s11128-024-04334-9

Quantum Information Processing (2024) 23:134
https://doi.org/10.1007/s11128-024-04334-9
Entanglement concentration of W state using linear optics
with a higher success probability
Fang-Fang Du1
· Ming Ma1 · Xue-Mei Ren1 · Gang Fan1
Received: 4 September 2023 / Accepted: 29 February 2024 / Published online: 29 March 2024
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024
Abstract
Entanglement concentration protocol (ECP), as a practical technique, can be use to
preclude degraded fidelity and improve security in long-haul quantum communication. We propose an efficient ECP for less-entangled unknown W states with simple
linear-optics elements and effective single-photon detectors, resorting to time-delay
degree of freedom. Moreover, in contrast to previous ECPs proposed for W states,
the scheme has the advantages over requiring less quantum resource without auxiliary
photons, comparatively simplified circuits, involving neither post-selection techniques
nor photon-number-resolving detectors to distinguish the parity outcome, and being
provided with a higher success probability by reusing the less-entangled states.
Keywords Entanglement concentration protocol · W states · Linear-optics elements ·
Time delay
1 Introduction
Quantum entanglement unquestionably plays a crucial role to motivate unexampled
insights into future applications in quantum information processing (QIP) [1–14],
especially quantum key distribution [15–18], quantum dense coding [19, 20], quantum
secret sharing [21], and quantum secure direct communication [22–35]. Unfortunately, the quality of nonlocal entangled quantum systems can be inevitably lowered
by the transmission loss and channel noise in tangible distribution process, which
will inevitably limit the transmission range and information integrity of QIP [36–
45]. For example, the maximally entangled quantum states [46] may degenerate into
mixed states or partially entangled pure states, resulting in incredible degradation
of the fidelity and downgrading security of long-distance quantum communication.
B Fang-Fang Du
Duff@nuc.edu.cn
1
Science and Technology on Electronic Test and Measurement Laboratory, North University of China,
Taiyuan 030051, China
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F. Du et al.
Entanglement purification protocol [47–59], and entanglement concentration protocol
(ECP) [60, 61] have been regarded as two efficacious techniques to overcome the above
difficulties. In especial, ECPs can be used to distill maximally entangled states from
non-maximally entangled pure states available for the imperfect entangled photonic
source or entangled decoherence from the quantum storage process.
The ECPs for photon systems[61–69, 74–76] have been continuously optimized and
developed, e.g., an efficient Kerr-effect-assisted ECP [64], photon-assisted ECPs [65,
66], hyperentanglement-assisted ECPs [67, 68], and linear-optics-assisted ECPs [69].
Sheng et al. set up two single-photon-assisted ECPs for partially entangled photon pairs
with known parameters by exploiting linear optics [70] and cross-Kerr-nonlinearities
[72, 73], respectively. Sequentially, by orchestrating auxiliary time-delay degree
of freedom, Jiang et al. [74] proposed two heralded and high-efficient ECPs for
less-entangled Bell and Greenberger-Horne-Zeilinger (GHZ) states with unknown
parameters with available linear optics and common single-photon detectors, where
postselection techniques or photon-number-resolving detectors are not required and
the success probability of ECP for Bell state is exactly heralded by the detection signatures without destroying the incident qubits. Further, they [75] also presented the
practically enhanced hyper-ECP for hyperentangled Bell States with a higher success
probability than the linear-optics-based hyper-ECP with unknown coefficients [76].
So far, the existing ECPs are mainly directed against distilling partially (hyper)entangled Bell states (or GHZ) states [77–79]. Recently, in view of the commendably
intrinsic robustness and meaningful applications of W state in QIP [80], the generation of W entanglement has attracted much attention both in theory and experiment.
Consequently, it is considerably valuable to discuss ECP in the partial-entangled W
states, e.g., single-photon-assisted [81] ECPs for photon systems in the less-entangled
W states [82], ECPs in W states with linear optics [83], cross-Kerr nonlinearity, and
cavity-assisted system, have been reported. Sheng et al. first proposed two-step ECPs
for arbitrary three-photon W states with known three parameters assisted by two single
photons [84]. The second protocol adopting the cross-Kerr nonlinearity through some
iteration rounds was higher success probability than the first one with linear optics.
In this paper, we propose a high-efficient ECP for unknown less-entangled W states
with linear optical elements, resorting to time-delay degree of freedom (DoF). Firstly,
compared with the existing ones [82–84], our scheme with unknown coefficients offers
a broader range of applications. Secondly, it requires neither photon number-resolving
detectors nor post-selection techniques. Thirdly, as the incorporating time-delay DoF
enables precise prediction of results from single-photon detectors, three parties can
acquire not only the partially entangled state with two new unknown parameters without time interval, but also the standard W state corresponding to time interval, where
the partially entangled can be utilized sufficiently with a higher success probability
through an iterative process.
2 ECP for unknown W states with linear optics
We present an efficient ECP for three-photon less-entangled W states with two
unknown parameters by some simple linear-optics elements. With the help of the
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Entanglement concentration of W state
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134
time-delay DoF, our ECP can be precisely heralded by the the results of detection
without recourse to the post-selection principle. Suppose that two three-photon systems in the maximally entangled W states |φ ABC and |φ A B C are generated from
two different entanglement sources S1 and S2 , respectively.
1
|φ ABC = √ (|HVH + |HHV + |VHH) ABC ,
3
1
|φ A B C = √ (|HVH + |HHV + |VHH) A B C .
3
(1)
Due to the influence of channel noise, the maximally entangled W states decay to
the less-entangled W states with two unknown parameters, respectively, i.e., from the
states |φ ABC and |φ A B C to that |φ ABC and |φ A B C , respectively,
|φ ABC = α|H A |V B |H C + β(|H A |H B |V C + |V A |H B |H C ),
|φ A B C = α|H A |V B |H C + β(|H A |H B |V C + |V A |H B |H C ). (2)
where the unknown parameters α and β meet the normalization relation α 2 +2β 2 = 1.
Thus the state of the composite system composed of six photons A, B, C, A , B and
C is
|1 = [α|H A |V B |H C + β(|H A |H B |V C + |V A |H B |H C )]
⊗[α|H A |V B |H C + β(|H A |H B |V C + |V A |H B |H C )]. (3)
Firstly, as shown in Fig. 1, two photons B and B undergo bit-flip operation through
◦
the HWP45 , i.e., |H |V . The state |1 of the composite system becomes
|2 = [α|H A |H B |H C + β(|H A |V B |V C + |V A |V B |H C )]
⊗[α|H A |H B |H C + β(|H A |V B |V C + |V A |V B |H C )]. (4)
Secondly, two photons A and B pass through a 50:50 beam splitter (BS), that is,
1
BS
| A −→ √ (| A + | B ),
2
1
BS
| B −→ √ (| A − | B ).
2
(5)
where represents H - or V -polarized state. Then the state |2 evolves into
|3 =
1
[α(|H A + |H B )|H B |H C + β(|H A + |H B )|V B |V C
2
+ β(|V A + |V B )|V B |H C ] ⊗ [α|H A (|H A − |H B )|H C + β|H A (|V A − |V B )|V C + β|V A (|V A − |V B )|H C .
(6)
For the convenience of distilling the maximally entangled W state shown in Eq.
(6) from the state |3 , Alice, Bob, and Charlie complete the following operations to
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F. Du et al.
Fig. 1 Schematic diagram of the ECP for unknown W states. S1 and S2 are two entanglement sources for
three-photon systems in the states |φABC and |φ A B C , respectively. BS is a 50:50 beam splitter. PBSi (i
= 1, 2, 3, 4, 5, 6, 7) is a polarized beam splitter which transforms the horizontal polarized component |H ◦
◦
and reflects the vertical polarized one |V . HWP45 and HWP22.5 represent half-wave plates oriented at
45◦ and 22.5◦ , respectively, completing the bit-flip and Hadmard operations. Di (i = 1, 2, 3, 4, 5, 6) is a
single-photon detector. The optical circle on the arms denotes time delay t0 or t1
clearly describe this process, according to the Hong-Ou-Mandel effect [85], the state
| 3 is rewritten
1
| 3 = √ [α 2 |HHHH A BCC (|HHAA − |HH B B ) + β 2 |HVHV A BCC 2
⊗ (|VVAA − |VV B B ) + β 2 |VVHH A BCC (|VVAA − |VV B B )
+ βα|HVVH A BCC (|HHAA − |HH B B )]
1
+ β 2 [|HVVV A BCC (|HVAA − |HV AB + |VH AB − |HV B B )
2
+ |VVVH A BCC (|HVAA − |HV AB + |VH AB − |HV B B )]
1
+ αβ[|HHHV A BCC (|HVAA − |HV AB + |VH AB − |HV B B )
2
+|VHHH A BCC (|HVAA − |HV AB + |VH AB − |HV B B )
+ |HVHH A BCC (|HVAA − |HV AB + |VH AB − |HV B B )].
(7)
Thirdly, after the photon A(B ) of Alice (Bob) pass through PBS1 (PBS2 ), the H polarized component and V -polarized one pass through the time delays t0 and t1 ,
respectively. Here t0 and t1 satisfy ω(t0 − t1 ) = 2nπ , where n is the nonzero integer.
Then the state | 3 becomes
1
|4 = √ [α 2 |H H H H A BCC (|Ht0 Ht0 A A − |Ht0 Ht0 B B )
2
+ β 2 |HVHV A BCC (|Vt1 Vt1 AA − |Vt1 Vt1 B B )
+ β 2 |VVHH A BCC (|Vt1 Vt1 AA − |Vt1 Vt1 B B )
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Entanglement concentration of W state
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134
+ βα|HVVH A BCC (|Ht0 Ht0 AA − |Ht0 Ht0 B B )]
1
+ β 2 [|HVVV A BCC (|Ht0 Vt1 AA − |Ht0 Vt1 AB 2
+ |Vt1 Ht0 AB − |Ht0 Vt1 B B )
+ |VVVH A BCC (|Ht0 Vt1 AA − |Ht0 Vt1 AB + |Vt1 Ht0 AB − |Ht0 Vt1 B B )]
1
+ αβ[|HHHV A BCC (|Ht0 Vt1 AA − |Ht0 Vt1 AB 2
+|Vt1 Ht0 AB − |Ht0 Vt1 B B )
+ |VHHH A BCC (|Ht0 Vt1 AA − |Ht0 Vt1 AB + |Vt1 Ht0 AB − |Ht0 Vt1 B B )
+ |HVHH A BCC (|Ht0 Vt1 AA − |Ht0 Vt1 AB + |Vt1 Ht0 AB − |Ht0 Vt1 B B )].
(8)
Fourthly, after two photons A and B traverse the PBS3 and PBS4 , respectively, Alice
◦
and Bob perform the Hadmard operation on photons A and B with a HWP22.5 , that
is,
◦
1
HWP22.5
|H −−−−−−→ √ (|H + |V ),
2
◦
1
HWP22.5
|V −−−−−−→ √ (|H − |V ).
2
(9)
Due to the Hong-Ou-Mandel effect, the final result evolved from |4 to
α 4 + 2β 4 +
|5 =
|ϕ A BC |H C [(|H A |H A + |V A |V A )DAA (0)
√
2 2
− (|H B |H B + |V B |V B )D B B (0)]
α 4 + 2β 4 −
+
|ϕ A BC |H C [2|H A |V A DAA (0) − 2|H B |V B D B B (0)]
√
2 2
√
3αβ
+ √ |φ+
A BC |H C [(|H t0 |H t1 ) A A − 2(|H t0 |V t1 ) A A
2 2
− (|V t0 |V t1 )AA − (|H t0 |H t1 ) B B + 2(|H t0 |V t1 ) B B + (|V t0 |V t1 ) B B ]
1
+ √ αβ|H A |V B |H C |V C [(|H t0 |H t0 )AA + 2(|H t0 |V t0 )AA
2 2
+ (|V t0 |V t0 )AA − (|H t0 |H t0 ) B B − 2(|H t0 |V t0 ) B B − (|V t0 |V t0 ) B B ]
1
+ √ β 2 |H A |V B |V C |V C (|H t0 |H t1 )AA − 2(|H t0 |V t1 )AA
2 2
− (|V t0 |V t1 )AA − (|H t0 |H t1 ) B B 123
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F. Du et al.
+ 2(|H t0 |V t1 ) B B + (|V t0 |V t1 ) B B ]
1
+ √ β 2 |V A |V B |H C |V C [(|H t0 |H t1 )AA − 2(|H t0 |V t1 )AA
2 2
− (|V t0 |V t1 )AA − (|H t0 |H t1 ) B B + 2(|H t0 |V t1 ) B B + (|V t0 |V t1 ) B B ].
(10)
where
1
|φ+
A BC = √ (|H A |V B |H C + |H A |H B |V C + |V A |H B |H C ).
3
|ϕ+
A BC = α |H A |H B |H C + β (|H A |V B |V C + |V A |V B |H C ),
|ϕ−
A BC = α |H A |H B |H C − β (|H A |V B |V C + |V A |V B |H C ), (11)
with α = √ 4α
2
α +2β 4
and β = √ 4β
2
α +2β 4
. Here D A A (0) (D B B (0)) represents that
there is no relative time delay with regard to the photon A (B ). That means that the
single-photon detector (D1 , D2 , D3 , or D4 ) responses without time interval.
Finally, after three photons A, B and C go away PBS5 , PBS6 , and PBS7 , respectively, they are measured on the basis (|H , |V ) with single-photon detectors D1 , D2 ,
D3 , D4 , D5 , or D6 . The relationship between the response outcomes of the singlephoton detector, the corresponding output states, the feed-forward operation on photon
B, and the success probability of the corresponding output states is shown in Table
1. It is worth noting that the measurement result of photon C can indicate whether
our ECP is successful or not. In detail, if the detector D6 responds, the ECP fails
with the probability |αβ|2 + 2|β|4 . Conversely, if the detector D5 responds, the ECP
successes. Moreover, if the detector pair (Di , D j ) (i, j = 1, 2, 3, 4) trigger with a
specific time interval |t0 − t1 |, the desired maximally entangled W state |φ+
A BC with
2
the success probability of P = 3|αβ| can get without arbitrary feed-forward operation. Otherwise, it means that detector pair (Di , D j ) fire without time interval. In
such case, they can get again the less-entangled W state |ϕ+
A BC with the probability
of (|α|4 + 2|β|4 )/2, as the state |ϕ−
can
be
transformed
into the one |ϕ+
A BC
A BC by applying π/2-phase shifter to operate on the polarized state |V B with the same
probability.
Obviously, after the state |ϕ+
A BC is similar to Eq. (4), which can be further
condensed to obtain the state |φ+
A BC in the next round. Therefore, the success probability of the ECP can be increased by recycling state |ϕ+
A BC , and its probability is
4
(|α|4 + 2|β|4 ) · 3|α β |2 = |α|3|αβ|
4 +2|β|4 in the second round. Figure 2 shows the success
probability P of our proposed scheme vs the parameter α (β = 21 (1 − α 2 )) in the
first round (n = 1) and second round (n = 2), respectively. The maximum success
probability has increased from 0.37 to 0.51.
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Entanglement concentration of W state
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Table 1 The relationship between the detector response, corresponding output results, and the operation
on photon B required to complete ECP. The operation σz = |H H | − |V V | can be accomplished with
a half-wave plate oriented at 0◦
Single-photon detectors
D1 , D2 , D3 , D4
D5
(D1 , D2 ), (D3 , D4 )
t
t
t
t
(D10 , D11 ), (D10 , D21 )
t0
t1
t0
t
(D2 , D2 ), (D3 , D31 )
t
t
t
t
(D30 , D41 ), (D40 , D41 )
t
Output results
Operation on photon B
Success probability
|ϕ+
A BC None
(|α|4 + 2|β|4 )
|ϕ−
A BC +
|φ A BC σz
None
3|αβ|2
t
(Di 0 , D j0 ) (i,j = 1,2,3,4) indicates that (Di , D j ) triggers with a time interval of |t0 − t1 |
Fig. 2 The success probability
P of our ECP vs the parameter α
in the first round (n = 1) and
second round (n = 2),
respectively
3 Discussion and summary
The impact of channel noise on the maximally entangled two-particle and three-particle
states (e.g., Bell states, GHZ states, or W states) cannot be negligible in the longdistance QIP. Fortunately, entanglement concentration, as a practical technique, can
be use to preclude degraded fidelity and improve security. As the environmental decoherence makes the information of the less-entangled state uncertain, the ECP with
unknown parameters is relatively more practical than that with known parameters.
However, the success probabilities of the existing ECPs for unknown less-entangled
states based on only linear optics are low originated from two fundamental challenges.
One is that PBSs can complete parity-check outcomes with the help of photon-numberresolving detectors, which are difficult in experiment by inefficiency. The other one is
unable to recycle the concentration process, as the photon pair coincide at the same
detector leading to the destructive measurements.
Nowadays, we have proposed the ECP for the W state with two unknown parameters assisted by time-delay DoF. At first, if the detector D6 to the photon C responds,
the ECP fails and this case is discarded. Otherwise, if the detector D5 responds, the
ECP successes. Moreover, as the detectors (Di , D j ) (i, j = 1, 2, 3, 4) to two photons
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F. Du et al.
A and B can accurately distinguish the case with absence of time delay (|Ht0 Ht0 AA ,
|Ht0 |Vt0 AA , |Vt0 |Vt0 AA , |Ht0 |Ht0 B B , |Ht0 |Vt0 B B , |Vt0 |Vt0 B B ) from the one
(|Ht0 |Ht1 AA , |Ht0 |Vt1 AA , |Vt0 |Vt1 AA , |Ht0 |Ht1 B B , |Ht0 |Vt1 B B , |Vt0 |Vt1 B B )
with occurrence of time interval |t0 − t1 |, the former is accompanied by the lessentangled W state |ϕ+
A BC with given feed-forward operation and the latter is the
standard W state |φ+
A BC without arbitrary feed-forward operation in sequence. Fur+
ther, the state |ϕ A BC can be reused in the next round of the ECP available for
increasing success probability.
In conclusion, we propose a high-efficient ECP for unknown less-entangled W
states by utilizing only linear optical elements, resorting to time-delay DoF. It has some
advantages compared with the existing ECPs with known coefficients [82–84]. Firstly,
our scheme with unknown coefficients except precise coefficients α and β offers a
broader range of applications. Secondly, auxiliary photon, photon number-resolving
detectors, and post-selection techniques do not require. Thirdly, as the incorporating
time-delay DoF enables precise prediction of results from single-photon detectors,
three parties can acquire not only the partially entangled |ϕ+
A BC state with two new
unknown parameters without time interval, but also the standard W state |φ+
A BC corresponding to time interval |t0 − t1 |, where the partially entangled |ϕ+
can
be
A BC recycled in the second round of the ECP improving the success probability from 0.37
to 0.51. Therefore, these characteristics make our ECP highly applicable with current
technology and promising for long-distance QIP applications and future advancements.
Declarations
Conflict of interest The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.
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