Bidirectional multi-qubit quantum teleportation in noisy channel aided with weak measurement
Department of Communication Engineering, School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China
† Corresponding author. E-mail:
sharon.yg@163.com
Project supported by the National Natural Science Foundation of China (Grant No. 61172071), the Scientific Research Program Funded by Shaanxi Provincial Education Department, China (Grant No. 16JK1711), the International Scientific Cooperation Program of Shaanxi Province, China (Grant No. 2015KW-013), and the Natural Science Foundation Research Project of Shaanxi Province, China (Grant No. 2016JQ6033).
1. IntroductionQuantum teleportation (QT) plays an important role in quantum computing and quantum communication, which utilizes pre-shared quantum entanglement and classical communication to transmit an unknown quantum state from one location to another. Since the first QT protocol was introduced by Bennnet et al. in 1993,[1] it has become a research hotspot. Many modified QT schemes have been proposed, such as controlled teleportation,[2–5] probabilistic teleportation,[6–9] and bidirectional quantum teleportation (BQT).[10–13]
Having the capability of transmitting quantum information between Alice and Bob in two directions simultaneously, BQT has drawn more and more attention in recent years. Huelga et al. discussed the possibility of using BQT to implement the nonlocal quantum gates.[10,11] In 2013, Zha et al. proposed a controlled BQT scheme by using a five-qubit cluster state to transmit arbitrary single-qubit states.[12] Other similar BQT protocols have been presented, by using different types of entanglement states as the teleportation channels, such as five-qubit composite GHZ–Bell state,[13] six-qubit cluster state,[14] and genuine five-qubit entangled state.[15] After that, asymmetric controlled BQT schemes were devised to transmit a single-qubit state and a two-qubit entanglement state.[16–18] Binayak and Arpan proposed a BQT protocol to transmit arbitrary two-qubit entanglement states via a ten-qubit entangled state.[19] Hassanpour and Houshmand presented a BQT scheme to transmit pure EPR states by using a composite GHZ state.[20]
However, current research work relating to the above BQT seldom thinks about the problem of quantum noise. In a practical situation, a quantum state will couple with the environment inevitably, which breaks the coherence of the quantum entanglement state and gives rise to quantum noise. This does occur during the entanglement distribution in BQT. So, it is very important to analyze the influence of the quantum noise in the process of BQT, thus to determine whether a BQT is successful in a certain circumstance.
On the other hand, to overcome the influence of decoherence on quantum entanglement, several methods have been proposed, including quantum entanglement purification,[21–23] quantum error coding,[24–26] and decoherence-free subspace.[27–29] In recent years, weak measurement and its reversing measurement were widely studied to protect quantum state and quantum entanglement. Korotkov and Jordan demonstrated that the weak measurement of a solid-state qubit could be recovered with a reversal procedure.[30] The similar recovery methods of a superconducting phase qubit and an arbitrary cavity field with finite photon number were discussed and demonstrated in Refs. [31] and [32]. Liao et al. proposed an alternative way based on a Hadamard gate and CNOT gate to recover an arbitrary pure two-qubit entanglement state that has undergone a weak measurement.[33] Korotkov and Keane,[34] Lee et al.,[35] and Kim et al.[36] demonstrated weak measurement and the reversing measurement can protect a single-qubit state and two-qubit entanglement state from decohering the amplitude damping (AD). In Refs. [37] and [38], it was demonstrated that the weak measurement method can increase the fidelities of a three-qubit GHZ state and a W-like state in an AD channel.
In this paper, we first propose a BQT scheme to transmit three-qubit GHZ states based on the work of Ref. [20], then we generalize this scheme to teleport multi-qubit GHZ states. As an example, we use the AD channel model to analyze the influence of quantum noise on this BQT scheme and give the fidelity of the result of the teleportation. Finally, we use the weak measurement and the reversing measurement on part of the qubits of the teleportation channel state before and after the entanglement distribution. The comparison between the results shows the weak measurement method is helpful for enhancing the fidelity of the bidirectional teleportation in an AD channel.
2. Bidirectional multi-qubit quantum teleportation in ideal environment2.1. Bidirectional quantum teleportation of three-qubit GHZ statesIn this section, we first discuss the bidirectional teleportation scheme to transport three-qubit GHZ states in a noiseless quantum channel. Suppose that Alice and Bob have the following three-qubit GHZ states to be teleported, respectively, which are described as
where
α0,
α1,
β0, and
β1 are the arbitrary complex numbers that satisfy
,
, respectively. The bidirectional teleportation procedure is composed of the following five steps.
Step 1 An eight-qubit composite GHZ state should be shared by Alice and Bob as the teleportation channel, which is generated as follows:
Here, qubits
,
,
, and
are held by Alice, and qubits
,
,
, and
are held by Bob. The total state of
, and
can be described as
Step 2 Alice performs a controlled-NOT (CNOT) operation on qubits
and
, and Bob performs a CNOT operation on qubits
and
, where
and
are control qubits and
and
are target qubits. Then, the state in Eq. (4) is changed into the following form:
Step 3 Alice and Bob perform a single-qubit
-basis measurement on qubits
and
respectively, and they perform a single-qubit
-basis measurement on qubits
and
respectively. After these measurements, one of the following sixteen results can be obtained with the same probability:
Alice and Bob announce the measurement results to each other in a classic channel. Suppose that the results are
, then the quantum state of the remaining ten qubits will collapse into the following state:
Step 4 Alice and Bob perform a single-qubit
-basis measurement on qubits
and
respectively, then they announce the measurement results. The results may be
or
. Suppose that the results are
, then the quantum state of the remaining eight qubits will collapse into
Step 5 Alice and Bob perform a single-qubit
-basis measurement on qubits
and
respectively, then they announce the measurement results. Suppose that the results are
, then the quantum state of the remaining six qubits will collapse into
Equation (
8) shows that after the above steps, Alice has reconstructed the initial state
on her qubits
, and
, and Bob has reconstructed the initial state
on his qubits
, and
. The bidirectional quantum teleportation of three-qubit GHZ state is completed successfully. If Alice and Bob obtain other results in Step 5, they can perform the corresponding unitary operations in Table
1 to reconstruct the quantum states
and
.
Table 1.
Table 1.
 | Table 1. Unitary operations of Alice and Bob. . |
3. Bidirectional quantum teleportation in noisy channelIn this section, we will discuss how the noise in a quantum channel influences the procedure of bidirectional teleportation. As an example, we assume that the channel model is an AD channel. It can model many practical physical processes, such as spontaneous emission and super-conduction with zero-temperature energy relaxation. In this model, the evolutions of a single-qubit system state and the environment can be described as the following quantum transformation:[39]
where subscripts
S and
E denote the system and the environment respectively, and
p is the amplitude damping factor. The Kraus operators of this channel are
We still consider the bidirectional teleportations of three-qubit GHZ states. Assume the noise mainly exists in the process of entanglement distribution in the quantum channel. To reduce the influence of quantum noise as much as possible, Bob should prepare the four-qubit GHZ state
and distribute qubit
to Alice. Similarly, Alice prepares the four-qubit GHZ state
and distribute qubit
to Bob. During the distribution, only qubits
and
undergo amplitude damping, the eight-qubit channel state in Eq. (3) evolves into the following form:
After the bidirectional teleportation described in Subsection 2.1, Alice can reconstruct the quantum state
on her qubits
, and
, and Bob can reconstruct the quantum state
on his qubits
, and
, which can be expressed as
where
and
are the normalization factors. Performing a partial trace over the environment in Eq. (
16), we can obtain the following reduced density matrix of
:
where
is described as
4. Bidirectional quantum teleportation aided with weak measurement and the reversing measurementAccording to Ref. [35], the weak measurement and the corresponding reversing measurement on a single-qubit system can be described as the following transformations:
where
and
are the strengths of the weak measurement and the reversing measurement, respectively;
denotes the weak measurement environment;
denotes the reversing measurement environment.
During the entanglement distribution, only qubit
of the four-qubit GHZ state
suffers amplitude damping, so Bob only needs to perform the weak measurement on qubit
before it is distributed, then
is changed into the following form:
After the weak measurement, qubit
is distributed in the amplitude damping channel, then the quantum state in Eq. (21) evolves into
where
is the normalization factor.
After the processes above, Alice receives qubit
and she performs the reversing measurement, then the quantum state in Eq. (22) is changed into
where
is the normalization factor.
With a similar derivation, the four-qubit GHZ state
will be changed into the following state:
Then, the eight-qubit teleportation channel state in Eq. (
3) is changed into the following form:
Using the shared channel state in Eq. (25), Alice and Bob perform the bidirectional teleportation described in Subsection 2.1, then Alice can reconstruct the quantum state on her qubits
,
, and
, which has the following form:
where
is the normalization factor. Similarly, Bob can reconstruct the quantum state on his qubits
, and
, which has the following form:
Performing a partial trace over the environment in Eq. (26), we can obtain the reduced density matrix of
as follows:
where
5. Analysis and discussion5.1. Fidelity of the bidirectional teleportation in AD channelThe fidelity is an important metric to display the distance between quantum states. The fidelity between states
and
is defined as
If state
is a pure quantum state, then the fidelity between
and an arbitrary quantum state ρ can be calculated as
In an AD channel, if Alice and Bob teleport the three-qubit GHZ states
and
by using the scheme in Subsection 2.1, then Alice can obtain a three-qubit quantum state with the density matrix
. According to Eqs. (2), (18), and (31), we can calculate the fidelity between
and
, which can be expressed as
The fidelity
varying with amplitude damping factor
p is shown in Fig.
1 with three different
states denoted as
, and
.
In Fig. 1, we choose
It can be seen from this figure that
decreases with the increase of
p. The quality of the teleportation deteriorates when the amplitude damping turns serious.
5.2. Fidelity of the bidirectional teleportation in AD channel aided with weak measurementAccording to Section 4, if Alice and Bob each perform a weak measurement before the entanglement distribution of qubits
and
, and perform a corresponding reversing measurement after the entanglement distribution, then Alice will obtain the teleportation result
. From Eqs. (2), (28), and (31), we can calculate the fidelity between
and
as
The comparison between
and
is given in Fig. 2. Here, we use the optimal reversing condition:
, and
. From Fig. 2 we can see that
and
both decrease with the increase of p. For each
,
, and
and a certain p,
is always bigger than
, showing the method aided with weak measurement and the reversing measurement can increase the fidelity of the bidirectional teleportation in the AD channel effectively.
The fidelities
and
varying with p and
are shown in Fig. 3. Here, we choose
. It can be seen from this figure that for a certain p,
is always bigger than
, and
increases with weak measurement strength
increasing.
From Eqs. (1), (17), (27), and (31), we can also obtain a similar result that the weak measurement and the reversing measurement can increase the teleportation fidelity of
in the AD channel. Because of the limitation of length, the details of the derivation process are omitted here.
6. ConclusionsIn this work, we present a BQT scheme to transmit three-qubit and multi-qubit GHZ states. This scheme mainly utilizes CNOT operation and single-qubit measurement, thus it has a higher practicability. Unlike previous research that usually discuss BQT in an ideal quantum channel, we intended to study the influence of quantum noise on BQT. As an example, we analyze our BQT scheme in an amplitude damping channel and obtain the fidelity of the teleportation, which decreases with the amplitude damping factor increasing. In order to suppress this influence, we use the weak measurement and the reversing measurement before and after the quantum entanglement distribution, demonstrating an effective enhancement of the teleportation fidelity. On the other hand, we find that a higher fidelity needs a stronger weak measurement strength, which leads to a smaller success probability. This is a contradiction we must consider, and the solution is expected to be found in future work.
