Heralded and high-efficient entanglement concentrations based on linear optics assisted by time-delay degree of freedom

Entanglement concentration is a critical technique to prevent degraded fidelity and security in long-distance quantum communication. We propose novel practical entanglement concentration protocols (ECPs) for less-entangled Bell and Greenberger-Horne-Zeilinger states with unknown parameters by solely using simple linear optics. We avoid the need for the post-selection principles or photon-number-resolving detector to identify the parity-check measurement completely by orchestrating auxiliary time degree of freedom, and the success of ECPs is exactly heralded by the detection signatures without destroying the incident qubits. Additionally, the outting incident photons kept are in the maximally entangled or the less-entangled state, and the success probability can be increased by recycling the latter. The heralded and the basic linear optical elements make our practical ECPs are accessible to experimental investigation with current technology.

Parameter-splitting is the current optimal strategy to implement ECP for less-entangled states with known parameters [39]. Post-selection principles are necessary for existing linear optical ECP for unknown less-entangled states [27][28][29]32] as polarizing beam splitters (PBSs) are employed to pick up the desired instances in which each of the spatial contains exactly one photon. The destructive photon-number-resolving detectors can be used to discard the case that the photon pair coincidence at one spatial. However, such sophisticated detectors are not likely to be available with current technology, which makes the linear optical ECPs cannot be accomplished simply. In addition, the recycling strategies are only introduced to increase the success probability of the cross-Kerr-based ECPs [31,32]. Hence, it is significant to investigate heralded and recyclable ECPs for partially entangled states without post-selection or photon-number-resolving detectors.
In this paper, we first present a heralded ECP for unknown less-entangled Bell states resorting to linear optical elements. Compared to the previous schemes, we avoid the need for photon-number-resolving detectors or postselection principles by introducing the time-delay degree of freedom (DOF). Our scheme is heralded unambiguously by the detection signatures, which makes our ECP much more practical. The incident photons where distillation fails are kept in the less-entangled Bell state, and employing the recycling strategies can improve the success probability of concentration from 0.5 to 0.75 in principle. Only the probability of approaching the target state in an open quantum system can reach unity by iteration due to quantum anti-Zeno effect [46,47]. Moreover, the program is also available for ECP for multi-photon less-entangled unknown GHZ states, and the schemes are later designed in detail. The presented architectures for ECPs with linear optics can be exactly realized with current experimental technology.

II. HERALDED ECP FOR UNKNOWN BELL STATE WITH LINEAR OPTICS
In the section, we present a heralded ECP for two-photon polarization less-entangled Bell states with unknown parameters using linear optics. By introducing the time-delay DOF to the detected photons, our ECP can be exactly heralded by the detection signatures. The entanglement concentration process does not rely on post-selection principle. Suppose two maximally entangled Bell states |φ AB and |φ A B are generated initially from S 1 and S 2 , respectively. Here The state of four-photon system composed of photons A, B, A , and B can be described as Then as shown in Fig. 1, photons A and B immediately pass through a 50:50 beam splitter (BS), resulting in the following transformations where |Γ represents the polarization state |H or |V . Considering Eq. (3), after passing the BS, |Φ 0 is transformed into Subsequently, photons A and A (B and B ) of the state |Φ 1 is sent to Alice (Bob), and owing to the noisy channels, |Φ 1 may decay to a partially less-entangled state where the unknown parameters α and β satisfy the normalization relation |α| 2 + |β| 2 = 1.
In order to distill the maximally entangled Bell state (|HH + |V V )/ √ 2 from |Φ 2 , the two distant parties, Alice and Bob, need to complete the operations shown in Fig. 1. To describe this process more clearly, combined with the Hong-Ou-Mandel effect, we rewrite Eq. (5) in the following normalized form Specifically, Alice flips the state of photon A by using a half-wave plate oriented at 45 • . That is, HWP 45 • completes the transformations |H Nextly, by using an unbalanced interferometer consisting of two PBSs, Alice (Bob) introduces time-delays t 0 and t 1 to the H-and V -polarization components of photon A (B ), respectively. Here t 0 and t 1 satisfy ω(t 0 − t 1 ) = 2nπ, where n is the nonzero integer. And then, the state |Φ 3 becomes After that, Alice (Bob) performs a Hadamard operation on photon A (B ) with a half wave plate (HWP). That is, HWP completes the transformations Those rotations convert |Φ 4 into where Here D AA(B B ) (0) represents that there is no relative time-delay between two photons A (B ). That is, there is no time interval between the reaction of the single-photon detector held by Alice or the single-photon detector held by Bob.
Finally, Alice (Bob) uses PBS 5 (PBS 6 ) and two single-photon detectors {D 1 , D 2 } ({D 3 , D 4 }) to complete the measurement on the outing photon A (B ) in the basis {|H , |V }. The relationship between the detection signatures, the corresponding output states, and the feed-forward operations on photon B is given in Tab. I. If detector pair (D i , D j ) (i, j = 1, 2, 3, 4) triggers with a time interval of |t 0 − t 1 |, they will get the desired maximally entangled state |φ + A B with a success probability of P = 2|αβ| 2 after applying the corresponding feed-forward operation shown in Tab. I. Otherwise, it means that detector pair (D i , D j ) fires without time interval. In such case, performing the feed-forward operation, they can get the normalization state |φ + 1 A B with a probability of |α| 4 + |β| 4 = 1 − 2|αβ| 2 .

III. HERALDED ECP FOR UNKNOWN GHZ STATE WITH LINEAR OPTICS
Our heralded ECP for polarization unknown Bell states with linear optics can be generalized to the case of multiphoton GHZ states. Suppose two maximally entangled GHZ states |ψ ABC and |ψ A B C are generated initially from S 1 and S 2 , respectively. Here The state of the six-photon system composed of photons A, B, C, A , B , and C is given by FIG. 3: Schematic diagram of the ECP for a three-photon GHZ states with unknown parameters. S1 and S2 are entanglement sources for |ψ ABC and |ψ A B C , respectively. The setups in dashed boxes held by Alice and Bob are shown in Fig. 1.
As shown in Fig. 3, photons A and B pass through a BS, and then photons pairs AA , BB , and CC of the state |Ψ 0 is sent to three distant parties Alice, Bob, and Charlie, respectively. Owing to the noisy channels, |Ψ 0 may decay to a partially less-entangled state where the unknown parameters α and β satisfy the normalization relation |α| 2 + |β| 2 = 1. The dashed boxes in Fig. 3 held by Alice and Bob are the same setups shown in Fig. 1. To be specific, Alice performs the σ x operation on photon A . After executing this operation, time delays t 0 and t 1 are introduced by Alice (Bob) to photons A (B ) by using the balanced interferometers, i.e., |H A → |H t0 A , |V A → |V t1 A (|H B → |H t0 B , |V B → |V t1 B ). Then, |Ψ 1 is converted to Then, as shown in Fig. 3, Alice, Bob, and Charlie lead photons A, B , and C to pass through HWP, respectively. These half-plate waves transform |Ψ 2 into where Finally, the outcomes of photons A, B , and C are measured by PBSs and single-photon detectors. Tab. II depicts the detection signatures and the output states. When the detectors held by Alice and Bob are triggered with a time interval of |t 0 − t 1 |, Alice, Bob, and Charlie will get the desired maximally entangled state |ψ + A BC with a success probability of 2|αβ| 2 , after performing the feed-forward operations on photon B. If the detectors held by Alice and Bob are triggered simultaneously, they will get the recyclable normalization state |ψ + 1 A BC with a probability of

FIG. 4:
Schematic diagram of the ECP for an arbitrary multi-photon GHZ state with unknown parameters. S1 and S2 are entanglement sources for two multi-photon GHZ states. The operations on photons C · · · Z (C · · · Z ) are identical.
linear optical elements or dark count will degrade the fidelity of the schemes. Recently, the effect of noise on the measurement is experimentally studied, leading to a reduction in the measurement precision. One effective way to solve this problem is to use the quantum Zeno effect, which can improve the measurement accuracy of entangled probes [48].
In summary, we have presented ECPs for Bell states and GHZ states with unknown parameters. The schemes are constructed by solely using linear optics, including HWP, PBS, and single-photon detector. Our protocols have several characteristics: First, the protocols can be exactly heralded by the detection signatures, and the photonnumber-resolving detections or post-selection principles are not required. Second, exact parameters α and β are not required. Third, the failed state has a good form for increasing the success probability of the protocols without resorting to the cross-Kerr media. Fourth, linear optical implementations of the heralded protocols are feasible in the experiment with current technology. These characteristics make our protocols more useful in long-distance quantum communication.