Characterizing entanglement robustness of N-qubit W superposition state against particle loss from quantum Fisher information

We investigate the entanglement robustness of N -qubit W superposition state against particle loss with quantum Fisher information. In the noninteracting environment, the quantum Fisher information of reduced W superposition state is found same as the W superposition state when the number of qubits larger than 5 , which shows very strong entanglement robustness compared to Twin-Fock state and Greenberger-Horne-Zeilinger state. Meanwhile, the effect of re-particle-loss is also studied and it presents a rapid decay of quantum Fisher information with the increase of losing qubits, indicating the weak entanglement robustness. In the interacting environment, the results show that the reduced W superposition state performs better than it in noninteracting environment. Alternatively, with the increase of qubits ( N > 5 ) the reduced W superposition state behaves much more robust than original state. To vividly approach the real noisy environment, the effects of different decoherence channels are also considered.

Lately, the N -qubit W superposition (WS) state composed of symmetric W state and its obverse is found to have better performance in quantum metrology [14] and three-qubit WS state behaves robust against particle loss [15][16][17][18][19]. How about an N -qubit WS state ?and how strong robustness it is, compared with the Greenberger-Horne-Zeilinger (GHZ) state, or Twin Fock (TF) state ?
Entanglement robustness can be defined in many ways, including the tolerance of noise [20], the minimum number of local measurements [21], the size of system [22], Schmidt robustness and the random Schmidt robustness [23], and etc.In this context, we mainly consider the effect of particle loss on entanglement robustness [24][25][26].Taking advantage of the * corresponding author: li8989971@163.comquantity, Quantum Fisher information (QFI) [27], we exhaustively characterize the entanglement robustness of N -qubit WS state against particle loss.
In the noninteracting environment, we analytically present the QFI of N -qubit WS state under particle loss, and find that the QFI of reduced WS state is same as it from original WS state when the number of qubits is N > 5, which indicates very strong entanglement robustness than GHZ state and TF state.Subsequently, the effect of re-particle-loss is considered and it shows a rapid decay of QFI, implying the weak entanglement robustness.Under the Ising-type interacting environment [28], the analytical QFI of N -qubit reduced WS state is obtained and with the increase of the number of qubits the reduced WS state shows better performance than original WS state.To simulate a real noisy environment, the different decoherence channels are considered in both environments and compared with the results in noninteracting environment it displays a similar variation trend but enhanced QFI in interacting environment.Specially, the 4-qubit reduced WS state is found to have same QFI under phase damping channel and depolarizing channel, whatever in noninteracting environment or interacting environment.
The paper is organized as follows.In Sec.II, some preliminaries are introduced, including the relationship between quantum Fisher information and metrological useful entanglement, the formula of QFI under noninteracting and interacting operations, and three traditional decoherence channels.In Sec.III and Sec.IV, we elaborately study the entanglement robustness of N -qubit WS state against particle loss with QFI, under noninteracting environment and interacting environment.Finally, the results are summarized in Sec.V.

II. MATERIALS AND METHODS
In this section, we first briefly review the knowledge that quantum Fisher information witnessing metrological useful entanglement, and then present the methods used for accessing QFI in noninteracting and interacting environments.At last, the general decoherence channels are introduced for evaluating the entanglement robustness of reduced WS state.

A. quantum Fisher information witnessing metrological useful entanglement
As an important quantity in parameter estimation theory [29,30], quantum Fisher information has attracted much attention in quantum metrology [31,32] and recently been utilized to witness metrological useful entanglement [33][34][35].Assuming a general quantum state ρ and an unitary linear transformation Ĥ, the relationship between phase sensitivity ∆θ and QFI F Q can be connected by quantum Cramér Rao lower bound [29,30], where the precision limit is bounded by the value of QFI.And evidently the larger the F Q , the higher the sensitivity [36].
From the point of view of statistical distinguishability [27,37], the separable state and entangled state have different statistical responses to the same external signal, which can be quantified by QFI, and thus we can exploit this property to witness entanglement.In other words, if the value of QFI is larger than it from product states, the given state ρ is found entangled [33], and vice versa.The larger QFI indicates much more entanglement exists in the quantum system, which is in turn helpful to high-precision quantum measurement through Eq. (4).Therefore, the QFI can be used to witness metrological useful entanglement or quantum resources [38].This feature has been applied to analyze several experimental results, including entanglement detection [39][40][41][42][43] and quantum metrology [44][45][46].

B. quantum Fisher information under noninteracting environment and interacting environment
In quantum mechanics, the dynamic evolution of a given initial quantum state ρ 0 under a general unitary Hamiltonian Ĥ is depicted by ρθ = e −iθ Ĥ ρ0 e iθ Ĥ , and the quantum Fisher information can be accessed by following formula [27], in which the diagonalized density matrix is expressed as ρ0 = k i=1 λ i |ψ i ψ i |.With respect to an ideal pure state, ρ0 = |ψ in ψ in |, the Eq. ( 5) is reduced to where ∆ 2 Ĥ = Ĥ2 − Ĥ 2 denotes the variance of Ĥ over the initial state.Hence, the key point of characterizing entanglement robustness against particle loss becomes the access of QFI from the given quantum system [40,46,[53][54][55].
Without loss of generality, we consider a general case that the given quantum state under an interacting Hamiltonian, which includes both the noninteracting term Ĥ0 = n .Here µ i means the inhomogeneous linear coupling and • n is the pauli matrix for the ith qubit and n is a vector.
With the formula Eq. ( 6), we have access to the QFI of a pure state under interacting Hamiltonian [49], Ĥ1 separately represent the noninteracting QFI, interacting QFI and covariant QFI.The explicit expressions are given as follows, Due to the permutation symmetry of N -qubit WS state, the calculation of QFI mainly focuses on the expectation value of interactive qubits, i.e., σ , and . Next we will exploit above formulas to present the specified QFI with respect to the given reduced WS state, and further analyze its entanglement robustness against particle loss, under both the noninteracting environment and interacting environment.

C. Decoherence channels
It is unavoidable for quantum system to be affected by the surrounding noises and further leads to decoherence.Amplitude damping, phase damping and depolarization are three traditional channels providing to describe the decoherence, and can be expressed in the Kraus representation [56].The density matrix after decoherence can be expressed by where Êk denotes the Kraus operator and satisfies the completeness relationship, The amplitude damping channel is generally used to describe the energy dissipation in quantum system, and its effect on a single qubit can be expressed by the Kraus operators, where p denotes the probability of decay from upper level |1 to the lower level |0 .The phase damping channel is adopted to model the inherently quantum non-dissipative physical situation where a system undergoes a loss of coherence without losing energy.A phase damping channel acting on a single qubit is described by the Kraus operators, The depolarizing channel describes the process where a system undergoes a symmetric decoherence.A depolarizing channel acting on a system of two qubits has the effect of leaving it untouched with probability 1 − p and of introducing a white noise that drives it into a maximal mixed state with probability p.Its effect on a single qubit can be expressed by the Kraus operators, According to the evolution of density matrix under different decoherence channels, the decohered density matrix of Nqubit reduced WS state is given by Then we diagonalize it and substitute the corresponding eigenvalues and eigenvectors into Eq.( 5) with noninteracting Hamiltonian Ĥ0 or interacting Hamiltonian Ĥ, the QFI of Nqubit reduced WS state is under different decoherence channels is obtained.After maximization over all direction n, the maximal QFI needed for evaluating entanglement robustness is achieved.

III. ROBUSTNESS AGAINST PARTICLE LOSS UNDER NONINTERACTING ENVIRONMENT
We first investigate the effect of particle loss on the entanglement robustness of N -qubit WS state under noninteracting environment.Due to the symmetry of WS state, we can trace any one qubit from original state to represent the reduced WS state, such as ρ ) ], where ρ (N) denotes the original N -qubit WS state and ρ (N ) t denotes the N -qubit reduced WS state, composed of N − 1 qubits.Here all WS states of different qubits are divided into three categories.
With respect to 3-qubit WS state, after one qubit traced out we have the 3-qubit reduced WS state and its diagonalized density matrix is given by where the basis states, ), are entangled.Obviously it is better than reduced GHZ state, for which it is completely unentangled [24].Under noninteracting environment or local operation, i.e., Ĥ0 with µ i = 1, the QFI of 3-qubit reduced WS state can be obtained by Eq. ( 5).The maximal value is 15 , which is smaller than the number of qubits involved and according to the entanglement criterion it is hard to witness entanglement [33].However, with some filtering procedures or entanglement distillations it is possible to obtain the useful entangled state |ϕ 1 or |ϕ 2 .
Analogously, we have managed the N -qubit WS state (N > 4) and the diagonalized density matrix is written as where the state  obtain the QFI of N -qubit reduced WS state and it is given by (22) Obviously, as the number of qubits N > 4, the maximal QFI is obtained and expressed by It is very interesting that the formula of F ρ (N ) t is same as it from N -qubit WS state [14], where the condition of the number of qubits involved is N > 5.This indicates the very strong entanglement robustness of N -qubit WS states against particle loss.
In Fig. 1a and Fig. 1b, we show the QFI of N -qubit WS state before and after trace, which clearly exhibits the strong entanglement robustness of WS state, as it shown by red dots.For comparison, we have also displayed the results from GHZ states (black squares) and TF states (blue triangles), which owns the relative larger QFI under perfect local operation [34] and performs weaker than WS state after trace.
To deeply explore the entanglement robustness of N -qubit WS state, the effect of re-particle-loss is studied in the same way, i.e., the calculation of QFI from the state whose qubits are traced one by one from the original WS state.In Fig. 1c, the 10-, 11-, 12-qubit WS state has been chosen as an example to display the change of QFI with respect to the number of tracing qubits.Meanwhile, the corresponding entanglement bounds are also shown by bottom colorful lines.Evidently, with increase of the number of losing qubits the entanglement decays rapidly, and if some filtering procedures can be involved, the situation maybe different.
Besides, we have also considered the decoherence effect and investigated the entanglement robustness of N -qubit reduced WS state under three different decoherence channels (amplitude damping, phase damping and depolarizing).In Fig. 2, the QFI of WS state (reduced WS state) with respect to p under three decoherence channels are shown and the variation trends are similar, where the dashed lines and solid lines respectively denote the results from WS state and reduced WS state.Obviously, a little noise will largely decrease the QFI of reduced WS state, as it shown in Fig. 2c and Fig. 2d where the QFI of ideal reduced state are F ρ (6) t = 16 and F ρ (8) t = 36.Additionally, it is found that the QFI of 4-qubit reduced WS state under phase damping channel is same as it under depolarizing channel.

IV. ROBUSTNESS AGAINST PARTICLE LOSS UNDER INTERACTING ENVIRONMENT
The interaction between nearby perfect local operations is inevitable [57,58].Therefore, in this section we investigate the entanglement robustness of N -qubit reduced WS state under interacting environment.In the following, we first present the analytical QFI of reduced WS state and then explore the effect of decoherence channels on entanglement robustness.
Similar to the previous classification of N -qubit WS states, 3-qubit WS state, 4-qubit WS state, 5, 6-qubit WS state and others are investigated separately.Here we take advantage of Ising-type interacting operation (V ij = ε δj,i+1+δj,i−1 2 ) to explore the entanglement robustness of WS state under particle loss, i.e., Eq. ( 7), in which it assumes m In the case of 3-qubit WS state, after one qubit traced out, we obtain the reduced state and its analytical QFI is expressed by In the same way, we obtain the analytical QFI of 4-qubit reduced WS state and it is written as With respect to the 5-, 6-qubit case, the values of QFI are respectively given by For the number of qubits involved N > 6, the QFI of N -qubit reduced WS state is analytically expressed by With respect to interaction strength ε, we have compared the QFI from different reduced WS states (dashed red lines) and WS states (dashed black lines) in Fig. 3.The results show that the QFI of reduced WS state is larger than it from original WS state when the number of qubits is N > 5, and vice versa.This means that the large-qubit WS state has the stronger entanglement robustness against particle loss than original WS state.Alternatively, we can also verify it analytically.In the limit of large qubits N ≫ 1 and stronger interaction strength ε ≫ 1, the Eq. ( 28) is reduced to and the optimal pauli matrix is given by σ z ∀i.It is evidently larger than the extreme value of QFI from original WS state, i.e., Eq. ( 44) in [14].Subsequently, we have investigated the effect of different decoherence channels on entanglement robustness.The results are shown in Fig. 4, where the QFI of 4-, 6-, 8-qubit reduced WS state with respect to ε and p are separately presented.With the increase of interaction strength ε, the QFI in Fig. 4 (a, c, e) is enhanced.Similarly, compared to the results from noninteracting environment shown in Fig. 2 (b, c, d), the QFI in Fig. 4 (b, d, f) is also enhanced and as same for the entanglement robustness.Additionally, the 4-qubit reduced WS state is still found special and its QFI under phase damping channel is same as it under depolarizing channel, as shown in Fig. 4a and Fig. 4b.

V. CONCLUSIONS
In summary, we have thoroughly characterized the entanglement robustness of N -qubit W superposition state against particle loss with quantum Fisher information.The results show that under noninteracting environment the reduced WS state behaves much more robust, especially for the case of qubits N > 5, where the QFI of reduced WS state is same as it from original WS state.However, under the re-particle-loss situation, it decays quickly and indicates the weak entanglement robustness.Considering the effect of different decoherence channels we find that very little noise will largely decreases the entanglement robustness.In the Ising-type interacting environment, the entanglement robustness of reduced  WS state is enhanced with the increasing interaction strength.As the number of qubits increase (N > 5), the reduced WS state is found to perform better than original one.Taking into account different decoherence channels it shows that the interacting environment will enhance the entanglement robustness to some extent.In particular, the 4-qubit reduced WS state is found to have the same QFI under phase damping channel and depolarizing channel, whatever in noninteracting environment or interacting environment.
Additionally, it is easy to deduce that the entanglement robustness of N -qubit WS state displays a size effect, that once the number of qubits N > 5 the reduced WS states behave same or stronger entanglement robustness than original ones.

Figure 1 .
Figure1.(color online) Quantum Fisher information from Nqubit quantum states under ideal situation and particle-loss situation.Black squares, blue triangles and red dots denote the QFI from GHZ state, TF state and WS state with no particle loss (a) and particle loss (b), respectively.(c) Black, red and blue dots separately represent the QFI of 10-, 11-, 12-qubit WS state with respect to the number of losing qubits N .

Figure 3 .
Figure 3. (color online) Quantum Fisher information from reduced WS states and original WS states in Ising-type interacting environment.Dashed red lines and black lines in (a-f) respectively denote the QFI of 3-, 4-, 5-, 6-, 8-, 10-qubit reduced WS state and original WS state, with respect to the interacting strength ε.
and the state |ψ 2 is the inverse of state |ψ 1 .Under the noninteracting environment, assuming σ y + cσ