# Complete analysis for arbitrary concatenated Greenberger-Horne-Zeilinger state assisted with photonic Faraday rotation

## Abstract

The concatenated Greenberger-Horne-Zeilinger (C-GHZ) state has great potential application in the future quantum network, for it is robust to the decoherence in a noisy environment. In the paper, we propose a complete C-GHZ state analysis protocol with the help of some auxiliary single atoms trapped in the low-quality cavities. In the protocol, we essentially make the parity check for the photonic states based on the photonic Faraday rotation effect, and complete the analysis task combined with the Hadamard operation and single qubit measurement. The success probability of our protocol can reach 100% in principle, and the number of physical qubit encoded in each logic qubit does not affect the analysis. Our analysis protocol may have its practical application in future long-distance quantum communication.

###### pacs:

03.67.Mn, 03.67.-a, 42.50.Dv## I Introduction

Entanglement is the key resource of quantum information processing (QIP). Entangled qubits are required for many important branches of QIP, such as quantum teleportation (1), quantum key distribution (QKD) (2), quantum secure direct communication (QSDC) (3); (4), and quantum repeaters (5). Recently, the multi-particle systems attract more and more attentions. For instance, the Greenberger-Horne-Zeilinger (GHZ) state is one of the most important resources in QIP. Due to the large information capacity, GHZ states play an important role in the foundations of quantum mechanics measurement theory and quantum communication (6); (7); (8); (9). In practical applications, the ideal quantum states are the maximally entangled states. However, due to the environmental noise, the decoherence problem is inevitable in practical applications. The decoherence greatly limits the building of high-quality quantum channel, even more, it may cause the quantum communication insecure. For dealing with the decoherence problem, people proposed large number of approaches, such as entanglement purification (10); (11); (12); (13); (14); (15); (16); (17); (18); (19), entanglement concentration (20); (21); (22); (23); (24); (25); (26), and entanglement amplification (27); (28); (29). Recently, a new kind of multi-particle quantum state, which is called the concatenated Greenberger-Horne-Zeilinger (C-GHZ) state has attracted high attention (30); (31); (32); (33); (34); (35). It is also called the macroscopic Schrödinger’s cat superposed state. For the common quantum states, people usually encode quantum qubit in physical qubit directly, while for the C-GHZ state, the parties encode many physical qubits in a logic qubit. The typical C-GHZ state can be written as

(1) |

Here, . and are the logic qubit number and physical qubit number in each logic qubit, respectively. This C-GHZ state shows similar features as
the common GHZ state. However, comparing with common GHZ state, the C-GHZ state has a highly attractive feature, that is, it is robust to the decoherence in a noisy environment (30); (32). Due to the robust feature, the C-GHZ state has great application potential in the future long-distance quantum communication. In 2014, Lu *et al.* demonstrated the first experiment to prepare the C-GHZ state with and in an optical system. They also verified that the C-GHZ state can tolerate more bit-flip and phase shift noise
than polarized GHZ state. Therefore, the C-GHZ state is useful for large-scale fibre-based quantum networks and
multipartite QKD schemes (32).

The quantum state analysis, which is the discrimination between the maximally entangled quantum states is quite important in various applications. The most common quantum state analysis is called the Bell-state analysis (BSA), which is for the two-particle entangled system. There are usually three different kinds of methods to realize the BSA. The first one is totally in linear optics (36); (37). However, the success probability of the BSA approaches with only linear optical elements can only reach 50%, so that the first kind of BSA approaches can not perform complete BSA (36); (37). The second kind of methods still requires the linear optical elements but resorts to the hyperentanglement (38); (39); (40); (41). For example, in 2003, Walborn *et al.*
once proposed a hyperentanglement-assisted BSA approach. In their approach,
the hyperentangled state is prepared in polarization and momentum degrees of freedom. They can realize the complete BSA for both the momentum and polarization entangled Bell-states (38). The third kind of methods adopt the nonlinear optical elements, such as the cross-Kerr nonlinearity and the quantum-dot system (42); (43); (44); (45); (46) to realize the complete BSA. For example, some groups adopt the cross-Kerr nonlinearity to construct the complete parity-check measurement (PCM) gate, which can distinguish the even parity states and from the odd parity states
and (42); (46). Although the analysis for GHZ states has been widely discussed (43); (47); (48); (49); (50); (51); (52), most analysis protocols can not deal with the C-GHZ states. Recently, Sheng and Zhou proposed two complete BSA protocols for the C-GHZ state with the help of the controlled-not (CNOT) gate and the cross-Kerr nonlinearity, respectively (53); (54). Unfortunately, the CNOT gate and the cross-Kerr nonlinearity are difficult to realize in current experimental condition, which limits the application of the two analysis protocols. Lee *et al.* proposed a partially BSA protocol for another type of logic-qubit entanglement in the linear optics (55).

On the other hand, the cavity quantum electrodynamics (QED) is a promising platform for performing quantum information tasks, due to the controllable interaction between atoms and photons. In 2009, the group of An successfully implemented QIP based on the photonic Faraday rotation (56). This method works in the low-quality (Q) cavities and only involves the virtual excitation of atoms. Therefore, it is insensitive to both the cavity decay and the atomic spontaneous emission. Following this scheme, various works based on the photonic Faraday rotation effect in the low-Q cavity have been presented, such as quantum logic gate (57), QIP in decoherence-free subspace (58), quantum teleportation (59), and entanglement detection (60); (61). Recently, with the help of Faraday rotation, Wei and Deng designed some compact quantum gates (62); (63). Their works proved that the universal quantum computation can be realized. Recently, we proposed a complete logic Bell-state analysis (LBSA) protocol with the help of the photonic Faraday rotation in low-Q cavity (64). Actually, the logic Bell-state is the special case of the C-GHZ state with . In this paper, we will put forward a complete analysis protocol for the C-GHZ state with arbitrary and based on the photonic Faraday rotation in low-Q cavity. Due to the attractive possible applications of the C-GHZ state, our analysis may be useful in the future QIP field.

This paper is organized as follows: In section 2, we will introduce the basic principle of the photonic Faraday rotation. In section 3, we will describe our complete analysis protocol for arbitrary C-GHZ state in detail. In section 4, we make a discussion and conclusion.

## Ii Basic principle of the photonic Faraday rotation

Our analysis protocol is based on the photonic Faraday rotation in low-Q cavity. In this way, we first introduce its basic principle briefly. As shown in Fig. 1, a three-level atom is trapped in the one-side low-Q cavity. The states and represent the two Zeeman sublevels of its degenerate ground state, and represents its excited state. A single photon pulse with frequency enters the cavity and reacts with the three-level atom. The transition between and is assisted with a left-circularly polarized photon (), and the transition between and is assisted with a right-circularly polarized photon (), respectively.

Based on the researches of Res. (56); (59); (65); (66), by solving the Langevin equations of motion for cavity and atomic lowering operators analytically, we can obtain the general expression of the reflection coefficient of the atom-cavity system in the form of

(2) |

Here, and are the cavity input operator and cavity output operator, respectively. and are the cavity damping rate and atomic decay rate, respectively. , and are the frequency of the cavity and the atom, respectively, and is the atom-cavity coupling strength. From Eq. (2), if the atom uncouples to the cavity, which makes , we can simplify to

(3) |

Eq. (3) can be written as a pure phase shift . On the other hand, in the interaction process, as the photon experiences an extremely weak absorption, we can consider that the output reflected photon only experiences a pure phase shift without any absorption. In this way, the expression of can be simplified to . Therefore, if the photon pulse takes action, the output photon state will convert to , otherwise, the single-photon would only sense the empty cavity, and the output photon state will convert to .

## Iii The complete C-GHZ state analysis with

In the section, we introduce the complete analysis protocol for the C-GHZ states under a simply case, say . For simplicity, we first suppose only two physical qubits encoded in each logic qubit, that is, . Under this case, we can write eight C-GHZ states as

(7) |

where

(8) |

The schematic drawing of our complete analysis protocol is shown in Fig. 2. The protocol includes two steps. In the first step, we first make each of the photons pass through a half-wave plate (HWP). In essence, the HWP plays the role of the Hadamard operation, which makes and . After the HWP, will not change, while will change to . In this way, after the HWPs, the eight C-GHZ states in Eq. (7) will evolve to

(9) |

The parties make four three-level atoms, here named atom ”1”, ”2”, ”3”, and ”4” trap in four low-Q cavities, respectively. The four atoms are prepared in the same states as . Then, the parties make the photons in and spatial modes pass through the cavities and interact with the atoms ”1” and , respectively. After the photons in and modes exiting the cavities, the parties make the photons in and modes enter another two cavities and interact with the atoms ”3” and ”4”, respectively. It is noticed that we should ensure that only a photon interacts with the atom at a time. In this way, we adopt the setup ”Delay” to exactly control the time of the photon entering the cavity. For example, as shown in Fig. 2, we first let the photon in mode interact with the atom ”1”. After the photon is reflected and exits the cavity, we let the photon in mode enter the cavity.

If the initial C-GHZ state is , the photon state combined with the four atom states can be written as

(10) | |||||

According to the input-output relation in Eq. (6), after the cavities, the states in Eq. (10) will evolve to

(11) | |||||

Similarly, we can also obtain the other six cases. If the initial logic GHZ state is , after the cavities, we can obtain

(12) | |||||

For , the photon state combined with four atom states will evolve to

(13) | |||||

Finally, combined with four atom states will evolve to

(14) | |||||

After all the photons exiting the cavities, we perform the Hadamard operations on the atoms ”1”, ”2”, ”3”, and ”4”, respectively. The Hadamard operation will make , and . After that, we measure the atom states of the four atoms. From Eqs. (11)-(14), it can be found that under the case that the initial photon state is , the measurement results of the atoms ”1” and ”2”, ”3” and ”4” are always the same, that is, if the measurement result of atom ”1” (”3”) is , the measurement result of atom ”2” (”4”) is also , while if the measurement result of atom ”1” (”3”) is , that of atom ”2” (”4”) must be . For , the measurement results of atom ”1” and ”2” are different, that is, if the measurement result of atom ”1” is , the measurement result of atom ”2” must be , and vice versa. On the other hand, the measurement results of atom ”3” and ”4” are the same. For the initial state of , both the measurement results of the atoms ”1” and ”2”, ”3” and ”4” are different. For the initial state of , the measurement results of atom ”1” and ”2” are the same, while those of atom ”3” and ”4” are different. Therefore, according to the measurement results of the four atoms, we can divide the eight C-GHZ states in Eq. (7) into four groups ,