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Entanglement concentration for photon systems assisted with single photons
Accepted Manuscript Title: Entanglement concentration for photon systems assisted with single photons Authors: Xiong Wang, Zhan-Ning Hu PII: DOI: Reference:
S0030-4026(18)31347-0 https://doi.org/10.1016/j.ijleo.2018.09.044 IJLEO 61483
To appear in: Received date: Accepted date:
8-7-2018 13-9-2018
Please cite this article as: Wang X, Hu Z-Ning, Entanglement concentration for photon systems assisted with single photons, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.09.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Entanglement concentration for photon systems assisted with single photons Xiong Wang*, Zhan-Ning Hu Department of Physics, School of Science, Tianjin Polytechnic University, Tianjin, 300387, People’s Republic of
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China
(*Corresponding author E-mail: [email protected])
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Abstract The entanglement concentration protocols (ECPs) are presented for concentrating arbitrary partially entangled Bell-like state, N-photon GHZ state and N-photon W state, respectively, and the corresponding maximally entangled state can be obtained finally with a certain probability. Besides, compared with other ECPs, the accurate coefficients of the initial partially entangled state in this ECP do not need to know in advance. Moreover, current ECPs can be repeated to acquire a higher success probability. The devices used in the schemes are linear which leads to flexible operations and improves the performance greatly in the present experiment and our ECPs may be useful in the long-distance quantum information processing.
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1. Introduction
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Keywords: linear optics; partially entangled state; entanglement concentration; repeating concentration; quantum communication
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Entanglement is an important source in the fields of quantum communication and quantum information, and is widely used in quantum computing [1], quantum key distribution [2-5], quantum teleportation [6-9]，quantum dense coding [10,11]，quantum secure direct communication [12-15], quantum secret sharing [16-20] and quantum state sharing [21-25] and so on. The maximally entangled states are usually required in above applications as an entanglement channel. The general process is to prepare the maximally entangled states locally and distribute it to distant parties. However, the entanglement degree of the states may be degraded due to the influence of channel noise during the storage and distribution process, and it will transform the maximally entangled states into non-maximally entangled states or mixed states. Hence, it is quite essential to reconstruct the maximally entangled states from the partially entangled states. In order to overcome this difficulty, two concepts, called entanglement purification [26-30] and entanglement concentration [31-44] came into being. When the initial maximally entangled state transforms into a mixed entangled state, at this point, Communication parties can use the entanglement purification to improve the fidelity of the systems. However, if the initial maximally entangled state becomes a partially entangled pure state, one can exploit entanglement concentration to distill the maximally entangled state from the non-maximally entangled pure state to reconstruct a subset of systems in a maximally entangled state. By and large, the entanglement purification is more general but the entanglement concentration is more efficient and most of the existing ECPs
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A
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are based on the entanglement concentration. In long-distance quantum communication, the photons are considered as the ideal candidate for quantum communication because of their advantages such as fast transmission, manipulate easily, and anti-interference strongly. In 1996, Bennett [31] et al. proposed the first entanglement concentration protocol, called Schmidt projection method. By performing some collective measurements on many particles, they extracted Einstein-Podolsky-Rosen (EPR) pairs. The success probability of this ECP is quite low because it is very hard to operate in the experiment. In 1999, Bose [32] et al. proposed an improved ECP based on the idea of entanglement swapping. Nevertheless, the collective Bell-state measurements are required which made it is also difficult to execute in experiment. Yamamoto [33] et al. and Zhao [34] et al. proposed two similar ECPs, independently. In their scheme, the parity check gate was made by using some PBSs. However, to ensure that each of the four output spatial modes of the two PBSs exits only one photon in the photon measurement, the communication parties need to employ the single photon detectors to probe the photons emitted from the four spatial modes. But the ideal single photon detector is still difficult to produce at the current experimental level. The protocols described above usually require two similar copies of partially entangled state. In addition, the above all ECPs pay attention to the concentration process of Bell-like state or GHZ state, there are few protocols for concentrating W state, which cannot be converted into GHZ state just using local operations and classical communication. For instance, in 2003, Cao [35] proposed an ECP for W-class state using the joint unitary transformation. In 2007, Zhang [36] et al. proposed an ECP for unknown triparticle W class states based on the appropriate Bell state measurements. In 2008, Sheng [37] et al. proposed an ECP using the cross-Kerr nonlinearities. The strong cross-Kerr media is required in their protocol for getting a higher efficiency. In 2012, Gu [38] proposed an interesting ECP for multiphoton W states with the help of single photon and this ECP only needs linear optics. Zhou [39] proposed an efficient ECP for electron-spin W state assisted with charge detection. In 2013, Sheng [40] et al. proposed a different ECP for W state using the quantum dot and optical microcavities and the advantage of this ECP is that during the whole concentration process, the less-entangled W state is not destroyed and only consumes some single photons at a cost. In 2015, Sheng [41] et al. proposed a special ECP for N-particle W state assisted by parity check gates. Du [42] et al. proposed a heralded ECP for photon systems with linear optical elements. Inspired by above, in this paper, we propose the ECPs for arbitrary Bell-like state, three-photon W state, N-photon W states, respectively, assisted with linear elements. Compare with traditional ECPs, our ECP has some advantages. First, the devices used in the concentration process are linear, which simplifies their realization in the experiment. Second, it only needs to implement a local operation on the auxiliary single photon to obtain the maximally entangled state from initial partially entangled state. Third, it only requires one copy of the partially entangled state, not two copies. Forth, current ECP reduces the requirements on the coefficients of partially entangled state, in other words, we do not need to know the accurate coefficients in advance but require the coefficients of ancillary single photon state are the same as initial partially entangled state. The paper is organized as follows: In Sec. 2, we will explain the concentration process of Bell-like state. In Sec. 3, the concentration process of three-photon W state is elucidated in detail and the corresponding success probability is calculated later on. In Sec. 4, the concentration process of N-photon W state is displayed here. In Sec. 5, a brief discussion and summary for the whole concentration process is arranged here.
2. Entanglement concentration for Bell-like state In this section, let us see the specific concentration process of the Bell-like state, shown in Fig. 1, the general form of Bell-like state is given by
a H
A0 B0
H
A0
B0
b V
V
A0
B0
.
(1)
a H
A1
A1
b V
A1
IP T
Here, suppose the two coefficients a and b are real numbers and satisfy the relation a2+b2=1. The photon A0 and B0 emitted from single photon source S0 are distributed to distant parties, Alice and Bob, respectively. At the same time, the assistant single photon emitted from source S1 is prepared by Alice in spatial mode A1 has the following form .
(2)
'A a V
A1
1
b H
A1
SC R
Before the two photons A0 and B0 pass through the PBS1, the single photon A1 is firstly sent to the HWP90, which completes the job H V . After that, the Eq. (2) will transform into .
(3)
Thus, the composite system composed of the three-photon can be described as
(a H
H
A0
a2 H
A0
ab( H
A0
b V
B0
H
B0
H
V
B0
U
' A1
A0 B0
V
A0
B0
b2 V
A1
) (a V
N
A0
A
A0 B0 A1
H
A1
M
V
A0
V
V
A1
H
B0
V
B0
b H
A1
)
A1
A1
).
(4)
PT
ED
Next, two photons A0 and B0 will pass through the PBS1. Obviously, there are two items will lead to the two photons emitted from the different output spatial modes, which means that both the two output spatial modes contain only one photon. The others will result the two photons emitted from the same spatial modes. If we choose the first case, after the PBS2, the composite system will project into
1
A4 B0 A5
ab( H
A4
H
B0
H
A5
V
V
A4
V
B0
A5
).
(5)
1 ' A B A ab / 2 ( H
A5
4 0 5
ab / 2 ( H
A5
H
B0
H
V
B0
A5
V
V
A5
B0
V
)H
B0
)V
A4
A4
.
(6)
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The corresponding success probability is P1=2a2b2. In addition, let the photon A4 pass through the HWP45, the Eq. (5) is transformed into
Finally, after the PBS3, by detecting the single photon polarization state in the basis H , V , the two-photon maximally entangled state can be obtained. That is to say, when the detection result is H , we can get
A5 B0
1 (H 2
A5
H
B0
V
A5
V
B0
).
(7)
Otherwise, if the detection result is V , we can get
1 (H 2
A5 B0
H
A5
B0
V
V
A5
B0
).
(8)
2
A0 B0 A1
a2 H
A0
H
B0
V
A1
b2 V
V
A0
H
B0
A1
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Eq. (7) and Eq. (8) are also the two-photon maximally entangled state. Alice or Bob, implements a local phase-flip operation on one of the two photons, Eq. (8) can be converted into Eq. (7). On the other side, if we select second case, in other words, the composite system will collapse to .
(9)
2
A5 B0
a2 H
A5
H
B0
b2 V
A5
If DV fires, we can get A5 B0
a 2 H
A5
H
B0
b2 V
V
B0
V
A5
.
B0
.
(10)
(11)
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After the PBS2, HWP45 and PBS3, respectively, just like the Eq. (7) and Eq. (8), when DH fires, we can get
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By using a phase-flip operation on one of the two photons, Eq. (11) can be converted into Eq. (10). It is easy to find that Eq. (10) has the similar form with Eq. (1) which means Eq. (10) can be concentrated by repeating our ECP. We only need to replace a2 and b2 with a ' a 2 / a4 b4 and b ' b2 / a 4 b4 , respectively, by this time, Alice needs to prepare another ancillary single photon with the form a ' H b ' V . After a series of operations, eventually, the maximally entangled
ED
Bell-like state can be distilled with the success probability P2=2(a4+b4)a'2b'2=2a4b4/(a4+b4). Similarly, the success probability of third round is P3=2a8b8/(a4+b4)(a8+b8), the success probability of Nth interaction is given by
(a 4 b4 )(a8 b8 )
PT
PN
N
2a 2 b 2
N
(a 2 b 2 ) N
N
.
(12)
Therefore, the total success probability for obtaining the maximally entangled Bell-like state is
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Ptotal P1 P2
N
PN
N 1
2a 2 b 2
N
(a 4 b4 )(a8 b8 )
(a 2 b 2 ) N
N
.
(13)
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By far, we have introduced the entanglement concentration protocol for Bell-like state. In fact, one can find that this ECP can be easily extended to concentrate N-photon GHZ state. The partially entangled N-photon GHZ state can be written as
A1 A2
AN
a H
A1
H
A2
H
AN
b V
A1
V
A2
V
AN
.
(14)
Where a2+b2=1. A1, A2, …, AN represent the N photons shared by N parties, Alice, Bob, Charlie and so on. After the ECP, the N-photon maximally entangled GHZ state can be obtained and has the same total success probability with Bell-like state if one of the N parties exploits an assistant single photon and implements this ECP in the same way as the situation of the partially entangled Bell-like state.
3. Entanglement concentration process for three-photon W class state In this section, we will illustrate the concentration process of three-photon W state and the corresponding schematic drawing is shown in Fig. 2. This ECP can divide into two steps, which are implemented by Bob and Charlie, respectively.
ABC
V
H
A
H
B
H
C
V
A
H
B
C
H
H
A
B
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Without loss of generality, we assume that the partially entangled three-photon W state shared by Alice, Bob and Charlie has the following form
V
,
C
(15)
B1
H
2 2
B1
2 2
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where the coefficients α, β, γ are real numbers and satisfy the relation α2+β2+γ2=1. The single photon source S1 on Bob's side emits the single photon with the form (16)
H
(17)
V
.
B1
B1
2
V
2
B1
2
N
1
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Firstly, make the single photon enter the HWP90, the single photon is transformed into
2
B1
,
B1
2
2 2
V
A
H
2 2
PT
1
ABC
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ABCB1
H
H
B
ED
A
The composite system composed of the four photons ABCB1 is
2
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2
2
H
A
A
H
V
B
C
V
B
V
H
B1
V
C
B1
H
C
B1
2 2 2 2 2 2
H V
H
V
A
H
A
H
B
H
A
H
B
B
V
C
V
C
C
B1
H
H
B1
B1
. (18)
Next, let two photons B and B1 pass through the PBS1, distinctly, there are three items will result them emitted from different output spatial modes and the others will cause them emitted from the
A
same spatial mode. In this time, if we pick up the first situation, the state
' ABCB1
2
2
V
2 2
A
H
H
A
B
H
H
B
C
V
H
C
B1
H
B1
.
2 2
H
A
ABCB1
V
B
will collapse to
H
C
V
B1
(19)
The success probability is P11 = (2α2β2+β2γ2)/(α2+β2). Here, the subscript 1 represents the first step of the concentration and the superscript 1 denotes the first round concentration executed by Bob.
The Eq. (19) can be rewritten as
' ABCB1
2 2
V
2
H
A
H
2 2 2
H
B
H
A
V
B
H
C
H
C
B1
B1
H
2 2 2
V
A
H
B
C
V
B1
.
(20)
After the PBS2 and HWP45, Eq. (20) will convert into
(
V
2 2 2
H
2 2 2
C
H
2 2 2
)H C
H
B5
C
V
B5
V
A
H
B5
H
A
H
2 2 2
H
A
B4
V
2 2 2 H
2 2 2
H
B5
(
V
A
C
H
A
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AB5CB4
H
B5
C
SC R
A
H
V
B5
C
)V
B4
.
(21)
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Finally, by detecting the single photon polarization state in the basis H , V . When the detection result is H , Bob will get the following state
2 2
2
V
H
A
B5
H
H
2 2 2
N
AB5C
C
2 2 2
H
V
A
H
B5
C
A
H
M
A
B5
V
C
.
(22)
2 2
2
V
PT
AB5C
ED
Otherwise, if the detection result is V , Bob will get the following state
2 2
H
H
A
B5
H
H
B5
C
V
C
2 2 2
H
AB5C5
B5
H
C
.
(23)
By performing a phase-flip operation on one of three photons, the state into 1
V
A
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2
A
AB5C
can be transformed
.
A
On the other hand, when we select the rest items, that is to say, the state
'' ABCB1
2 2 2
V
2 2
A
H
H
A
B
H
H
B
C
V
V
2
B
2 2
1
C
V
B1
.
H
A
ABCB1
V
B
will collapse to
H
C
H
B1
(24)
Following the same rules mentioned above, after the PBSs and HWP45, if DHB clock, Bob will get
1
AB5C
2
2 2
V
A
H
2 2
H
A
H
B5
H
C
V
B5
2
C
H
2 2
A
V
B5
H
C
.
(25)
AB5C
2
2 2
V
2 2
A
V
H
A
B5
H
H
B5
2
C
H
C
H
2 2
A
.
V
B5
H
C
(26)
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1
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If DVB clock, Bob will get
Only a phase-flip operation required can convert Eq. (26) into Eq. (25). Apparently, the state shown in Eq. (26) is a partially entangled state. If we utilize ' 2 / 4 4 2 2 ,
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' 2 / 4 4 2 2 , ' / 4 4 2 2 to substitute 2 / 2 2 , 2 / 2 2 , / 2 2 , respectively, and it can be found that Eq. (26) has the same form with Eq. (15). At
A
this moment, Bob prepares a single photon with the following form
'
H
M
'B
'2 '2
1
B1
'
'2 '2
V
B1
.
(27)
ED
With the same principle introduced above, after a series of operations, we can acquire the maximally entangled state. The success probability of the second round is
2 4 4 2 4 2 , P 2 ( 2 )( 4 4 )
PT
2 1
(28)
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Similarly, the success probability of the third round, …, the Nth round can be described as
P13
2 8 8 6 8 2 , ( 2 2 )( 4 4 )( 8 8 )
(29)
......
2 2 2 2 N
P1 N
N
( 2 2 )( 4 4 )
N
2 2 . ( 2 2 )
2
N
N
N
(30)
Here, the N denotes the interaction numbers of first step concentration process. The total success probability of the first step concentration process is
P1 P11 P12
P1N P1N .
(31)
N 1
When Bob implements the first step concentration process successfully, he will inform Charlie to
carry out the second step concentration process. To be specific, the single photon source S2 on Charles's side emits the single photon with the form
C1
H
2 2
C1
V
2 2
.
C1
(32)
Charlie lets the single photon pass through the HWP90 and the associated state can be written as
AB5C
'C1
2
V
2 2 2 2 2
A
H
2 2 2 2 2
H
2 2 2 2 2
H
B5
H
A
A
H
V
B5
C
V
B5
V
H
C1
V
C
C1
H
C
2
2
V
A
V
2 2 2 2 2
C1
H
2 2 2 2 2
H
A
H
H
B5
B5
H
C
V
2 2 2 2 2
A
C1
IP T
AB5CC1
H
V
C
SC R
B5
C
H
H
C1
C1
moment, when Charlie picks up the former, the state
V
2 2 2 2 2
ED
A
H
2 2 2 2 2
will collapse to
N
A
AB5CC1
M
1
AB5CC1
U
. (33) It is evidence that there are three items will result the two photons emitted from different output spatial modes and the others will cause the two photon are in the same output spatial mode. At the
H
2 2 2 2 2
H
A
A
H
B5
V
H
B5
H
C1
H
C
V
B5
H
C
C
V
C1
C1
.
(34)
PT
With the success probabilityP21=3α2γ2/(γ2+2α2)(γ2+α2). Eq. (34) can be rewritten as
AB5CC1
1 (V 3
CC E
1
H
A
H
A
H
V
B5
H
B5
C
H
C
V
C1
A
After passing through PBS5 and HWP45, the state 1
1
AB5C4C5
1 (V 3 H H
A
A
A
H V
H
B5
B5
B5
V
H
C5
C1
V
H
B5
C
H
C1
(35) will be transformed into
H
)H
C4
H C V 5
A
).
AB5CC1
C5
H
A
A
V
B5
1 (V 3 H
5B
H
A
C5
H
H )C 5
B5
H
V. C
C5
(36)
4
Finally, Charlie will detect the single photon polarization state in the basis H , V . When the
detection result is
1
, he will get
H
AB5C5
1 (V 3
A
H
H
B5
C5
H
A
V
B5
H
H
C5
H
A
V
B5
C5
).
(37)
).
(38)
If the detection result is V , he will get
1
AB5C5
1 (V 3
A
H
H
B5
C5
H
A
V
B5
H
H
C5
H
A
V
B5
C5
AB5C5
.
However, when Charlie chooses the latter in Eq. (33), the state
2 2 2 2 2
V
2
H
2 2 2 2 2 2
H
A
A
H
B5
H
V
A
B5
H
A
H
C
V
B5
V
C
C
C1
V
C1
H
C1
.
(39)
M
2 2 2 2 2
will collapse to
U
AB5CC1
2
N
2
AB5CC1
AB5C5
SC R
can be transformed into 1
IP T
Charlie can carry out a local phase-flip operation on one of the three photons and the state 1
The processing method is similar to Bob's. Charlie can replace the coefficients of Eq. (39) with
ED
'' 2 / 2 4 4 , '' 2 / 2 4 4 , '' 2 / 2 4 4 , respectively. Meanwhile, he prepares a single photon with the following form
''C
PT
1
''
'' '' 2
2
H
C1
''
''2 ''2
V
C1
.
(40)
CC E
He can repeat the concentration process described above and the success probability of second round is
P2 2
3 4 4 , ( 2 2 )( 4 4 )( 2 2 2 )
(41)
Analogously, the success probability of the third round, …, the Nth round can be written as
A
3 8 8 , P2 2 ( 2 )( 4 4 )( 8 8 )( 2 2 2 ) 3
(42)
......
3 2 2 M
P2 M
( 2 2 )( 4 4 )
M
( 2 2 )( 2 2 2 ) M
M
Thus, the total success probability of the second concentration process is
.
(43)
P2 P2 . M
(44)
M 1
The total success probability of our ECP for obtaining the maximally three-photon W state is
N =1
M 1
N M . P PP 1 2 P1 P2
(45)
IP T
4. Concentration process of arbitrary partially entangled N-photon W class state
A1 A2
AN
1 V
H
A1
N H
A1
H
A2
H
AN
V
A2
2 H .
AN
A1
V
H
A2
AN
U
SC R
The ECP in Section 3 is still workable if we directly extend the present protocol to concentrate the maximally N-photon W class state from the partially entangled N-photon W class state. The schematic diagram is shown in Fig. 3 and the arbitrary partially entangled N-photon W class state has the following form
(46)
2
N 2 1.
A
2
M
and satisfy this relation 1 2
1
1=
2
ED
2 1
PT
2=
CC E
N 1
N
Here, the subscripts A1, A2, …, AN represent the N photons shared by N parties, say Alice, Bob, … respectively. Every participant holds one photon. The coefficients 1 , 2 , , N are real numbers
H 1
2
1
3 2 1
H 1
2
2 22 2 1
3 32 2 1
V 1,
(47)
V 1,
(48)
......
1
=
H 1
12 N 2
N 12 N 2
V
N
.
(49)
A
The following process is similar to the concentration of partially entangled three-photon W state. In the end, all parties simultaneously select the cases which the two photons emitted from different output spatial modes of the first PBS. After that, let this photons severally pass through the adjacent PBS and HWP45 and all parties will measure the single photon polarization state in the basis H , V . When all detection results are H , then the maximally entangled N-photon W class state can be obtained
W
A1 A2
AN
1 (V N H
The corresponding success probability is
A1
A1
H
H
A2
H
A2
V
AN
AN
).
H
A1
V
A2
H
AN
(50)
Pt PP 1 2
i 1
j 1
k 1
PN 1 P1i P2 j PN 1k .
(51)
In addition, when all parties choose the remaining items, they can repeat the concentration process which is similar to above.
5. Discussion and summary
CC E
PT
ED
M
A
N
U
SC R
IP T
By far, we have completely explained the concentration process for Bell-like state, partially entangled three-photon W state, partially entangled N-photon W state, respectively, and finally obtained the maximally entangled state with a certain probability. It should be noted that the key to success of our ECP is selecting the cases where the two input photons emitted from different spatial modes of the first PBS. For other items, it can be used in next round concentration, which can increase the total success probability. It is well known that the linear optical method is a promising approach for constructing quantum networks and the computational power of passive and active linear optical elements have been investigated extensively, which shows that linear optical elements are enough to implement reliable quantum computation and provide important network primitives. Compared with other ECPs, the ECPs we propose have some merits. First of all, the devices used in the concentration process are linear, which simplifies their realization in the experiment. Second, it only needs to implement a local operation on the auxiliary single photon to obtain the maximally entangled state from initial partially entangled state. Third, it only requires a partially entangled state and an auxiliary single photon, not two partially entangled states. Forth, current ECP reduces the requirements on the coefficients of partially entangled state, in other words, we do not need to know the accurate coefficients in advance but require the coefficients of ancillary single photon state are the same as initial partially entangled state. As a matter of fact, in the practical concentration process, the accurate coefficients can be acquired by measuring enough amounts of the initial less-entangled samples [43]. Last but not least, the resource in this ECP can be utilized adequately by repeating the concentration process for the remaining items and can achieve a higher success probability. The curves of the success probability of our ECP can be plotted from Eq. (13), (45) for the Bell-like state and three-photon W state, respectively, which is shown in Fig. 4 and Fig. 5. It should be noted that the relation between the total success probability of obtaining the maximally entangled state from the partially entangled Bell-like state and the entanglement E is illustrated in Fig. 4. For the partially entangled state
A0 B0
a H
A
H
B0
b V
A0
V
B0
, E = min {2|a|2, 2|b|2}, if |a|< |b|, E = 2|a|2 [30, 32]. It is obvious
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that total success probability Pt depends on the entanglement and the interaction numbers strongly, according to Fig. 4. With the development of the E, the success probability also enlarges. When the interaction numbers N=8, the total success probability Pt is very close to 1 and the maximal value of total success probability is equal to the entanglement E in theory. In summary, we have proposed the ECPs for Bell-like state, partially entangled three-photon W state, partially entangled N-photon W state, respectively. During the whole concentration process, we just need one copy of the corresponding partially entangled state. Following some ideas in the
work by Sheng [44] et al., with the assist of single photons, the relevant maximally entangled state can be finally obtained. It needs to be point that the key to success is selecting the case that both the output spatial modes of the first PBS contain exactly one photon and only one photon. By repeating the concentration process using the remaining items, we can get a higher success probability. These advantages make our ECPs more practical and flexible in current experiment.
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Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant No. 11447153.
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References [1] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, et al. “Cloud Quantum Computing of an Atomic Nucleus,” Phys. Rev. Lett. 120, 210501 (2018). [2] P. Papanastasiou, C. Weedbrook, S. Pirandola. “Continuous-variable quantum key distribution in uniform fast-fading channels,” Phys. Rev. A, 97 032311 (2018). [3] W. Q. Liu, P. Huang, J. Y. Peng, et al. “Integrating machine learning to achieve an automatic parameter prediction for practical continuous-variable quantum key distribution,” Phys. Rev. A 97, 022316 (2018). [4] U. Vazirani and T. Vidick. “Fully Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 113, 140501 (2014). [5] C. Lupo, C. Ottaviani, P. Papanastasiou, et al. “Parameter Estimation with Almost No Public Communication for Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. 120, 220505 (2018). [6] D. B. Uwmeester, J. W. Pan, K. Mattle, et al. “Experimental quantum teleportation,” Nature, 390, 575-579 (1997). [7] X. L. Wang, X. D. Cai, Z. E. Su, et al. “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516-519 (2015). [8] Y. H. Li, X. L. Li, L. P. Nie, et al. “Quantum Teleportation of Three and Four-Qubit State Using Multi-qubit Cluster States,” Int. J. Theory Phys. 55(3), 1820-1823 (2016). [9] A. E. Ulanov, D. Sychev, A. A. Pushkina, et al. “Quantum Teleportation Between Discrete and Continuous Encodings of an Optical Qubit,” Phys. Rev. Lett. 118, 160501 (2017). [10] J. Liu, Z. W. Mo, S. Q. Sun. “Controlled Dense Coding Using the Maximal Slice States,” Int. J. Theory Phys. 55(4), 2182-2188 (2016). [11] M. B. Tian, G. F. Zhang. “Improving the capacity of quantum dense coding by weak measurement and reversal measurement,” Quantum Inf. Process. 17:19 (2018). [12] W. Zhang, D. S. Ding, Y. B. Sheng, et al. “Quantum Secure Direct Communication with Quantum Memory,” Phys. Rev. Lett. 118, 220501 (2017). [13] S. Hassanpour, M. Houshmand. “Efficient controlled quantum secure direct communication based on GHZ-like states,” Quantum Inf. Process. 14(2), 739-753 (2015). [14] F. Z. Wu, G. J. Yang, H. B. Wang, et al. “High-capacity quantum secure direct communication with two-photon six-qubit hyperentangled states,” Sci. China Phys. Mech. 60, 120313 (2017). [15] Y. Q. Cui, J. G. Gao. “Bidirectional Quantum Secure Direct Communication in Trapped Ion Systems,” 55(3), 1706-1709 (2016).
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[16] R. Rahaman and M. G. Parker. “Quantum scheme for secret sharing based on local distinguishability,” Phys. Rev. A 91, 022330 (2015). [17] H. W. Qin, Y. W. Dai. “Proactive quantum secret sharing,” Quantum Info. Process. 14(11), 4237-4244 (2015). [18] S. Mishra, C. Shukla, A. Pathak, et al. “An Integrated Hierarchical Dynamic Quantum Secret Sharing Protocol,” Int. J. Theory phys. 54(9), 3143-3154 (2015). [19] H. W. Qin, Y. W. Dai. “Dynamic quantum secret sharing by using d-dimensional GHZ state,” Quantum Inf. Process. 16:64 (2017). [20] G. Gao, Y. Wang, and D. Wang. “Cryptanalysis of a semi-quantum secret sharing scheme based on Bell states,” Mod. Phys. Lett. B 32(9), 1850117 (2018). [21] Z. H. Huang. “Quantum State Sharing of an Arbitrary Three-Qubit State by Using a Seven-Qubit Entangled State,” Int. J. Theory Phys. 54(9), 3438-3441 (2015). [22] W. L. Li, C. Li, and H. S. Song. “Quantum synchronization and quantum state sharing in an irregular complex network,” Phys. Rev. E 95, 022204 (2017). [23] J. Y. Peng, M. Q. Bai, and Z. W. Mo. “Hierarchical and probabilistic quantum state sharing via a non-maximally entangled |χ〉state,” Chinese Phys. B 23(1), 010304 (2014). [24] M. Jiang, D. Y. Dong. “An efficient scheme for multi-party quantum state sharing of an arbitrary multi-qubit state with one GHZ channel,” Quantum Inf. Process. 12(2), 841-851 (2013). [25] Z. Dou, G. Xu, X. B. Chen, et al. “ A secure rational quantum state sharing protocol,” Sci. China Inform. Sci. 61:022501 (2018). [26] C. H. Bennett, G. Brassard, S. Popescu, et al. “Purification of noisy entanglement and faithful teleportation via noisy channel,” Phys. Rev. Lett. 76, 722 (1996) [27] Y. B. Sheng, F. G. Deng, and H. Y. Zhou. “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr Nonlinearity,” Phys. Rev A 77, 042308 (2008). [28] C. Wang, Y. Zhang, R. Zhang. “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express 19, 25685-25695 (2011). [29] Y. B. Sheng and L. Zhou. “Deterministic polarization entanglement purification using time-bin entanglement,” Laser Phys. Lett. 11, 085203 (2014). [30] B. S. Shi, Y. K. Jiang and G. C. Guo. “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000). [31] C. H. Bennett, H. J. Bernstein, S. Popescu, et al. “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996). [32] S. Bose, V. Vedral and P. L. Knight. “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194 (1999). [33] T. Yamamoto, M. Koashi, N. Imoto. “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001). [34] Z. Zhao, J. W. Pan, M. S. Zhan. “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001). [35] Z. L. Cao and M. Yang. “Entanglement distillation for three-particle W class states,” J. Phys. B-At. Mol. Opt. 36 4245 (2003). [36] L. H. Zhang, M. Yang, Z. L. Cao. “Entanglement concentration for unknown W class states,” Phys. A 374, 611-616 (2007)
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Figure captions
Fig. 1. The schematic diagram of the concentration process for arbitrary Bell-like state. S0 and S1 are the single photon source. PBSi (i=1, 2, 3) represents the polarization beam splitter, which transmits the photon in the horizontal polarization H and reflects the photon in the vertical polarization V . HWP90 denotes the 90。half wave plate and it can accomplish the job H V , HWP45 represents 45。 half wave plate whose function is given by
) , V 1/ 2 ( H V ) . DH (DV) denotes the photon detection. Aj (j=0, 1, 2, 3, 4, 5) represents
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H 1/ 2 ( H V
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the spatial mode of PBS.
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Fig. 2.
A schematic diagram for the concentration process of the three-photon W class state. A, B and C denote
the photon held by Alice, Bob and Charlie, respectively. The dotted lines connecting A, B, and C represent the partially entanglement three-photon W state shared by them.
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A Schematic diagram for the concentration process of arbitrary partially entangled N-photon W class
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Fig. 3.
state. The basic devices used here are the same with the concentration process of partially entangled three-photon W state. The dotted lines connect the Alice, Bob, …, represent the partially entangled N-photon W state
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There are some single photon sources on Alice's side, Bob's side, …, respectively which emit the single photon with nether forms
Fig. 4. The relation between the total success probability Pt for obtaining the maximally entangled Bell-like state from the partially entangled Bell-like state and the entanglement E. It can be found that the entanglement E and the interaction numbers N have great effect on the total success probability. The maximal value of Pt is almost equal to 1 when N=8.
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Fig. 5.
The total success probability Pt of acquiring the maximally entangled three-photon W state from the
partially entangled three-photon W state versus the initial coefficient α when the interaction numbers N=M=1, 2, 3, 4, 6, respectively. Here, we choose 2 1/ 3 and the variation range of α is (0, 2 / 3) . It is evident that with the increase of interaction numbers, the total success probability Pt also magnifies. The maximal value of
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Pt occurs when =1/ 3 .
S0030-4026(18)31347-0 https://doi.org/10.1016/j.ijleo.2018.09.044 IJLEO 61483
To appear in: Received date: Accepted date:
8-7-2018 13-9-2018
Please cite this article as: Wang X, Hu Z-Ning, Entanglement concentration for photon systems assisted with single photons, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.09.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Entanglement concentration for photon systems assisted with single photons Xiong Wang*, Zhan-Ning Hu Department of Physics, School of Science, Tianjin Polytechnic University, Tianjin, 300387, People’s Republic of
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(*Corresponding author E-mail: [email protected])
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Abstract The entanglement concentration protocols (ECPs) are presented for concentrating arbitrary partially entangled Bell-like state, N-photon GHZ state and N-photon W state, respectively, and the corresponding maximally entangled state can be obtained finally with a certain probability. Besides, compared with other ECPs, the accurate coefficients of the initial partially entangled state in this ECP do not need to know in advance. Moreover, current ECPs can be repeated to acquire a higher success probability. The devices used in the schemes are linear which leads to flexible operations and improves the performance greatly in the present experiment and our ECPs may be useful in the long-distance quantum information processing.
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1. Introduction
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Keywords: linear optics; partially entangled state; entanglement concentration; repeating concentration; quantum communication
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Entanglement is an important source in the fields of quantum communication and quantum information, and is widely used in quantum computing [1], quantum key distribution [2-5], quantum teleportation [6-9]，quantum dense coding [10,11]，quantum secure direct communication [12-15], quantum secret sharing [16-20] and quantum state sharing [21-25] and so on. The maximally entangled states are usually required in above applications as an entanglement channel. The general process is to prepare the maximally entangled states locally and distribute it to distant parties. However, the entanglement degree of the states may be degraded due to the influence of channel noise during the storage and distribution process, and it will transform the maximally entangled states into non-maximally entangled states or mixed states. Hence, it is quite essential to reconstruct the maximally entangled states from the partially entangled states. In order to overcome this difficulty, two concepts, called entanglement purification [26-30] and entanglement concentration [31-44] came into being. When the initial maximally entangled state transforms into a mixed entangled state, at this point, Communication parties can use the entanglement purification to improve the fidelity of the systems. However, if the initial maximally entangled state becomes a partially entangled pure state, one can exploit entanglement concentration to distill the maximally entangled state from the non-maximally entangled pure state to reconstruct a subset of systems in a maximally entangled state. By and large, the entanglement purification is more general but the entanglement concentration is more efficient and most of the existing ECPs
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are based on the entanglement concentration. In long-distance quantum communication, the photons are considered as the ideal candidate for quantum communication because of their advantages such as fast transmission, manipulate easily, and anti-interference strongly. In 1996, Bennett [31] et al. proposed the first entanglement concentration protocol, called Schmidt projection method. By performing some collective measurements on many particles, they extracted Einstein-Podolsky-Rosen (EPR) pairs. The success probability of this ECP is quite low because it is very hard to operate in the experiment. In 1999, Bose [32] et al. proposed an improved ECP based on the idea of entanglement swapping. Nevertheless, the collective Bell-state measurements are required which made it is also difficult to execute in experiment. Yamamoto [33] et al. and Zhao [34] et al. proposed two similar ECPs, independently. In their scheme, the parity check gate was made by using some PBSs. However, to ensure that each of the four output spatial modes of the two PBSs exits only one photon in the photon measurement, the communication parties need to employ the single photon detectors to probe the photons emitted from the four spatial modes. But the ideal single photon detector is still difficult to produce at the current experimental level. The protocols described above usually require two similar copies of partially entangled state. In addition, the above all ECPs pay attention to the concentration process of Bell-like state or GHZ state, there are few protocols for concentrating W state, which cannot be converted into GHZ state just using local operations and classical communication. For instance, in 2003, Cao [35] proposed an ECP for W-class state using the joint unitary transformation. In 2007, Zhang [36] et al. proposed an ECP for unknown triparticle W class states based on the appropriate Bell state measurements. In 2008, Sheng [37] et al. proposed an ECP using the cross-Kerr nonlinearities. The strong cross-Kerr media is required in their protocol for getting a higher efficiency. In 2012, Gu [38] proposed an interesting ECP for multiphoton W states with the help of single photon and this ECP only needs linear optics. Zhou [39] proposed an efficient ECP for electron-spin W state assisted with charge detection. In 2013, Sheng [40] et al. proposed a different ECP for W state using the quantum dot and optical microcavities and the advantage of this ECP is that during the whole concentration process, the less-entangled W state is not destroyed and only consumes some single photons at a cost. In 2015, Sheng [41] et al. proposed a special ECP for N-particle W state assisted by parity check gates. Du [42] et al. proposed a heralded ECP for photon systems with linear optical elements. Inspired by above, in this paper, we propose the ECPs for arbitrary Bell-like state, three-photon W state, N-photon W states, respectively, assisted with linear elements. Compare with traditional ECPs, our ECP has some advantages. First, the devices used in the concentration process are linear, which simplifies their realization in the experiment. Second, it only needs to implement a local operation on the auxiliary single photon to obtain the maximally entangled state from initial partially entangled state. Third, it only requires one copy of the partially entangled state, not two copies. Forth, current ECP reduces the requirements on the coefficients of partially entangled state, in other words, we do not need to know the accurate coefficients in advance but require the coefficients of ancillary single photon state are the same as initial partially entangled state. The paper is organized as follows: In Sec. 2, we will explain the concentration process of Bell-like state. In Sec. 3, the concentration process of three-photon W state is elucidated in detail and the corresponding success probability is calculated later on. In Sec. 4, the concentration process of N-photon W state is displayed here. In Sec. 5, a brief discussion and summary for the whole concentration process is arranged here.
2. Entanglement concentration for Bell-like state In this section, let us see the specific concentration process of the Bell-like state, shown in Fig. 1, the general form of Bell-like state is given by
a H
A0 B0
H
A0
B0
b V
V
A0
B0
.
(1)
a H
A1
A1
b V
A1
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Here, suppose the two coefficients a and b are real numbers and satisfy the relation a2+b2=1. The photon A0 and B0 emitted from single photon source S0 are distributed to distant parties, Alice and Bob, respectively. At the same time, the assistant single photon emitted from source S1 is prepared by Alice in spatial mode A1 has the following form .
(2)
'A a V
A1
1
b H
A1
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Before the two photons A0 and B0 pass through the PBS1, the single photon A1 is firstly sent to the HWP90, which completes the job H V . After that, the Eq. (2) will transform into .
(3)
Thus, the composite system composed of the three-photon can be described as
(a H
H
A0
a2 H
A0
ab( H
A0
b V
B0
H
B0
H
V
B0
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' A1
A0 B0
V
A0
B0
b2 V
A1
) (a V
N
A0
A
A0 B0 A1
H
A1
M
V
A0
V
V
A1
H
B0
V
B0
b H
A1
)
A1
A1
).
(4)
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Next, two photons A0 and B0 will pass through the PBS1. Obviously, there are two items will lead to the two photons emitted from the different output spatial modes, which means that both the two output spatial modes contain only one photon. The others will result the two photons emitted from the same spatial modes. If we choose the first case, after the PBS2, the composite system will project into
1
A4 B0 A5
ab( H
A4
H
B0
H
A5
V
V
A4
V
B0
A5
).
(5)
1 ' A B A ab / 2 ( H
A5
4 0 5
ab / 2 ( H
A5
H
B0
H
V
B0
A5
V
V
A5
B0
V
)H
B0
)V
A4
A4
.
(6)
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The corresponding success probability is P1=2a2b2. In addition, let the photon A4 pass through the HWP45, the Eq. (5) is transformed into
Finally, after the PBS3, by detecting the single photon polarization state in the basis H , V , the two-photon maximally entangled state can be obtained. That is to say, when the detection result is H , we can get
A5 B0
1 (H 2
A5
H
B0
V
A5
V
B0
).
(7)
Otherwise, if the detection result is V , we can get
1 (H 2
A5 B0
H
A5
B0
V
V
A5
B0
).
(8)
2
A0 B0 A1
a2 H
A0
H
B0
V
A1
b2 V
V
A0
H
B0
A1
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Eq. (7) and Eq. (8) are also the two-photon maximally entangled state. Alice or Bob, implements a local phase-flip operation on one of the two photons, Eq. (8) can be converted into Eq. (7). On the other side, if we select second case, in other words, the composite system will collapse to .
(9)
2
A5 B0
a2 H
A5
H
B0
b2 V
A5
If DV fires, we can get A5 B0
a 2 H
A5
H
B0
b2 V
V
B0
V
A5
.
B0
.
(10)
(11)
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After the PBS2, HWP45 and PBS3, respectively, just like the Eq. (7) and Eq. (8), when DH fires, we can get
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By using a phase-flip operation on one of the two photons, Eq. (11) can be converted into Eq. (10). It is easy to find that Eq. (10) has the similar form with Eq. (1) which means Eq. (10) can be concentrated by repeating our ECP. We only need to replace a2 and b2 with a ' a 2 / a4 b4 and b ' b2 / a 4 b4 , respectively, by this time, Alice needs to prepare another ancillary single photon with the form a ' H b ' V . After a series of operations, eventually, the maximally entangled
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Bell-like state can be distilled with the success probability P2=2(a4+b4)a'2b'2=2a4b4/(a4+b4). Similarly, the success probability of third round is P3=2a8b8/(a4+b4)(a8+b8), the success probability of Nth interaction is given by
(a 4 b4 )(a8 b8 )
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PN
N
2a 2 b 2
N
(a 2 b 2 ) N
N
.
(12)
Therefore, the total success probability for obtaining the maximally entangled Bell-like state is
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N
PN
N 1
2a 2 b 2
N
(a 4 b4 )(a8 b8 )
(a 2 b 2 ) N
N
.
(13)
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By far, we have introduced the entanglement concentration protocol for Bell-like state. In fact, one can find that this ECP can be easily extended to concentrate N-photon GHZ state. The partially entangled N-photon GHZ state can be written as
A1 A2
AN
a H
A1
H
A2
H
AN
b V
A1
V
A2
V
AN
.
(14)
Where a2+b2=1. A1, A2, …, AN represent the N photons shared by N parties, Alice, Bob, Charlie and so on. After the ECP, the N-photon maximally entangled GHZ state can be obtained and has the same total success probability with Bell-like state if one of the N parties exploits an assistant single photon and implements this ECP in the same way as the situation of the partially entangled Bell-like state.
3. Entanglement concentration process for three-photon W class state In this section, we will illustrate the concentration process of three-photon W state and the corresponding schematic drawing is shown in Fig. 2. This ECP can divide into two steps, which are implemented by Bob and Charlie, respectively.
ABC
V
H
A
H
B
H
C
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A
H
B
C
H
H
A
B
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Without loss of generality, we assume that the partially entangled three-photon W state shared by Alice, Bob and Charlie has the following form
V
,
C
(15)
B1
H
2 2
B1
2 2
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where the coefficients α, β, γ are real numbers and satisfy the relation α2+β2+γ2=1. The single photon source S1 on Bob's side emits the single photon with the form (16)
H
(17)
V
.
B1
B1
2
V
2
B1
2
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Firstly, make the single photon enter the HWP90, the single photon is transformed into
2
B1
,
B1
2
2 2
V
A
H
2 2
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1
ABC
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H
H
B
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The composite system composed of the four photons ABCB1 is
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2
H
A
A
H
V
B
C
V
B
V
H
B1
V
C
B1
H
C
B1
2 2 2 2 2 2
H V
H
V
A
H
A
H
B
H
A
H
B
B
V
C
V
C
C
B1
H
H
B1
B1
. (18)
Next, let two photons B and B1 pass through the PBS1, distinctly, there are three items will result them emitted from different output spatial modes and the others will cause them emitted from the
A
same spatial mode. In this time, if we pick up the first situation, the state
' ABCB1
2
2
V
2 2
A
H
H
A
B
H
H
B
C
V
H
C
B1
H
B1
.
2 2
H
A
ABCB1
V
B
will collapse to
H
C
V
B1
(19)
The success probability is P11 = (2α2β2+β2γ2)/(α2+β2). Here, the subscript 1 represents the first step of the concentration and the superscript 1 denotes the first round concentration executed by Bob.
The Eq. (19) can be rewritten as
' ABCB1
2 2
V
2
H
A
H
2 2 2
H
B
H
A
V
B
H
C
H
C
B1
B1
H
2 2 2
V
A
H
B
C
V
B1
.
(20)
After the PBS2 and HWP45, Eq. (20) will convert into
(
V
2 2 2
H
2 2 2
C
H
2 2 2
)H C
H
B5
C
V
B5
V
A
H
B5
H
A
H
2 2 2
H
A
B4
V
2 2 2 H
2 2 2
H
B5
(
V
A
C
H
A
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AB5CB4
H
B5
C
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A
H
V
B5
C
)V
B4
.
(21)
U
Finally, by detecting the single photon polarization state in the basis H , V . When the detection result is H , Bob will get the following state
2 2
2
V
H
A
B5
H
H
2 2 2
N
AB5C
C
2 2 2
H
V
A
H
B5
C
A
H
M
A
B5
V
C
.
(22)
2 2
2
V
PT
AB5C
ED
Otherwise, if the detection result is V , Bob will get the following state
2 2
H
H
A
B5
H
H
B5
C
V
C
2 2 2
H
AB5C5
B5
H
C
.
(23)
By performing a phase-flip operation on one of three photons, the state into 1
V
A
CC E
2
A
AB5C
can be transformed
.
A
On the other hand, when we select the rest items, that is to say, the state
'' ABCB1
2 2 2
V
2 2
A
H
H
A
B
H
H
B
C
V
V
2
B
2 2
1
C
V
B1
.
H
A
ABCB1
V
B
will collapse to
H
C
H
B1
(24)
Following the same rules mentioned above, after the PBSs and HWP45, if DHB clock, Bob will get
1
AB5C
2
2 2
V
A
H
2 2
H
A
H
B5
H
C
V
B5
2
C
H
2 2
A
V
B5
H
C
.
(25)
AB5C
2
2 2
V
2 2
A
V
H
A
B5
H
H
B5
2
C
H
C
H
2 2
A
.
V
B5
H
C
(26)
SC R
1
IP T
If DVB clock, Bob will get
Only a phase-flip operation required can convert Eq. (26) into Eq. (25). Apparently, the state shown in Eq. (26) is a partially entangled state. If we utilize ' 2 / 4 4 2 2 ,
N
U
' 2 / 4 4 2 2 , ' / 4 4 2 2 to substitute 2 / 2 2 , 2 / 2 2 , / 2 2 , respectively, and it can be found that Eq. (26) has the same form with Eq. (15). At
A
this moment, Bob prepares a single photon with the following form
'
H
M
'B
'2 '2
1
B1
'
'2 '2
V
B1
.
(27)
ED
With the same principle introduced above, after a series of operations, we can acquire the maximally entangled state. The success probability of the second round is
2 4 4 2 4 2 , P 2 ( 2 )( 4 4 )
PT
2 1
(28)
A
CC E
Similarly, the success probability of the third round, …, the Nth round can be described as
P13
2 8 8 6 8 2 , ( 2 2 )( 4 4 )( 8 8 )
(29)
......
2 2 2 2 N
P1 N
N
( 2 2 )( 4 4 )
N
2 2 . ( 2 2 )
2
N
N
N
(30)
Here, the N denotes the interaction numbers of first step concentration process. The total success probability of the first step concentration process is
P1 P11 P12
P1N P1N .
(31)
N 1
When Bob implements the first step concentration process successfully, he will inform Charlie to
carry out the second step concentration process. To be specific, the single photon source S2 on Charles's side emits the single photon with the form
C1
H
2 2
C1
V
2 2
.
C1
(32)
Charlie lets the single photon pass through the HWP90 and the associated state can be written as
AB5C
'C1
2
V
2 2 2 2 2
A
H
2 2 2 2 2
H
2 2 2 2 2
H
B5
H
A
A
H
V
B5
C
V
B5
V
H
C1
V
C
C1
H
C
2
2
V
A
V
2 2 2 2 2
C1
H
2 2 2 2 2
H
A
H
H
B5
B5
H
C
V
2 2 2 2 2
A
C1
IP T
AB5CC1
H
V
C
SC R
B5
C
H
H
C1
C1
moment, when Charlie picks up the former, the state
V
2 2 2 2 2
ED
A
H
2 2 2 2 2
will collapse to
N
A
AB5CC1
M
1
AB5CC1
U
. (33) It is evidence that there are three items will result the two photons emitted from different output spatial modes and the others will cause the two photon are in the same output spatial mode. At the
H
2 2 2 2 2
H
A
A
H
B5
V
H
B5
H
C1
H
C
V
B5
H
C
C
V
C1
C1
.
(34)
PT
With the success probabilityP21=3α2γ2/(γ2+2α2)(γ2+α2). Eq. (34) can be rewritten as
AB5CC1
1 (V 3
CC E
1
H
A
H
A
H
V
B5
H
B5
C
H
C
V
C1
A
After passing through PBS5 and HWP45, the state 1
1
AB5C4C5
1 (V 3 H H
A
A
A
H V
H
B5
B5
B5
V
H
C5
C1
V
H
B5
C
H
C1
(35) will be transformed into
H
)H
C4
H C V 5
A
).
AB5CC1
C5
H
A
A
V
B5
1 (V 3 H
5B
H
A
C5
H
H )C 5
B5
H
V. C
C5
(36)
4
Finally, Charlie will detect the single photon polarization state in the basis H , V . When the
detection result is
1
, he will get
H
AB5C5
1 (V 3
A
H
H
B5
C5
H
A
V
B5
H
H
C5
H
A
V
B5
C5
).
(37)
).
(38)
If the detection result is V , he will get
1
AB5C5
1 (V 3
A
H
H
B5
C5
H
A
V
B5
H
H
C5
H
A
V
B5
C5
AB5C5
.
However, when Charlie chooses the latter in Eq. (33), the state
2 2 2 2 2
V
2
H
2 2 2 2 2 2
H
A
A
H
B5
H
V
A
B5
H
A
H
C
V
B5
V
C
C
C1
V
C1
H
C1
.
(39)
M
2 2 2 2 2
will collapse to
U
AB5CC1
2
N
2
AB5CC1
AB5C5
SC R
can be transformed into 1
IP T
Charlie can carry out a local phase-flip operation on one of the three photons and the state 1
The processing method is similar to Bob's. Charlie can replace the coefficients of Eq. (39) with
ED
'' 2 / 2 4 4 , '' 2 / 2 4 4 , '' 2 / 2 4 4 , respectively. Meanwhile, he prepares a single photon with the following form
''C
PT
1
''
'' '' 2
2
H
C1
''
''2 ''2
V
C1
.
(40)
CC E
He can repeat the concentration process described above and the success probability of second round is
P2 2
3 4 4 , ( 2 2 )( 4 4 )( 2 2 2 )
(41)
Analogously, the success probability of the third round, …, the Nth round can be written as
A
3 8 8 , P2 2 ( 2 )( 4 4 )( 8 8 )( 2 2 2 ) 3
(42)
......
3 2 2 M
P2 M
( 2 2 )( 4 4 )
M
( 2 2 )( 2 2 2 ) M
M
Thus, the total success probability of the second concentration process is
.
(43)
P2 P2 . M
(44)
M 1
The total success probability of our ECP for obtaining the maximally three-photon W state is
N =1
M 1
N M . P PP 1 2 P1 P2
(45)
IP T
4. Concentration process of arbitrary partially entangled N-photon W class state
A1 A2
AN
1 V
H
A1
N H
A1
H
A2
H
AN
V
A2
2 H .
AN
A1
V
H
A2
AN
U
SC R
The ECP in Section 3 is still workable if we directly extend the present protocol to concentrate the maximally N-photon W class state from the partially entangled N-photon W class state. The schematic diagram is shown in Fig. 3 and the arbitrary partially entangled N-photon W class state has the following form
(46)
2
N 2 1.
A
2
M
and satisfy this relation 1 2
1
1=
2
ED
2 1
PT
2=
CC E
N 1
N
Here, the subscripts A1, A2, …, AN represent the N photons shared by N parties, say Alice, Bob, … respectively. Every participant holds one photon. The coefficients 1 , 2 , , N are real numbers
H 1
2
1
3 2 1
H 1
2
2 22 2 1
3 32 2 1
V 1,
(47)
V 1,
(48)
......
1
=
H 1
12 N 2
N 12 N 2
V
N
.
(49)
A
The following process is similar to the concentration of partially entangled three-photon W state. In the end, all parties simultaneously select the cases which the two photons emitted from different output spatial modes of the first PBS. After that, let this photons severally pass through the adjacent PBS and HWP45 and all parties will measure the single photon polarization state in the basis H , V . When all detection results are H , then the maximally entangled N-photon W class state can be obtained
W
A1 A2
AN
1 (V N H
The corresponding success probability is
A1
A1
H
H
A2
H
A2
V
AN
AN
).
H
A1
V
A2
H
AN
(50)
Pt PP 1 2
i 1
j 1
k 1
PN 1 P1i P2 j PN 1k .
(51)
In addition, when all parties choose the remaining items, they can repeat the concentration process which is similar to above.
5. Discussion and summary
CC E
PT
ED
M
A
N
U
SC R
IP T
By far, we have completely explained the concentration process for Bell-like state, partially entangled three-photon W state, partially entangled N-photon W state, respectively, and finally obtained the maximally entangled state with a certain probability. It should be noted that the key to success of our ECP is selecting the cases where the two input photons emitted from different spatial modes of the first PBS. For other items, it can be used in next round concentration, which can increase the total success probability. It is well known that the linear optical method is a promising approach for constructing quantum networks and the computational power of passive and active linear optical elements have been investigated extensively, which shows that linear optical elements are enough to implement reliable quantum computation and provide important network primitives. Compared with other ECPs, the ECPs we propose have some merits. First of all, the devices used in the concentration process are linear, which simplifies their realization in the experiment. Second, it only needs to implement a local operation on the auxiliary single photon to obtain the maximally entangled state from initial partially entangled state. Third, it only requires a partially entangled state and an auxiliary single photon, not two partially entangled states. Forth, current ECP reduces the requirements on the coefficients of partially entangled state, in other words, we do not need to know the accurate coefficients in advance but require the coefficients of ancillary single photon state are the same as initial partially entangled state. As a matter of fact, in the practical concentration process, the accurate coefficients can be acquired by measuring enough amounts of the initial less-entangled samples [43]. Last but not least, the resource in this ECP can be utilized adequately by repeating the concentration process for the remaining items and can achieve a higher success probability. The curves of the success probability of our ECP can be plotted from Eq. (13), (45) for the Bell-like state and three-photon W state, respectively, which is shown in Fig. 4 and Fig. 5. It should be noted that the relation between the total success probability of obtaining the maximally entangled state from the partially entangled Bell-like state and the entanglement E is illustrated in Fig. 4. For the partially entangled state
A0 B0
a H
A
H
B0
b V
A0
V
B0
, E = min {2|a|2, 2|b|2}, if |a|< |b|, E = 2|a|2 [30, 32]. It is obvious
A
that total success probability Pt depends on the entanglement and the interaction numbers strongly, according to Fig. 4. With the development of the E, the success probability also enlarges. When the interaction numbers N=8, the total success probability Pt is very close to 1 and the maximal value of total success probability is equal to the entanglement E in theory. In summary, we have proposed the ECPs for Bell-like state, partially entangled three-photon W state, partially entangled N-photon W state, respectively. During the whole concentration process, we just need one copy of the corresponding partially entangled state. Following some ideas in the
work by Sheng [44] et al., with the assist of single photons, the relevant maximally entangled state can be finally obtained. It needs to be point that the key to success is selecting the case that both the output spatial modes of the first PBS contain exactly one photon and only one photon. By repeating the concentration process using the remaining items, we can get a higher success probability. These advantages make our ECPs more practical and flexible in current experiment.
IP T
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant No. 11447153.
A
CC E
PT
ED
M
A
N
U
SC R
References [1] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, et al. “Cloud Quantum Computing of an Atomic Nucleus,” Phys. Rev. Lett. 120, 210501 (2018). [2] P. Papanastasiou, C. Weedbrook, S. Pirandola. “Continuous-variable quantum key distribution in uniform fast-fading channels,” Phys. Rev. A, 97 032311 (2018). [3] W. Q. Liu, P. Huang, J. Y. Peng, et al. “Integrating machine learning to achieve an automatic parameter prediction for practical continuous-variable quantum key distribution,” Phys. Rev. A 97, 022316 (2018). [4] U. Vazirani and T. Vidick. “Fully Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 113, 140501 (2014). [5] C. Lupo, C. Ottaviani, P. Papanastasiou, et al. “Parameter Estimation with Almost No Public Communication for Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. 120, 220505 (2018). [6] D. B. Uwmeester, J. W. Pan, K. Mattle, et al. “Experimental quantum teleportation,” Nature, 390, 575-579 (1997). [7] X. L. Wang, X. D. Cai, Z. E. Su, et al. “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516-519 (2015). [8] Y. H. Li, X. L. Li, L. P. Nie, et al. “Quantum Teleportation of Three and Four-Qubit State Using Multi-qubit Cluster States,” Int. J. Theory Phys. 55(3), 1820-1823 (2016). [9] A. E. Ulanov, D. Sychev, A. A. Pushkina, et al. “Quantum Teleportation Between Discrete and Continuous Encodings of an Optical Qubit,” Phys. Rev. Lett. 118, 160501 (2017). [10] J. Liu, Z. W. Mo, S. Q. Sun. “Controlled Dense Coding Using the Maximal Slice States,” Int. J. Theory Phys. 55(4), 2182-2188 (2016). [11] M. B. Tian, G. F. Zhang. “Improving the capacity of quantum dense coding by weak measurement and reversal measurement,” Quantum Inf. Process. 17:19 (2018). [12] W. Zhang, D. S. Ding, Y. B. Sheng, et al. “Quantum Secure Direct Communication with Quantum Memory,” Phys. Rev. Lett. 118, 220501 (2017). [13] S. Hassanpour, M. Houshmand. “Efficient controlled quantum secure direct communication based on GHZ-like states,” Quantum Inf. Process. 14(2), 739-753 (2015). [14] F. Z. Wu, G. J. Yang, H. B. Wang, et al. “High-capacity quantum secure direct communication with two-photon six-qubit hyperentangled states,” Sci. China Phys. Mech. 60, 120313 (2017). [15] Y. Q. Cui, J. G. Gao. “Bidirectional Quantum Secure Direct Communication in Trapped Ion Systems,” 55(3), 1706-1709 (2016).
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Figure captions
Fig. 1. The schematic diagram of the concentration process for arbitrary Bell-like state. S0 and S1 are the single photon source. PBSi (i=1, 2, 3) represents the polarization beam splitter, which transmits the photon in the horizontal polarization H and reflects the photon in the vertical polarization V . HWP90 denotes the 90。half wave plate and it can accomplish the job H V , HWP45 represents 45。 half wave plate whose function is given by
) , V 1/ 2 ( H V ) . DH (DV) denotes the photon detection. Aj (j=0, 1, 2, 3, 4, 5) represents
U
H 1/ 2 ( H V
CC E
PT
ED
M
A
N
the spatial mode of PBS.
A
Fig. 2.
A schematic diagram for the concentration process of the three-photon W class state. A, B and C denote
the photon held by Alice, Bob and Charlie, respectively. The dotted lines connecting A, B, and C represent the partially entanglement three-photon W state shared by them.
IP T SC R U N A M
A Schematic diagram for the concentration process of arbitrary partially entangled N-photon W class
ED
Fig. 3.
state. The basic devices used here are the same with the concentration process of partially entangled three-photon W state. The dotted lines connect the Alice, Bob, …, represent the partially entangled N-photon W state
A
CC E
PT
There are some single photon sources on Alice's side, Bob's side, …, respectively which emit the single photon with nether forms
Fig. 4. The relation between the total success probability Pt for obtaining the maximally entangled Bell-like state from the partially entangled Bell-like state and the entanglement E. It can be found that the entanglement E and the interaction numbers N have great effect on the total success probability. The maximal value of Pt is almost equal to 1 when N=8.
IP T SC R
Fig. 5.
The total success probability Pt of acquiring the maximally entangled three-photon W state from the
partially entangled three-photon W state versus the initial coefficient α when the interaction numbers N=M=1, 2, 3, 4, 6, respectively. Here, we choose 2 1/ 3 and the variation range of α is (0, 2 / 3) . It is evident that with the increase of interaction numbers, the total success probability Pt also magnifies. The maximal value of
A
CC E
PT
ED
M
A
N
U
Pt occurs when =1/ 3 .