion and the concurrence measure. We also present some explicit formulae describing the values of CCNR criterion and concurrence for exemplary short spin chains. Utilizing the obtained results, we indicate that analyzing the level of entanglement allows to detect the noise or deviation in the transfer process, in comparison to the perfect transfer where only operat…
We investigate the mutual information and entanglement of stationary state of two locally driven qubits under the influence of collective dephasing. It is shown that both the mutual information and the entanglement of two qubits in the stationary state exhibit damped oscillation with the scaled action time $γ{T}$ of the local external driving field. It means that we can control both the entanglement and total correlation of the stationary state of two qubits by adjusting the action time of the driving field. We also consider the influence of collective dephasing on entanglement of two qutrits …
We investigate the state space of bipartite qutrits. We construct an analog to the "magic" tetrahedron for bipartite qubits--a magic simplex W. It is formed by all convex combination of nine Bell states which are constructed using the Weyl operators. Due to the high symmetry it is enough to consider certain typical slices through W. Via optimal entanglement witnesses we find regions of bound entangled states inside W.
We classify biqutrit and triqutrit pure states under stochastic local operations and classical communication. By investigating the right singular vector spaces of the coefficient matrices of the states, we obtain explicitly two equivalent classes of biqutrit states and twelve equivalent classes of triqutrit states respectively.
We investigate the geometric characterization of pure state bipartite entanglement of $(2\times{D})$- and $(3\times{D})$-dimensional composite quantum systems. To this aim, we analyze the relationship between states and their images under the action of particular classes of local unitary operations. We find that invariance of states under the action of single-qubit and single-qutrit transformations is a necessary and sufficient condition for separability. We demonstrate that in the $(2\times{D})$-dimensional case the von Neumann entropy of entanglement is a monotonic function of the minimum sq…
The effect of decoherence on a qubit-qutrit system under the influence of global and multilocal decoherence in non-inertial frames is investigated. It is shown that the entanglement sudden death (ESD) can be avoided in non-inertial frames in the presence of dephasing and bit-trit phase flip channels for entire range of decoherence. However, ESD behaviour is seen for higher level of decoherence in case of phase flip and bit-trit flip channels. Irrespective of the entanglement degradation caused by the Unruh effect, no ESD occurs for the dephasing environment. Furthermore, flipping channels have…
We investigate the effects of nonlinear couplings and external magnetic field on the thermal entanglement in a two-spin-qutrit system by applying the concept of negativity. It is found that the nonlinear couplings favor the thermal entanglement creating. Only when the nonlinear couplings $|K|$ are larger than a certain critical value does the entanglement exist. The dependence of the thermal entanglement in this system on the magnetic field and temperature is also presented. The critical magnetic field increases with the increasing nonlinear couplings constant $|K|$. And for a fixed nonlinear …
The possibility of immunized and improving the entanglement of accelerated systems via local filtering is discussed. The maximum bounds of entanglement depend on the dimensions of the accelerated and the filtered subsystems. If the small dimensional subsystem is accelerated and the large dimensional subsystem is filtered, one can get a long-lived entanglement. Moreover, if the larger subsystem is accelerated, then by filtering any subsystem, the upper bounds of entanglement of the filtered state are larger then that for the initial states.For any accelerated subsystem, the entanglement always …
Among the many facets of quantum correlations, bound entanglement has remained one the most enigmatic phenomena, despite the fact that it was discovered in the early days of quantum information. Even its detection has proven to be difficult, let alone its precise quantitative characterization. In this work, we present the exact quantification of entanglement for a two-parameter family of highly symmetric two-qutrit mixed states, which contains a sizable part of bound entangled states. We achieve this by explicitly calculating the convex-roof extensions of the linear entropy as well as the conc…
The dynamics of entanglement in some hybrid qubit-qutrit systems under the influence of global, collective, local and multilocal depolarizing noise is studied. It is shown that the depolarizing noise can be used to induce entanglement. A critical point exists under every coupling of the system with the environment at which all the states are equally entangled. Furthermore, it is seen that no ESD occurs when either only the qubit is coupled to its local environment or the system is coupled to multilocal or global environment. This is an important result for various quantum information processin…
