Counterfactual entanglement swapping enables high-efficiency entanglement distribution

Qi Guo; Liu-Yong Cheng; Hong-Fu Wang; Shou Zhang

doi:10.1364/OE.26.027314

1. Introduction

Counterfactual quantum communication is defined as the communication without any particles traveling in the transmission channel [1], which is carried out based on interaction-free measurement [2,3] and chained quantum Zeno effect [4]. Due to the highly counterintuitive fact, more and more people have been paying attention on the counterfactual quantum information processing in recent years, and many counterfactual protocols has been proposed, such as, counterfactual quantum cryptography [5–9], counterfactual quantum information transfer [10–15], counterfactual quantum computation [16–18], and other tasks [19, 20]. We suggested a counterfactual entanglement distribution protocol without transmitting any separated or entangled particles a few years ago [21]. Subsequently, this entanglement distribution protocol was generalized to the multipartite situation [22]. These counterfactual protocols above indicated that relevant nonlocal quantum information tasks can be achieved without exchanging particles among distant quantum nodes. With the development of theoretical research, some progress has also been made in experiment [23–25]. Especially, in 2017, Cao et al. firstly demonstrated the counterfactual communication in experiment [25], which provided preliminary experimental support for theoretical works based on counterfactual communication. Counterfactual quantum information protocols may open mew ways for nonlocal quantum information processing and also cause deep discussion about quantum counterfactuality [15,26–32].

Entanglement swapping is one of the most fascinating processes of quantum mechanics, which indicates that a direct interaction is not necessary for producing entanglement between distant systems [33,34]. The procedure of the entanglement swapping can be described as following. Two pairs of entangled particles are prepared by two distant parties Alice and Bob, respectively. Then one particle from each entangled pairs is sent to Charlie. When Charlie projects the two particles her received onto an entangled state (e.g. Bell measurement), Alice’s and Bob’s particles will be entangled although they have never interacted with each other. In the past few decades, with the development of quantum information technique, entanglement swapping was demonstrated in experiment [35–37], and became the central process in quantum repeaters for overcoming the qubit decoherence in long-range quantum communication [38,39].

On the other hand, researchers have tried to combine entanglement swapping with other quantum effects, whose purpose is not just to propose more counterintuitive entanglement swapping schemes, but also to explore the conditions and methods for the production of quantum entanglement. For example, Peres was inspired by the Wheeler’s delayed-choice experiment to put forward the radical idea of delayed-choice entanglement swapping [40], which predicted entanglement can be “produced a posteriori, after the entangled particles have been measured and may no longer exist”. This gedanken experiment was realized experimentally in 2012 [41]. In 2013, Megidish et al. proposed a entanglement swapping protocol between photons separated temporally [42], i.e., photons have never coexisted, which was aimed at demonstrating that the quantum nonlocality does not apply only to the situation of spacelike separation, but also to the situation of timelike separation.

Here, we try to combine entanglement swapping with counterfactual quantum communication, and construct a counterfactual entanglement swapping protocol. The protocol will demonstrate that two pairs of entanglement separated spatially can be swapped without transmitting any particles, which will be a mind-boggling quantum phenomenon. Moreover, the present protocol is nondestructive, i.e., every particle in entangled states will not be detected, thus two pairs of nonlocal entanglement can be obtained in each entanglement swapping process.

2. The procedure of counterfactual entanglement swapping

The fundamental feature of the counterfactual quantum communication compared with interaction-free measurement is that the absorption or passing of a photon in a transmission channel is controlled by a two-dimension quantum superposition state instead of classical means. Therefore, before starting the entanglement swapping procedure, let us briefly introduce the quantum control device used here [11, 20], i.e., the atom-cavity quantum electrodynamics system, which has been extensively studied theoretically [43, 44] and experimentally [45, 46]. The basic setup diagram is shown in Fig. 1. CM₁ and CM₂ compose a single-side cavity, in which a single ⁸⁷Rb atom is trapped. The ground state |g〉, the lower excited state |e〉, and the upper excited state |u〉 are respectively selected as the states of the ⁸⁷Rb atom |5²S_1/2, F = 1, m_F = 1〉, |5²S_1/2, F = 2, m_F = 2〉, and |5²P_3/2, F = 3, m_F = 3〉. The transition between the states |e〉 and |u〉 is resonant with the cavity mode, and the transition |g〉 ↔ |e〉 is driven by a pair of Raman lasers [45, 47]. According to the input-output theory, the references [45, 46] have shown that, when a photon which is resonant with the empty cavity interacts with the atom-cavity system, it experiences a phase shift depending on the atom-photon coupling strength. For the atom in the state |g〉, the atom-photon coupling strength is 0, thus the photon will be reflected with a π phase shift. For the atom in the state |e〉, the strong coupling leads to a normal-mode splitting [48], so that the photon is directly reflected without π phase shift. By adding a 50:50 beam splitter (BS) and a mirror (MR) outside the cavity, a Michelson interferometer can be constructed. Clearly, if the atom is in the state |g〉, the incoming photon will be absorbed by the detector D; if the atom is in the state |e〉, the photon will return back to the input port. Therefore, whether the photon is absorbed depends on the quantum state of the atom.

Fig. 1 The quantum device for controlling the absorption or passing of a photon. MR: normal mirror. BS: 50:50 beam splitter. D: conventional photon detector. CM₁ and CM₂ compose a single-side cavity.

Download Full Size | PDF

Now we begin to introduce the process of the counterfactual entanglement swapping. The two nonlocal parties Alice and Bob independently possess two entangled atoms, atoms 1 and 2 (3 and 4) in the state ${| ϕ 〉}_{12 (34)} = \frac{1}{\sqrt{2}} ({| g g 〉}_{12 (34)} + {| e e 〉}_{12 (34)})$ belongs to Alice (Bob). We will show that, by the third party Charlie’s operations, atoms 1(2) and 3(4) will be entangled without transmitting particles in channels. The schematic diagram of the scheme is shown in Fig. 2. The nested Michelson-type interferometer is constructed between Charlie and Alice (Bob). The two optical paths SM_1a(b) → MR_1a(b) and SM_1a(b) →cavity form the outer Michelson-type interferometer, and the inner Michelson-type interferometer formed by the two optical paths SM_2a(b) → MR_2a(b) and SM_2a(b) →cavity is inserted in one of the arms of the outer interferometer. Initially, atoms 1 and 3 are placed in the cavities, and Charlie launches horizontally polarized photons 1 and 2 |H〉₁|H〉₂ to enter the nested interferometer.

Fig. 2 Schematic of the counterfactual entanglement swapping. PBS: polarizing beam splitter. SM: switchable mirror. SPR: switchable polarization rotator, where the arrow means SPR can only rotate the photon comes from the SM side. D: conventional photon detector. HWP: half-wave plate oriented at 22.5°. BS: 50:50 beam splitter. OD: optical delay line. AO: absorbing object used to absorb the photon from PBS_2a(b).

Download Full Size | PDF

Therefore, the joint initial state is

| ψ 〉 = \frac{1}{2} {(| g g 〉 + | e e 〉)}_{12} {(| g g 〉 + | e e 〉)}_{34} {| H 〉}_{1} {| H 〉}_{2} .

Because Bob has the same setup with Alice, for simplicity, we only take atom 1, atom 2 and photon 1 (

| φ 〉 = \frac{1}{\sqrt{2}} (| g g 〉 + {| e e 〉)}_{12} {| H 〉}_{1})

to introduce the process in the nested interferometer in the following, and atoms 3, 4 and photon 2 will evolves to the same form. The photon 1 goes toward the nested interferometer via the optical circulator. The switchable mirror SM_1a(b) is initially switched off (transmits the photon), but it remains on (reflects the photon) once the photon enters the interferometer until the photon finishes M cycles in the outer interferometer. The switchable polarization rotator SPR_1a(b) is used to rotate the photon coming from SM_1a(b) by an angle ϑ (ϑ = π/2M), i.e., |H〉 → cos ϑ|H〉 + sin ϑ|V〉 and |V〉 → cos ϑ|V〉 − sin ϑ|H〉. Thus, after the photon 1 passing through SPR_1a, |φ〉 evolves to

| φ 〉 \to \frac{1}{\sqrt{2}} cos ϑ {(| g g 〉 + | e e 〉)}_{12} {| H 〉}_{1} + \frac{1}{\sqrt{2}} sin ϑ {(| g g 〉 + | e e 〉)}_{12} {| V 〉}_{1} .

Then the component |V〉 goes toward the inner interferometer and the component |H〉 goes to the optical delay line OD_1a, so we can only take the second term in Eq. (2),

| ϕ 〉 \equiv \frac{1}{\sqrt{2}} (| g g 〉 + {| e e 〉)}_{12} {| V 〉}_{1}

, to show the N inner cycles. The SM_2a(b) transmits the photon initially, and keeps reflecting the photon during the N inner cycles. SPR_2a(b) has the same action with SPR_1a(b), but the rotating angle is θ = π/2N. So after passing through SPR_2a, the component |ϕ〉 becomes

| ϕ 〉 \to \frac{1}{\sqrt{2}} {(| g g 〉 + | e e 〉)}_{12} (cos θ {| V 〉}_{1} - sin θ {| H 〉}_{1}) .

Due to PBS_3a, the component |V〉 above stays at Charlie’s site, and the component |H〉 will enter Alice’s quantum control device. From the quantum control mechanism in Fig. 1, we know if atom 1 in the state |g〉, the photon will be absorbed by the detector D₁, and for the atom state |e〉, the photon will return back to SM_2a. Therefore, if the photon is not absorbed by D₁, at the end of the first inner cycle, the state becomes

| ϕ 〉 \to \frac{1}{\sqrt{2}} cos θ {(| g g 〉 + | e e 〉)}_{12} {| V 〉}_{1} - \frac{1}{\sqrt{2}} sin θ {| e e 〉}_{12} {| H 〉}_{1} .

Clearly, the state above is not normalized, that’s because it is only the component unabsorbed. Then repeat the process of the inner interferometer for N cycles. If the photon is still not absorbed after the Nth inner cycle, now the state will be

| ϕ 〉 \to \frac{1}{\sqrt{2}} [{cos}^{N} θ {| g g 〉}_{12} {| V 〉}_{1} + cos (N θ) {| e e 〉}_{12} {| V 〉}_{1} - sin (N θ) {| e e 〉}_{12} {| H 〉}_{1}] .

For

θ = \frac{π}{2 N}

,

| ϕ 〉 \to \frac{1}{\sqrt{2}} ({cos}^{N} \frac{π}{2 N} {| g g 〉}_{12} {| V 〉}_{1} - {| e e 〉}_{12} {| H 〉}_{1}) .

Therefore, after the N inner cycles, the whole state of the system can be obtained by substituting the equation above into Eq. (2),

| φ 〉 \to \frac{1}{\sqrt{2}} cos ϑ {(| g g 〉 + | e e 〉)}_{12} {| H 〉}_{1} + \frac{1}{\sqrt{2}} sin ϑ ({cos}^{N} \frac{π}{2 N} {| g g 〉}_{12} {| V 〉}_{1} - {| e e 〉}_{12} {| H 〉}_{1}) .

Then the components from the inner interferometer and MR_1a combine by PBS_2a. The |H〉 component from the inner interferometer will be absorbed by the absorbing object (AO), and other components will return to SM_1a. Therefore, if the photon is not absorbed by D₁ or AO, after the first outer cycle the system state will become

| φ 〉 \to \frac{1}{\sqrt{2}} cos ϑ {(| g g 〉 + | e e 〉)}_{12} {| H 〉}_{1} + \frac{1}{\sqrt{2}} sin ϑ {cos}^{N} \frac{π}{2 N} {| g g 〉}_{12} {| V 〉}_{1} .

Repeat the whole process above M cycles. If the photon still exist, the system state can be expressed as

| φ 〉 \to \frac{1}{\sqrt{2}} (x_{M} {| g g 〉}_{12} {| H 〉}_{1} + y_{M} {| e e 〉}_{12} {| H 〉}_{1} + z_{M} {| g g 〉}_{12} {| V 〉}_{1}),

where the probability amplitudes x_M, y_M, and z_M depend on the following recursion relations

\begin{array}{l} x_{M} = (x_{M - 1} cos ϑ - z_{M - 1} sin ϑ), \\ y_{M} = y_{M - 1} cos ϑ, \\ z_{M} = (z_{M - 1} cos β_{1} + x_{M - 1} sin β_{1}) {cos}^{N} θ, \end{array}

where ϑ = π/2M, θ = π/2N, and the initial conditions are x₁ = y₁ = cos ϑ and z₁ = sin ϑ cos^Nθ. Note that now atoms 3, 4 and photon 2 have also evolved to the same state as Eq. (9),

| φ^{'} 〉 \to \frac{1}{\sqrt{2}} (x_{M} {| g g 〉}_{34} {| H 〉}_{2} + y_{M} {| e e 〉}_{34} {| H 〉}_{2} + z_{M} {| g g 〉}_{34} {| V 〉}_{2}),

Therefore, after M outer cycles, the joint state of the whole system evolves to

\begin{array}{l} | ψ 〉 & \to | φ 〉 \otimes | φ^{'} 〉 \\ = \frac{1}{2} (x_{M}^{2} {| g g 〉}_{12} {| g g 〉}_{34} {| H 〉}_{1} {| H 〉}_{2} + x_{M} y_{M} {| g g 〉}_{12} {| e e 〉}_{34} {| H 〉}_{1} {| H 〉}_{2} + x_{M} z_{M} {| g g 〉}_{12} {| g g 〉}_{34} {| H 〉}_{1} {| V 〉}_{2} \\ + y_{M} x_{M} {| e e 〉}_{12} {| g g 〉}_{34} {| H 〉}_{1} {| H 〉}_{2} + y_{M}^{2} {| e e 〉}_{12} {| e e 〉}_{34} {| H 〉}_{1} {| H 〉}_{2} + y_{M} z_{M} {| e e 〉}_{12} {| g g 〉}_{34} {| H 〉}_{1} {| V 〉}_{2} \\ + z_{M} x_{M} {| g g 〉}_{12} {| g g 〉}_{34} {| V 〉}_{1} {| H 〉}_{2} + z_{M} y_{M} {| g g 〉}_{12} {| e e 〉}_{34} {| V 〉}_{1} {| H 〉}_{2} + z_{M}^{2} {| g g 〉}_{12} {| g g 〉}_{34} {| V 〉}_{1} {| V 〉}_{2}, \end{array}

Now we numerically evaluate the variation trend of the coefficients above with the values of N and M, as shown in Fig. 3. It’s clear that $x_{M}^{2} \to 0$ , $y_{M}^{2} \to 1$ , $z_{M}^{2} \to 1$ x_My_M → 0, x_Mz_M → 0, and y_Mz_M → 1 for large values of N and M, for example, $x_{M}^{2} = 0.0001$ , $y_{M}^{2} = 0.9518$ , $z_{M}^{2} = 0.9616$ , x_My_M = 0.0119, x_Mz_M = 0.0119, and y_Mz_M = 0.9567 for M = 50 and N = 1600. Therefore, with the increase of N and M, Eq. (12) will be approximated as

\begin{array}{l} | ψ 〉 & \to & \frac{1}{2} ({| e e 〉}_{12} {| e e 〉}_{34} {| H 〉}_{1} {| H 〉}_{2} + {| e e 〉}_{12} {| g g 〉}_{34} {| H 〉}_{1} {| V 〉}_{2} \\ + {| g g 〉}_{12} {| e e 〉}_{34} {| V 〉}_{1} {| H 〉}_{2} + {| g g 〉}_{12} {| g g 〉}_{34} {| V 〉}_{1} {| V 〉}_{2}) . \end{array}

When the photons 1 and 2 leave the nested interferometers, Charlie performs parity check on their polarization states. The nondestructive parity measurement has been demonstrated by introducing nonlinear process [49,50] or nondestructive detection [45]. Here we take the photons’ even parity (the same polarization) for the downstream process (odd parity can also been used for the further processing), i.e.,

| ψ 〉 \to \frac{1}{\sqrt{2}} ({| e e 〉}_{12} {| e e 〉}_{34} {| H 〉}_{1} {| H 〉}_{2} + {| g g 〉}_{12} {| g g 〉}_{34} {| V 〉}_{1} {| V 〉}_{2}) .

Then the two photon pass through half-wave plates (HWP_1a(b)), PBS_1a(b), and will be detected by detectors. The action of HWP_1a(b) is to perform the transformations {

| H 〉 \to (1 / \sqrt{2}) (| H 〉 + | V 〉)

,

| V 〉 \to (1 / \sqrt{2}) (| H 〉 - | V 〉)

}. By simple calculation, it can be obtained that if (D_1H, D_2H) or (D_1V, D_2V) click, the atoms’ state will collapse to

| ψ 〉 \to \frac{1}{\sqrt{2}} ({| g g 〉}_{12} {| g g 〉}_{34} + {| e e 〉}_{12} {| e e 〉}_{34}),

if (D_1H, D_2V) or (D_1V, D_2H) click, the atoms’ state will be

| ψ 〉 \to \frac{1}{\sqrt{2}} (- {| g g 〉}_{12} {| g g 〉}_{34} + {| e e 〉}_{12} {| e e 〉}_{34}) .

Take Eq. (15) for example, Alice and Bob respectively perform Hadamard transformations on the atoms 1 and 3 by using the driving laser field, i.e., {

| g 〉 \to (1 / \sqrt{2}) (| g 〉 + | e 〉)

,

| e 〉 \to (1 / \sqrt{2}) (| g 〉 - | e 〉)

. The state becomes

\begin{array}{l} | ψ 〉 \to & \frac{1}{2 \sqrt{2}} ({| g g 〉}_{12} {| g g 〉}_{34} + {| g g 〉}_{12} {| e g 〉}_{34} + {| e g 〉}_{12} {| g g 〉}_{34} + {| e g 〉}_{12} {| e g 〉}_{34} \\ + {| g e 〉}_{12} {| g e 〉}_{34} - {| g e 〉}_{12} {| e e 〉}_{34} - {| e e 〉}_{12} {| g e 〉}_{34} + {| e e 〉}_{12} {| e e 〉}_{34}) . \end{array}

Fig. 3 The coefficients in Eq. (12) versus the different values of N and M. (a) $x_{M}^{2} \to 0$ , (b) x_My_M → 0, (c) x_Mz_M → 0, (d) $y_{M}^{2} \to 1$ , (e) y_Mz_M → 1, (f) $z_{M}^{2} \to 1$ , with the increase of M and N.

Download Full Size | PDF

Then Charlie let horizontally polarized photons 3 and 4 |H〉₃|H〉₄ to enter the nested interferometer. This process is the same with the one of the photon 1 and 2 in the nested interferometer, so the result can be derived by repeating the process from Eq. (1) to Eq. (12) with Eq. (17) as initial condition. For simplicity, we here omit the derivation process. The final state after photons 3 and 4 finishing M outer cycles contains the same coefficients with Eq. (12), thus, according to the analysis in Fig. 3, the state approximately evolves to

\begin{array}{l} | ψ 〉 \to & \frac{1}{2 \sqrt{2}} ({| e g 〉}_{12} {| e g 〉}_{34} {| H 〉}_{3} {| H 〉}_{4} + {| e e 〉}_{12} {| e e 〉}_{34} {| H 〉}_{3} {| H 〉}_{4} \\ + {| g g 〉}_{12} {| e g 〉}_{34} {| V 〉}_{3} {| H 〉}_{4} - {| g e 〉}_{12} {| e e 〉}_{34} {| V 〉}_{3} {| H 〉}_{4} \\ + {| e g 〉}_{12} {| g g 〉}_{34} {| H 〉}_{3} {| V 〉}_{4} - {| e e 〉}_{12} {| g e 〉}_{34} {| H 〉}_{3} {| V 〉}_{4} \\ + {| g g 〉}_{12} {| g g 〉}_{34} {| V 〉}_{3} {| V 〉}_{4} + {| g e 〉}_{12} {| g e 〉}_{34} {| V 〉}_{3} {| V 〉}_{4}) . \end{array}

By now photons 3 and 4 have left the nested interferometer, and Charlie performs parity check on them. Here we also take the even parity for example,

\begin{array}{l} | ψ 〉 \to & \frac{1}{2} ({| e g 〉}_{12} {| e g 〉}_{34} {| H 〉}_{3} {| H 〉}_{4} + {| e e 〉}_{12} {| e e 〉}_{34} {| H 〉}_{3} {| H 〉}_{4} \\ + {| g g 〉}_{12} {| g g 〉}_{34} {| V 〉}_{3} {| V 〉}_{4} + {| g e 〉}_{12} {| g e 〉}_{34} {| V 〉}_{3} {| V 〉}_{4}) . \end{array}

Then photons will go toward HWP_1a(b), PBS_1a(b), and detectors. Straightforwardly, (D_1H, D_2H) or (D_1V, D_2V) clicking means the atoms’ state will collapse to

| ψ 〉 \to \frac{1}{2} ({| g 〉}_{1} {| g 〉}_{3} + {| e 〉}_{1} {| e 〉}_{3}) ({| g 〉}_{2} {| g 〉}_{4} + {| e 〉}_{2} {| e 〉}_{4}),

while, for (D_1H, D_2V) or (D_1V, D_2H) clicking, the state is

| ψ 〉 \to \frac{1}{2} (- {| g 〉}_{1} {| g 〉}_{3} + {| e 〉}_{1} {| e 〉}_{3}) ({| g 〉}_{2} {| g 〉}_{4} + {| e 〉}_{2} {| e 〉}_{4}),

Obviously, entanglement swapping has been achieved. During the whole process, all the photons have never entered the channel between Charlie and Alice (Bob), that’s because if the photon enters the channel, it will be absorbed by D₁₍₂₎ or AO and can not arrive the last detectors. We will demonstrate the probability that a photon enters the channels approaches to 0 for large M and N in next section. Therefore, this is a counterfactual entanglement swapping protocol without transmitting any particles. Moreover, by the counterfactual process, Alice and Bob share two pairs of entanglement, which means this scheme can realize high-efficiency entanglement distribution.

3. Discussion and conclusions

For clarity, in the description of the present scheme above, we have assumed perfect experiment devices and ideal experiment conditions. However, in practice, many imperfections will inevitably affect the performance of the scheme. Now we begin to analyze these influence factors and the experimental feasibility of the counterfactual entanglement swapping. The most significant factor for all the counterfactual quantum information schemes is the number of the outer (inner) cycle M(N). This factor reflects in Eqs. (13) and (18) in this paper, which are approximated results that M(N) is infinite. Therefore, we can evaluate the effect of M(N) on the present scheme by analyzing the probability of obtaining the states in Eqs. (13) and (18) and their fidelity. Take Eq. (13) for example, it is the ideal state, and the real state for finite M(N) is the state of Eq. (12). Thus we can calculate the fidelity of the real state and the probability of obtaining the ideal state. The fidelity is F = [𝒞(y_M + z_M)²/4]², where 𝒞 is the normalized coefficient of Eq. (12), and the success probability is $P = {(y_{M}^{2} + z_{M}^{2})}^{2} / 4$ . Then we numerically evaluate F and P via Eq. (10) with different N and M, and the results are shown in Figs. 4(a) and 4(b), respectively. Clearly, both the fidelity and the success probability are close to 1 for large N and M. For example, F = 0.9992 and P = 0.8870 for N = 1000 and M = 30, and F = 0.9996 and P = 0.9129 for N = 1500 and M = 50. The higher the success probabilities of Eqs. (13) and (18) are, the lower the probability that photons pass through the channels is. Therefore, the probability of photons passing through the channel can be reduced by multiple cycles of the nested interferometer, and the present scheme will be achieved counterfactually.

Fig. 4 The fidelity (a) and the probability (b) of obtaining Eqs. (13) or (18) versus values of outer and inner cycles M and N.

Download Full Size | PDF

The present scheme has high requirement on the precision of switchable polarization rotators. The rotation angles of SPR_1a(b) and SPR_2a(b) should be exactly set according to the values of M and N, respectively, i.e., ϑ = π/2M and θ = π/2N. So the errors introduced by SPRs will directly reduce the performance of the present scheme. For the convenience of discussion, we suppose all the switchable polarization rotators have the same error coefficient s defined as [1] s = MΔϑ/ϑ = NΔθ/θ, where Δϑ(Δθ) indicates the error of the rotation angle that a photon passes through the switchable polarization rotator in outer (inner) interferometer. Therefore, in order to analyze the effect of the error s, we need to replace ϑ(θ) with ϑ + Δϑ(θ + Δθ) during the whole process, then recalculate the fidelity and the success probability. The numerical results are plotted in Figs. 5(a) and 5(b), which show that the performance of the scheme is significantly affected by s when M and N are small, and with the increase of M and N, the influence of s decreases.

Fig. 5 The fidelity (a) and the success probability (b) of the scheme versus the error coefficient s of the SPR for the different values of M and N.

Download Full Size | PDF

Photon loss in transmission channels is another critical factor that affects the performance of the scheme. Especially, the scheme requires photons traveling many times in the nested interferometer, hence it is sensitive to photon loss. To quantitatively analyze this influence factor, we assume two channels in the present scheme have the same photon loss rates γ, which is defined as the probability that the photon is absorbed by other objects in channels rather than D₁₍₂₎ or AO during every inner cycle. Therefore, in an inner cycle from Eq. (3) to Eq. (4), the probability that the photon is not absorbed is $\frac{1}{2} (1 - γ) {sin}^{2} θ$ . Then rederive the whole process in the previous section by considering γ. Through tedious calculations, the fidelity and the success probability containing γ can be obtained, whose numerical results for different γ, M, and N are shown in Figs. 6(a) and 6(b). As expected, both the fidelity and the success probability decrease with the increase of γ. What needs to be specially noted is that the fidelity and the success probability increase with the increase of M and N only for little γ. However, when γ is about greater than 10%, the fidelity and the success probability will reduce rapidly for large M and N, which is the opposite of the effect of the SPR’s error. Therefore, in the practical experiment, the values of M and N should be chosen by considering the trade-off between these two factors. Besides these factors discussed above, the present scheme also requires the strong couple between the cavity and the transition |e〉 ↔ |u〉, which has been deeply studied by using the input-output theory in [43–46]. The the probability of getting a reflected photon from such atom-cavity system has reached 66% in experiment [45], and the coupling strength can be further improved by whispering-gallery-mode microresonator [51].

Fig. 6 The fidelity (a) and the success probability (b) of the scheme versus the photon loss rate γ for the different values of M and N.

Download Full Size | PDF

In conclusion, we have combined counterfactual quantum communication with entanglement swapping to propose a counterfactual entanglement swapping scheme, in which two pairs of entanglement separated spatially can be swapped without transmitting any particles between them. Moreover, through an entanglement swapping process, one can obtain two pairs nonlocal entanglement, i.e., the high-efficiency entanglement distribution is achieved. We have also analyzed the experiment feasibility of the present scheme. The possible negative effects from relevant experimental conditions have been evaluated numerically, and the results showed that the scheme can be implemented with high fidelity and probability. The counterfactual scheme maybe provides an alternative method for entanglement swapping that can be used for quantum repeater in large-scale quantum communication network, and can also be regarded as a further evidence of quantum counterfactuality.

Funding

National Natural Science Foundation of China (11604190, 61465013, 11465020, 11704235, 61801280); Fund for Shanxi “1331 Project” Key Subjects Construction.

References

1. H. Salih, Z. H. Li, M. Al-Amri, and M. S. Zubairy, “Protocol for direct counterfactual quantum communication,” Phys. Rev. Lett. 110, 170502 (2013). [CrossRef] [PubMed]

2. A. C. Elitzur and L. Vaidman, “Quantum mechanical interaction-free measurements”, Found. Phys. 23, 987–997 (1993). [CrossRef]

3. P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-free measurement,” Phys. Rev. Lett. 74, 4763–4766 (1995). [CrossRef] [PubMed]

4. P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-efficiency quantum interrogation measurements via the quantum zeno effect,” Phys. Rev. Lett. 83, 4725–4728 (1999). [CrossRef]

5. T.-G. Noh, “Counterfactual quantum cryptography,” Phys. Rev. Lett. 103, 230501 (2009). [CrossRef]

6. Y. Liu, L. Ju, X. L. Liang, S. B. Tang, G. L. Shen Tu, L. Zhou, C. Z. Peng, K. Chen, T. Y. Chen, Z. B. Chen, and J. W. Pan, “Experimental demonstration of counterfactual quantum communication,” Phys. Rev. Lett. 109, 030501 (2012). [CrossRef] [PubMed]

7. Z. Q. Yin, H. W. Li, W. Chen, Z. F. Han, and G. C. Guo, “Security of counterfactual quantum cryptography,” Phy. Rev. A 82, 042335 (2010). [CrossRef]

8. Z. Q. Yin, H. W. Li, Y. Yao, C. M. Zhang, S. Wang, W. Chen, G. C. Guo, and Z. F. Han, “Counterfactual quantum cryptography based on weak coherent states,” Phy. Rev. A 86, 022313 (2012). [CrossRef]

9. J. L. Zhang, F. Z. Guo, F. Gao, B. Liu, and Q. Y. Wen, “Private database queries based on counterfactual quantum key distribution,” Phys. Rev. A 88, 022334 (2013). [CrossRef]

10. Q. Guo, L. Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual quantum-information transfer without transmitting any physical particles,” Sci. Rep. 5, 8416 (2015). [CrossRef] [PubMed]

11. Z. H. Li, M. Al-Amri, and M. S. Zubairy, “Direct counterfactual transmission of a quantum state,” Phys. Rev. A 92, 052315 (2015). [CrossRef]

12. N. Gisin, “Optical communication without photons,” Phys. Rev. A 88, 030301 (2013). [CrossRef]

13. D. R. M. Arvidsson-Shukur and C. H. W. Barnes, “Quantum counterfactual communication without a weak trace,” Phys. Rev. A 94, 062303 (2016). [CrossRef]

14. Z. H. Li, M. Al-Amri, and M. S. Zubairy, “Direct quantum communication with almost invisible photons,” Phys. Rev. A 89, 052334 (2014). [CrossRef]

15. D. R. M. Arvidsson-Shukur, A. N. O. Gottfries, and C. H. W. Barnes, “Evaluation of counterfactuality in counterfactual communication protocols,” Phys. Rev. A 96, 062316 (2017). [CrossRef]

16. O. Hosten, M. T. Rakher, J. T. Barreiro, N. A. Peters, and P. G. Kwiat, “Counterfactual quantum computation through quantum interrogation,” Nature (London) 439, 949–952 (2006). [CrossRef]

17. L. Vaidman, “Impossibility of the counterfactual computation for all possible outcomes,” Phys. Rev. Lett. 98, 160403 (2007). [CrossRef] [PubMed]

18. F. Kong, C. Ju, P. Huang, P. Wang, X. Kong, F. Shi, L. Jiang, and J. Du, “Experimental realization of high-efficiency counterfactual computation,” Phys. Rev. Lett. 115, 080501 (2015). [CrossRef] [PubMed]

19. A. Shenoy-Hejamadi and R. Srikanth, “Counterfactual distribution of Schrödinger cat states,” Phys. Rev. A 92, 062308 (2015). [CrossRef]

20. Q. Guo, S. Zhai, L.-Y. Cheng, H.-F. Wang, and S. Zhang, “Counterfactual quantum cloning without transmitting any physical particles,” Phys. Rev. A 96, 052335 (2017). [CrossRef]

21. Q. Guo, L. Y. Cheng, L. Chen, H. F. Wang, and S. Zhang, “Counterfactual entanglement distribution without transmitting any particles,” Opt. Express 22, 8970–8984 (2014). [CrossRef] [PubMed]

22. Y. Chen, X. Gu, D. Jiang, L. Xie, and L. Chen, “Tripartite counterfactual entanglement distribution,” Opt. Express 23, 21193–21203 (2015). [CrossRef] [PubMed]

23. G. Brida, A. Cavanna, I.P. Degiovanni, M. Genovese, and P. Traina, “Experimental realization of counterfactual quantum cryptography,” Laser Phys. Lett. 9, 247–252 (2012). [CrossRef]

24. C. Liu, J. Liu, J. Zhang, and S. Zhu, “Improvement of reliability in multi-interferometer-based counterfactual deterministic communication with dissipation compensation,” Opt. Express 26, 2261–2269 (2018). [CrossRef] [PubMed]

25. Y. Cao, Y.-H. Li, Z. Cao, J. Yin, Y.-A. Chen, H.-L. Yin, T.-Y. Chen, X. Ma, C.-Z. Peng, and J.-W. Pan, “Direct counterfactual communication via quantum Zeno effect,” Proc. Natl. Acad. Sci. USA 114, 4920 (2017). [CrossRef] [PubMed]

26. L. Vaidman, “Comment on ‘Protocol for direct counterfactual quantum communication’,” Phys. Rev. Lett. 112, 208901 (2014). [CrossRef]

27. H. Salih, Z. H. Li, M. Al-Amri, and M. S. Zubairy, “Salih et al. Reply,” Phys. Rev. Lett. 112, 208902 (2014). [CrossRef]

28. L. Vaidman, “Counterfactuality of ‘counterfactual’ communication,” J. Phys. A: Math. Theor. 48, 465303 (2015). [CrossRef]

29. L. Vaidman, “Comment on ‘Direct counterfactual transmission of a quantum state’,” Phys. Rev. A 93, 066301 (2016). [CrossRef]

30. Y. Aharonov and L. Vaidman, “Modification of ‘Counterfactual communication protocols’ which makes them truly counterfactual,” arXiv:1805.10634 (2018).

31. H. Salih, W. McCutcheon, and J. Rarity, “Do the laws of physics prohibit counterfactual communication?” arXiv:1806.01257 (2018).

32. H. Salih, “From a quantum paradox to provably counterfactual disembodied transport: counterportation,” arXiv:1807.06586 (2018).

33. M. Zukowski, A. Zeilinger, M. A. Horne, and A. Ekert, “‘Event-Ready-Detectors’ Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993). [CrossRef] [PubMed]

34. S. Bose, V. Vedral, and P. L. Knight, “Multiparticle generalization of entanglement swapping,” Phys. Rev. A 57, 822–829 (1998). [CrossRef]

35. J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998). [CrossRef]

36. A. M. Goebel, C. Wagenknecht, Q. Zhang, Y.-A. Chen, K. Chen, J. Schmiedmayer, and J.-W. Pan, “Multistage entanglement swapping,” Phys. Rev. Lett. 101, 080403 (2008). [CrossRef] [PubMed]

37. E. Megidish, T. Shacham, A. Halevy, L. Dovrat, and H. S. Eisenberg, “Resource efficient source of multiphoton polarization entanglement,” Phys. Rev. Lett. 109, 080504 (2012). [CrossRef] [PubMed]

38. H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932 (1998). [CrossRef]

39. L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London) 414, 413–418 (2001). [CrossRef]

40. A. Peres, “Delayed choice for entanglement swapping,” J. Mod. Opt. 47, 139–143 (2000). [CrossRef]

41. X.-s. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, C. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nat. Phys. 8, 479–484 (2012). [CrossRef]

42. E. Megidish, A. Halevy, T. Shacham, T. Dvir, L. Dovrat, and H. S. Eisenberg, “Entanglement swapping between photons that have never coexisted,” Phys. Rev. Lett. 110, 210403 (2013). [CrossRef] [PubMed]

43. L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004). [CrossRef] [PubMed]

44. J. Cho and H.-W. Lee, “Generation of atomic cluster states through the cavity input-output process,” Phys. Rev. Lett. 95, 160501 (2005). [CrossRef] [PubMed]

45. A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349 (2013). [CrossRef] [PubMed]

46. A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature (London) 508, 237–240 (2014). [CrossRef]

47. D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett. 96, 063001 (2006). [CrossRef] [PubMed]

48. A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the vacuum rabi spectrum for one trapped atom,” Phys. Rev. Lett. 93, 233603 (2004). [CrossRef] [PubMed]

49. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-not gate,” Phys. Rev. Lett. 93, 250502 (2004). [CrossRef]

50. P. Kok, W. J. Munro, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]

51. C. Junge, D. O’Shea, J. Volz, and A. Rauschenbeutel, “Strong coupling between single atoms and nontransversal photons,” Phys. Rev. Lett. 110, 213604 (2013). [CrossRef] [PubMed]

Counterfactual entanglement swapping enables high-efficiency entanglement distribution

Abstract

1. Introduction

2. The procedure of counterfactual entanglement swapping

3. Discussion and conclusions

Funding

References

Cited By

Figures (6)

Equations (21)

Optics Express