Entanglement of Formation for Quantum States

We investigate the entanglement of formation for a class of high-dimensional quantum mixed states. We present a kind of generalized concurrence for a class of high-dimensional quantum pure states such that the entanglement of formation is a monotonically increasing convex function of the generallzed concurrence, from the monotonicity and convexity the entanglement of formafion for a class of high-dimensional mixed states has been calculated analytically,     

Degree of Entanglement and Violation of Bell Inequality by Two-Spin-1/2 States

Bell inequality is violated by the quantum mechanical predictions made from an entangled state of the composite system. In this paper we examine this inequality and entanglement measures in the construction of the coherent states for two-qubit pure and mixed states. we find a link to some entanglement measures through some new parameters (amplitudes of coherent states). Conditions for maximal entanglement and separability are then established for both pure and mixed states. Finally, we analyze and compare the violation of Bell inequality for a class of mixed states with the degree of
entanglement by applying the formalism of Horodecki et al.     

Quantum Entanglement Swapping in a Non-Markovian Environment

We propose an entanglement swapping scheme for mixed states in a classical non-Markovian noises, which is modelled as the so-called Ornstein-Uhlenbeck processes. The two mixed states before entanglement swapping are two X-type mixed states, which are caused by the environment-induced decoherence on the initially Bell states. This is more practical than the pure state case in quantum information processing. The fidelity and concurrence of the post-swapping states are discussed.     

Detecting entanglement by the mean value of spin on a quantum computer

We implement a protocol to determine the degree of entanglement between a qubit and the rest of the system on a quantum computer. The protocol is based on results obtained in paper [Frydryszak et al. (2017) [23]]. This protocol is tested on a 5-qubit superconducting quantum processor called ibmq-ourense provided by the IBM company. We determine the values of entanglement of the Schrödinger cat and the Werner states prepared on this device and compare them with the theoretical ones. In addition, a protocol for determining the entanglement of rank-2 mixed states is proposed. We apply this protocol to the mixed state which consists of two Bell states prepared on the ibmq-ourense quantum device.     

Generalized measures of quantum correlations for mixed states

The exponential speedup achieved in certain quantum algorithms based on mixed states with negligible entanglement has renewed the interest on alternative measures of quantum correlations. Here we discuss a general measure of quantum correlations for composite systems based on generalized entropic functions, defined as the minimum information loss due to a local measurement. For pure states, the present measure becomes an entanglement entropy, i.e., it reduces to the generalized entropy of the reduced state. However, for mixed states it can be nonzero in separable states, vanishing just for states diagonal in a general product basis, like the quantum discord. Quadratic measures of quantum correlations can be derived as particular cases of the present formalism. The minimum information loss due to a joint local measurement is also considered. The evaluation of these measures in a simple yet relevant case is also discussed.     

Entanglement or separability: the choice of how to factorize the algebra of a density matrix

Quantum entanglement has become a resource for the fascinating developments in quantum information and quantum communication during the last decades. It quantifies a certain nonclassical correlation property of a density matrix representing the quantum state of a composite system. We discuss the concept of how entanglement changes with respect to different factorizations of the algebra which describes the total quantum system. Depending on the considered factorization a quantum state appears either entangled or separable. For pure states we always can switch unitarily between separability and entanglement, however, for mixed states a minimal amount of mixedness is needed. We discuss our general statements in detail for the familiar case of qubits, the GHZ states, Werner states and Gisin states, emphasizing their geometric features. As theorists we use and play with this free choice of factorization, which for an experimentalist is often naturally fixed. For theorists it offers an extension of the interpretations and is adequate to generalizations, as we point out in the examples of quantum teleportation and entanglement swapping.     

Entanglement in four qubit states: Polynomial invariant of degree 2, genuine multipartite concurrence and one-tangle

Characterization of the multipartite mixed state entanglement is still a challenging problem. This is due to the fact that the entanglement for the mixed states, in general, is defined by a convex-roof extension. That is the entanglement measure of a mixed state ρ of a quantum system can be defined as the minimum average entanglement of an ensemble of pure states. In this paper, we show that polynomial entanglement measures of degree 2 of even-N qubits X states is in the full agreement with the genuine multipartite (GM) concurrence. Then, we plot the hierarchy of entanglement classification for four qubit pure states and then using new invariants, we classify the four qubit pure states. We focus on the convex combination of the classes whose at most the one of the invariants is non-zero and find the relationship between entanglement measures consist of non-zero-invariant, GM concurrence and one-tangle. We show that in many entanglement classes of four qubit states, GM concurrence is equal to the square root of one-tangle.     

Entanglement persistency of multiphoton entangled states

We experimentally demonstrate the entanglement persistency when losing photons in three- and four-photon polarization-entangled states. The entanglement properties of the mixed states obtained from multiphoton spontaneous parametric down-conversion are studied via witness and positive partial transpose approaches. Together with a quantification of the bipartite entanglement such analysis enables intuitive understanding of novel multiparty quantum communication protocols.     

Dynamical correlations of negativity and entropy for pure and mixed states in two coupled quartic oscillators

Various measures of entanglement have triggered considerable interest in the relationship between entanglement measures and other well-known quantities. As a demonstration, the dynamical correlation of negativity and entropy is studied in two coupled quartic oscillators for initial pure and mixed states that are respectively taken to be the products and mixed density matrices of coherent states and squeezed states on each oscillator. The correlation with energy is also considered. It is shown that for the initial pure states with a small magnitude, two negativities are positively correlated with the von Neumann entropy while they are anti-correlated with the energy of each oscillator in the weak coupling regime. For mixed states with a small magnitude the two negativities and the mutual entropy exhibit dominantly positive correlation, whereas those three quantities are dominantly anti-correlated with the sum of energies of two oscillators in the case of weak interactions. Such correlation behaviors in the mixed state with small magnitudes are most striking at the same step in maximal and minimal values and in oscillation. The differences in entropies and negativities between coherent states and squeezed states are discussed. These are useful for quantum entanglement and quantum information processing.     

Evaluating the geometric measure of multiparticle entanglement

We present an analytical approach to evaluate the geometric measure of multiparticle entanglement for mixed quantum states. Our method allows the computation of this measure for a family of multiparticle states with a certain symmetry and delivers lower bounds on the measure for general states. It works for an arbitrary number of particles, for arbitrary classes of multiparticle entanglement, and can also be used to determine other entanglement measures.     

Quantum correlations in non-Markovian environments

We have studied the analytical Markovian and non-Markovian dynamics of quantum correlations, such as entanglement, quantum discord and Bell nonlocalities for three noisy qubits. Quantum correlation as measured by quantum discord is found to be immune to death contrary to entanglement and Bell nonlocality for initial GHZ- or W-type mixed states.     

Fidelity and entanglement breaking properties of qutrit channels

We investigate fidelity and entanglement breaking properties of quantum qutrit channels. We focus on channel fidelity evaluated for pure initial states and entanglement fidelity for purified mixed states (or pure entangled qutrit states) and use negativity as an entanglement measure for qutrits. We analyze properties of qutrit gates and channels based on affine transformations of qutrit Bloch vectors. We employ channel complete positivity constraints into the discussion of fidelity and entanglement behaviour.     

Entanglement sudden death in qubit-qutrit systems

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.     

Uniqueness of the entanglement measure for bipartite pure states and thermodynamics

We apply the axiomatic approach to thermodynamics presented by Giles to derive a unique measure of entanglement for bipartite pure states. This implies that local manipulations of entanglement in quantum information theory and adiabatic transformations of states in thermodynamics have the same underlying mathematical structure. We discuss possible extensions of our results to mixed and multipartite states.     

The Elusive Source of Quantum Speedup

We discuss two qualities of quantum systems: various correlations existing between their subsystems and the distinguishability of different quantum states. This is then applied to analysing quantum information processing. While quantum correlations, or entanglement, are clearly of paramount importance for efficient pure state manipulations, mixed states present a much richer arena and reveal a more subtle interplay between correlations and distinguishability. The current work explores a number of issues related with identifying the important ingredients needed for quantum information processing. We discuss the Deutsch-Jozsa algorithm, the Shor algorithm, the Grover algorithm and the power of a single qubit class of algorithms. In the latter, a quantity called discord is seen to be more important than entanglement. One section is dedicated to cluster states where entanglement is crucial, but its precise role is highly counter-intuitive. Here we see that the notion of distinguishability becomes a more useful concept.     

Experimental Investigation of the Robustness of a New Bell-Type Inequality of Triphoton GHZ States in Open Systems

Knowing the level of entanglement robustness against quantum bit loss or decoherence mechanisms is an important issue for any application of quantum information. Fidelity of states can be used to judge whether there is entanglement in multi-particle systems. It is well known that quantum channel security in QKD can be estimated by measuring the robustness of Bell-type inequality against noise. We experimentally investigate a new Bell-type inequality (NBTI) in the three-photon Greenberger–Horne–Zeilinger (GHZ) states with different levels of spin-flip noise. The results show that the fidelity and the degree of violation of the NBTI decrease monotonically with the increase of noise intensity. They also provide a method to judge whether there is entanglement in three-particle mixed states.     

Entanglement cost under positive-partial-transpose-preserving operations

We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this entanglement measure. As examples we discuss general Werner states and arbitrary bipartite Gaussian states. Then we prove that for the antisymmetric Werner state PPT cost and PPT entanglement of distillation coincide. This is the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible, which points towards a unique entanglement measure under PPT operations.     

Fluctuations of quantum entanglement

It is emphasized that quantum entanglement determined in terms of the von Neumann entropy operator is a stochastic quantity and, therefore, can fluctuate. The rms fluctuations of the entanglement entropy of two-qubit systems in both pure and mixed states have been obtained. It has been found that entanglement fluctuations in the maximally entangled states are absent. Regions where the entanglement fluctuations are larger than the entanglement itself (strong fluctuation regions) have been revealed. It has been found that the magnitude of the relative entanglement fluctuations is divergent at the points of the transition of systems from an entangled state to a separable state. It has been shown that entanglement fluctuations vanish in the separable states.     

Entanglement witnesses of four-qubit tripartite separable quantum states

A quantum entangled state is easily disturbed by noise and degenerates into a separable state. Compared to the entanglement with bipartite quantum systems, less progress has been made for the entanglement with multipartite quantum systems. For tripartite separability of a four-qubit system, we propose two entanglement witnesses, each of which corresponds to a necessary condition of tripartite separability. For the four-qubit GHZ state mixed with a W state and white noise, we prove that the necessary conditions of tripartite separability are also sufficient at W states side.     

Linear response of tripartite entanglement to infinitesimal noise

Recent experimental progress in prolonging the coherence time of a quantum system prompts us to explore the behavior of quantum entanglement at the beginning of the decoherence process. The response of the entanglement under an infinitesimal noise can serve as a signature of the robustness of entangled states. A crucial problem of this topic in multipartite systems is to compute the degree of entanglement in a mixed state. We find a family of global noise in three-qubit systems, which is composed of four W states. Under its influence, the linear response of the tripartite entanglement of a symmetrical three-qubit pure state is studied. A lower bound of the linear response is found to depend completely on the initial tripartite and bipartite entanglement. This result shows that the decay of tripartite entanglement is hastened by the bipartite one.