† Corresponding author. E-mail:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11474168 and 61401222), the Qing Lan Project of Jiangsu Province of China, the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20151502), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

We put forward two efficient entanglement concentration protocols (ECPs) for arbitrary less-entangled NOON state. Both ECPs only require one pair of less-entangled NOON state and an auxiliary photon. In the first ECP, the auxiliary photon is shared by two parties, while in the second ECP, the auxiliary photon is only possessed by one party, which can increase the practical success probability by avoiding the transmission loss and simplify the operations. Moreover, both ECPs can be used repeatedly to get a high success probability. Based on the above features, our two ECPs, especially the second one, may be useful in the quantum information processing.

Entanglement plays a significant role in the quantum information field. For example, quantum teleportation,^{[1,2]} quantum densecoding,^{[3]} quantum secret sharing,^{[4]} quantum computation,^{[5–7]} communication,^{[8–10]} protocols^{[11–22]} all require entanglement to set up the quantum entanglement channels. Recently, a special quantum state, which is the so-called NOON state, has attracted much attention.^{[23–29]} The NOON state can be written with the form of

*N*particles are in the spatial mode

*a*and none are in the spatial mode

*b*, plus none in the mode

*a*and all in the mode

*b*. The NOON state has provided many important applications. For example, it has been proved that the NOON state can serve as an important resource for Heisenberg-limited metrology and quantum lithography, where it has sensitivity for optical interferometry and can approach the Heisenberg limit of 1/

*N*.

^{[30–32]}Meanwhile, the NOON state also shows the de Broglie wavelength of

*λ*/

*N*for

*N*-photon interference.

^{[26,33–35]}It is worth noting that the power of the NOON state lies in its entanglement. In all applications, the ideal NOON state should be the maximally entangled NOON state as in Eq. (

The method for recovering the pure less-entangled state into the maximally entangled state is called the entanglement concentration. In 1996, Bennett *et al*. proposed the first entanglement concentration protocol (ECP), which is known as the Schmidt projection method.^{[36]} Since then, various interesting ECPs have been put forward, successively.^{[37–78]} For example, in 1999, Bose *et al*. proposed an ECP based on entanglement swapping,^{[37]} which was later improved by Shi *et al*.^{[38]} In 2001, Zhao *et al.* and Yamamoto *et al.* proposed two similar concentration protocols based on polarizing beam splitters (PBSs) independently.^{[39,40]} In 2008, Sheng *et al*. developed their protocols with the help of the cross-Kerr nonlinearity.^{[41]} So far, most of the ECPs are focused on the two-particle entanglement, especially the encoding system in the polarization degree of freedom of the optical system. They cannot deal with the less-entangled NOON state. Recently, we have proposed an ECP for the less-entangled NOON state.^{[44]} In the protocol, two pairs of less-entangled NOON states are required. After performing this ECP, one pair of maximally entangled NOON state can be obtained. As the NOON state is precious, this protocol is not optimal.

In the present paper, we put forward two efficient ECPs for arbitrary less-entangled NOON state. Different from the previous ECP,^{[44]} both of our ECPs only require one pair of less-entangled NOON state and a single auxiliary photon to complete the concentration task and can reach the same success probability as Ref. [44]. In the two ECPs, we adopt the cross-Kerr nonlinearity to construct the quantum nondemolition detection (QND) gate, which means the obtained maximally entangled NOON state can be kept for other applications. Moreover, both the ECPs can be used repeatedly to get a high success probability. The second ECP is more optimal than the first one, for it only requires the local single auxiliary photon, which can effectively avoid the photon loss during the transmission process. Moreover, one party can perform the whole task which can simplify the practical experiment largely.

This paper is organized as follows: In Sections 2 and 3, we explain the basic principles of the first and the second ECPs, respectively. Some discussions are made in Section 4, and a summary is given in the last section.

*N*

The cross-Kerr nonlinearity is the key element of both ECPs. Therefore, we would like to make a brief introduction about it first. The cross-Kerr nonlinearity has been widely used in the quantum information field. It has played an important role in the construction of CNOT gate,^{[79,80]} Bell-state analysis,^{[81,82]} and so on.^{[83–92]} In particular, the cross-Kerr nonlinearity is a powerful tool in the construction of the QND gate, which is widely used in the quantum entanglement concentration.^{[41–43,55,57]} The cross-Kerr nonlinearity can be described by its Hamiltonian as

*ħ*

*χ*is the coupling strength of the nonlinearity, which depends on the cross-Kerr material.

^{[79,80,93]}

*n̂*

_{a}and

*n̂*

_{b}are the photon number operators for the spatial modes

*a*and

*b*, respectively. During the cross-Kerr interaction, a laser pulse in the coherent state |

*α*〉 interacts with photons through a proper cross-Kerr material. If the single photon is presented, it can induce a phase shift

*θ*to the coherent state. The cross-Kerr interaction can be described as

*θ*=

*χt*, and

*t*means the interaction time for the signal with the nonlinear material. According to Eq. (

Now, we start to explain the first ECP for arbitrary less-entangled NOON state. The principle of the ECP is shown in Fig. *N*-photon NOON state |*ϕ*〉_{a1b1} in the spatial modes *a*_{1} and *b*_{1}, which is generated by the photon source *S*_{1}. |*ϕ*〉_{a1b1} can be written as

*α*and

*β*are the entanglement coefficients of the NOON state, where |

*α*|

^{2}+ |

*β*|

^{2}= 1, and

*α*≠

*β*.

A single-photon entanglement source *S*_{2} emits an auxiliary photon and sends it to Alice and Bob in the spatial modes *a*_{2} and *b*_{2}, which creates a single-photon entangled state of the form

*α*and

*β*, which are not difficult to know by measuring a sufficient amount of the sample of entangled NOON state.

In this way, the whole (*N* + 1)-photon system can be described as

According to Fig. *b*_{1} and *b*_{2} pass through two cross-Kerr nonlinearities, respectively. Then, the (*N* + 1)-photon system combined with the coherent state can evolve to

*N*,0,0,1〉

_{a1b1a2b2}and |0,

*N*,1,0〉

_{a1b1a2b2}can make the coherent state pick up the phase shifts of

*θ*and −

*θ*, respectively, while the other two items make the coherent state pick up no phase shift. As ±

*θ*is indistinguishable in the homodyne measurement, Bob selects the items corresponding to the phase shift of ±

*θ*, and discards the other items. Under this case, the state in Eq. (

*αβ*|

^{2}.

Then, Alice and Bob make the photons in the modes *a*_{2} and *b*_{2} pass through the 50:50 BS, which makes

*a*

_{j},

*b*

_{j}, and

*d*

_{j}(

*j*= 1,2) are the creation operators for the spatial mode

*a*

_{j},

*b*

_{j}, and

*d*

_{j}, respectively. The creation operators obey the rules that

*Φ*

_{1}〉

_{a1b1a2b2}can ultimately evolve to

Finally, the output photons in the spatial modes *d*_{1} and *d*_{2} are detected by the single-photon detectors D_{1} and D_{2}, respectively. If the detector D_{1} fires, the state in Eq. (

_{2}fires, the state in Eq. (

Both |*ϕ*_{1}〉_{a1b1} and *N*-photon NOON states and there is only a phase difference between them. *ϕ*_{1}〉_{a1b1} by the phase flip operation. So far, the first concentration protocol is completed. In the protocol, with the help of the QND gate constructed by the cross-Kerr nonlinearity, Alice and Bob can distill the maximally entangled *N*-photon NOON state with the success probability *P* = 2|*αβ*|^{2}.

Moreover, we will prove that the ECP can be reused to further concentrate the discarded items. In the first ECP, the discarded items which make the coherent state pick up no phase shift can be written as

*a*

_{2}and

*b*

_{2}pass through the BS, and detect the output photons in the

*d*

_{1}and

*d*

_{2}modes. After that, they can finally obtain

It can be found that |*ϕ*_{2}〉_{a1b1} has a similar form to the initial NOON state in Eq. (*ϕ*_{2}〉_{a1b1} is a new less-entangled *N*-photon NOON state and can be reconcentrated for the next round. In the second concentration round, the single photon source *S*_{2} creates an auxiliary single-photon entangled state with the form of

*b*

_{1}and

*b*

_{2}pass through the QND, and the whole (

*N*+ 1)-photon system combined with the coherent state can be described as

*θ*, and the state in Eq. (

*N*-photon NOON state, Alice and Bob also make the photons in the spatial modes

*a*

_{2}and

*b*

_{2}pass through the BS, and detect the photons in the

*d*

_{1}and

*d*

_{2}modes. In this way, the state in Eq. (

*N*-photon NOON state. It can be calculated that in the second concentration round, the success probability is

Similarly, by making the discarded items in the second concentration round pass through the BS and detect the output photons, the parties can ultimately obtain a new less-entangled NOON state as

*k*-th concentration round (

*k*= 1, 2, 3, …), our ECP can be used repeatedly to further concentrate the less-entangled NOON state.

*N*

The schematic diagram of our second ECP is shown in Fig. *N*-photon NOON state |*ϕ*〉_{a1b1} with the form of Eq. (*S*_{1}. Here, the single-photon source *S*_{2} emits an auxiliary photon and only sends it to Bob in the spatial mode *b*_{2}. Bob makes this auxiliary photon pass through a variable beam splitter (VBS) with the transmittance of *t*, which can create a single-photon entangled state between the spatial modes *c*_{1} and *c*_{2} with the form of

*b*

_{2}will couple with the vacuum state in the another input mode of the VBS. However, the vacuum state does not affect the final result, which can be omitted for simplicity. In this way, the whole (

*N*+ 1)-photon system can be descried as

Bob makes the photons in the spatial mode *b*_{1} and *c*_{1} pass through the QND gate, and the whole (*N* + 1)-photon system combined with the coherent state can evolve to

It can be seen that the items *θ*, while both the items *θ* and discards the other items. Then, |*Ψ*〉_{a1b1c1c2} in Eq. (

*P*′ = |

*α*|

^{2}(1−

*t*) + |

*β*|

^{2}

*t*.

Then, Bob makes the photons in the spatial modes *c*_{1} and *c*_{2} pass through the 50:50 BS, which makes

*Ψ*

_{2}〉

_{a1b1c1c2}can evolve to

*e*

_{1}and

*e*

_{2}modes are detected by the detectors D

_{1}and D

_{2}, respectively. It is obvious that if D

_{1}fires, |

*Ψ*

_{3}〉

_{a1b1e1e2}will collapse to

_{2}fires, |

*Ψ*

_{3}〉

_{a1b1e1e2}will collapse to

*ψ*

_{2}〉

_{a1b1}in Eq. (

*ψ*

_{1}〉

_{a1b1}with the phase shift operation. According to Eq. (

*t*=

*α*

^{2}, equation (

*t*=

*α*

^{2}, the success probability of this ECP is

Moreover, we can also prove that the second ECP can be used repeatedly. When *t* = *α*^{2}, the discarded items in the first concentration round can be written as

*c*

_{1}and

*c*

_{2}pass through the BS, and detects the photons in the

*e*

_{1}and

*e*

_{2}modes. |

*Ψ*

_{4}〉

_{a1b1c1c2}can finally evolve to the state in Eq. (

*N*-photon NOON state and can be reconcentrated in the next concentration round.

In the second concentration round, the single photon source *S*_{2} provides a new auxiliary single photon to Bob. Bob makes it pass through another VBS with the transmittance of *t*_{2}, where the subscript “2” means in the second concentration round. A new single-photon entangled state with the similar form of Eq. (*b*_{1} and *c*_{1} pass through the QND gate. Then, the whole (*N* + 1)-photon system combined with the coherent state can be written as

*θ*, and the above state can evolve to

*α*|

^{4}(1−

*t*

_{2}) + |

*β*|

^{4}

*t*

_{2}.

Next, Bob makes the photons in the spatial modes *c*_{1} and *c*_{2} pass through the BS, and detects the output photons. In this way,

If we can find a suitable VBS with the form

Similarly, by making the discarded items in the second concentration round pass through the BS, the discarded items can also evolve to a new less-entangled NOON state, which can be reconcentrated in the next round. Therefore, by choosing the suitable VBSs with the transmittance

*k*” means the iteration time, the second ECP can also be used repeatedly to distill the maximally entangled NOON state.

Finally, we would like to calculate the total success probability of these two ECPs. According to the description mentioned above, both ECPs have the same success probability. In each concentration round, the success probability can be written as

In theory, the ECPs can be reused indefinitely, and the total success probability equals the sum of the probability in each concentration round, which can be described as

It can be found that if the initial entangled state is the maximally entangled NOON state, where *P*_{total} = 1/2 + 1/4 + 1/8 + ··· + 1/2^{k} + ··· = 1, while if *α* ≠ *β*, the *P*_{total} <1. Figure *P*_{total} as a function of the entanglement coefficient *α*, under the iteration number *k* = 1, 3, 5, respectively. It can be found that *P*_{total} largely depends on the initial entanglement state. The higher initial entanglement leads to the greater *P*_{total}. Meanwhile, increasing the iteration number can largely increase the *P*_{total}. For example, when *P*_{total} ≈ 0.500 under *k* = 1, while *P*_{total} ≈ 0.968 under *k* = 5.

In the paper, we presented two efficient ECPs for distilling the maximally entangled *N*-photon NOON state from an arbitrary less-entangled NOON state. It is interesting to compare these two ECPs with our previous ECP for the *N*-photon NOON state.^{[44]} In the ECP of Ref. [44], the parties need to consume two pairs of less-entangled NOON states to distill one pair of maximally entangled NOON state, while the current two ECPs only require one pair of less-entangled NOON state and an auxiliary single photon, which can reach the same high success probability as that in Ref. [44]. zhouqip1. As the NOON state is quite precious, the current two ECPs are more economical. On the other hand, the previous ECP in Ref. [44] zhouqip1 require the two parties to take the coincidence measurement for a pair of NOON state, which will increase the operation difficulty, especially when the photon number *N* is large. In the current two ECPs, we only need to measure the auxiliary single photon, which can simplify the operation largely. Therefore, the two current ECPs are more advantageous than the previous one.

Comparing the two current ECPs, it looks that if we change the spatial modes *c*_{1} and *c*_{2} to *b*_{1} and *b*_{2}, and make the transmittance of the VBS meet *t* = 0.5, the second ECP can be transformed to the first ECP. However, it is noteworthy that the second ECP is more optimal than the first one. In the first ECP, the auxiliary photon should be shared by Alice and Bob in two distant locations. It requires the entangled photon source, which is precious under current experimental condition. Moreover, after passing through the QND gate, the two parties need to make the coincidence measurement of the auxiliary single-photon entangled state, which is difficult in practical experiment. Moreover, in the first ECP, the auxiliary photon in Alice’s hand should be sent back and will be transmitted for a second time. As pointed out by Ref. [94], the decoherence and photon loss are unavoidable during the transmission process due to the environmental noise. It will limit the application of the first ECP in the practical applications. Fortunately, in the second ECP, the auxiliary photon entanglement can be created locally with the help of the VBS and only possessed by Bob. Only Bob needs to operate this concentration process, which makes this ECP more advantageous in practical applications.

In the second ECP, the VBS is the key element for creating the local single-photon entanglement. According to the initial coefficients of the less-entangled NOON state, the parties should provide the VBSs with the transmittance of

*k*-th concentration round. Certainly, in the first ECP, the single-photon entanglement created by

*S*

_{2}can also be reviewed as the single-photon source plus the VBS. The VBS is a common optical element in current technology. In 2012, Osorio

*et al*. reported their experimental results about the heralded photon amplification for quantum communication with the help of the VBS.

^{[95]}amplification They adjusted the transmittance of VBS to increase the probability of the single photon |1〉 from a mixed state

*η*

_{t}|1〉〈1| + (1−

*η*

_{t})|0 〉〈0|. In their protocol, they adjust the splitting ratio of VBS from 50:50 to 90:10 to increase the visibility from (46.7 ± 3.1%) to (96.3 ± 3.8%). Based on their results, it is believed that our requirement for VBS can be realized under current experimental condition.

Finally, we will discuss another key element of both ECPs, that is, the cross-Kerr nonlinearity. Although the cross-Kerr nonlinearity has been widely discussed in quantum information processing,^{[80,81,83–85,96]} it has been regarded as a controversial topic in practical experiment for a long time.^{[97–99]} First, the qubit states may degrade to the mixed states in the homodyne detection process.^{[100,101]} Second, the natural cross-Kerr nonlinearity is extremely weak, so that it is difficult to discriminate two overlapping coherent states in homodyne detection. Fortunately, the decoherence can be extremely weak simply by an arbitrary strong coherent state associated with a displacement *D* (−*α*) performed on the coherent state.^{[100]} On the other hand, it is possible to obtain an observable cross-Kerr phase shift with the help of the weak measurement.^{[102]} As early as 2003, Hofmann showed that with the help of a single two-level atom trapped in a one-sided cavity, we can obtain a phase shift as large as *π*.^{[103]} Meanwhile, as shown in Ref. [104], large cross-Kerr nonlinearities were also obtained in a double-quantum-well structure with a four-level, double-type configuration. The “giant” cross-Kerr effect with phase shift of 20° per photon has been observed in the current experiment.^{[105]} Recent work of the Xiao group also showed that the Rydberg atom system can generate large cross phase between photons.^{[106]} According to the recent theoretical and experimental works based on cross-Kerr nonlinearity, the cross-Kerr nonlinearity may provide its practical application in the future quantum information processing.

In summary, we put forward two efficient ECPs for distilling the maximally entangled *N*-photon NOON state from arbitrary less-entangled NOON state. In both ECPs, we only require one pair of less-entangled NOON state and an auxiliary single photon to complete the task. Moreover, with the help of the cross-Kerr nonlinearity, both ECPs can be used repeatedly. In this way, they can both obtain high success probability. Comparing with our previous ECP of the NOON state, the two current ECPs are more economical and easy to operate. In the first ECP, the auxiliary single photon should be shared by the two parties, and the two parties need to take the coincidence measurement for the auxiliary photon. The second ECP is more optimal than the first one. In the second ECP, the auxiliary single photon is only possessed by Bob benefitting from the VBS, which can reduce the requirement for the single-photon source. Meanwhile, as all the operations are local, the second ECP is easier to operate than the first one. In particular, in the second ECP, as the auxiliary single photon does not need to transmit in a long-distance channel, the second ECP can effectively reduce the docoherence and photon loss in the transmission process. In this way, the second ECP is more advantageous in practical applications. Based on the features described above, our two ECPs, especially the second ECP may be useful and convenient in quantum information processing.

**Reference**

1 | Phys. Rev. Lett. 70 1895 |

2 | Phys. Rev. A 58 4394 |

3 | Phys. Rev. Lett. 69 2881 |

4 | Phys. Rev. A 59 1829 |

5 | Phys. Rev. A 87 022305 |

6 | Phys. Rev. Lett. 110 190501 |

7 | Opt. Express 21 17671 |

8 | Phys. Rev. A 65 032302 |

9 | Phys. Rev. A 68 042317 |

10 | Phys. Rev. A 71 044305 |

11 | Chin. Sci. Bull. 59 2541 |

12 | Chin. Sci. Bull. 58 2927 |

13 | Chin. Sci. Bull. 58 2329 |

14 | Sci. Chin.-Phys. Mech. Astron. 57 1210 |

15 | Sci. Chin.-Phys. Mech. Astron. 57 1696 |

16 | Chin. Phys. B 24 080306 |

17 | Chin. Phys. B 23 090309 |

18 | Chin. Phys. B 23 050306 |

19 | Chin. Phys. Lett. 30 060302 |

20 | Chin. Phys. Lett. 31 024206 |

21 | Chin. Phys. Lett. 32 040303 |

22 | Int. J. Theor. Phys. 52 4461 |

23 | Phys. Rev. Lett. 79 3865 |

24 | Phys. Rev. Lett. 98 223601 |

25 | Nature 429 161 |

26 | Nature 429 158 |

27 | Science 316 726 |

28 | Chin. Sci. Bull. 59 2829 |

29 | Phys. Rev. A 91 042332 |

30 | Phys. Rev. Lett. 85 2733 |

31 | Phys. Rev. A 54 R4649 |

32 | Phys. Rev. A 73 013814 |

33 | Phys. Rev. Lett. 87 013602 |

34 | Phys. Rev. A 73 032308 |

35 | Phys. Rev. A 81 033823 |

36 | Phys. Rev. A 53 2046 |

37 | Phys. Rev. A 60 194 |

38 | Phys. Rev. A 62 054301 |

39 | Phys. Rev. A 64 014301 |

40 | Phys. Rev. A 64 012304 |

41 | Phys. Rev. A 77 062325 |

42 | Phys. Rev. A 85 012307 |

43 | Phys. Rev. A 85 042302 |

44 | Quantum. Inform. Process. 12 1307 |

45 | Phys. Rev. A 85 022311 |

46 | Phys. Rev. A 86 012323 |

47 | Quantum. Inform. Process. 12 1885 |

48 | J. Opt. Soc. Am. B 30 71 |

49 | Quantum Inf. Process. 12 2087 |

50 | Opt. Commun. 313 217 |

51 | Chin. Phys. B 22 020307 |

52 | Phys. Rev. A 88 012302 |

53 | Laser Phys. Lett. 10 115201 |

54 | Entropy 15 1776 |

55 | Sci. China-Phys. Mech. Astron. 58 040303 |

56 | |

57 | Int. J. Theor. Phys. 53 1337 |

58 | Chin. Phys. Lett. 31 050303 |

59 | Quantum Inform. Process. 14 1305 |

60 | Chin. Phys. B 21 030307 |

61 | Opt. Commun. 340 80 |

62 | Quantum Inform. Process. 14 1265 |

63 | Laser Phys. Lett. 12 036001 |

64 | Quantum Inform. Process. 13 1025 |

65 | Phys. Rev. A 91 062302 |

66 | Chin. Phys. B 23 050308 |

67 | Quantum Inform. Process. 13 1595 |

68 | Sci. China-Phys. Mech. Astron. 58 060301 |

69 | Int. J. Theor. Phys. 53 3026 |

70 | Quantum Inform. Process. 14 2077 |

71 | Quantum Inform. Process. 12 2577 |

72 | Int. J. Theor. Phys. 53 80 |

73 | Int. J. Theor. Phys. 52 3965 |

74 | Quantum Inform. Process. 13 1967 |

75 | Chin. Phys. B 23 020313 |

76 | Chin. Phys. B 22 110303 |

77 | Laser Phys. Lett. 11 125201 |

78 | J. Opt. Soc. Am. B 29 1685 |

79 | Phys. Rev. Lett. 93 250502 |

80 | Phys. Rev. A 79 022301 |

81 | Phys. Rev. A 71 060302 |

82 | J. Opt. Soc. Am. B 29 1029 |

83 | Phys. Rev. A 79 052323 |

84 | Phys. Rev. A 85 033814 |

85 | Phys. Rev. A 83 053826 |

86 | |

87 | Opt. Commun. 308 304 |

88 | Chin. Phys. B 24 050304 |

89 | Phys. Rev. A 72 034305 |

90 | Sci. Rep. 5 12792 |

91 | Chin. Phys. B 23 060306 |

92 | Chin. Phys. B 22 090309 |

93 | Phys. Rev. A 71 033819 |

94 | Nature 414 413 |

95 | Phys. Rev. A 86 023815 |

96 | Phys. Rev. A 80 042310 |

97 | Phys. Rev. A 81 043823 |

98 | Phys. Rev. A 73 062305 |

99 | New J. Phys. 9 16 |

100 | Phys. Rev. A 73 052320 |

101 | Phys. Rev. A 74 060302 |

102 | Phys. Rev. Lett. 107 133603 |

103 | J. Opt. B: Quantum Semiclass. Opt. 5 218 |

104 | Opt. Express 19 23364 |

105 | Phys. Rev. Lett. 111 053601 |

106 | Phys. Rev. Lett. 112 133606 |