Multipartite quantum correlations among atoms in QED cavities

We study the nonlocality dynamics for two models of atoms in cavity quantum electrodynamics (QED); the first model contains atoms in a single cavity undergoing nearest-neighbor interactions with no initial correlation, and the second contains atoms confined in n different and noninteracting cavities, all of which were initially prepared in a maximally correlated state of n qubits corresponding to the atomic degrees of freedom. The nonlocality evolution of the states in the second model shows that the corresponding maximal violation of a multipartite Bell inequality exhibits revivals at precise times, defining, nonlocality sudden deaths and nonlocality sudden rebirths, in analogy with entanglement. These quantum correlations are provided analytically for the second model to make the study more thorough. Differences in the first model regarding whether the array of atoms inside the cavity is arranged in a periodic or open fashion are crucial to the generation or redistribution of quantum correlations. This contribution paves the way to using the nonlocality multipartite correlation measure for describing the collective complex behavior displayed by slightly interacting cavity QED arrays.     

Spectra and Eigenstates of Spin Chain Hamiltonians

We prove that translationally invariant Hamiltonians of a chain of n qubits with nearest-neighbour interactions have two seemingly contradictory features. Firstly in the limit \({n \rightarrow \infty}\) we show that any translationally invariant Hamiltonian of a chain of n qubits has an eigenbasis such that almost all eigenstates have maximal entanglement between fixed-size sub-blocks of qubits and the rest of the system; in this sense these eigenstates are like those of completely general Hamiltonians (i.e., Hamiltonians with interactions of all orders between arbitrary groups of qubits). Secondly, in the limit \({n \rightarrow \infty}\) we show that any nearest-neighbour Hamiltonian of a chain of n qubits has a Gaussian density of states; thus as far as the eigenvalues are concerned the system is like a non-interacting one. The comparison applies to chains of qubits with translationally invariant nearest-neighbour interactions, but we show that it is extendible to much more general systems (both in terms of the local dimension and the geometry of interaction). Numerical evidence is also presented that suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains.     

Thermal Entanglement Between Atoms in the Four-Cavity Linear Chain Coupled by Single-Mode Fibers

Natural thermal entanglement between atoms of a linear arranged four coupled cavities system is studied. The results show that there is no thermal pairwise entanglement between atoms if atom-field interaction strength f or fiber-cavity coupling constant J equals to zero, both f and J can induce thermal pairwise entanglement in a certain range. Numerical simulations show that the nearest neighbor concurrence C_AB is always greater than alternate concurrence C_AC in the same condition. In addition, the effect of temperature T on the entanglement of alternate qubits is much stronger than the nearest neighbor qubits.     

Monogamy Relations in Terms of Tsallis-Q Entropy in any Dimensional System

Monogamy of entanglement is a fundamental property of multipartite entangled states. In this article, due to the convexity of Trρ^q with respect to q when q ≥ 1, we give a monogamy-like relation in terms of Tsallis-q entanglement entropy of assistance (TqEEA) for pure states over an n- partite any dimensional system and monogamy-like relations in terms of Tsallis-q entanglement entropy (TqEE) for mixed states for any dimensional system, we also give a lower bound for the TqEE of a four-partite pure state. At last, we show that the generalized W-class states satisfy the polygamy relation in terms of TqEE when q = 2.     

Entanglement measure for pure <Emphasis Type="Italic">M</Emphasis> ⊗ <Emphasis Type="Italic">N</Emphasis> bipartite quantum states

We propose an entanglement measure for pure M ? N bipartite quantum states. We obtain the measure by generalizing the equivalent measure for a 2 ? 2 system, via a 2 ? 3 system, to the general bipartite case. The measure emphasizes the role Bell states have, both for forming the measure and for experimentally measuring the entanglement. The form of the measure is similar to the generalized concurrence. In the case of 2 ? 3 systems, we prove that our measure, which is directly measurable, equals the concurrence. It is also shown that, in order to measure the entanglement, it is sufficient to measure the projections of the state onto a maximum of M(M ? 1)N(N ? 1)/2 Bell states.     

Maximizing Complementary Quantities by Projective Measurements

In this work, we study the so-called quantitative complementarity quantities. We focus in the following physical situation: two qubits (q _A and q _B ) are initially in a maximally entangled state. One of them (q _B ) interacts with a N-qubit system (R). After the interaction, projective measurements are performed on each of the qubits of R, in a basis that is chosen after independent optimization procedures: maximization of the visibility, the concurrence, and the predictability. For a specific maximization procedure, we study in detail how each of the complementary quantities behave, conditioned on the intensity of the coupling between q _B and the N qubits. We show that, if the coupling is sufficiently “strong,” independent of the maximization procedure, the concurrence tends to decay quickly. Interestingly enough, the behavior of the concurrence in this model is similar to the entanglement dynamics of a two qubit system subjected to a thermal reservoir, despite that we consider finite N. However, the visibility shows a different behavior: its maximization is more efficient for stronger coupling constants. Moreover, we investigate how the distinguishability, or the information stored in different parts of the system, is distributed for different couplings.     

Quantification of Quantum Entanglement in a Multiparticle System of Two-Level Atoms Interacting with a Squeezed Vacuum State of the Radiation Field

We quantify multiparticle quantum entanglement in a system of N two-level atoms interacting with a squeezed vacuum state of the electromagnetic field. We calculate the amount of quantum entanglement present among one hundred such two-level atoms and also show the variation of that entanglement with the radiation field parameter. We show the continuous variation of the amount of quantum entanglement as we continuously increase the number of atoms from N = 2 to N = 100. We also discuss that the multiparticle correlations among the N two-level atoms are made up of all possible bipartite correlations among the N atoms.     

A Total Measure of Multi-Particle Quantum Correlations in Atomic Schrödinger Cat States

We propose a total measure of multi-particle quantum correlation in a system of N two-level atoms (N qubits). We construct a parameter that encompasses all possible quantum correlations among N two-level atoms in arbitrary symmetric pure states and define its numerical value to be the total measure of the net atom-atom correlations. We use that parameter to quantify the total quantum correlations in atomic Schrödinger cat states, which are generated by the dispersive interaction in a cavity. We study the variation of the net amount of quantum correlation as we vary the number of atoms from N=2 to N=100 and obtain some interesting results. We also study the variation of the net correlation, for fixed interaction time, as we increase the number of atoms in the excited state of the initial system, and notice some interesting features. We also observe the behaviour of the net quantum correlation as we continuously increase the interaction time, for the general state of N two-level atoms in a dispersive cavity.     

Protecting Distribution Entanglement for Two-Qubit State Using Weak Measurement and Reversal

Protection of entanglement from disturbance of the environment is an essential task in quantum information processing. We investigate the effect of the weak measurement and reversal (WMR) on the protection of the entanglement for an arbitrarily entangled two-qubit pure state from these three typical quantum noisy channels, i.e., amplitude damping channel, phase damping channel and depolarizing quantum channel. Given the parameters of the Bell-like initial qubits’ state |ψ〉 = a|00〉 + d|11〉, it is found that the WMR operation indeed helps for protecting distributed entanglement from the above three noisy quantum channels. But for the Bell-like initial qubits’ state |?〉 = b|01〉 + c|10〉, the WMR operation only protects entanglement in the amplitude damping channel, not for the phase damping and depolarizing quantum channels. In addition, we discuss how the concurrence and the success probability behave with adjusting the weak or the reversal weak measurement strength.     

The Dynamics of Quantum Entanglement and General Quantum Correlations in the Double Tavis-Cumming Model

We study the dynamics of quantum correlations involving entanglement and discord of two pairs of two-level atoms in cavity QED. In the model, two atoms A and C are coupled with a single-mode cavity field via Tavis-Cumming interaction at one location, and the same for B and D at another location. The two locations are connected by the entanglement of the atoms AB and CD while there are no any direct interactions between them. Through comparing the robustness of entanglement and discord of the atoms in various initial conditions of cavities, it is shown the discord is more robust than the entanglement and would be useful in quantum information technology.     

Control of Quantum Echo and Bell State Swapping of Two Atomic Qubits in the Two-Mode Vacuum Field Environment

We investigate quantum echo control and Bell state swapping for two atomic qubits (TAQs) coupling to two-mode vacuum cavity field (TMVCF) environment via two-photon resonance. We discuss the effect of initial entanglement factor ?? and relative coupling strength R=g₁/g₂ on quantum state fidelity of TAQs, and analyze the relation between three kinds of quantum entanglement(C(ρ_a),C(ρ_f),S(ρ_a)) and quantum state fidelity, then reveal physical essence of quantum echo of TAQs. It is shown that in the identical coupling case R=1, periodic quantum echo of TAQs with π cycle is always produced, and the value of fidelity can be controlled by choosing appropriate ?? and atom-filed interaction time. In the non-identical coupling case R≠1, quantum echoes with periods of π, 2π and 4π can be formed respectively by adjusting R. The characteristics of quantum echo results from the non-Markovianity of TMVCF environment, and then we propose Bell state swapping scheme between TAQs and two-mode cavity field.     

Preservation of Quantum Fisher Information and Geometric Phase of a Single Qubit System in a Dissipative Reservoir Through the Addition of Qubits

In this paper, we have investigated the preservation of quantum Fisher information (QFI) of a single-qubit system coupled to a common zero temperature reservoir through the addition of noninteracting qubits. The results show that, the QFI is completely protected in both Markovian and non-Markovian regimes by increasing the number of additional qubits. Besides, the phenomena of QFI display monotonic decay or non-monotonic with revival oscillations depending on the number of additional qubits N ??1 in a common dissipative reservoir. If N < N _c (a critical number depending on the reservoirs parameters), the behavior of QFI with monotonic decay occurs. However, if N ≥ N _c , QFI exhibits non-monotonic behavior with revival oscillations. Moreover, we extend this model to investigate the effect of additional qubits and the initial conditions of the system on the geometric phase (GP). It is found that, the robustness of GP against the dissipative reservoir has been demonstrated by increasing gradually the number of additional qubits N ??1. Besides, the GP is sensitive to the initial parameter ??, and possesses symmetric in a range regime [0,2π].     

Constraints on <Emphasis Type="Italic">SU</Emphasis>(2) ⊗ <Emphasis Type="Italic">SU</Emphasis>(2) invariant polynomials for a pair of entangled qubits

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices \(\mathfrak{P}_ +\). Since elements of \(\mathfrak{P}_ +\) are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, \(\mathfrak{P}_ + \in \mathbb{R}^{15}\). We define \(\mathfrak{P}_ +\) explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.     

Optimal dimensionality for quantum cryptography

A comparison of two protocols for generating a cryptographic key composed from d-valued symbols, one using a string of independent qubits and another utilizing d-level systems prepared in states belonging to d + 1 mutually unbiased bases, is performed. The protocol based on qubits is shown to be optimal for quantum cryptography, since it provides higher security and a higher key generation rate.     

Entanglement of three qubits

For the formulation of Bell inequalities, it is important to include not just N-site correlation functions, but also (N-n)-site correlation functions. In this article, we focus on a three-qubit Bell inequality, which has been shown to be a good candidate for generalizing Gisin’s theorem to three qubits. The three-qubit Bell inequality can be used to detect the W-type entanglement in a proposed experiment.     

Entanglement and Berry Phase in a Parameterized Three-Qubit System

In this paper, we construct a parameterized form of unitary \(\breve {R}_{123}(\theta _{1},\theta _{2},\varphi )\) matrix through the Yang-Baxterization method. Acting such matrix on three-qubit natural basis as a quantum gate, we can obtain a set of entangled states, which possess the same entanglement value depending on the parameters ?? ₁ and ?? ₂. Particularly, such entangled states can produce a set of maximally entangled bases Greenberger-Horne-Zeilinger (GHZ) states with respect to ?? ₁ = ?? ₂ = π/2. Choosing a useful Hamiltonian, one can study the evolution of the eigenstates and investigate the result of Berry phase. It is not difficult to find that the Berry phase for this new three-qubit system consistent with the solid angle on the Bloch sphere.     

Quantum Discord of 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-Dimensional Bell-Diagonal States

In this study, using the concept of relative entropy as a distance measure of correlations we investigate the important issue of evaluating quantum correlations such as entanglement, dissonance and classical correlations for 2 ⁿ -dimensional Bell-diagonal states. We provide an analytical technique, which describes how we find the closest classical states(CCS) and the closest separable states(CSS) for these states. Then analytical results are obtained for quantum discord of 2 ⁿ -dimensional Bell-diagonal states. As illustration, some special cases are examined. Finally, we investigate the additivity relation between the different correlations for the separable generalized Bloch sphere states.     

String order and degenerate entanglement spectrum of S = 1 / 2 and S = 1 Heisenberg bond-alternating chains

Generalized string orders and entanglement spectrum of S = 1/2 and S = 1 Heisenberg bond-alternating chains have been investigated by the infinite time-evolving block decimation (iTEBD) method. Generalized string order parameters with appropriate θ are capable of distinguishing all the topological phases. Central charges c ? 1 and critical exponents β ?1/12 indicate all the topological QPTs belong to the Gaussian universality class. Interestingly, odd- and even-fold degeneracies of the entanglement spectrum are observed. Even-fold (doubly) degenerate entanglement spectra and the typical two-fold degenerate lowest-lying level are found to exist in both the spin-1/2 dimer and the S = 1 Haldane phases. However, odd-fold degenerate entanglement spectra with three-fold degenerate lowest-lying level are observed in both the S = 1 dimer and the S = 2 Haldane phase. The degeneracy of the lowest-lying entanglement spectrum level, which can be understood by entanglement spectra in the dimer limit (J ₁ = 0), is adopted to estimate the lowest boundary of the bipartite entanglement. The entanglement spectrum and the generalized string orders are valuable for uncovering the underlying features of these symmetry-protect topological (SPT) states. Similar entanglement spectrum shows that the S = 1 (S = 2) Haldane phase is essentially the same as the S = 1/2 (S = 1) dimer phase.     

Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy,Relative Entropy Distance and Energy Constraints

We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: first, a tight analysis of the Alicki–Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications, we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entanglement, E_R, and its regularization \({E_R^{\infty}}\), as well as of the entanglement of formation, E_F. Using a novel “quantum coupling” of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, \({E_C=E_F^{\infty}}\). Second, we derive analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.     

Global geometric entanglement and quantum phase transition in three-leg spin-3/2 Heisenberg tubes

Based on the tensor network representations, we have developed an efficient scheme to calculate the global geometric entanglement as a multipartite entanglement measure for the three-leg spin tubes. From the geometric entanglement, the phase diagram of a spin-3 / 2 isosceles triangle spin tube has been investigated varying the base interaction α. Two Berezinsky-Kosterlitz-Thouless phase transitions are estimated to be α_c1 ? 0.68 and α_c2 ? 3.85, respectively. Then, even though the spin tube is in gapless spin liquid phases for α<α_c1 and α >α_c2, the geometrical structure difference between the groundstate wavefunctions for the two regions is found to reflect the global geometric entanglement that contains bipartite and multipartite contributions. Further, the phase transition points from the von Neumann entropies and fidelity are consistent with that from the geometric entanglement. As a result, the global geometric entanglement can be used to explore a geometrical nature of quantum phases as well as an indicator for quantum phase transitions in many-body lattice systems.