Quantum entanglement states are the critical resources needed in quantum computation and quantum communication, such as quantum teleportation,^[1–3] quantum dense coding,^[4–6] quantum key distribution,^[7–10] quantum secure direct communication,^[11–13] and quantum secret sharing.^[14–16] However, a quantum state will couple with the environment inevitably, which brings quantum noise and breaks the coherence of quantum entanglement state, leading to the degradation or even the failure of quantum tasks.

Several methods have been proposed to overcome the influence of decoherence on quantum entanglements. Quantum entanglement purification is one of the most direct methods. The purification uses proper local operations and classical communications to extract a sub-ensemble with higher purity from a quantum ensemble polluted by noise.^[17–19] Quantum error coding is another important method to solve the decoherence problem, which is used to recover the initial quantum information from the coded quantum messages by inserting some redundant information.^[20–22] Recently, a strategy based on the so-called decoherence-free subspace has been proposed. This strategy utilizes a noiseless sub-space of the quantum Hilbert space to construct the quantum code that can avoid the decoherence.^[23–25]

Weak measurement is a type of quantum measurement that would not cause the quantum system to fully collapse, thus the weak-measured system may be recovered through some reversal operations. The probabilistic reversing operation of an unsharp measurement was discussed in the context of quantum error correction.^[26] Some researchers employed an additional weak measurement to implement the recovery and have experimentally demonstrated their methods on a single superconducting qubit, single photonic qubit and arbitrary cavity field states with finite photons.^[27–29] Al-Amri et al. proposed an alternative way based on Hadamard and Controlled-NOT (CNOT) gates that can shorten the reversal time.^[30] This approach was also discussed in Ref. [31] to recover an arbitrary pure two-qubit entanglement state that has undergone a weak measurement. Recently, Korotkov and Keane,^[32] Kim et al.,^[33] and Lee et al.^[34] demonstrated that the weak measurement and the measurement reversal can protect the single qubit state and two-qubit entanglement state from the amplitude damping decoherence.

With the rapid development of quantum technology, multipartite communication will become an important form of quantum communication, whose realization is based on multi-qubit quantum entanglement states.^[35–38] Current research about decoherence suppression for multi-qubit entanglement usually focuses on a few special quantum states, such as Greenberger–Horne–Zeilinger (GHZ) state and W state.^[39–41] Based on Refs. [32]–[34], Liao et al.^[42] used the weak measurement method and quantum measurement reversal to preserve the entanglement of three-qubit GHZ state and three-qubit W state, demonstrating that this method was capable of enhancing the negativity of the entanglement and the fidelity of teleportation. The three-qubit W-like state is a more general quantum entanglement state that has been used in many quantum communication schemes.^[43–45] However, research on decoherence suppression for W-like state is rather insufficient.

In this paper, we use the weak measurement and the reversal measurement to cope with the decoherence of three-qubit W-like state in the amplitude damping channel, and demonstrate that it is effective to improve the quality of the entanglement. Through the theoretical derivation, we obtain a more concise result to show the effect of the decoherence suppression. After the weak measurement and the reversal measurement, the initial damping factor of the amplitude damping channel is changed into another smaller damping factor, and we call it the equivalent damping factor. Unfortunately, if we want to obtain a smaller equivalent damping factor, we must increase the measurement strengths, which will eventually reduce the total success probability of the decoherence suppression. Further, to enhance the success probability, we propose a method of iterated measurements and analyze the influence of the number of the iterations on the success probability. Finally, we present a method called “half iteration” that is optimal for increasing the success probability when it obtains the same equivalent damping factor as other methods.

Amplitude damping is an important type of quantum decoherence, which can model many practical physical processes, such as spontaneous emission and super-conduction with zero-temperature energy relaxation. In an amplitude damping channel, the evolutions of a single qubit system state and the environment can be described as the following quantum transformation:^[46]

The three-qubit W-like state is described as

The coefficients α, β, and γ are complexes that satisfy |α|² + |β|² + |γ|² = 1. The subscripts A, B, and C denote different qubits of the W-like state. After suffering the amplitude damping, the evolution of the W-like state can be expressed as the following operator-sum representation:

Here, ρ₀ is the density matrix of the initial pure state |W_l〉, and are the Kraus operators on qubits A, B, and C, respectively. For the simplicity of analysis, we assume that each qubit of the W-like state undergoes amplitude damping with the same damping factor λ, then we obtain the following result:

According to Eq. (5), we have ρ₀=ρ₁|_{λ = 0}. Compare ρ₁ with ρ₀, then it will be easy to obtain the following equation:

That is to say, after suffering the amplitude damping, the W-like state lies in the initial state with probability (1 − λ) and turns into state |000〉 with probability λ.

From Eq. (6), the Bures fidelity of ρ₁ to ρ₀ can be calculated as

In this part, we will discuss the effects of the weak measurement and the reversal measurement on the W-like state undergoing amplitude damping.

The weak measurement on a three-qubit system can be described as a tensor product of three non-unitary quantum operations, expressed as

Before suffering amplitude damping, the weak measurement is performed on the initial W-like state. Then, the three qubits of the W-like state are distributed to the three remote users through amplitude damping channel. After that, the users carry out the three-qubit reversal measurement described as the following operation:

Considering the assumption that the three qubits of the W-like state have the same amplitude damping factor λ, let p₁ = p₂ = p₃ = p, then p_r1 = p_r2 = p_r3 = p_r. After the weak measurement, the amplitude damping and the reversal measurement above, the initial pure W-like state will evolve into a mixed state having the following density matrix:

The total success probability of the weak measurement and the reversal measurement is

Comparing Eq. (11) with Eq. (5), we can derive the following result:

Equation (14) means that for the W-like state, the weak measurement and the reversal measurement change the initial amplitude damping channel into an equivalent amplitude damping channel with a smaller damping factor as

Here, λ_ev is called the equivalent damping factor. The Bures fidelity of ρ₂ to ρ₀ is

Obviously, F(ρ₂,ρ₀) is higher than F(ρ₁,ρ₀), showing that the weak measurement and the reversal measurement can suppress the decoherence of W-like state effectively.

The equivalent damping factor λ_ev of the weak measurement and the reversal measurement, varying with p and λ is shown in Fig. 1. It can be seen from Fig. 1 that, given a specific λ, a smaller λ_ev can be obtained in the case of a stronger weak measurement strength p; while given a specific p, λ_ev increases with the increase of λ.

Fig. 1.

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	Fig. 1. Equivalent damping factor of the weak measurement and the reversal measurement.

The total success probability P_S of the weak measurement and the reversal measurement is shown in Fig. 2. From this figure we can see that P_S decreases with p and λ increasing. This means that if we want to obtain a smaller λ_ev by increasing p, then we will obtain a smaller success probability, and P_S will be much smaller in the case of larger λ, which leads to a great consumption of the quantum entanglement resources.

Fig. 2.

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	Fig. 2. Total success probability of the weak measurement and the reversal measurement.

According to this problem, we try to find whether the success probability could be raised to some extent and thus to improve the overall performance of the decoherence suppression scheme above. This will be discussed below.

We consider Eq. (15) again and change it into the following expression:

Equation (17) means that a smaller (1 − p_r) will lead to a smaller λ_ev in the case of a given λ. Based on this thinking, if the weak measurement and the reversal measurement are carried out several times, a smaller measurement strength pair (p, p_r) would be needed to obtain an equal λ_ev, which may lead to a larger success probability.

If we perform a weak measurement successfully with strength p, then we repeat the weak measurement once with the same strength p; and if the corresponding reversal measurement is successful with strength p_r, we repeat the reversal measurement once with the same strength p_r. A pair of repeated operations above is called one iteration. The total measurements with one iteration are described as

After the iterated weak measurement, the amplitude damping and the iterated reversal measurement, the pure W-like state evolves into a mixed state whose density matrix (denoted as ρ₃) has a similar form to Eq. (11). The corresponding elements of the matrix ρ₃ are

The total success probability of the operations above is

Then, we can obtain the following result:

Comparing Eq. (22) with Eq. (17), if we want to obtain the same equivalent damping factors through using the measurements without iteration as the iterated measurements, the reversal measurement strength must meet the following conditions:

From Eq. (23), considering the optimal reversing condition and p_r = p + λ (1 − p), the weak measurement strength should satisfy

When we choose the measurement strength pair (p, p_r) described in Eqs. (23) and (24) to perform the measurements without iteration, the total success probability can be obtained according to Eq. (13).

Figure 3 shows the comparison between the success probabilities obtained using the method without iteration and those using the method with one iteration when both methods obtain the same equivalent damping factors, and the corresponding equivalent damping factors are shown in Fig. 4. In Figs. 3 and 4, the X axis means the initial amplitude damping factor and the Y axis represents the iterated weak measurement strength. Obviously, using iterated measurements has a higher success probability.

Fig. 3.

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	Fig. 3. Comparison between the success probabilities obtained by the method with one iteration and those by the method without iteration.

Fig. 4.

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	Fig. 4. Equivalent damping factors obtained by the method with one iteration and by the method without iteration.

On the other hand, if we want to obtain the same success probabilities by using the two methods, then the following condition should be satisfied according to Eqs. (13) and (20):

Let X = 1 − p_r and equation (25) can be changed into the following form:

To make Eq. (26) hold true, the condition X ≥ Y must be satisfied. Then, from Eqs. (17) and (22), we have

The results above show that the measurements with one iteration can increase the success probability of the decoherence suppression and enhance the entanglement fidelity of the three-qubit W-like state compared with the measurements without iteration.

In this paper, we show theoretically that decoherence of the three-qubit W-like state undergoing amplitude damping can be suppressed by weak measurement and the reversal measurement. Given an initial damping factor λ, this method is capable of reducing λ to a smaller equivalent damping factor λ_ev, which is related to the measurement strength. By increasing the measurement strength, we can obtain a smaller λ_ev, but the total success probability of the measurements will decrease quickly at the same time. To solve this problem, we propose a method of iterated measurements and demonstrate that it is useful to increase the success probability when its equivalent damping factor is equal to the λ_ev of the measurements without iteration. With increasing the iteration number, the equivalent damping factor becomes smaller and the success probability also becomes smaller. We find that the iteration number bigger than three has no practical significance because of the low yield, whereas another method called half iteration is a better option that has the optimal overall effects.