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Entanglement detection is one of the important topics in quantum information theory. Wootters formula can be seen as a high level criterion for entanglement detection or entanglement quantification. However, it is restricted to two-qubit system, and its generalization is also limited. Based on a thorough study of a three-qubit Greenberger-Horne-Zeilinger (GHZ) state mixed with a W state and white noise, we find a connection between Wootters formula and the matched entanglement witness. Further study shows that for a four-qubit GHZ state mixed with a Dicke state and white noise, there is a new kind of Wootters formula.

We use matched quantum entanglement witnesses to study the separable criteria of continuous variable states. The witness can be written as an identity operator minus a Gaussian operator. The optimization of the witness then is transformed to an eigenvalue problem of a Gaussian kernel integral equation. It follows a separable criterion not only for symmetric Gaussian quantum states, but also for non-Gaussian states prepared by photon adding to or/and subtracting from symmetric Gaussian states. Based on Fock space numeric calculation, we obtain an entanglement witness for more general two-mode states. A necessary criterion of separability follows for two-mode states and it is shown to be necessary and sufficient for a two mode squeezed thermal state and the related two-mode non-Gaussian states. We also connect the witness based criterion with Werner-Wolf criterion and refine the Werner-Wolf criterion., Comment: 11pages, 2 figures

Quantum entanglement is one of the most important resources in quantum information. A certain degree of white noise will make the entangled state separable, so finding the minimal white noise tolerance of a specific state is important. Based on a detailed study of the tripartite separability of a four-qubit W state mixed with a Dicke state and white noise, we obtain the necessary and (or) sufficient conditions either numerically or analytically. In particular, we give the analytical formula at W state side, our results show that the sufficient condition of tripartite separability agrees with the necessary condition well.

Starting from explicitly constructing an entanglement witness for the mixture of a Dicke state with white noise (noisy Dicke state), we provide an entanglement criterion for any multiqubit state. The criterion is necessary and sufficient for noisy $W$ states of three, four, and five qubits. We demonstrate an entanglement criterion for any four-qubit state. The criterion is necessary and sufficient for the generalized noisy four-qubit Dicke states. We present the sufficient criterion of full separability for a noisy $2k$ qubit Dicke state with $k$ excitations. Then we generalize the entanglement witness and criterion to the mixture of white noise and generic $n$-qubit Dicke state with $k$ excitations based on strong numeric evidence. Apart from noisy Dicke states, we also consider the entanglement properties of simultaneously amplitude damped and dephased Dicke states, and depolarized multiqubit $W$ 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.

With a two-step method of optimizing entanglement witness, we propose a set of necessary and sufficient entanglement criteria for four-qubit symmetric Greenberger–Horne–Zeilinger (GHZ) diagonal states. The criterion set contains four criteria. Two of them are linear with density matrix elements. The other two criteria are nonlinear with density matrix elements. The criterion set has a nest structure. A proper subset of the criteria is necessary and sufficient for entanglement of the corresponding subset of states. We illustrate the nest structure of criterion set with highly symmetric GHZ diagonal state set, its subset the general Werner state set and its superset the symmetric GHZ diagonal state set.

We use the generalized Wootters formula, the positive partial transpose (PPT) criterion and the matched entanglement witness, to detect entanglement of three-qubit GHZ and W superposition state and its decayed states. It shows that the results of the generalized Wootters formula in the part near the W state are tight. In the other parts, the PPT criterion is superior to the generalized Wootters formula. Furthermore, we investigate the relationship between entanglement and coherence.

We analytically prove that the necessary criterion coincides with the sufficient criterion for the full separability of three-qubit Greenberger---Horne---Zeilinger (GHZ) diagonal states. Our result closes the discussion on the separability of three-qubit GHZ diagonal states. Based on the criterion, the corresponding entanglement is exactly calculable for some GHZ diagonal states and is tractable for the others in terms of the relative entropy of entanglement.

Entanglement criteria for multipartite entangled states are obtained by matching witnesses to multipartite entangled states. The necessary and sufficient criterion of separability for three qubit X states is given as an example to illustrate the procedure of finding a criterion. The result is utilized to obtain the noise tolerance of W state. The necessary and sufficient criteria of three partite separability and full separability for four qubit noisy cluster states, three partite separability for four qubit noisy GHZ states are obtained.

We derive the realignment entanglement criterion for non-Gaussian states prepared by two mode symmetric Gaussian states undergoing phase damping. The entanglement detecting ability is compared with that of second moment criterion and Fock space criterion of positive partial transpose. New non-Gaussian entangled states are detected.

A Multipartite entangled state has many different kinds of entanglement specified by the number of partitions. The most essential example of multipartite entanglement is the entanglement of multi-qubit Greenberger-Horne-Zeilinger (GHZ) state in white noise. We explicitly construct the entanglement witnesses for these states with stabilizer generators of the GHZ states. For a $N$ qubit GHZ state in white noise, we demonstrate the necessary and sufficient criterion of separability when it is divided into $k$ parties with $N\leq 2k-1$ for arbitrary $N$ and $k$. The criterion covers more than a half of all kinds of partial entanglement for $% N $-qubit GHZ states in white noise. For the rest of multipartite entanglement problems, we present a method to obtain the sufficient conditions of separability. As an application, we consider $N$ qubit GHZ state as a codeword of the degenerate quantum code passing through depolarizing channel. We find that the output state is neither genuinely entangled nor fully separable when the quantum channel capacity reduces from positive to zero., Comment: 10 pages, 1 figure

We investigate how entanglement can be transferred between continuous variable and qubit systems. We find that a two-mode squeezed vacuum state and a continuous variable Werner state can be converted to the product states of an infinitive number of two-qubit states while keeping the entanglement. The reverse process is also possible. The proposed conversion model is theoretically viable but difficult to realize in the laboratory.

We analyze the entanglement condition of a new kind of non-Gaussian quantum state, which is prepared by photon number subtraction from the two mode squeezed thermal state. Fock space criterion and Shchukin–Vogel criterion are applied, and they give the same condition which is also that of the input state.

Quantum distortion operator is introduced based on canonical operators. As the lower bound of quantum information rate distortion, the entanglement information rate distortion is achieved by Gaussian map for Gaussian source. The entanglement information rate-distortion function is calculated for one-mode Gaussian source, it is achieved by a nontrivial unitary transformation. The rate distortion is achievable at zero distortion point. In contrast to the distortion defined via fidelity, our definition of the distortion makes it possible to calculate the entanglement information rate-distortion function for Gaussian source.

The evolution of the optical NOON state is investigated. The relative entropy of entanglement is exactly obtained for the NOON state simultaneously damped by symmetric amplitude damping and phase damping. The distillable entanglement of the phase-damped state is also obtained. The phase measurement performance of the damped state is studied according to the full measurement results of the quantum interferometry output. The phase sensitivity does not rely on the phase under estimate for the symmetrically amplitude-damped state.

Hypergraph states as real equally weighted pure states are important resources for quantum codes of non-local stabilizer. Using local Pauli equivalence and permutational symmetry, we reduce the 32768 four qubit real equally weighted pure states to 28 locally inequivalent hypergraph states and several graph states. The calculation of geometric entanglement supplemented with entanglement entropy confirms that further reduction is impossible for true hypergraph states., 6 pages,2 figures, Accepted by J. Phys. A

We provide the entanglement criterion of computable cross norm and realignment for any two-mode continuous-variable state with permutational symmetry. The entanglement criterion for a non-Gaussian state is obtained from its Gaussian kernel by employing the technique of functional analysis. The entanglement conditions turn out to be necessary and sufficient for the non-Gaussian states prepared by one-photon subtracting from or adding to symmetric Gaussian states. The entanglement criterion of non-Gaussian states evolved in thermal noise and amplitude damping environment also is obtained.

We reduce the necessary and sufficient biseparable conditions of the four qubit cluster diagonal state to concise forms. Only 4 out of the 15 parameters are proved to be relevant in specifying the genuine entanglement of the state. Using the relative entropy of entanglement as the entanglement measure, we analytically find the genuine entanglement of all the four qubit cluster diagonal states. The formulas of the genuine entanglement are of five kinds, for seven different parameter regions of entanglement., Comment: 7 pages, 4 figures

The relative entropy of entanglement, the global robustness of entanglement and the geometric measure are equal for stabilizer quantum code words. The entanglement upper bounds can be get from the generator with Pauli measurement. Using an iterative algorithm can get the exact value of the entanglement.

The number of in equivalent classes of up to 8-qutrit graph states is 1002, with 239 decomposable graphs and 763 indecomposable graphs. Apparently, the former can be decomposed to several indecomposable parts. For qubit graph states, the upper and lower bounds of entanglement have been given. If the two bounds dont coincide, the entanglement can be calculated by iterative method. For qutrit graph states, their lower bounds can be found through the local complementation operations. But its hard to find the upper bound because the qutrit graph states have 2 basis states which are GHZ state and W state. Fortunately, the iterative method can also be used here. In this paper, we calculated the entanglement of all the1002 graph states. Through researching the results, we found the similar feature with graph qubit states that the decimal part of entanglement is stable.

An entanglement bound based on local measurements is introduced for multipartite pure states. It is the upper bound of the geometric measure and the relative entropy of entanglement. It is the lower bound of minimal measurement entropy. For pure bipartite states, the bound is equal to the entanglement entropy. The bound is applied to pure tripartite qubit states and the exact tripartite relative entropy of entanglement is obtained for a wide class of states., 8 pages, 1 figure, accepted by Physical Review A

The number of inequivalent classes of 10-qubit graph states is 3132. All 3132 local inequivalent graphs are classified as two categories: graphs with equal bound, graphs with unequal bounds. We will use different methods to calculate the entanglement of graph states of ten qubits for the different categories. For graph states with different entanglement bounds, the closest product states are given, and they have different structural characters.

We introduce graph state basis diagonalization to calculate the coherent information of a quantum code passing through a Pauli channel. The scheme is 5000 times faster than the best known one for some concatenated repetition codes, providing us a practical constructive way of approaching the quantum capacity of a Pauli channel. The calculation of the coherent information of non-additive quantum code can also be greatly simplified in graph state basis., Comment: 5 pages

The entanglement of graph states can be calculated by adopting Iterative algorithm. The number of inequivalent classes of 9-qubit graph states is 440. All 440 local inequivalent graphs are classified as two categories: graphs with equal upper LOCC entanglement bound and lower bipartite entanglement bound, graphs with unequal bounds. The later may display non-integer entanglement. After determining the entanglement, the closest product state can also be obtained. As for graph states with different entanglement values, the closest product states have different structural characteristics. The precision of iteration algorithm of the entanglement is less than 10−14.

In quantum information theory, von Neumann entropy plays an important role. The entropies can be obtained analytically only for a few states. In continuous variable system, even evaluating entropy numerically is not an easy task since the dimension is infinite. We develop the perturbation theory systematically for calculating von Neumann entropy of non-degenerate systems as well as degenerate systems. The result turns out to be a practical way of the expansion calculation of von Neumann entropy., 7 pages

The quantum information transfer between a single photon and a two-level atom is considered as a part of a quantum channel. The channel is a degradable channel even when there are decays of the atomic excited state and the single photon state, as far as the total excitation of the combined initial state does not exceed 1. The single letter formula for quantum capacity is obtained., 4 pages

We give the time evolution solution of simultaneous amplitude and phase damping for any continuous variable state. For the simultaneous amplitude and phase damping of two-mode x-p symmetric entangled Gaussian states, two analytical conditions of the separability are given. One is the sufficient condition of separability. The other is the condition of positive partial transpose (PPT) separability where the Peres-Horodecki criterion is applied. Between the two conditions there may exist bound entanglement. The simplest example is the simultaneous amplitude and phase damping of a two-mode squeezed vacuum state. The damped state is non-Gaussian.

We investigate how quantum state can be converted between continuous variable and qubits systems. Non-linear Jaynes-Cumings interaction Hamiltonian is introduced to accomplish the conversion. Detail analysis on the conversion of thermal state exhibits that pretty good fidelity can be achieved., Comment: 6 pages

For two gaussian states with given correlation matrices, in order that relative entropy between them is practically calculable, I in this paper describe the ways of transforming the correlation matrix to matrix in the exponential density operator. Gaussian relative entropy of entanglement is proposed as the minimal relative entropy of the gaussian state with respect to separable gaussian state set. I prove that gaussian relative entropy of entanglement achieves when the separable gaussian state is at the border of separable gaussian state set and inseparable gaussian state set. For two mode gaussian states, the calculation of gaussian relative entropy of entanglement is greatly simplified from searching for a matrix with 10 undetermined parameters to 3 variables. The two mode gaussian states are classified as four types, numerical evidence strongly suggests that gaussian relative entropy of entanglement for each type is realized by the separable state within the same type.For symmetric gaussian state it is strictly proved that it is achieved by symmetric gaussian state., 12 pages, 3 figures

We consider the entanglement transformations of the protocols that convert quantum qubit system to continuous variable system and convert it back. Here the quantum D/A conversion protocol maps finite-dimensional input states onto continuous variable states while the quantum A/D conversion protocol maps two mode squeezed state onto entangled state in 2x2 dimensions. Quantum carrier is investigated when pure qubit state interacts with coherent state.

Multipartite entanglement is quite difficult to deal with. In this paper, I will consider a particular kind of tripartite Gaussian states, which can be produced as the initial pure tripartite states undergo different noise for each party. I will make use of the relative entropy of entanglement to evaluate the entanglement properties of the states based on the fact that the separability of tripartite Gaussian states is well known. In addition, the procedure of calculating the relative entropy between Gaussian states have developed recently. All these make the evaluating the relative entropy of entanglement of tripartite Gaussian states possible.

We in this paper strictly prove that some block diagonalizable two qubit entangled state with six none zero elements reaches its quantum relative entropy entanglement by the a separable state having the same matrix structure. The entangled state comprises local filtering result state as a special case., Comment: 5 pages

I propose the multi-mode squeezed thermal state based on the multi-mode pure entangled state. The correlation matrix of the state is characterized by two parameters. I then analysis the separable condition for this state, and calculating the relative entropy of the state with respect to the same kind of fully separable state in order to provide an upper bound of the relative entropy of entanglement. The bound is compared with the other bounds which were obtained with reduced state., 9 pages, 3 figures

The upper and lower bounds of entanglement of formation are given for two mode squeezed thermal state. The bounds are compared with other entanglement measure or bounds. The entanglement distillation and the relative entropy of entanglement of infinitive squeezed state are obtained at the postulation of hashing inequality., 3 figures

We study the entanglement conditions of two quite general classes of two-mode non-Gaussian states. Using computable cross norm and realignment criterion, we obtain the sufficient conditions of entanglement for coherently added or subtracted two-mode squeezed thermal states, and the sufficient condition of entanglement for any photon number entangled state (PNES) evolving in a thermal noise and amplitude damping channel.

The entanglement of graph states up to eight qubits is calculated in the regime of iteration calculation. The entanglement measures could be the relative entropy of entanglement, the logarithmic robustness or the geometric measure. All 146 local inequivalent graphs are classified as two categories: graphs with identical upper LOCC entanglement bound and lower bipartite entanglement bound, graphs with unequal bounds. The late may displays non-integer entanglement. The precision of iteration calculation of the entanglement is less than $10^{-14}$.