# Efficient W state entanglement concentration using quantum-dot and optical microcavities

###### Abstract

We present an entanglement concentration protocols (ECPs) for less-entangled W state with quantum-dot and microcavity coupled system. The present protocol uses the quantum nondemolition measurement on the spin parity to construct the parity check gate. Different from other ECPs, this less-entangled W state with quantum-dot and microcavity coupled system can be concentrated with the help of some single photons. The whole protocol can be repeated to get a higher success probability. It may be useful in current quantum information processing.

###### pacs:

03.67.Bg, 42.50.Pq, 78.67.Hc, 78.20.Ek## I introduction

Entanglement plays an important role in quantum information processing book (); rmp (). It has many practical applications, such as quantum teleportation teleportation (); cteleportation (), quantum cryptograph Ekert91 (); cryp (), quantum dense coding densecoding (), quantum secure direct communication QSDC1 (); QSDC2 (); QSDC3 () and other quantum information protocols QSTS1 (); QSTS2 (); QSTS3 (); QSS1 (); QSS2 (); QSS3 (). In order to achieve such tasks, the legitimate uses, say the sender Alice and the receiver Bob should first share the entanglement in distant locations. In a long-distance quantum communication, quantum repeaters are unusually used to sent the quantum signals over an optical fiber or a free space repeater1 (); DLCZ (). Unfortunately, in a practical transmission, the entangled quantum system cannot avoid the channel noise from the environment. It will make the quantum system decoherence. That is, a maximally entangled state will become a less-entangled state or a mixed entangled state.

Quantum concentration is to distill some maximally entangled state from an ensemble in a pure less-entangled state C.H.Bennett2 (); swapping1 (); swapping2 (); zhao1 (); zhao2 (); Yamamoto1 (); Yamamoto2 (); bose (); cao1 (); cao2 (); shengpra2 (); shengpra3 (); shengqic (); wangxb (); shengpla (); zhang (); dengconcentration (); shengwstateconcentration (); shengsinglephotonconcentration (). In 1996, Bennett et al. proposed an entanglement concentration protocol (ECP), which is called Schmidit projection method C.H.Bennett2 (). Bose et al. proposed an ECP based on the swapping bose (). ECPs based on linear optical elements were proposed by Yamamoto et al. and Zhao et al., respectively zhao1 (); zhao2 (); Yamamoto1 (); Yamamoto2 (). In 2008, an ECP based on cross-Kerr nonlinearity was proposed shengpra2 (). In 2011, an ECP based on the quantum-dot in an optical cavity was proposed. This kind of ECPs are all used to concentrate a less-entangled state to a maximally entangled state . Unusually, in each concentration step, they choose two similar copies of less-entangled states and after performing the ECP, at least one pair of maximally entangled state can be obtained with certain probability. The most advantage of this ECP is that they do not need to know the exact coefficients and . On the other hand, there is another kind of ECP which can be used to concentration a less-entangled W state into a maximally entangled W state. For example, in 2003, Cao and Yang proposed an ECP for W-class state using joint unitary transformationcao (). Zhang et al. proposed an ECP with the help of collective Bell-state measurement zhanglihua (). The ECPs for a special less-entangled W state were proposed in both linear optical system and cavity QED system wanghf1 (); wanghf2 (). In 2011, Yildiz proposed an optimal ECP for asymmetric W states of the form yildiz ()

(1) |

On the other hand, the ECPs described above are unusually based on the universal qubit say and C.H.Bennett2 () or the optical system, which and Yamamoto1 (); Yamamoto2 (); zhao1 (); zhao2 (). Here and represent the horizontal and vertical polarization of the photon, respectively. Recently, there is a novel candidate for qubit which is a single spin coupled to an optical microcavity based on a charged qantum-dot cavity (); hu1 (); hu2 (); hu3 (); hu4 (). For example, Hu et al. proposed an deterministic photon entangler using a charged quantum dot inside a microcavityhu1 (). They also proposed an entanglement beam splitter and discussed the loss-resistant state teleportation and entanglement swapping using a quantum dot spin in an optical microcavity hu2 (); hu3 (); hu4 (). Bonato et al. discussed the CNOT gate and Bell-state analysis in the weak-coupling cavity QED regime cavity (). Wang et al. proposed two entanglement purification protocols based on the hybrid entangled state using quantum-dot and microcavity coupled system wangc2 (); wangc3 (). Recently, a efficient quantum repeater protocol was proposed wangtiejun (). Inspired by the novel works of Hu and Wang, we propose an ECP for the less-entangled W state exploiting the quantum-dot and microcavity coupled system. In this protocol, the less-entangled W state of the spin in the cavity QED system can be concentrated into a maximally entangled W state with some ancillary single photons. This protocol is quite different from the others. First, we can concentrate the arbitrary less-entangled W state. Second, we do not need two copies of less-entangled pairs. Third, the ECP can be performed between different degrees of freedoms, that is we use the single photons to concentrate the less-entangled state in spin. Fourth, by repeating this ECP, it can reach a higher success probability.

This paper is organized as follows: In Sec. II, we describe the theoretical model for our ECP. We call it a hybrid parity check gate based on photon and electron coupled systems. In Sec. III, we explain our ECP based on the parity check gate. In Sec. IV, we discuss the efficiency and errors on a practical implementation. In Sec. V, we present a discussion and summary.

## Ii hybrid parity check gate

Before we start to explain this ECP. We first describe the basic element for this protocol. It is also shown in Refs. hu2 (); cavity (); wangc2 (); wangc3 (). As shown in Fig. 1, the system is composed of a single charged quantum-dot in micropillar microcavites. The charge exciton consists of two electrons bound in one hole and the excitation with negative charges can created by the optical excitation of the system. Therefore, if we consider a photon entrances into the cavity from the input mode and it will interact with the electron in the coupling cavity. Interestingly, the left circularly polarized photon only couples with the electron in the spin up state to the exciton in the state because of the Pauli’s exclusion principle for two electrons. On the other hand, the right circularly polarized photon only couples with the electron of the spin down in the state . Here the and are the spin direction of the heavy hole spin state. In Ref. hu2 (), Hu et al. discussed that such system essentially is an entanglement beam splitter which directly splits an initial hybrid product state of photon and spin into two entangled states via transmission and refection in a deterministic way. They denoted the transmission and reflection operators as

(2) |

From Fig. 1, we consider a photon is in the state with and the electron spin is . Here the superscript means the photon’s propagation direction is along the z axis. Both the polarization of the photon and propagation direction are flipped into . In the same way, the photon and electron interaction in quantum dot and microcavity coupled systems can be fully described as

(3) |

So if the initial input state is the photon-spin product state , it will be converted into the two constituent hybrid entangled state in the transmission port, say output2 mode and in the reflection port, say output1 mode respectively, with the success probability of 100%, in principle. In the following, we denote the transmission port as output2 mode and the reflection port as output1 for simple, shown in Fig. 1. A parity check gate has been widely in current quantum processing. Entanglement purification and concentration all need such elements. An optical parity check gate, such as polarization beam splitter, can convert the product state into two constituent entangled state and . Compared with the polarization beam splitter in optical system, it essentially acts the same role of the parity check gate, but with different degrees of freedom. So we call it a hybrid parity check gate.

## Iii ECP for less-entangled W state

Now we start to explain our protocol. From Fig. 2, the less-entangled W state are shared by Alice, Bob and Charlie. It can be written as:

(4) | |||||

Here and subscripts 1, 2 and 3 mean spin 1, spin 2 and spin 3 respectively. Suppose that the three parties know the initial coefficients , , and . Alice first prepare a single photon of the form

(5) |

The subscript means the photon coupled with the spin 1. She sends the state to the cavity from the input mode. The initial less-entangled W state combined with the single photon evolve as

(6) | |||||

Interestingly, from Eq. (6), if the photon is transmitted and in the output2, the original state collapses to

(7) | |||||

Then Alice lets her photon pass through the HWP and PBS. The HWP makes

(8) |

and the PBSs transmit the polarization photon and reflect polarization photon.

Finally, if the single-photon detector D fires, they will get

(9) | |||||

It can be rewritten as

(10) | |||||

Finally, if the single-photon detector D fires, they will get

(11) | |||||

In order to get , one of the three parties, says Alice, Bob or Charlie should perform a local operation of phase rotation on her or his spin. The success probability is

(12) |

On the other hand, after passing through the microcavity, if the photon is reflected and in the output1, then the Eq. (6) collapses to

(13) | |||||

Following the same principle described above, if the D fires, they will obtain

(14) | |||||

if the D fires, they will obtain

(15) | |||||

In order to get , one of the parties, says Alice, Bob or Charlie should perform a local operation of phase rotation on her or his spin.

It is interesting to compare with . only has two different coefficients, and , and the initial coefficient disappears. But still has three different coefficients. We denote , and . So if Alice obtains , it is successful. Then she asks Charlie to continue this ECP. Otherwise, she has to repeat this ECP in a second round. That is, she prepares another single photon of the form

(16) |

Then she lets this single photon entrance into the microcavity and couple with the spin. After the photon passing through the microcavity, following the same principle, if this single photon is in the output2 and detected by D or D, the concentration is successful. Otherwise, if it is in the output1 and detected by D or D, the concentration is a failure. Alice should prepare a third single photon and restart to perform this ECP until it is successful. So the success probability in the second round is

(17) |

the success probability in the third round is

(18) |

If it is repeated for times, the success probability is

The total success probability for Alice is

(20) |

If Alice is successful, then Charlie start to perform this ECP. His concentration step is analogy with Alice. In detail, he first prepares a single photon of the form

(21) |

Charlie lets his single photon entrance the microcavity and couple with the spin. Then the state combined with evolves as

From Eq. (LABEL:evolve2), after the photon passing through the microcavity, if the photon is in the output1, then above state collapses to

It can be rewritten as

Finally, after the photon passing through the HWP and PBS, if D fires, the remained state is essentially the maximally entangled W state

(25) | |||||

If D fires, the remained state is

(26) | |||||

They can obtain by performing a local operation of the phase rotation on one of the spin.

On the other hand, if the photon is in the output2 and leads the D fire, the Eq. (LABEL:evolve2) collapses to

(27) | |||||

It can be written as

(28) | |||||

Otherwise, if D fires, the Eq. (LABEL:evolve2) collapses to

(29) | |||||

They can also obtain , by performing a local operation of phase rotation on one of the spin. Eqs. (28) and (29) are both lesser-entangled W states, which have the same form of . That is, if Charlie obtains Eq. (28), he can repeat this ECP and obtain the maximally entangled W state in a second round. They can also obtain the success probability for Charlie as

(30) | |||||

The total success probability for Charlie is

(31) |

## Iv Success probability and experiment feasibilities

Thus far, we have briefly explained this ECP. We can calculate the success probability for both Alice and Charlie. Suppose that Alice and Charlie repeat this ECP for times, the total success probability is

(32) |

If both Alice and Charlie perform this protocol only one time, the total success probability