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Local conversion of four Einstein-Podolsky-Rosen photon pairs into four-photon polarization-entangled decoherence-free states with non-photon-number-resolving detectors

Hong-Fu Wang, Shou Zhang, Ai-Dong Zhu, X. X. Yi, and Kyu-Hwang Yeon

Open Access Open Access

Abstract

We propose a linear-optics-based scheme for local conversion of four Einstein-Podolsky-Rosen photon pairs distributed among five parties into four-photon polarization-entangled decoherence-free states using local operations and classical communication. The proposed setup involves simple linear optical elements and non-photon-number-resolving detectors that can only distinguish between the presence and absence of photons, and no information on the exact number of photons can be obtained. This greatly simplifies the experimental realization for linear optical quantum computation and quantum information processing.

© 2011 Optical Society of America

1. Introduction

Entanglement of bipartite system or multipartite system is not only the essential ingredient for testing local hidden variable theories against quantum mechanics, but also the key physical resource for quantum information processing, such as quantum teleportation [1], quantum computing [2, 3], quantum key distribution [4], quantum secret sharing [5, 6], quantum dense coding [7, 8], quantum secure direct communication [9, 10], etc.. In practice, however, the coherent superposition of quantum state is easily destroyed because of the uncontrolled coupling between a quantum system and the environment, which greatly reduces the quantum efficiencies in quantum information processing. To overcome the disadvantage, several strategies have been proposed to deal with quantum decoherence, such as exploiting quantum error-correction and dynamical decoupling techniques, encoding logical qubits into a decoherence-free subspace in which the structure of the state is symmetry to the system-environment interaction such that the state is absolutely collective-noise-influence independent, etc.. The decoherence-free subspace is inherently immune to decoherence resulting in that it is robust to perturbing error processes, thus it is considered to be ideally suited for concatenation in a quantum error correction codes [11]. Therefore, the decoherence-free states are very useful for long-distance quantum communication and quantum computation. The decoherence-free states, which consiste of N qubits, originally proposed by Kempe et al. [12], are invariant under any identical unitary transformation on each of the qubits. For N = 2 qubits, it exists only one decoherence-free singlet state

|ζ〉=12(|01〉12−|10〉12).
For N = 4, we obtain the four-qubit entangled decoherence-free states of the form
|ξ〉=δ|χ0〉+γ|χ1〉,
whose dimension is 2, with
|χ0〉=12(|01〉12−|10〉12)(|01〉34−|10〉34),|χ1〉=13[|0011〉1234+|1100〉1234−12(|01〉12+|10〉12)(|01〉34+|10〉34)].
The state (2) has novel property that it is sufficient to fully protect an arbitrary logical qubit against collective decoherence in contrast to the state (1), which results in that it is very useful for quantum information processing. In 2004 Bourennane et al. [13] have experimentally reported a method to generate four-photon polarization-entangled decoherence-free states via a spontaneous parametric down-conversion source. Moreover, they verified the immunity of the states by quantum state tomography of the encoded qubit. However, in Bourennane et al.’s scheme, it required to first generate two sources |χ0〉 and |χ1〉 using spontaneous parametric down-conversions, then coherently overlapped two four-photon sources to generate the desired superposition states (2), which is slightly complicated to implement. Very recently, Zou et al. [14] and Gong et al. [15] proposed schemes to generate four-photon polarization-entangled decoherence-free states based on linear optical elements and post-selection strategy. However, those schemes mentioned in Refs. [13–15] work in the destructive way because the generated four-photon polarization-entangled decoherence-free states cannot be used for further quantum information processing and quantum computation when a four-photon coincidence measurement was made on the photonic states.

In this paper, we propose a simple linear optical scheme to locally convert four Einstein-Podolsky-Rosen (EPR) photon pairs distributed among five parties into the four-photon polarization-entangled decoherence-free states with non-photon-number-resolving detectors that can only distinguish the vacuum and nonvacuum Fock number states. With the help of one of the honest parties, say Frank, the other four parties Alice, Bob, Charlie, and Dick can reliably share the four-photon polarization-entangled decoherence-free states in a nondestructive way with the certain success probability via local operations and classical communication (LOCC). The experimental realization of the present scheme would be an important step towards long-distance and long range quantum networking with presently available linear optical technology.

2. Converting four Einstein-Podolsky-Rosen photon pairs into four-photon polarization-entangled decoherence-free states

We assume that four EPR photon pairs, |ψ−〉aa′ ⊗ |ϕ+〉bb′ ⊗ |ϕ+〉cc′ ⊗ |ψ−〉dd′, are distributed such that Alice holds photon a, Bob holds photon b, Charlie holds photon c, Dick holds photon d, and Frank holds photons a′, b′, c′, and d′. The state of the system is written as

|Ψ〉T=|ψ−〉aa′⊗|ϕ+〉bb′⊗|ϕ+〉cc′⊗|ψ−〉dd′=14(|HV〉aa′−|VH〉aa′)⊗(|HH〉bb′+|VV〉bb′)⊗(|HH〉cc′+|VV〉cc′)⊗(|HV〉dd′−|VH〉dd′),
where |H〉 (|V〉) denotes the horizontal (vertical) polarization state of a photon, and |H〉 and |V〉 correspond the logical zero and one states, respectively, |0〉 ≡ |H〉 and |1〉 ≡ |V〉.

To help Alice, Bob, Charlie, and Dick to share the four-photon polarization-entangled decoherence-free states, Frank first sends his two photons in modes a′ and d′ to a beam splitter (BS), whose reflectivity and transmissivity are independent of polarizations, as shown in Fig. 1. The action of BS is given by the transformation

aa′β†→μaa1β†+1−μaa2β†,ad′β†→μaa2β†−1−μaa1β†,
where μ and 1−μ are the transmission and reflection coefficients of BS, β = H,V, and aa1β† and aa2β† denote the creation operators of the input and output modes a1 and a2. Giving
|Ψ〉′=−14{μ(−|HHVV〉ada1a2+|HVVH〉ada1a2+|VHHV〉ada1a2−|VVHH〉ada1a2)+(1−μ)(|HHVV〉ada1a2−|HVHV〉ada1a2−|VHVH〉ada1a2+|VVHH〉ada1a2)+μ(1−μ)[|HH〉ad(|2V〉a1−|2V〉a2)−(|HV〉ad+|VH〉ad)(|HV〉a1−|HV〉a2)+|VV〉ad(|2H〉a1−|2H〉a2)]}⊗(|HHHH〉bb′cc′+|HHVV〉bb′cc′+|VVHH〉bb′cc′+|VVVV〉bb′cc′).
Then let modes a1 and b′ be mixed at the PBS1 and modes a2 and c′ be mixed at the PBS2, the state of the total system is given by
|Ψ〉′​′=−14(|φ〉1+|φ〉2+|φ〉3+|φ〉4+|φ〉5),
with
|φ1〉=μ(|HVHV〉abcd|VVHH〉b1b2c1c2−|HVVH〉abcd|VVVV〉b1b2c1c2              +|VHVH〉abcd|HHVV〉b1b2c1c2−|VHHV〉abcd|HHHH〉b1b2c1c2)              +(1−μ)(|HVVH〉abcd|VVVV〉b1b2c1c2−|HHVV〉abcd|HHVV〉b1b2c1c2               −|VVHH〉abcd|VVHH〉b1b2c1c2+|VHHV〉abcd|HHHH〉b1b2c1c2),|φ2〉=(1−2μ)(|HHHH〉abcd|HV〉b2|HV〉c2+|HHVH〉abcd|HV〉b2|V〉c1|V〉c2               +|HVHH〉abcd|V〉b1|V〉b2|HV〉c2+|VHVV〉abcd|H〉b1|H〉b2|HV〉c1                +|VVHV〉abcd|HV〉b1|H〉c1|H〉c2+|VVVV〉abcd|HV〉b1|HV〉c1)|φ3〉=μ(|HHHV〉abcd|HV〉b2|H〉c1|H〉c2+|HHVV〉abcd|HV〉b2|HV〉c1               |HVVV〉abcd|V〉b1|V〉b2|HV〉c1+|VHHH〉abcd|H〉b1|H〉b2|HV〉c2              +|VVHH〉abcd|HV〉b1|HV〉c2+|VVVH〉abcd|HV〉b1|V〉c1|V〉c2)|φ4〉=(μ−1)(|HHHV〉abcd|H〉b1|H〉b2|HV〉c2+|HVHV〉abcd|HV〉b1|HV〉c2              +|HVVV〉abcd|HV〉b1|V〉c1|V〉c2+|VHHH〉abcd|HV〉b2|H〉c1|H〉c2              +|VHVH〉abcd|HV〉b2|HV〉c1+|VVVH〉abcd|V〉b1|V〉b2|HV〉c1),|φ5〉=μ(1−μ)[|HH〉ad(|2V〉b2−|2V〉c2)−|HV〉ad(|HV〉b1b2−|HV〉c1c2)                −|VH〉ad(|HV〉b1b2−|HV〉c1c2)+|VV〉ad(|2H〉b1−|2H〉c1)]                ⊗(|HHHH〉bcb2c2+|HVHV〉bcb2c1+|VHVH〉bcb1c2+|VVVV〉bcb1c2).
Next the photons in modes bi and ci (i = 1, 2) pass through a half-wave plate (HWP), whose action is given by the transformation
|H〉i→HWP12(|H〉i+|V〉i),|V〉i→HWP12(|H〉i−|V〉i),
and a polarizing beam splitter (PBS), which transmits the horizontal polarization and reflect vertical polarization, respectively.

 figure: Fig. 1

Fig. 1 Schematic diagram of converting four EPR photon pairs distributed among five parties, Alice, Bob, Charlie, Dick, and Frank into the four-photon polarization-entangled decoherence-free states. Where PBSi (i = 1,2,3,4,5,6) denote polarizing beam splitters, HWP is half-wave plate, BS is beam splitter, and Djα (j = 1,2,3,4, α = H,V) are non-photon-number-resolving detectors.

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Finally, Frank detects the photons in output modes bj and cj (j = 5, 6, 7, 8) using non-photon-number-resolving detectors, which we consider here are the realistic detectors commonly used in photonic experiments. The feature of this kind of detector is that it cannot resolve the number of the detected photons but instead tell us whether photons exist in a detection event with nonunit probability ηd. Usually the dark count of the detector is considerable low and hence here is neglected. The positive-operator-valued-measure (POVM) describing a non-photon-number-resolving detector is [16]

Πoff=∑n=0∞(1−ηd)n|n〉〈n|,Πclick=∑n=0∞[1−(1−ηd)n]|n〉〈n|.
After the procedure of detection, the state of photons in modes a, b, c, and d is given by
ρoutk=Trb5,b6,b7,b8,c5,c6,c7,c8[Πclick1αΠclick2αΠclick3αΠclick4αΠoff1βΠoff2βΠoff3βΠoff4β(|Ψ〉rr〈Ψ|)]Tra,b,c,d,b5,b6,b7,b8,c5,c6,c7,c8[Πclick1αΠclick2αΠclick3αΠclick4αΠoff1βΠoff2βΠoff3βΠoff4β(|Ψ〉rr〈Ψ|)]=12μ2−2μ+1|Ψoutk〉〈Ψoutk|,
where k ∈ {1,2,3,4}, α ≠ β ∈ {H,V}, and |Ψ〉r is the resulting state of the total system in the output modes, with
|Ψoutk〉=(±1,42,3|HVVH〉abcd±2,41,3|HVHV〉abcd±3,41,2|VHVH〉abcd−|VHHV〉abcd)+(1−μ)(±2,31,4|HVVH〉abcd±1,23,4|HHVV〉abcd±1,32,4|VVHH〉abcd+|VHHV〉abcd),
where ±λ,νδ,γ indicates that the signs of the states in |Ψoutδ〉 and |Ψoutγ〉 are “+”, and the signs of the states in |Ψoutλ〉 and |Ψoutν〉 are “−”; |Ψout1〉 corresponds to that photon detectors {D1H, D2H, D3H, D4H} (or {D1V, D2V, D3V, D4V}, or {D1H, D2H, D3V, D4V}, or {D1V, D2V, D3H, D4H}) detect photons and the others do not register any photon; |Ψout2〉 corresponds to that photon detectors {D1V, D2H, D3H, D4H} (or {D1H, D2V, D3H, D4H}, or {D1V, D2H, D3V, D4V}, or {D1H, D2V, D3V, D4V}) detect photons; |Ψout3〉 corresponds to that photon detectors {D1H, D2H, D3V, D4H} (or {D1V, D2V, D3V, D4H}, or {D1H, D2H, D3H, D4V}, or {D1V, D2V, D3H, D4V}) detect photons; |Ψout4〉 corresponds to that photon detectors {D1V, D2H, D3V, D4H} (or {D1H, D2V, D3V, D4H}, or {D1V, D2H, D3H, D4V}, or {D1H, D2V, D3H, D4V}) detect photons. The states |Ψout2〉, |Ψout3〉, and |Ψout4〉 can be easily transformed into state |Ψout1〉 by applying local transformations to photons 1, 2, 3, and 4 appropriately.

In Eq. (12), we rewrite the state |Ψout1〉 as

|Ψout1〉=3μ−12(|HV〉ab−|VH〉ab)(|HV〉cd−|VH〉cd)−(1−μ)[|VVHH〉abcd+|HHVV〉abcd−12(|HV〉ab+|VH〉ab)(|HV〉cd+|VH〉cd)]=121−3μ(1−μ)[(3μ−1)|χ0〉−3(1−μ)|χ1〉],
where |χ0〉 and |χ1〉 are given by Eq. (3). In this way the desired four-photon polarization-entangled decoherence-free states (2) is successfully distributed among Alice, Bob, Charlie, and Dick, with
δ=3μ−121−3μ(1−μ),γ=3(μ−1)21−3μ(1−μ).
The overall probability of success for obtaining the state |Ψout1〉 is
P=ηd4(2μ2−2μ+1)4,
with ηd being the quantum efficiency of photon detector. In Fig.2 we drew the 3-dimensional plot for the probability of success P as a function of the quantum efficiency ηd of photon detector and the experimental parameter μ of beam splitter.

 figure: Fig. 2

Fig. 2 The overall probability of success for the generation of four-photon polarization-entangled decoherence-free states, P, vs ηd and μ.

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3. Analysis and discussion

We now briefly discuss the feasibility of the present scheme. First, as resources, experimental realization of our scheme requires the consumption of entangled photon pairs distributed among five parties. Therefore, we require reliable sources of deterministic entangled photon pairs to serve as the qubits of interest. Experimentally, the highly entangled photon pairs can be generated by spontaneous parametric down conversion (SPDC) [17–20]. Yamamoto et al. [17] have experimentally demonstrated a robust and faithful entanglement-distribution scheme that used a state-independent encoding into decoherence-free subspaces (DFSs). They realized a high-fidelity distribution of two-photon entangled pair over a fibre communication channel, and also demonstrated that the scheme was robust against fragility of the reference frame. Walther et al. [18] have demonstrated a method to convert a GHZ state to an arbitrarily good approximation to a W state with a trade off between the fidelity of the final state and the success probability based on local positive operator valued measurements and LOCC. Tashima et al. [19] have proposed a scheme for direct transformation of two EPR photon pairs distributed among three parties into a three-photon W state via LOCC, using a polarization-dependent beam splitter and postselection, with the probability of success of 30%. Six-photon experiments for graph states entanglement were already able to detect sixfold coincidences at a rate of 40 per minute and with two-photon visibilities beyond 90% (93% in the H/V basis and 91% in the +/− basis) [20], which corresponds, respectively, to the expected generation rate and fidelity of the heralded EPR entangled photon pair if the same parameters were used. Second, in an ideal experimental condition, the probability of success in the present scheme is ηd48≤P<ηd44. It is worth pointing out that if we set μ = 0 or μ = 1 in Eq. (15), a maximal probability of success, Pmax=ηd44, can be obtained. In this case, however, one can see from Eq. (13) that the desired four-photon polarization-entangled decoherence-free states cannot be distributed successfully among Alice, Bob, Charlie, and Dick when μ = 1 is chosen. While when μ = 0 is chosen, a product state of two EPR pairs, |ψ−〉ac ⊗ |ψ−〉bd, will be obtained. Therefore, the action of the BS in Fig. 1 is inefficient for both the cases of μ = 0 and μ = 1 and the maximal probability of success of the present scheme does not exceed ηd4/4. The minimal probability of success of the present scheme is Pmin=ηd48 when we choose μ=12, and in this case the BS in Fig. 1 is a 50/50 balanced beam splitter. The value of the parameter μ is thus chosen as 0 < μ < 1. Third, the photon detectors used in our scheme are non-photon-number-resolving detectors that only can distinguish the vacuum and nonvacuum Fock number states, a sophisticated single-photon detector distinguishing one or two photon states is unnecessary. For scalable linear optics quantum computation, the required quantum efficiency ηd of the single-photon detectors is extremely high, e.g., for gate success with probability p ≃ 0.99, ηd ≥0.999987 [21]. Although experiments for single-photon detectors have made tremendous progress, such detectors still go beyond the current experimental technologies. This greatly decreases the high-quality requirements of photon detectors in practical realization. Therefore, our scheme is simple and feasible and is within the reach of current linear optical technology.

4. Conclusions

In conclusion, we have proposed a scheme, based only on linear optics and standard non-photon-number-resolving detectors, that allowed to locally convert four EPR photon pairs distributed among five parties into the heralded four-photon polarization-entangled decoherence-free states via LOCC. Our scheme does not need photon-number resolution in the detection process, since the scheme inherently suppresses situations in which more than one photon is emitted into each of the detection modes. On the other hand, our scheme is inherently robust to detector inefficiency since a click from each of the detectors is never recorded if one photon is lost, each click of the detectors should correspond exactly to the existence of one photon. We hope that our work will be useful for future quantum computation and communication networks.

Acknowledgments

This work was supported by the Talent Program of Yanbian University of China under Grant No. 950010001, and the National Natural Science Foundation of China under Grant Nos. 61068001, 11165015, 11064016, 61078011, and 10935010.

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Figures (2)
Fig. 1
Fig. 1 Schematic diagram of converting four EPR photon pairs distributed among five parties, Alice, Bob, Charlie, Dick, and Frank into the four-photon polarization-entangled decoherence-free states. Where PBSi (i = 1,2,3,4,5,6) denote polarizing beam splitters, HWP is half-wave plate, BS is beam splitter, and Djα (j = 1,2,3,4, α = H,V) are non-photon-number-resolving detectors.

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Fig. 2
Fig. 2 The overall probability of success for the generation of four-photon polarization-entangled decoherence-free states, P, vs ηd and μ.

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Equations (15)

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| ζ 〉 = 1 2 ( | 01 〉 12 − | 10 〉 12 ) .
| ξ 〉 = δ | χ 0 〉 + γ | χ 1 〉 ,
| χ 0 〉 = 1 2 ( | 01 〉 12 − | 10 〉 12 ) ( | 01 〉 34 − | 10 〉 34 ) , | χ 1 〉 = 1 3 [ | 0011 〉 1234 + | 1100 〉 1234 − 1 2 ( | 01 〉 12 + | 10 〉 12 ) ( | 01 〉 34 + | 10 〉 34 ) ] .
| Ψ 〉 T = | ψ − 〉 a a ′ ⊗ | ϕ + 〉 b b ′ ⊗ | ϕ + 〉 c c ′ ⊗ | ψ − 〉 d d ′ = 1 4 ( | H V 〉 a a ′ − | V H 〉 a a ′ ) ⊗ ( | H H 〉 b b ′ + | V V 〉 b b ′ ) ⊗ ( | H H 〉 c c ′ + | V V 〉 c c ′ ) ⊗ ( | H V 〉 d d ′ − | V H 〉 d d ′ ) ,
a a ′ β † → μ a a 1 β † + 1 − μ a a 2 β † , a d ′ β † → μ a a 2 β † − 1 − μ a a 1 β † ,
| Ψ 〉 ′ = − 1 4 { μ ( − | H H V V 〉 a d a 1 a 2 + | H V V H 〉 a d a 1 a 2 + | V H H V 〉 a d a 1 a 2 − | V V H H 〉 a d a 1 a 2 ) + ( 1 − μ ) ( | H H V V 〉 a d a 1 a 2 − | H V H V 〉 a d a 1 a 2 − | V H V H 〉 a d a 1 a 2 + | V V H H 〉 a d a 1 a 2 ) + μ ( 1 − μ ) [ | H H 〉 a d ( | 2 V 〉 a 1 − | 2 V 〉 a 2 ) − ( | H V 〉 a d + | V H 〉 a d ) ( | H V 〉 a 1 − | H V 〉 a 2 ) + | V V 〉 a d ( | 2 H 〉 a 1 − | 2 H 〉 a 2 ) ] } ⊗ ( | H H H H 〉 b b ′ c c ′ + | H H V V 〉 b b ′ c c ′ + | V V H H 〉 b b ′ c c ′ + | V V V V 〉 b b ′ c c ′ ) .
| Ψ 〉 ′ ​ ′ = − 1 4 ( | φ 〉 1 + | φ 〉 2 + | φ 〉 3 + | φ 〉 4 + | φ 〉 5 ) ,
| φ 1 〉 = μ ( | H V H V 〉 a b c d | V V H H 〉 b 1 b 2 c 1 c 2 − | H V V H 〉 a b c d | V V V V 〉 b 1 b 2 c 1 c 2                             + | V H V H 〉 a b c d | H H V V 〉 b 1 b 2 c 1 c 2 − | V H H V 〉 a b c d | H H H H 〉 b 1 b 2 c 1 c 2 )                             + ( 1 − μ ) ( | H V V H 〉 a b c d | V V V V 〉 b 1 b 2 c 1 c 2 − | H H V V 〉 a b c d | H H V V 〉 b 1 b 2 c 1 c 2                               − | V V H H 〉 a b c d | V V H H 〉 b 1 b 2 c 1 c 2 + | V H H V 〉 a b c d | H H H H 〉 b 1 b 2 c 1 c 2 ) , | φ 2 〉 = ( 1 − 2 μ ) ( | H H H H 〉 a b c d | H V 〉 b 2 | H V 〉 c 2 + | H H V H 〉 a b c d | H V 〉 b 2 | V 〉 c 1 | V 〉 c 2                               + | H V H H 〉 a b c d | V 〉 b 1 | V 〉 b 2 | H V 〉 c 2 + | V H V V 〉 a b c d | H 〉 b 1 | H 〉 b 2 | H V 〉 c 1                                 + | V V H V 〉 a b c d | H V 〉 b 1 | H 〉 c 1 | H 〉 c 2 + | V V V V 〉 a b c d | H V 〉 b 1 | H V 〉 c 1 ) | φ 3 〉 = μ ( | H H H V 〉 a b c d | H V 〉 b 2 | H 〉 c 1 | H 〉 c 2 + | H H V V 〉 a b c d | H V 〉 b 2 | H V 〉 c 1                               | H V V V 〉 a b c d | V 〉 b 1 | V 〉 b 2 | H V 〉 c 1 + | V H H H 〉 a b c d | H 〉 b 1 | H 〉 b 2 | H V 〉 c 2                             + | V V H H 〉 a b c d | H V 〉 b 1 | H V 〉 c 2 + | V V V H 〉 a b c d | H V 〉 b 1 | V 〉 c 1 | V 〉 c 2 ) | φ 4 〉 = ( μ − 1 ) ( | H H H V 〉 a b c d | H 〉 b 1 | H 〉 b 2 | H V 〉 c 2 + | H V H V 〉 a b c d | H V 〉 b 1 | H V 〉 c 2                             + | H V V V 〉 a b c d | H V 〉 b 1 | V 〉 c 1 | V 〉 c 2 + | V H H H 〉 a b c d | H V 〉 b 2 | H 〉 c 1 | H 〉 c 2                             + | V H V H 〉 a b c d | H V 〉 b 2 | H V 〉 c 1 + | V V V H 〉 a b c d | V 〉 b 1 | V 〉 b 2 | H V 〉 c 1 ) , | φ 5 〉 = μ ( 1 − μ ) [ | H H 〉 a d ( | 2 V 〉 b 2 − | 2 V 〉 c 2 ) − | H V 〉 a d ( | H V 〉 b 1 b 2 − | H V 〉 c 1 c 2 )                                 − | V H 〉 a d ( | H V 〉 b 1 b 2 − | H V 〉 c 1 c 2 ) + | V V 〉 a d ( | 2 H 〉 b 1 − | 2 H 〉 c 1 ) ]                                 ⊗ ( | H H H H 〉 b c b 2 c 2 + | H V H V 〉 b c b 2 c 1 + | V H V H 〉 b c b 1 c 2 + | V V V V 〉 b c b 1 c 2 ) .
| H 〉 i → HWP 1 2 ( | H 〉 i + | V 〉 i ) , | V 〉 i → HWP 1 2 ( | H 〉 i − | V 〉 i ) ,
Π off = ∑ n = 0 ∞ ( 1 − η d ) n | n 〉 〈 n | , Π click = ∑ n = 0 ∞ [ 1 − ( 1 − η d ) n ] | n 〉 〈 n | .
ρ out k = Tr b 5 , b 6 , b 7 , b 8 , c 5 , c 6 , c 7 , c 8 [ Π click 1 α Π click 2 α Π click 3 α Π click 4 α Π off 1 β Π off 2 β Π off 3 β Π off 4 β ( | Ψ 〉 r r 〈 Ψ | ) ] Tr a , b , c , d , b 5 , b 6 , b 7 , b 8 , c 5 , c 6 , c 7 , c 8 [ Π click 1 α Π click 2 α Π click 3 α Π click 4 α Π off 1 β Π off 2 β Π off 3 β Π off 4 β ( | Ψ 〉 r r 〈 Ψ | ) ] = 1 2 μ 2 − 2 μ + 1 | Ψ out k 〉 〈 Ψ out k | ,
| Ψ out k 〉 = ( ± 1 , 4 2 , 3 | H V V H 〉 a b c d ± 2 , 4 1 , 3 | H V H V 〉 a b c d ± 3 , 4 1 , 2 | V H V H 〉 a b c d − | V H H V 〉 a b c d ) + ( 1 − μ ) ( ± 2 , 3 1 , 4 | H V V H 〉 a b c d ± 1 , 2 3 , 4 | H H V V 〉 a b c d ± 1 , 3 2 , 4 | V V H H 〉 a b c d + | V H H V 〉 a b c d ) ,
| Ψ out 1 〉 = 3 μ − 1 2 ( | H V 〉 a b − | V H 〉 a b ) ( | H V 〉 c d − | V H 〉 c d ) − ( 1 − μ ) [ | V V H H 〉 a b c d + | H H V V 〉 a b c d − 1 2 ( | H V 〉 a b + | V H 〉 a b ) ( | H V 〉 c d + | V H 〉 c d ) ] = 1 2 1 − 3 μ ( 1 − μ ) [ ( 3 μ − 1 ) | χ 0 〉 − 3 ( 1 − μ ) | χ 1 〉 ] ,
δ = 3 μ − 1 2 1 − 3 μ ( 1 − μ ) , γ = 3 ( μ − 1 ) 2 1 − 3 μ ( 1 − μ ) .
P = η d 4 ( 2 μ 2 − 2 μ + 1 ) 4 ,
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