# Hyperentanglement purification for two-photon six-qubit quantum systems^{1}

^{1}

## Abstract

Recently, two-photon six-qubit hyperentangled states were produced in experiment and they can improve the channel capacity of quantum communication largely. Here we present a scheme for the hyperentanglement purification of nonlocal two-photon systems in three degrees of freedom (DOFs), including the polarization, the first-longitudinal-momentum, and the second longitudinal momentum DOFs. Our hyperentanglement purification protocol (hyper-EPP) is constructed with two steps resorting to parity-check quantum nondemolition measurement on the three DOFs and SWAP gates, respectively. With these two steps, the bit-flip errors in the three DOFs can be corrected efficiently. Also, we show that using SWAP gates is a universal method for hyper-EPP in the polarization DOF and multiple longitudinal momentum DOFs. The implementation of our hyper-EPP is assisted by nitrogen-vacancy centers in optical microcavities, which could be achieved with current techniques. It is useful for long-distance high-capacity quantum communication with two-photon six-qubit hyperentanglement.

###### pacs:

03.67.Bg, 03.67.Pp, 03.65.Yz, 03.67.Hk## I Introduction

Quantum entanglement plays a critical role in quantum information processing (1). It is the key resource in quantum communication, such as quantum teleportation (2), quantum dense coding (3); (4), quantum key distribution (5); (6), quantum secret sharing (7), quantum secure direct communication (8); (9), and so on. Maximally entangled states can be used as the quantum channel for teleporting an unknown state of a quantum particle, without moving the particle itself (2). It can also be used to carry two bits of information by moving only one two-level particle (3), not two or more. The two parties of quantum communication can create a private key with entangled photon pairs in an absolutely secure way (5); (6). Moreover, they can exchange the secret message directly and securely without distributing the private key if they share maximally entangled photon pairs (8); (9).

Hyperentanglement, a state of a quantum system being simultaneously
entangled in multiple degrees of freedom (DOFs), has attracted much
attention as it can improve both the channel capacity and the
security of quantum communication largely, beat the channel
capacity of linear photonic superdense coding (10), assist
the complete analysis of Bell states (11); (12); (13); (14), and
so on. With the barium borate (BBO) crystal, photon pairs
produced by spontaneous parametric down conversion (PDC) can be in a
hyperentangled state. Many theoretical and experimental schemes for
the generation of hyperentangled states have been proposed and
implemented in optical systems
(15); (18); (17); (19); (20); (23); (21); (16); (22),
such as in polarization-momentum DOFs (18),
polarization-orbital-angular momentum DOFs (19), multipath
DOFs (20), and so on. In 2009,
Vallone *et al.* (23) demonstrated experimentally the
generation of a two-photon six-qubit hyperentangled state in three
DOFs.

Although the interaction between a photon and its environment is
weaker than other quantum systems, it still inevitably suffers from
channel noise, such as thermal fluctuation, vibration, imperfection
of an optical fiber, and birefringence effects. The interaction
between the photons and the environment will make the entangled
photon pairs in less entangled states or even in mixed states,
which will decrease the security and the efficiency of quantum
communication. Entanglement purification is used to obtain a subset
of high-fidelity nonlocal entangled quantum systems from a set of
those in mixed entangled states (24); (25); (26); (27); (28). In
1996, Bennett *et al.* (24) introduced the entanglement purification protocol (EPP) for quantum systems in a
Werner state (29) with quantum controlled-NOT gates. In
2001, Pan *et al.* (26) proposed an EPP with linear
optical elements. In 2002, Simon and Pan (27) presented an
EPP for a PDC source, not an ideal entanglement source. In 2008,
Sheng *et al.* (28) proposed an efficient EPP for
polarization entanglement from a PDC source, assisted by
nondestructive quantum nondemolition detectors (QND) with cross-Kerr
nonlinearity. In 2010, Sheng and Deng (30) introduced the
original deterministic EPP for two-photon systems, which is far
different from the conventional EPPs (24); (25); (26); (27); (28)
as it works in a deterministic way, not a probabilistic one.
Subsequently, some interesting deterministic EPPs were proposed
(32); (31); (33); (34). In 2003, Pan *et al.* (35)
demonstrated the entanglement purification of photon pairs using
linear optical elements. Recently, Ren and Deng (36)
proposed the original hyperentanglement purification protocol
(hyper-EPP) for two-photon four-qubit systems in mixed
polarization-spatial hyperentangled Bell states with polarization
bit-flip errors and spatial-mode bit-flip errors assisted by
nonlinear optical elements.

A nitrogen vacancy (NV) center in diamond is an attractive candidate for quantum information processing because of its long-lived coherence time at room temperature and optical controllability. The coherence time of a diamond NV center can continue for 1.8 ms (37). In a diamond NV center, the electron spin can be exactly populated by the optical pumping with 532-nm light, and it can be easily manipulated (38); (39); (40); (41) and readout (42); (43) by using the microwave excitation. Many interesting approaches based on an NV center in diamond coupled to an optical cavity have been proposed in theory (44); (45); (46); (47); (48) and implemented in experiment (49); (50); (51); (52); (53).

In this paper, we present a hyper-EPP for nonlocal photon systems entangled in three DOFs assisted by nitrogen-vacancy centers in optical microcavities, including the polarization DOF, the first-longitudinal-momentum DOF, and the second-longitudinal-momentum DOF. Our protocol is completed by two steps. The first step resorts to parity-check quantum nondemolition measurement on the polarization DOF (P-QND) and the two longitudinal-momentum DOFs (S-QND). The second step resorts to the SWAP gate between the polarization states of two photons (P-P-SWAP gate) and the SWAP gate between the polarization state and the spatial state of one photon (P-F-SWAP gate and P-S-SWAP gate). Also, we show that using the SWAP gates is a universal method for hyper-EPP in the polarization DOF and multiple longitudinal-momentum DOFs. Our hyper-EPP can effectively improve the entanglement of photon systems in long-distance quantum communication.

This paper is organized as follows: We give the interface between a circularly polarized light and a diamond nitrogen-vacancy center confined in an optical microcavity in Sec. II. Subsequently, we present the polarization-spatial parity-check QND of two-photon six-qubit systems in Sec. III, and then, we give the principle of our SWAP gate between the polarization states of two photons in Sec. IV.1 and that of our SWAP gate between the polarization state and the spatial state of one photon in Sec. IV.2. In Sec.V, we propose an efficient hyper-EPP for mixed two-photon six-qubit hyperentangled Bell states. In Sec. VI, we discuss the expansion for purifying the two-photon systems in the polarization DOF and multiple longitudinal momentum DOFs. A discussion and a summary are given in Sec. VII. In addition, some other cases for our hyper-EPP are discussed in the Appendix.

## Ii The interface between a circularly polarized light and a diamond nitrogen-vacancy center confined in an optical microcavity

A diamond NV center consists of a vacancy adjacent to a substitutional nitrogen atom. The NV center is negatively charged with two unpaired electrons located at the vacancy. The energy-level structure of the NV center coupled to the cavity mode is shown in Fig. 1(a). The ground state is a spin triplet with the splitting at 2.87 GHz in a zero external field between levels and owing to spin-spin interaction. The excited state, which is one of the six eigenstates of the full Hamiltonian including spin-spin and spin-orbit interactions in the absence of any perturbation, is labeled as , where and are the orbital states with the angular momentum projections and along the NV axis, respectively. We encode the qubits on the sublevels as the ground states, and take as an auxiliary excited state. The transitions and in the NV center are resonantly coupled to the right (R) and the left (L) circularly polarized photons with the identical transition frequency, respectively. The two transitions take place with equal probability.

Let us consider the composite unit, a diamond NV center confined inside a single-sided resonant microcavity, as shown in Fig. 1(b). The Heisenberg equations of motion for this system can be written as (54)

(1) |

Here and are the annihilation operator of the cavity mode and the transition operator of the diamond NV center. represents the inversion operator of the NV center. , , and are the frequencies of the cavity mode, the photon, and the diamond NV center level transition, respectively. , , and are the coupling strength between a diamond NV center and a cavity, the decay rate of a diamond NV center, and the damping rate of a cavity, respectively. is the vacuum input field with the commutation relation .

In the weak excitation limit, , the reflection coefficient for the NV-cavity unit is (55); (56)

(2) |

When the diamond NV center is uncoupled to the cavity or coupled to an empty cavity, that is, , the reflection coefficient can be written as

(3) |

If , the reflection coefficients can be written as

(4) |

That is, when , the change of the input photon can be summarized as

(5) |

Considering the condition , we can obtain the approximate evolution of the system composed of a diamond NV center and a photon as follows (45):

(6) |

## Iii Polarization-spatial parity-check QND of two-photon six-qubit systems

A two-photon six-qubit hyperentangled Bell state can be described as follows:

(7) |

Here the subscripts A and B represent the two photons. and represent the horizontal and the vertical polarizations of photons, respectively. and represent the right and the left spatial modes of a photon in the first-longitudinal-momentum DOF, respectively. and denote the external and the internal spatial modes of a photon in the second-longitudinal-momentum DOF, respectively, shown in Fig. 2. This two-photon six-qubit hyperentangled Bell state can be produced by two 0.5-mm-thick type I BBO crystal slabs aligning one behind the other and an eight-hole screen (23), shown in Fig. 2. When a continuous-wave (cw) vertically polarized laser beam interacts through spontaneous parametric downconversion (SPDC) with the two BBO crystal slabs, the nonlinear interaction between the laser beam and the BBO crystal leads to the production of the degenerate photon pairs which are entangled in polarization and belong to the surfaces of two emission cones. As shown in Fig. 2, the insertion of an eight-hole screen allows us to achieve the double longitudinal-momentum entanglement.

Generally speaking, an arbitrary two-photon six-qubit hyperentangled Bell state for a photon pair can be written as

(8) |

where is one of the four Bell states for a two-photon system in the polarization DOF,

(9) |

is one of the four Bell states for a two-photon system in the first-longitudinal-momentum DOF,

(10) |

and is one of the four Bell states for a two-photon system in the second-longitudinal-momentum DOF,

(11) |

The polarization-spatial parity-check QND is used to distinguish the odd-parity mode (, and ) from the even-parity mode (, and ) of the hyperentangled Bell states in both the polarization and the two longitudinal-momentum DOFs.

The schematic diagram of the polarization parity-check QND is shown in Fig. 3, which consists of some circularly polarizing beam splitters CPBS, an NV center, and some half-wave plates Z. The NV center is prepared in the initial state . If the two-photon system is in a hyperentangled Bell state, one can inject the photons and into the quantum circuit sequentially. The evolutions of the system consisting of the two photons and the NV center are described as

(12) |

where . By measuring the state of the NV center in the orthogonal basis , one can distinguish the even-parity Bell states from the odd-parity Bell states of the two-photon system in the polarization DOF without affecting the states of the two-photon systems in the spatial-mode DOFs. That is, if the NV center is projected into the state , the polarization state of the hyperentangled two-photon system is . Otherwise, the polarization state of the hyperentangled two-photon system is .

The schematic diagram of the spatial-mode parity-check QND is shown in Fig. 4. It consists of two NV centers and some half-wave plates . Each of the two NV centers is prepared in the initial state with . One can inject the photons and into the quantum circuit sequentially. The evolutions of the system composed of the two photons and the two NV centers are described as:

(13) |

By measuring the states of two NV centers in the orthogonal basis , one can distinguish the even-parity states from the odd-parity states in the first-longitudinal-momentum DOF and the even-parity states from the odd-parity states in the second longitudinal momentum DOF, without affecting the states of the two-photon system in the polarization DOF. That is, if is projected into the state , the two-photon system is in the state in the first-longitudinal-momentum DOF and if is projected into the state , the two-photon system is in the state in the second-longitudinal-momentum DOF, respectively. If is projected into the state , the two-photon system is in the state in the first-longitudinal-momentum DOF and if is projected into the state , the two-photon system is in the state in the second-longitudinal-momentum DOF, respectively.

## Iv SWAP gates

### iv.1 SWAP gate between the polarization states of two photons

The SWAP gate between the polarization states of two photons (P-P-SWAP gate) is used to swap the polarization states of photon and photon without affecting their spatial-mode states. The initial states of two photons in the polarization DOF and two longitudinal-momentum DOFs are

(14) |

Here and are the polarization states of the photons and , respectively. and represent the states of the photons and in both the first-longitudinal-momentum DOF and the second-longitudinal-momentum DOF, respectively. The schematic diagram of the P-P-SWAP gate is shown in Fig. 5. Suppose that the NV center is prepared in the initial state , the SWAP gate works with the following steps.

First, one injects the photons and into the quantum circuit sequentially, and lets photon pass through the circularly polarizing beam splitter , , NV center, , and in sequence. After performing a Hadamard operation on the NV center [], the whole state of the system composed of two photons and one NV center is transformed from to . Here

(15) |

Second, one performs Hadamard operations on photons and with the half-wave plates and . The state of the whole system is changed into

(16) |

Third, one lets photon pass though , , NV center, , , and , and the state of the whole system is changed from to . Here

(17) |

Finally, by performing a Hadamard operation on the NV center, the state of the whole system is transformed into

(18) |

By measuring the NV center in the orthogonal basis , the polarization state of photon is swaped with the polarization state of photon without disturbing their states in the spatial-mode DOFs. If the NV center is projected into state , phase-flip operations are performed on the polarization DOF of photons and . After the phase-flip operations, the state of the two photons is transformed into

(19) |

Here, is the objective state of the P-P-SWAP gate. If the NV center is projected into state , the objective state can be obtained directly without phase-flip operations.

### iv.2 SWAP gate between the polarization state and the spatial state of one photon

The schematic diagram of our SWAP gate between the polarization state and the spatial-mode state in the first-longitudinal-momentum DOF of one photon (P-F-SWAP) is shown in Fig. 6(a), which is constructed with linear optical elements such as and half-wave plate . One can inject photon , which is in the state described as Eq. (14), into the quantum circuit shown in Fig. 6(a). After photon passes through the quantum circuit shown in Fig. 6(a), the state of photon is transformed into

(20) |

Here is the objective state of the P-F-SWAP gate. The schematic diagram of our SWAP gate between the polarization state and the spatial-mode state in the second longitudinal-momentum DOF of one photon (P-S-SWAP gate) is shown in Fig. 6(b). After photon , whose initial state is described as Eq. (14), passes through the circuit shown in Fig. 6(b), the state is changed into

(21) |

Here, is just the objective state of the P-S-SWAP gate.

## V Efficient hyper-EPP for mixed two-photon six-qubit hyperentangled Bell states

In the practical transmission of photons in hyperentangled Bell states for high-capacity quantum communication, both the bit-flip error and the phase-flip error will occur on the photon systems. Although a phase-flip error cannot be directly purified, it can be transformed into a bit-flip error using a bilateral local operation (24); (25); (26); (27); (28). If a bit-flip error purification has been successfully solved, phase-flip errors also can be solved perfectly. In this way the two parties in quantum communication, say Alice and Bob, can purify a general mixed hyperentangled state. Below, we only discuss the purification of the two-photon six-qubit hyperentangled mixed state with bit-flip errors in the three DOFs.

Two identical two-photon six-qubit systems in mixed hyperentangled Bell states in the polarization DOF and two longitudinal-momentum DOFs with bit-flip errors can be described as follows:

(22) |

Here the subscripts and represent two photon pairs shared by the two parties in quantum communication , say Alice and Bob. Alice holds the photons and , and Bob holds the photons and . , , and represent the probabilities of states , , and in the mixed states, respectively.

The initial state of the system composed of the two identical two-photon six-qubit subsystems can be expressed as . It can be viewed as a mixture of maximally hyperentangled pure states. In the polarization DOF, it is a mixture of the states , , , and with the probabilities , , , and , respectively. In the first-longitudinal-momentum DOF, it is a mixture of the states , , , and with the probabilities , , , and , respectively. In the second-longitudinal-momentum DOF, it is a mixture of the states , , , and with the probabilities , , , and , respectively.

Our hyper-EPP for the nonlocal two-photon six-qubit systems in hyperentangled Bell states with bit-flip errors in the polarization DOF and the two longitudinal-momentum DOFs can be achieved with two steps in each round. The first step is completed with polarization and spatial-mode parity-check QNDs introduced in Sec. III. The second step of our hyper-EPP scheme is completed with the SWAP gates introduced in Sec. IV. We discuss the principles of these two steps in detail as follows.

### v.1 The first step of our hyper-EPP with polarization and spatial-mode parity-check QND

The principle of the first step of our hyper-EPP is shown in Fig. 7. Alice performs the P-S-QNDs on photons and performs the Hadamard operations on the polarization DOF and the spatial-mode DOFs of photon . Bob performs the same operations on photons .

First, Alice and Bob perform the P-S-QNDs on the two two-photon systems and , respectively. Based on the results of the P-S-QNDs on the identical two-photon systems, the states can be classified into eight cases. In case (1), the two identical two-photon systems and are in the same parity-mode in all of the three DOFs, including the polarization DOF, the first-longitudinal-momentum DOF, and the second-longitudinal-momentum DOF. This case corresponds to the states or in the polarization DOF, the states or in the first-longitudinal-momentum DOF, and the states or in the second-longitudinal-momentum DOF. The classification of the eight cases is shown in Table 1.

Case | DOF | Parity-mode | Bell states |
---|---|---|---|

p | Same | or | |

(1) | F | Same | or |

S | Same | or | |

p | different | or | |

(2) | F | Different | or |

S | Different | or | |

p | Different | or | |

(3) | F | Same | or |

S | Same | or | |

p | Same | or | |

(4) | F | Different | or |

S | Same | or | |

p | Same | or | |

(5) | F | Same | or |

S | Different | or | |

p | Different | or | |

(6) | F | Different | or |

S | Same | or | |

p | Different | or | |

(7) | F | Same | or |