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Entanglement swapping and testing quantum steering into the past via collective decay

We propose a scheme to realize entanglement swapping via superradiance, entangling two distant cavities without a direct interaction. The successful Bell-state-measurement outcomes are performed naturally by the electromagnetic reservoir, and we show how, using a quantum trajectory method, the non-local properties of the state obtained after the swapping procedure can be verified by the steering inequality. Furthermore, we discuss how the unsuccessful measurement outcomes can be used in an experiment of delayed-choice entanglement swapping. An extension of testing the quantum steering inequality with the observers at three different times is also considered


I. INTRODUCTION
Entanglement swapping 1 is a procedure to create entanglement between two qubits which have never directly interacted with each other, and has been demonstrated experimentally by Pan et al. 2 . Together with quantum memories 3 , one can, in principle, use entanglement swapping to build quantum repeaters 4 to overcome the decoherence problem in quantum communication 5 . In fact, entanglement swapping can also be viewed as a special example of quantum teleportation 6 if the unknown state is replaced by an entangled state 7 . Very recently, the notion of delayed-choice entanglement swapping was experimentally demonstrated by Ma et al. 8 , an idea first considered in Peres' 9 and Cohen's 10 gedanken experiments. The intriguing feature of this mechanism is that the entanglement is made 'a posteriori', after the entangled qubits have been measured.
Another route to create entanglement between two qubits is to make use of the common environment 11 , i.e. via which they can effectively interact. Taking this one step further, Chen et al. 12 combined the concept of teleportation and a common environment and proposed a method to teleport charge qubits via a common reservoir using the super-radiance effect. The necessary Bellstate measurements are performed naturally by the collective decay, i.e. the sub-and superradiance channels 13 . An analysis of the fidelity and the success probability was further examined 14 using the quantum trajectory method.
In this work, we propose a scheme to accomplish entanglement swapping via collective decay, effectively swapping the entanglement between two atoms into entanglement between two distant cavities. The steering inequality 15 is utilized to verify the non-local properties of the final state obtained between the two distant cav- First, the entanglement is generated between the j-th atom and its corresponding cavity. The next step is to trap the two atoms, so that they decay collectively. If the measurement outcome is a single photon, the entanglement swapping can be achieved.
ities. We further point out that the unsuccessful outcomes, naturally occurring from the superradiance effect we employ here, may be useful in a delayed-choice entanglement-swapping experiment. Finally, we consider the scenario of quantum steering into the past when the observers perform their measurements at three different times.

II. ENTANGLEMENT SWAPPING VIA SUPERRADIANCE
It is well known that highly-entangled states can be 'naturally' generated via collective spontaneous decay 13 . For two identical qubits interacting with a common photon reservoir, separated by a distance shorter than the emitted radiation wavelength, entanglement appears in the two intermediate states, |S 0 = (|+− − |−+ )/ √ 2 and |T 0 = (|+− + |−+ )/ √ 2, of the two decay channels 16 from the excited state |T 1 = |++ to the ground state |T −1 = |−− . Armed with this, we now propose how to accomplish the entanglement swapping protocol through such collective decay processes. We first consider two identical two-level atoms each passing through a separate cavity, as shown in Fig. 1. At this stage the atom-cavity system 1 and atom-cavity system 2 are well separated from each other. In the strong atom-cavity coupling regime the interactions between the atoms and their respective cavities can be written as 17 where j = 1 or 2, g j is the atom-cavity coupling strength, b ± j and the Pauli matrices σ j,± are the cavity photon and the atom operators respectively. With the appropriate preparation of the initial states and control of the passage times through the cavities, the singlet entangled states are prepared between the j-th atom and its corresponding cavity. Here, |0 c,j (|1 c,j ) and |+ j (|− j ) represent the j-th cavity with no (one) photon and the j-th atom in its excited (ground) state, respectively. Our goal is to use these entangled atom-cavity states and the phenomena of collective decay, as well as postselection, to perform entanglement swapping and entangle the photonic states of the two distant cavities. To achieve this, the next step is to remove the atoms from the cavities and trap the two atoms close together so that they experience superradiant collective-decay processes in the common electromagnetic environment. This is a collective decay phenomenon which is enhanced when the inter-atom distance is much shorter than the wavelength of the emitted photon. One can also have a similar enhanced effect by placing the two atoms at the anti-nodes of a cavity 11 . Before the collective decay occurs, the total wavefunction of the combined atom-cavity systems can be written as Assuming that all photonic decay processes from the two atoms can be observed (inefficient detection is introduced below), there are four possible outcomes due to the collective decay: zero photons emitted (|T −1 12 ), two photons emitted (|T 1 12 ), and finally just one photon via sub-radiant channel (|S 0 12 ), or one photon via superradiant channel (|T 0 12 ). If the measurement outcome is a single photon, i.e. via either |S 0 12 or |T 0 12 , then entanglement swapping can be achieved, provided that the sub-and super-radiant decay can be distinguished. As pointed out in Ref. [12], the momentum of the emitted photon − → k depends on the separation of the two atoms − → r , i.e. − → k · − → r = 0 or π corresponds to the emission of a super-or sub-radiant photon, respectively. Therefore, to distinguish between the sub-and super-radiant photons one can place the detectors at the appropriate angles.
One can use the recently proposed steering inequality 15 or Bell-CHSH inequality 18 to verify the non-local properties of the state obtained after the swapping procedure. The density operator of the state obtained after the postselection measurement can be easily calculated by using the quantum trajectory method 14 , where it can be described as a probabilistic mixture of different measurement outcomes: where i denotes the events, or measurements, of photodetection and |ψ i (t) is the pure state conditioned on this event. For example, the total master equation of the evolution of two two-atoms, and their collective decay phenomena, without post-selection, can be written as H is any Hermitian Hamiltonian evolution (e.g., dipoledipole interactions between the atoms, or external magnetic fields). Here we neglect such terms and set H = 0. The photon-emission-event operators are We assume the two atoms are separated by a distance d and that the wavelength of the emitted light is λ. Thus, γ is the spontaneous emission rate for a single atom and Γ = sin(2πd/λ) (2πd/λ) γ. To obtain the post-selected state, one can unravel the evolution (see Ref. [14] for a full description of a similar derivation for a teleportation scheme driven by superradiance), and then the un-normalized state of the two-atoms is given by: with the effective non-Hermitian Hamiltonian written as As pointed out in Ref. [14], the sub-and super-radiant outcomes can also be distinguished in a statistical sense.
In this case, one notes that the state |S 0 12 (|T 0 12 ) favors the longer (shorter) emission times. Following the derivation in Ref. [14], one can define a crossover time t * 1 , such that the swapped state should be corrected (i.e., |0 c,2 |1 c,1 + |1 c,2 |0 c,1 → |0 c,2 |1 c,1 − |1 c,2 |0 c,1 ) for a single photon emission at time t 1 < t * 1 . For t 1 > t * 1 , there is no need to correct the swapped state. We now assume that after the atoms are brought together one waits a time T and retains the state when exactly one photon, emitted by the atoms, was detected in the period T (with detection efficiency η). After averaging over all such detection events (distinguished by the statistical scheme mentioned above), and tracing out the atomic states 14 , the remaining two-cavity state is Here, To test the non-local properties of the resultant cavity state ρ c (η, T ), we use both the steering inequality 15 and the maximum value of a Bell inequality violation 18 . For the steering inequality, the correlated measurements observed by Alice and Bob on the two cavities are described by the probability distribution P (B i = b, A i = a) with b = ±1,or 0, and a = ±1. If the two cavities are not entangled, the steering inequality is written as 15 where N (= 2 or 3) is the number of mutually-unbiased measurements that Alice implements on her qubit (cavity), and with Alice's expectation value for a measured (conditioned on Bob's result) defined as (17) For the maximum value of a Bell inequality violation, the Bell operator associated with the CHSH inequality has the following form 18 , where σ i are the standard Pauli matrices. If the two cavities are not entangled, the CHSH inequality of the two-cavity state ρ obeys The maximum value of the CHSH inequality is given by Thus, violations of the steering parameter S N or the maximum value of the CHSH inequality B max can mean that the resultant cavity state is entangled. In Fig. 2(a), we plot the steering parameter S N and the maximum value of the CHSH inequality B max for various photon detection efficiencies. When decreasing the photon detection efficiency η below 0.79, the steering inequality (dashed curves) is still violated while the Bell-CHSH one (solid curves) is not. For the inter-atomic distance d, the range for the violation of the steering parameter is larger than that for the mean value of the Bell observable. In Fig. 2(b), we also plot S N and B max as a function of the waiting time T . As seen, if the waiting time T is too short, it is possible that the inequalities are not violated. This coincides with the results obtained in Ref. [14]: The longer the waiting time T , the higher the success probability. Besides, the results in Fig. 2(a) and (b) all show that the steering inequality has better tolerance in examining the non-local properties of the entangled states 15 .

III. DELAYED-CHOICE ENTANGLEMENT SWAPPING
From Eq. (3), naively one knows there is a 50% chance that this protocol may fail, i.e., one may obtain the outcomes |T −1 12 or |T 1 12 . This drawback is actually useful if one wishes to implement delayed-choice entanglement swapping. To illustrate this, let us start with Peres' 9 original gedanken experiment. The joint state of a pair of singlets (particles a and b and particles c and d) takes the form where |ψ + ad = (|↑ a |↓ d + |↓ a |↑ d )/ √ 2 and |φ ± ad = (|↑ a |↑ d ± |↓ a |↓ d )/ √ 2 are the symmetric Bell states (and likewise for particles b and c). In the normal scheme of delayed-choice entanglement swapping, particles a and d are sent to Alice and Bob. The third observer, Eve, performs the Bell-state measurement on particles b and c after Alice and Bob have measured the values of their spin components (randomly chosen along arbitrary directions). With Eq. (22), Alice and Bob can then sort their data into four subsets according to the measurement outcomes of Eve. If Alice and Bob test the Bell's inequality with only the data in one of the subsets, they would find the inequality is readily violated.
Very recently, Ma et al. 8 experimentally demonstrated the delayed-choice entanglement swapping protocol using photons. In their experiment, instead of choosing all of the four Bell-state measurements, they performed measurements which either project onto entangled states or onto the separable states (|↑ a |↑ d or |↓ a |↓ d ). This allows them to a posteriori decide on whether Alice and Bob's states are entangled or separable. Returning to our entanglement swapping scheme in Eq. (3), although there is a 50% chance that the swapping may fail, the unsuccessful results (|T −1 12 and |T 1 12 ) are just those that project the particles into the separable states 8 , and therefore can be used in the experiment of delayed-choice entanglement swapping. An additional advantage here, over that of a scheme based on purely photonic degrees of freedom, is that we can make use of other internal metastable states of the atoms to postpone the collective decay (Fig. 3). When the atoms b and c are trapped, laser pulses can be applied to store the information in metastable states with a much longer dephasing/decay time. After Alice and Bob have performed their measurements on their own particles (which here are now the photonic degrees of freedom in the cavities) at time t 1 , the reverse pulses are applied to the atoms b and c to continue the collective decay at a later time t 2 .

IV. TESTING QUANTUM STEERING INTO THE PAST
In our scheme, the Bell measurements are made 'automatically' by the common photon reservoir. Therefore, the delayed choice is intrinsically random, i.e. in the stochastic sense of spontaneous decay. However, a loophole may arise when Alice and Bob happen to choose their measurements both along the z-axis. If FIG. 5: (Color online) A simplified version of testing quantum steering into the past. After the atom 1 is trapped and the information is stored in its metastable state, one then performs a measurement on cavity c1 at time t1. The next step is to reset the cavity and let atom 2 pass through it. When atom 2 arrives at the trap, the reverse pulse is applied to atom 1, so that a collective decay can occur at time t2. Then, the measurement is performed on the cavity c2 at time t3.
their measurement outcomes are the same (|0 c,1 |0 c,2 or |1 c,1 |1 c,2 ), from Eq. (3) they would know in advance that the later joint-measurement settings can only be the separable ones, and could never be the entangled ones. To overcome this, the relative temporal order of the three observers should be changed: t 1 (Alice) < t 2 (Eve) < t 3 (Bob) as shown in Fig. 4. In this case, even if Alice chooses her setting along the z-axis, the wavefunction of Eq. The later joint atomic-measurement driven by the common photon reservoir can still either project it into the separable state (|T −1 12 ) or the entangled states (|S 0 12 , |T 0 12 ). Actually, such an arrangement can be viewed as a special kind of quantum steering if the atomic regime and Bob's cavity are considered as a black box 15 . This means Bob can steer Alice's state by his measurement, which is conditioned on the random choice of the collective decay. Remember that Alice's measurement is performed before Bob's measurement. Similar to the Bell's test in Ref. [8], one can therefore test 'quantum steering into the past' by using the steering inequality 15 .
The above proposal may also be further simplified by using only one cavity with two atoms. As shown in Fig.  5, one first lets the atom 1 pass through the cavity, and as it does so the cavity plays the role of cavity c1 in the above scheme. After the atom 1 has exited the cavity and is trapped, and its state has been transferred to an internal metastable state, one then performs a measurement on the photonic state in the cavity c1 and records the data. The next step is to 'reset' the cavity and let atom 2 pass through it, so that the cavity now plays the role of the cavity c2. When atom 2 arrives at the trap, the atom 1 is transferred back to its original state so that the collective decay can occur. Then, a measurement is again performed on the cavity c2.
Following the above procedure one can obtain the violation of the steering inequality and may have the illusion of "self-entanglement" since the measurement data is produced by the same cavity. However, this is not the case because when we reset the cavity, it (of course) becomes a new system. The use of a single cavity simplifies a possible experimental implementation and emphasizes the fact that the relative temporal order of the three observers' events is irrelevant 8,9 .

V. SUMMARY
In summary, we have proposed a scheme to accomplish entanglement swapping via superradiance. We outlined how the swapping protocol could be combined with collective decay to entangle two distant cavities. After post-selection and averaging over all single-photon events, the non-local properties of the two-cavity state were analyzed by both the steering and Bell-CHSH inequalities. We found that the steering inequality has a better tolerance in verifying the non-local properties of the swapping state. Furthermore, we have also pointed out that the unsuccessful events in our scheme can be used in a delayed-choice entanglement swapping protocol.