Deterministic conversion of a four-photon GHZ state to a W state via homodyne measurement

Wen-Xue Cui; Shi Hu; Hong-Fu Wang; Ai-Dong Zhu; Shou Zhang

doi:10.1364/OE.24.015319

1. Introduction

Quantum entanglement is one of the most important resources of QIP task, such as quantum teleportation [1,2], quantum key distribution (QKD) [3,4], quantum secret sharing [5,6], quantum secure direct communication (QSDC) [7, 8], etc.. Although the property of entanglement has been well understood for bipartite system, the entanglement of multipartite system is also appealing and the preparation and application of multipartite entanglement has attracted much attention. For a multipartite system, the two most common classes of entangled states are the W state and the Greenberger-Horne-Zeilinger (GHZ) state, which are impossible to convert a GHZ state into an W state only by local operations and classical communication (LOCC) [9,10]. For GHZ state, which plays a crucial role in engineering entanglement [11, 12], quantum secret sharing [13] and hyperentanglement concentration [14], the entanglement completely destroys with the loss of any one of the qubits. However, the W state is robust against the loss of one qubit, that is, the remaining qubits can be still entangled with each other when one qubit is discarded [9]. Therefore, the W state as a resource is widely used in the development of quantum computing and quantum information science [15, 16]. Recently, some studies have sought to implement the conversion between GHZ state and W state. In 2005, Walther et al. described a probabilistic method to convert a GHZ state into an approximate W state based on partial quantum measurement (POVMs) and experimentally realized the scheme in the 3-qubit case [17]. In 2009, Tashima et al. proposed and experimentally demonstrated how to transform two Einstein-Podolsky-Rosen Photon pairs into a three-photon W state [18]. Then we propsoed a scheme for converting four EPR photon pairs distributed among five parties into the heralded four-photon polarization-entangled decoherence-free states via LOCC [19]. In 2013, Song et al. proposed a scheme for converting a four-atom W state to a Greenberger-Horne-Zeilinger state via a dissipative process [20]. All these two classes of entangled states can perform different tasks in QIP, and there have been many experimental realizations on the preparation of GHZ state and more about W state [21, 22]. Naturally, the question wether it is possible to convert a GHZ state into an exact W state deterministically thus arises.

It is well known that photons are ideal quantum information carrier as they travel with high speed and are negligibly affected by decoherence. Qubit is usually encoded on the vertical polarization and horizontal polarization states of photons, and quantum information tasks can be achieved by using linear optical elements. However, as the probabilistic nature of linear optical quantum gates, the quantum information tasks can not be achieved deterministically in linear optical system. Cross-Kerr nonlinearity is a promising way for deterministic optical QIP. In 1995, Chuang and Yamamoto [23] first proposed an implementation of a quantum computer to solve Deutsch’s problem and realize error correction with the cross-Kerr nonlinearity. In 2003, Munro et al. [24] proposed a scheme for a highly efficient photon-number quantum-nondemolition detector (QND) with single-photon resolving based on the cross-Kerr nonlinearity. Due to strong Kerr nonlinearities are difficult to achieve in experiment, in their follow-on work, they realized nearly deterministic quantum gate with strong coherent state and weak-Kerr nonlinearity with electromagnetically induced transparency (EIT) [24–27]. Recently, more and more schemes for QIP have been proposed with cross-Kerr nonlinearities [28–32]. Our scheme is basd on the cross-Kerr nonlinearity that having a brief rundown of the cross-Kerr nonlinear interaction between probe mode and signal mode is needed, as shown in Fig. 1. The interaction Hamiltonian using creation and annihilation operators can be written as [33]

{\hat{H}}_{QND} = \bar{h} χ {\hat{a}}_{s}^{†} {\hat{a}}_{s} {\hat{a}}_{p}^{†} {\hat{a}}_{p} .

where

a_{s}^{†} (a_{p}^{†})

and â_s (â_p) represent the creation and annihilation operators of the signal mode (probe mode), χ is the nonlinearity strengh. Consider a coherent beam in the state |α〉 with a signal pulse in the Fock state |Φ〉 = a|0〉 + b|1〉. Here, |0〉 and |1〉 represent the number of the photons. After interaction with the cross-Kerr interaction medium, the state evolves as

\begin{array}{l} U_{ck} | Φ 〉 | α 〉 & = e^{i {\hat{H}}_{QND} t / \bar{h}} (a | 0 〉 + b | 1 〉) | α 〉 \\ = a | 0 〉 | α 〉 + b | 1 〉 | α e^{i θ} 〉 . \end{array}

where θ = χt and t is the interaction time. From Eq. (2), it is obvious that the signal photon state is unaffected after the interaction, but the coherent state picks up a θ phase shift directly proportional to the number of the photons in the signal mode. So, the photon numbers in signal mode can be distinguished through the measurement on the phase of probe beam.

Fig. 1 Cross-Kerr nonlinear interaction model.

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2. Converting a four-photon GHZ state to a W state

In this section, we illustrate how to convert a four-photon GHZ state to a W state in a deterministic way by using cross-Kerr nonlinearities, linear optics elements, and homodyne measurement. The schematic setup is shown in Fig. 2. Assume that four photons are initially prepared in the GHZ state

{| ϕ 〉}_{0} = \frac{1}{2} ({| H 〉}_{1} {| H 〉}_{2} {| H 〉}_{3} {| H 〉}_{4} + {| V 〉}_{1} {| V 〉}_{2} {| V 〉}_{3} {| V 〉}_{4}) .

In the FS basis representation, the state in Eq. (3) is rewritten as

\begin{array}{l} {| ϕ 〉}_{0} & = \frac{1}{2 \sqrt{2}} ({| F 〉}_{1} {| F 〉}_{2} {| F 〉}_{3} {| F 〉}_{4} + {| F 〉}_{1} {| S 〉}_{2} {| F 〉}_{3} {| S 〉}_{4} + {| S 〉}_{1} {| F 〉}_{2} {| F 〉}_{3} {| S 〉}_{4} \\ + {| S 〉}_{1} {| S 〉}_{2} {| F 〉}_{3} {| F 〉}_{4} + {| F 〉}_{1} {| F 〉}_{2} {| S 〉}_{3} {| S 〉}_{4} + {| F 〉}_{1} {| S 〉}_{2} {| S 〉}_{3} {| F 〉}_{4} \\ + {| S 〉}_{1} {| F 〉}_{2} {| S 〉}_{3} {| F 〉}_{4} + {| S 〉}_{1} {| S 〉}_{2} {| S 〉}_{3} {| S 〉}_{4}) \otimes | α 〉 . \end{array}

where |F〉 and |S〉 are the superposition of horizontal and vertical polarization states [34], namely,

| F 〉 = 1 / \sqrt{2} (| H 〉 + | V 〉)

and

| S 〉 = 1 / \sqrt{2} (| H 〉 - | V 〉)

.

Fig. 2 Schematic diagram for conversion from a GHZ state to a W state. And the qubit flip (QF) device is composed of two HV-PBS and a phase shifter (PS). Here, the FS-PBS, which transmits F-polarized photons and reflects S-polarized photons is the polarizing beam splitter (PBS) in the |F〉 and |S〉 basis. HV-PBS, which transmits H-polarized photons and reflects V-polarized photons, is the PBS in the in the |H〉 and |V〉 basis. PS denotes a phase shifter, which is to realize the transformation: |H〉 → |H〉, |V〉 → −|V〉.

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The photons in modes 1, 2, 3, 4 respectively pass through a FS-PBS, which transmits F-polarized photons and reflects S-polarized photons. Then the photon in modes 1′ passes through a qubit-flip (QF) device, which is to complete the transformation {|F〉 → |S〉, |S〉 → |F〉}. Next the |F〉 polarization states of photons in modes a′, b′, c′, d′ interacts with cross-Kerr nonlinearities, respectively. After the photons pass through the remaining FS-PBS, the whole system is evolved into the following state

\begin{array}{l} {| ϕ 〉}_{2} & = \frac{1}{2 \sqrt{2}} [({| S 〉}_{a} {| F 〉}_{b} {| F 〉}_{c} {| F 〉}_{d} + {| F 〉}_{a} {| S 〉}_{b} {| F 〉}_{c} {| S 〉}_{d} + {| S 〉}_{a} {| F 〉}_{b} {| S 〉}_{c} {| S 〉}_{d} \\ + {| F 〉}_{a} {| F 〉}_{b} {| F 〉}_{c} {| S 〉}_{d}) + | α e^{3 θ} 〉 + ({| F 〉}_{a} {| S 〉}_{b} {| S 〉}_{c} {| S 〉}_{d} + {| S 〉}_{a} {| F 〉}_{b} {| F 〉}_{c} {| F 〉}_{d} \\ + {| S 〉}_{a} {| S 〉}_{b} {| F 〉}_{c} {| S 〉}_{d} + {| S 〉}_{a} {| S 〉}_{b} {| S 〉}_{c} {| F 〉}_{d}) | α e^{θ} 〉] . \end{array}

We now take advantage of X-quadrature homodyne measurement [35, 36] which will project the probe mode to position space. With α real, the state as shown in Eq. (5) will be collapsed to

{| ϕ 〉}_{X} \equiv {〈 X | ϕ 〉}_{3} = \frac{1}{N} [2 f (x, α cos 3 θ) e^{i ϕ_{{(X)}_{1}}} {| φ 〉}_{1} + 2 f (x, α cos θ) e^{i φ_{{(X)}_{2}}} {| φ 〉}_{2}],

where

\begin{array}{l} f (x, β) & = exp [- \frac{1}{4} {(x - 2 β)}^{2}] / {(2 π)}^{1 / 4}, \\ ϕ_{{(X)}_{1}} & = α sin 3 θ (x - 2 α cos 3 θ) Mod 2 π, \\ ϕ_{(X) 2} & = α sin θ (x - 2 α cos θ) Mod 2 π, \\ {| φ 〉}_{1} & = \frac{1}{2} ({| S 〉}_{a} {| F 〉}_{b} {| F 〉}_{c} {| F 〉}_{d} + {| F 〉}_{a} {| S 〉}_{b} {| F 〉}_{c} {| S 〉}_{d} + {| F 〉}_{a} {| F 〉}_{b} {| S 〉}_{c} {| F 〉}_{d} + {| F 〉}_{a} {| F 〉}_{b} {| F 〉}_{c} {| S 〉}_{d}), \\ {| φ 〉}_{2} & = \frac{1}{2} ({| F 〉}_{a} {| S 〉}_{b} {| S 〉}_{c} {| S 〉}_{d} + {| S 〉}_{a} {| F 〉}_{b} {| S 〉}_{c} {| S 〉}_{d} + {| S 〉}_{a} {| S 〉}_{b} {| F 〉}_{c} {| S 〉}_{d} + {| S 〉}_{a} {| S 〉}_{b} {| S 〉}_{c} {| F 〉}_{d}), \\ N & {[4 f^{2} (x, α cos 3 θ) + 4 f^{2} (x, α cos θ)]}^{1 / 2} . \end{array}

With the homodyne measurement of the state of Eq. (5), the conversion from a four-photon GHZ state to a four-photon W state is achieved deterministically, with the probability of success of 1.0.

3. Discussion

In this section, we first discuss the effects of X-quadrature homodyne measurement on the fidelity of the W state and the experimental feasibility of the proposed scheme. To qualify the performance of the present scheme, we plot the effects of homodyne measurement on the fidelity of the converted W state, as shown in Fig. 3. One can see from Fig. 3(a) that the fidelity of the W state increases with the increase of the amplitude of coherent state α and the increase of the phase shift angle θ, here we set δ = 10. In Fig. 3(b), we can see that the fidelity of the W state increases with the increase of the amplitude of coherent state α, but decreases with increasing the distance between the measurement result and the peak δ, here we set θ = 0.01. Figure 3(c) shows that the fidelity of the W state increases with the increase of the phase shift angle θ, while decreases with the increase of the distance between the measurement result and the peak δ, here we set α = 40000. In the following we consider f(x, αcosnθ) (n = 1, 3) as a function of the position component x, and the curves of f(x, αcosnθ) is shown in Fig. 4. It is totally to see that f(x, αcosnθ) are Gaussian curves with the peak located at 2αcosnθ. The distance of the two peaks are 2d = 2α(cosθ − cos3θ), which are commonly reported as the distinguish abilities of the homodyne measurement [37]. We make a detailed analysis that the result of the homodyne measurement x is near the peak of 2 f(x, αcos3θ). Without loss of generality, we consider that the result of measurement is x₁ = 2αcos3θ + δ₁. In this case, the W state in Eq. (6) will be projected to |φ〉₁, with the fidelity

F_{(X_{1})} = | {}_{1}{〈 φ | ϕ 〉}_{X} | \approx \frac{1}{\sqrt{1 + e^{- 2 d_{1} (d_{1} - δ_{1})}}},

from which one could calculate the fidelity easily if the value of α and θ are given out. For example, setting θ = 0.01 and α = 40000, it can be obtained that F_(x₁) = 0.9999 with δ₁ ≈ 0.9958d₁.

Fig. 3 (a) The effects of the homodyne measurement on the fidelity of the W state versus the amplitude of coherent state α and the distance between the measurement results and the phase shift angle θ, here we set δ = 10. (b) The effects of the homodyne measurement on the fidelity of the W state versus the amplitude of coherent state α and the distance between the measurement results and the peak δ, here we set θ = 0.01. (c) The effects of the homodyne measurement on the fidelity of the W state versus the phase shift angle θ and the distance between the measurement results and the peak δ, here we set α = 40000.

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Fig. 4 Curves of the homodyne measurement result X with α = 40000 and θ = 0.01.

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The small overlap between two adjacent Gaussian cures is shown in Fig. 4, f(x, αcosθ) and f(x, α cos3θ), amount to the error rate of the homodyne measurement. The error probability is given by $P_{error} = \frac{1}{2} erfc ([d / 2 \sqrt{2}])$ [36], here d is the distance between the adjacent curves. setting α = 40000 and θ = 0.01, P_error with an approximate value equals to 10⁻¹⁶, which indicates that the probability of success approaches to unity. This shows that it is still possible to operate in the regime of weak cross-Kerr nonlinearity.

In what follows, we analyze and discuss the decoherence in our scheme based on the weak cross-Kerr nonlinearity. In ideal circumstances, the output state of the W state should be the pure entangled state. With the decoherence effects in the nonlinear media, however, the realistic outcome state will be a mixed state. There are two reasons for the photon losses, which may occur both in the probe filed (p) and in the signal modes (1, 2, 3, and 4). One can see that the probability of the photon losses in signal modes becomes lower with the increase of the initial amplitude because the interaction time t = θ/χ becomes shorter. Therefore, it is necessary to consider the factor of losing photons in the probe filed mode. The master equation of decoherence effects for describing dissipation is as follows [38]

\frac{\partial ρ}{\partial t} = \hat{J} ρ + \hat{L} ρ, \hat{J} ρ = γ a ρ a^{†}, \hat{L} ρ = - \frac{γ}{2} (a^{†} a ρ + ρ a^{†} a) .

where a and a^† denote the annihilation and creation operators of the coherent state, γ is the decay constant, and ρ represents the density matrix of the system. The formal solution of the master equation (9) can be described as ρ(t) = D̃(t)ρ(0), where

\tilde{D} (t) = exp [(\hat{J} + \hat{L}) t] = exp (\hat{L} t) exp [\frac{\hat{J}}{γ} 1 - e^{- γ t}]

is the decoherence superoperator. We should note that the decoherence process (D̃(t)) caused by the nonlinear interaction between photons and the coherent state are companied with a unitary evolution (Ũ(t)), which can be given by Ũ(t)ρ (0)= U(t)ρ (0)U(t)^†, where U(t) is the unitary superoperator. We assume that the Ũ and D̃ alternately and continuously execute on the system with a short time Δt = t/N, i.e., the interval t is divided into N (N ∼ ∞) parts. As mentioned above, we consider the evolution of the initial state of the system |ϕ〉₁, and the system evolves to [39–41]

\begin{array}{l} ρ (t) & = {[\tilde{D} (Δ t) \tilde{U} (Δ t)]}^{N} ρ (0) \\ = \sum_{j, k = 1, 3} e^{B_{j, k}} (| ϕ_{j} 〉 〈 ϕ_{k} |) \otimes | A a e^{i j θ} 〉 〈 A a e^{i k θ} |, \end{array}

with

A = - γ t / 2, B_{j, k} = α^{2} (1 - A^{\frac{2}{N}}) \sum_{n = 1}^{N} A^{\frac{2 (n - 1)}{N}} [e^{i (J - k) \frac{n θ}{N}} - 1] .

where the j, k represent the multiple of θ. It is clear from the Eq. (10) that the photon losses in the probe filed mode reduce its amplitude and also lead the original pure state to a mixed state, with the homodyne measurement of the dissipated coherence state, which can be described by

| x 〉 ρ (t) 〈 x | = \sum_{j, k = 1, 3} C_{j, k} (| ϕ_{j} 〉 〈 ϕ_{k} |) f (x, j θ) f (x, k θ) e^{i [μ_{j k} + φ (x, j θ) - φ (x, k θ)]},

where

\begin{array}{l} C_{j, k} & = e^{Re (B_{j k})}, μ_{j k} = Im (B_{j k}), \\ f (x, n θ) & = exp [- \frac{1}{4} {(x - 2 A α cos n θ)}^{2})] / {(2 π)}^{1 / 4} (n = 1, 3), \\ φ (x, n θ) & = (x - 2 A α cos n θ) A α sin n θ (n = 1, 3) . \end{array}

Then we consider the partial overlap between two adjacent Gaussian curves inducing a measurement error rate and the fidelity of the output state, which is similar to that discussed earlier. Here the decoherence coefficient A should be taken into account, as shown in Fig. 5. In Fig. 5(a), we plot the measurement error rate P_error versus the decoherence coefficient A and the amplitude α of the coherent state. It would seem obviously that the measurement error rate decreases with the increase of the the decoherence coefficient A and the amplitude α of the coherent state, where we set θ = 0.01. If setting α = 40000, A = 0.9, P_error ≈ 10⁻¹³; α = 40000, A = 0.5, P_error ≈ 10⁻⁵, and the two measurement outcomes can be distinguished with the error rate less than 10⁻⁴. Figure 5(b) shows the fidelity which are the function of the decoherence coefficient A and the amplitude α of the coherent state, where we set δ = 0.1d₁, θ = 0.01. One can see that the fidelity of the process of X-quadrature homodyne measurement increases with the increase of the decoherence coefficient A and the amplitude α of the coherent state. By calculation, we obtain that α = 40000, A = 0.5, F = 0.9999; α = 60000, A = 0.3, F = 0.9999, that is to say, the amplitude α of the coherent state can make up for the influence from the decoherence coefficient A.

Fig. 5 (a) The measurement error rate versus the decoherence coefficient A and the amplitude α of the coherent state, here we set θ = 0.01. (b) The fidelity of the output state versus the decoherence coefficient A and the amplitude α of the coherent state, where we set the δ = 0.1d₁, θ = 0.01.

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On the other hand, the feasibility of the present scheme depends on the cross-Kerr medium, X-quadrature homodyne measurement, and linear optical elements. In practice, the nonlinearity of cross-Kerr medium is extremely small with θ < 10⁻¹⁸ [42, 43]. Recently, it has been suggested that the Kerr nonlinearity can be improved to θ ≈ 10⁻¹² with electromagnetically induced transparency (EIT) [24–27], which could be effectively used in our scheme compensated by using a strong coherent pulse. X-quadrature homodyne measurement, which is also a standard tool of continuous variable experiment, has considerably high efficiency [44]. However, during the course of considering the experiment, many factors will affect the experimental performance. Such as self-phase modulation (SPM), dispersion, molecular vibrations in Kerr media, etc. In Ref. [45], Shapiro et al. analyzed the phase noise on Kerr medium detailedly. SPM can be suppressed by operating in the slow-response regime, so phase noise is caused mainly by coupling to localized noise oscillators.

4. Conclusion

In conclusion, we have proposed a deterministic scheme for converting the four-photon GHZ state to W state by using weak cross-Kerr nonlinearities, linear optics elements, and homodyne measurement. We also investigate the effects of the quadrature homodyne measurements on the fidelity of the W state and analyze the experimental feasibility of the proposed scheme. This may be useful to future optical QIP.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11264042, 11465020, 61465013, 11564041, and the Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project under Grant No. 20160519022JH.

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Deterministic conversion of a four-photon GHZ state to a W state via homodyne measurement

Abstract

1. Introduction

2. Converting a four-photon GHZ state to a W state

3. Discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (13)

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