Quantum Inf Process (2014) 13:1025–1034DOI 10.1007/s11128-013-0707-5

Hybrid entanglement concentration using quantum dotand microcavity coupled system

Chuan Wang · Cong Cao · Ling-yan He ·Chuan-lin Zhang

Received: 12 November 2012 / Accepted: 25 November 2013 / Published online: 10 December 2013© Springer Science+Business Media New York 2013

Abstract We present two hybrid entanglement concentration protocols based onquantum dots (QDs) and optical microcavity coupled systems. The system is the-oretically analyzed and used for photon and electron hybrid entanglement generation.Also, the proposed system can be further used for parity check that allows a quantumnondemolition measurement on the spin parity. By performing parity check processon electron spins, the entangled state can be concentrated into maximally entangledstate efficiently.

Keywords Hybrid entanglement · Concentration · Quantum dot · Microcavity

1 Introduction

Quantum entanglement in hybrid systems is important for quantum repeaters andlong-distance quantum communications. And the ability to distribute highly entan-gled multi-photon states between two distant parties is an essential requirement inquantum communications, as the entanglement distribution process will inevitablybe disturbed by the channel noise, and finally, less entangled state will be obtained.In 1996, Bennett et al. [1] proposed a theoretical protocol to distill a subset systemin a maximally entangled state from a set of systems in a less entangled pure state,called entanglement concentration protocol (ECP). In the proposed protocol, the twocommunication parties can obtain information about the coefficients by performing

C. Wang (B)State Key Laboratory of Information Photonics and Optical Communications, School of Science,Beijing University of Posts and Telecommunications, Beijing 100876, Chinae-mail: wangchuan82@gmail.com

C. Cao · L. He · C. ZhangSchool of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

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the collective and nondestructive measurement. In recent years, significant advanceshave been achieved in the field of entanglement concentration. In 2000, Shi et al. [2]proposed an ECP using a two-particle collective unitary evaluation. In 2001, two ECPsusing polarization beam splitters were proposed by Yamamoto et al. [3] and Zhao etal. [4], respectively. The experimental realization has also been demonstrated in therelated work [5,6]. In 2008, Sheng et al. [7] proposed an efficient ECP using cross-Kerrnonlinearity, and they also designed an ECP for single-photon entangled systems [8].Later, various ECPs were proposed by researchers [9–13]. In 2011, we [14] proposedan efficient entanglement purification and concentration protocol based on electron-spin entangled state using quantum dot and microcavity systems. In 2012, the optimalprotocol of ECP with the help of cross-Kerr nonlinearities was proposed in which onlyone copy of the less entangled state is used, and the efficiency is improved [15,16].Recently, Ren et al. [17] proposed a hyperentanglement concentration protocol usinglinear optics, and they further developed the solid-state-based hyperentanglement con-centration exploiting the diamond nitrogen vacancy centers inside photonic-crystalcavities [18].

Electron spin based on solid-state system is one of the most promising candidatesfor quantum information processing (QIP). Recently, a variety of QIP protocols basedon the single-electron-charged self-assembled GaAs/InAs quantum dot (QD) in a two-side micropillar system has been discussed [19–24]. Based on the optical selectionrules, an exciton with negative charge can be created on the QD spins in this system.The optical excitation dynamics of the cavity field operator a and the trion X− dipoleoperator σ− can be written as [25,26]

da

dt= −

[i(ωC − ω) + κ + κs

2

]a − gσ− − √

κ ain − √κ a′

in + H (1)

dσ−dt

= −[i(ωX− − ω) + γ

2

]σ− − gσz a + G (2)

where the coefficient g represents the coupling strength between the microcavity andQD. H and G are the noise operators related to reservoirs, and γ

2 demonstrates theexciton dipole decay rate. κ and κs are the cavity decay rate and the cavity leaky rate,respectively. The frequencies of the input photon, cavity mode and the spin-dependentoptical transition are represented by ω, ωC and ωX− , respectively.

The above Heisenberg equations of motion can be solved, and the reflection andtransmission coefficients of this system can be obtained and described as [20]

r(ω) = [i(ωX− − ω) + γ2 ][i(ωc − ω) + κs

2 ] + g2

[i(ωX− − ω) + γ2 ][i(ωc − ω) + κ + κs

2 ] + g2,

t (ω) = −κ[i(ωX− − ω) + γ2 ]

[i(ωX− − ω) + γ2 ][i(ωc − ω) + κ + κs

2 ] + g2. (3)

On the resonant condition of the system with ωc = ωX− = ω0, by taking g = 0, thereflection and transmission coefficients r and t for the uncoupled cavity system canbe written as

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r0(ω) = i(ω0 − ω) + κs2

i(ω0 − ω) + κs2 + κ

,

t0(ω) = −κ

i(ω0 − ω) + κs2 + κ

. (4)

For example, if the electron spin in QD is in the upward direction | ↑〉 with angularmomentum sz = 1/2, the exciton will be excited between the electron spin and theheavy hole with spin angular momentum s = −3/2 by the circularly polarized photonwith angular momentum S = −1. Otherwise, the exciton will be excited betweenthe electron with spin −1/2 and the heavy holes with spin +3/2 by the S = +1circularly polarized photon. So, the photon oscillated with the QD spin will feel acoupled cavity, and the nonoscillated photon will feel an uncoupled system. And therules of polarization state changing under the interaction between the photon and thespin of QD in the microcavity can be described as

|R↑,↑〉 → |L↓,↑〉, |L↑,↑〉 → −|L↑,↑〉,|R↓,↑〉 → −|R↓,↑〉, |L↓,↑〉 → |R↑,↑〉,|R↑,↓〉 → −|R↑,↓〉, |L↑,↓〉 → |R↓,↓〉,|R↓,↓〉 → |L↑,↓〉, |L↓,↓〉 → −|L↓,↓〉.

(5)

where |L〉 and |R〉 represent the state of the left and right circularly polarized photonswith angular momentum sz = ±1, respectively. The superscript arrows in the photonstate indicate the propagation direction, and the arrows represent the direction of theelectrons.

Recently, the proposed photon-spin interaction setup has been widely used in QIP[27,28]. The hybrid system combining with photons and electron spins is a good candi-date for long-distance quantum information processing, such as quantum repeater forquantum communication. As in the photon-spin hybrid quantum system, the photonicqubits are easily manipulated and the electron-spin qubits are perfect for informationstorage and processing. In 2006, Waks et al. [19] designed a dipole and microcavitycoupled system in which the cavity can be switched perfectly even in the bad cavitylimit, and quantum repeaters can be designed in this weak coupling regime efficiently.Hu et al. proposed several interesting protocols based on such QD and microcavitycoupled system, such as the quantum logic gate [20], photon entangler [21], entangle-ment swapping [23] and so on. Bonato et al. [24] proposed a system that consists ofa single-electron-charged quantum dot in a resonator which exhibits good interactionbetween photon and electron spin. In 2011, Young et al. experimentally observed thereflection spectroscopy in QD and microcavity coupled system [29], and the possibleuse for circuit QED system was discussed.

In this paper, we propose a conventional ECP based on hybrid entangled state usinga single-electron-charged self-assembled GaAs/InAs quantum dot in a micropillarsystem. Then, we introduce the parity check gate of QDs and single photons. Byexploiting an ancillary photon that passes through the QD and microcavity system,the state 1√

2(|R↑,↑〉−|L↓,↓〉) can be obtained between the two distant parties. Then,

we generalize another ECP using only one pair of nonmaximally entangled state and

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one ancillary QD. The success probability of the generalized protocol can be increasedto unity by iterance.

2 Entanglement concentration on hybrid entangled states

In the practical QIP, the maximally hybrid entangled state may inevitably suffer fromthe environment noise and become less entangled. Suppose that the two pairs of non-maximally entangled hybrid entangled states shared by the two users Alice and Bobare as follows:

|Φa〉 = α|R↑,↑〉1,2 − β|L↓,↓〉1,2, (6)

|Φb〉 = α|R↑,↑〉3,4 − β|L↓,↓〉3,4, (7)

where the coefficients α and β in the states are unknown for the users, the particlesmarked with 1 and 3 are owned by Alice, and the particles marked with 2 and 4 areowned by Bob. Also, an incident photon in the state |L↓〉 is prepared beforehand.

Figure 1 illustrates the principle of photon polarization and electron-spin hybridentanglement concentration process. In the first step of ECP, the two nonlocal usersperform a bit-flip operation on photon 4 and electron 3, respectively. The bit-flipoperation will change the polarization state of the photons |L〉 and |R〉. And the bit-flip operation on the electron will change the spins | ↑〉 to | ↓〉. Then the hybridentangled state of the two pairs of composite system becomes

|Φ ′in〉 = α2| ↑↓, R, L〉 + β2| ↓↑, L , R〉 − αβ| ↑↑, R, R〉 − αβ| ↓↓, L , L〉 (8)

During the second step of ECP, one ancillary photon is sent into the input port of themicrocavity. After the interaction between the ancillary photon and the electron oneby one, the output detectors will record the photons in different spatial modes whichreveal the parity of the electrons. For example, when a photon with left polarized stateis sent through the input mode of the two cavities, and the two electron spins are in thesame state, then the state of the input photon will be changed, and it will trigger D2

Fig. 1 Schematic diagram showing the principle of hybrid entanglement concentration process. S1 and S2represent two hybrid entanglement sources. D1 and D2 are single-photon detectors.

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detector. Otherwise, the state of the photon will remain unchanged and be detected bythe detector D1. The evolution of the photon-electron state in the parity check processcan be written as:

|L↓〉| ↑〉1| ↑〉2 ⇒ −|R↓〉| ↑〉1| ↑〉2 , (9)

|L↓〉| ↑〉1| ↓〉2 ⇒ |L↑〉| ↑〉1| ↓〉2 ,

where the superscript arrows in the photon state represent the direction along or in theopposite direction of the spin. By detecting the output mode of the input photon, onecan distinguish the spin parity of electron systems. If the two electrons are in the evenparity, the two users will keep the two entangled pairs. Otherwise, they will drop thephoton and electron pairs. Then the state of the remaining composite system after theconcentration process can be described as

|Φ ′out〉 = 1√

2

(| ↑,↑, R, R〉 + | ↓,↓, L , L〉). (10)

In the proposed ECP, the ancillary photon is sent through the two microcavities andwill be transmitted or reflected by the QD and cavity system. This process is called theparity check gate (PCG) operation on the spin of the two QDs. For example, if the twospins are in the same state, the PCG operation will reveal the even parity result andthe ancillary will be reflected once. Otherwise, if the two spins are in different states,the PCG operation will reveal to us the odd parity result and the ancillary photon willeither be reflected twice or pass through the cavities twice.

For the remaining two pairs, the state of the composite system can be described as

|Φ ′out〉 = 1

2√

2

[| ↑〉(|+e〉 + |−e〉)|R〉(∣∣+p〉 + |−p〉

)

+ | ↓〉(|+e〉 − |−e〉)|L〉(|+p〉 − |−p〉

)]

= 1

2√

2

(| ↑, R〉 + | ↓, L〉)(|+,+〉e,p + |−,−〉e,p)

+ (| ↑, R〉 − | ↓, L〉)(|+,−〉e,p + |−,+〉e,p). (11)

Alice and Bob both perform single particle measurement on spin 3 and photon 4 onthe |±〉e = 1√

2(| ↑〉± | ↓〉) and |±〉p = 1√

2(|L〉± |R〉) measuring basis, respectively.

If the measurement results on Alice’s and Bob’s sides are |+,+〉e,p or |−,−〉e,p, thetwo spins 1 and 2 are in the maximally entangled state 1√

2(| ↑, R〉+| ↓, L〉). However,

if the results are not in the same state, one of them performs a phase-flip operation onthe electron and reverts to the original state 1√

2(| ↑, R〉 + | ↓, L〉). After that, the two

nonlocal users can obtain a maximally spin entangled state by single qubit operationand single particle detection.

The yield Y of maximally entangled states in ECP is |αβ|2, which is defined as theratio of the number of maximally entangled pairs and the number of originally lessentangled pairs. In the above analysis, we did not take into account the influences of

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Fig. 2 The yield is altered with the coupling constant of QD and cavity system and the coefficient α ∈[0, 1√

2

]

the coupling strength between QDs and microcavities. Actually, this aspect plays asignificant role in the realization of entanglement concentration. In our ECP process,the yield varies with the coupling strength g. The relation between the yield that twoparties obtain a hybrid entangled pair in the maximally entangled state from a partiallyentangled state and the coupling strength and the probability amplitude α ∈ [0, 1√

2]

is shown in Fig. 2.As shown in Fig. 2, the yield of ECP increases with the coupling strength. When

the coupling strength is larger than 0.5, it is enough to obtain an optimal yield. The keyelement in the present ECP protocol is the parity check process, and we construct itwith QD and microcavity system. At present, the implementation of a large couplingstrength between QD and microcavity is still difficult in experiment. In recent experi-ments [30], the coupling strength has been reported to be g = 0.5 in microcavity withdiameter d = 1.5µm, which is enough for us to get an optimal yield in ECP.

3 Optimal hybrid entanglement concentration

In the above section, the two nonlocal users can concentrate on one pair of maxi-mally entangled state using two pairs of entangled states with unknown superposi-tion coefficients. In this section, we will propose an optimal ECP on hybrid entan-glement by exploiting ancillary single quantum dot and one pair of hybrid entan-gled state with known superposition coefficients. Compared with the previous one,the optimal protocol uses only one pair of less entangled state in each concentrat-ing step, which allows us to obtain high efficiency. And the maximally entangle-ment between QD electron and single photon can be distilled between nonlocal usersefficiently.

Assume that the initial nonmaximally hybrid entangled state can be described as

|Φa〉 = α|R↑,↑〉1,2 − β|L↓,↓〉1,2, (12)

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where the electron marked with 1 and the photon marked with 2 are shared by twononlocal users. Different from the protocol discussed above, the two amplitudes α

and β are known between the users. Then one local user prepares a QD in the state|φ〉 = α| ↑〉 + β| ↓〉 for concentration. The composite system can be described asfollows:

|Φa〉 ⊗ |φ〉 = α2|R↑,↑,↑〉1,2,e + β2|L↓,↓,↓〉1,2,e

−αβ(|R↑,↑,↓〉1,2,e + |L↓,↓,↑〉1,2,e). (13)

The concentration process only needs one step parity check operations on thetwo QD electrons performed by one local user. In detail, the hybrid state collapsesto (|R↑,↑,↓〉1,2,e + |L↓,↓,↑〉1,2,e)/

√2 if the two electrons are in the odd par-

ity after checking. Otherwise, the two nonlocal users share the state α2|R↑,↑,↑〉1,2,e + β2|L↓,↓,↓〉1,2,e. In the first case, we can expand the third particle by theorthogonal basis |±〉 = 1

2 (| ↑〉 ± | ↓〉), and the state can be described as

1√2

(|R↑,↑〉1,2(|+〉 − |−〉)e + |L↓,↓〉1,2

(|+〉 − |−〉)e

). (14)

Then, one user measures the ancillary QD in the |±〉 basis, and the maximally hybridentangled state can be concentrated to (|R↑,↑〉 + |L↓,↓〉)/√2 if the ancillary QD isin the state |+〉 or to the state (|R↑,↑〉 − |L↓,↓〉)/√2 if the ancillary QD is in thestate |−〉.

However, the second case (α2|R↑,↑,↑〉1,2,e + β2|L↓,↓,↓〉1,2,e) can also be usedfor ECP. If we project the ancillary QD onto the basis |±〉, we can obtain the state(α2|R↑,↑〉1,2 ± β2|L↓,↓〉1,2). Then, one local user can prepare another ancillaryQD in the state (α2| ↑〉 ± β2| ↓〉) for the second-round ECP. The concentrationprocess can be repeated by the process discussed above. As discussed in Ref. [16], thesuccess probability of the protocol of obtaining one pair of maximally entangled stateis

P =n∑

i=1

i∏m

(1

α2m + β2m

)2|αβ|2i . (15)

Here n represents the repeated times. The relation between the success probabilitythat the two parties obtain the maximally entangled state from a pair in the partiallyentangled state and the coefficient α is shown in Fig. 3. As n increases, the successprobability approaches to unity. Compared with the above hybrid ECP, the concen-tration on a partially entangled pure state with known probability amplitudes is moreefficient than that on a state with unknown probability amplitudes [17].

4 Experiment feasibility

In this study, the basic element to complete the ECP task is the electron-spin paritycheck process, and such process relies on the practical implementation of the cou-

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Fig. 3 The success probability is altered with the coefficient of the nonmaximally entangled state α ∈ [0, 1].Here the dot dashed line, the dashed line and the solid line represent the concentration process is repeatedone time, two times and four times, respectively

pled system. We have designed a ECP protocol by exploiting the QD and microcavitysystem which can also be used to construct a quantum network, as the nonlocal dis-tributed quantum dot can be used to generate maximally entangled states. And variousquantum information protocols can also be accomplished. However, the realization ofour protocol relies on the spin coherence time and the strong coupling between theQD and microcavities. Much experimental work has been done on the area of strongcoupling between QD and microcavities. A recent experimental study has demon-strated the strong coupling that has been observed in various systems. Yoshie et al.[31] experimentally realized the strong coupling of a single QD in a nanocavity. Thevacuum Rabi splitting is observed, which exceeds the decoherence linewidth of boththe cavity and the QD. Moreover, the coupling strength has reached g/κ = 2.4 withthe quality factor Q = 4 × 104 that has been reported in a recent experiment [32].As the coupling strength g/κ approaches to larger than 2, the numerical simulationshows that under the resonant condition with ω0 = ωc = ωX− , the performance ofthe scheme fulfills the requirements to carry out entanglement concentration and QIPtasks.

In current experiments, the electron-spin coherence time ranges to microsec-onds [33,34], and the gate operation time using optical spin control can be lim-ited within a picosecond timescale. The optical coherence time increasing to sev-eral hundred picoseconds in QDs has been reported [35,36]. Also, Gallardo et al.[37] realized the optical coupling of two distant quantum dots to the microcav-ity. The results prospect an experimental step toward the realization of quantuminformation processing using solid-state systems. Recent experiments have realizedthe dynamic control of a QD strongly coupled to a photonic-crystal cavity, andthe electro-optic switching of a probe laser coherently coupled to the cavity couldspeed up to as high as 150 MHz [38]. These results show that the QD and micro-cavity coupled device can be integrated in an on-chip optical or quantum networkwhich will be an essential building block for future quantum information processingdevices.

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5 Summary

In summary, exploiting a quantum interface between a single photon and the spinstate of a QD, we proposed two hybrid entanglement concentration protocols. Themaximal entanglement can be distilled between nonlocal users efficiently, and thishigh-fidelity entanglement can be widely used in QIP. Also, the electrons entanglementconcentration can be further applied in various branches of QIP, such as quantumcommunication [39] and entanglement analysis [40,41], which are also very importantin quantum information science and technology.

Acknowledgments This work is supported by the National Fundamental Research Program Grant No.2010CB923202, China National Natural Science Foundation Grant No. 61205117, Beijing Higher Educa-tion Young Elite Teacher Project No. YETP0456 and the Open Research Fund Program of the State KeyLaboratory of Low-Dimensional Quantum Physics, Tsinghua University Grant No. KF201301.

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