Bidirectional Quantum Controlled Teleportation by Using Five-Qubit Entangled State As a Quantum Channel

Bidirectional Quantum Controlled Teleportation by Using Five-qubit Entangled State as a Quantum Channel Moein Sarvaghad-Moghaddam1,*, Ahmed Farouk2, Hussein Abulkasim3 to each other simultaneously. Unfortunately, the proposed Abstract— In this paper, a novel protocol is proposed for protocol required additional quantum and classical resources so implementing BQCT by using five-qubit entangled states as a that the users required two-qubit measurements and applying quantum channel which in the same time, the communicated users global unitary operations. There are many approaches and can teleport each one-qubit state to each other under permission prototypes for the exploitation of quantum principles to secure of controller. The proposed protocol depends on the Controlled- the communication between two parties and multi-parties [18, NOT operation, proper single-qubit unitary operations and single- 19, 20, 21, 22]. While these approaches used different qubit measurement in the Z-basis and X-basis. The results showed that the protocol is more efficient from the perspective such as techniques for achieving a private communication among lower shared qubits and, single qubit measurements compared to authorized users, but most of them depend on the generation of the previous work. Furthermore, the probability of obtaining secret random keys [23, 2]. In this paper, a novel protocol is Charlie’s qubit by eavesdropper is reduced, and supervisor can proposed for implementing BQCT by using five-qubit control one of the users or every two users. Also, we present a new entanglement state as a quantum channel. In this protocol, users method for transmitting n and m-qubits entangled states between may transmit an unknown one-qubit quantum state to each other Alice and Bob using proposed protocol. in the same time. The proposed BQCT protocol only use Controlled-NOT operation, proper single-qubit unitary Index Terms— Bidirectional Quantum Teleportation, One- operations and single-qubit measurement in the Z-basis and X- qubit State, Entanglement State, Five-qubit Channel. basis. In addition, controller as a supervisor can select the control of one of the users or every two users. Finally, this I. INTRODUCTION method is generalized for transmitting n and m-qubits entangled uantum information theory represents a fundamental states between users. Q principle to understand the nature of quantum mechanics. The rest of the paper is organized as follows. Section II presents The most used quantum principles are quantum teleportation the proposed method in details. Preparation of five-qubit and dense coding. In quantum teleportation (QT), the quantum entangled state has been demonstrated in section III. The information can be transmitted between remote parties based on proposed work compared to previous work in section IV. both classical communication and maximally shared quantum Finally, Section V concludes the paper. entanglement among the distant parties [13, 14, 15, 16]. The main idea of QT was proposed when Bennett et al. [8] utilized the principle of Einstein-Podolsky-Rosen (EPR) pair as a II. METHOD quantum channel. After that, Zha et al. [9] demonstrated a new Consider a one-qubit state that Alice and Bob would like to version of QT called as Bidirectional Controlled Quantum Teleportation (BCQT) by employing a five-qubit cluster state teleport with each other, simultaneously, in the presence of as a quantum channel. Subsequently, many development and Charlie as controller or supervisor is given by (1): 2 2 research in BQCT area have been represented [10-16]. BCQT |Φ⟩퐴 = 훼0|0⟩ + 훼1|1⟩ where |훼0| + |훼1| = 1 (1) 2 2 protocol allowed the communicated users, in the same time, to |Φ⟩퐵 = 훽0|0⟩ + 훽1|1⟩ where |훽0| + |훽1| = 1 teleport the secret messages to each other under permission of The protocol consists of the following steps: a controller. A BQCT protocol Using the six-qubit state as a quantum channel has been introduced in [12,13] for Step 1: A quantum channel is shared among Alice, Bob, and transmitting one-qubit state in which the operation contained Charlie with five qubits. According to the distribution of two-qubit measurements and so are not optimal. In [17], a Charlie’s qubit in the channel (c), the structure of shared BCQT protocol is proposed using the five-qubit state as a channel can change as one of four states shown in Table I. As quantum channel for transmitting one qubit state between users stated in this Table; Charlie can encode the four states with two *Corresponding Author Ahmed Farouk is with Department of Physics and Computer Science, Moein Sarvaghad-Moghaddam is with the Young Researchers and Elite Wilfrid Laurier University, Waterloo, Canada (email: [email protected] ). Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran (e-mail: Hussein Abulkasim is with Faculty of Science, Assiut University, New [email protected]). Valley branch, Egypt Faculty of Science, South Valley University, Qena, Egypt ([email protected]). classic bits. Where the qubits 푎0, 푎1 and 푏0, 푏1 belong to Alice results to each other. Then, they apply X unitary operation to and Bob, respectively. Also, the qubit c regard to Charlie. their unmeasured qubits ((푏0)(푎1)) (see Table III). The state of the unmeasured qubits will be transferred to Eq. (5). 훼0훽0|00000⟩ + 훼0훽1|01101⟩ + 훼1훽0|10010⟩ + 훼1훽1|11111⟩ (5) TABLE I Step 5: Single-qubit measurements are applied in the X-basis DIFFERENT CHANNELS CREATED USING OF DISTRIBUTION CHARLIE’S QUBIT. on sending qubits (퐴,퐵) by Alice and Bob. The other qubits Encode Channel |휓⟩ (푎0)(푏0)(푎1)(푐)(푏1) collapsed to one of four possible states with the same 0 0 1 (|00000⟩ + |00101⟩ + |11000⟩ probability (see Table IV). 2 + |11101⟩) Step 6: Alice and Bob notify their measurement results to each 0 1 1 other. Then, they apply Z unitary operation to their unmeasured (|00000⟩ + |00111⟩ + |11000⟩ 2 qubits (푏 , 푎 ) as shown in Table V. The state of the + |11111⟩) 0 1 1 0 1 unmeasured qubits will be in the state of Eq. (6). (|00000⟩ + |00101⟩ + |11010⟩ 2 + |11111⟩) (훼0훽0|000⟩ + 훼0훽1|011⟩ + 훼1훽0|100⟩ + 훼1훽1|111⟩)(푏0)(푎1)(푐) (6) 1 1 1 (|00000⟩ + |00111⟩ + |11010⟩ 2 + |11101⟩) TABLE III For example, suppose that the channel shared among Alice, Bob APPLYING X UNITARY OPERATION and Charlie is encoded with 01 as shown in Eq. (2). Continue Alice’s Result Bob’s Result Unitary Operation on (푏0)(푎1) of the protocol is explained using the channel described in Eq. 0 0 퐼⨂퐼 (2). 0 1 퐼⨂푋 1 0 푋⨂퐼 1 (2) |휓⟩ = (|00000⟩ + |00111⟩ + |11000⟩ 1 1 푋⨂푋 (푎0)(푏0)(푎1)(푐)(푏1) 2 + |11111⟩) TABLE IV The general state of system is stated by Eq. (3). THE MEASUREMENT RESULTS BASED ON X AND COLLAPSED STATES. (3) |휙⟩(푎0)(푏0)(푎1)(푐)(푏1)퐴퐵 = |휓⟩(푎0)(푏0)(푎1)(푐)(푏1)⨂ |Φ⟩퐴⨂|Φ⟩퐵 Alice’s Bob’s The collapsed state of qubits (푏0)(푎1)(푐) Result Result Step 2: In this step, a Controlled-NOT gate is applied by Alice + + 훼0훽0|000⟩ + 훼0훽1|011⟩ + 훼1훽0|100⟩ + 훼1훽1|111⟩ and Bob with A and B as control inputs and 푎 , 푏 as target 0 1 + − 훼0훽0|011⟩ − 훼0훽1|000⟩ + 훼1훽0|111⟩ − 훼1훽1|100⟩ inputs, respectively. The general state of the system is stated by Eq. (4). − + 훼0훽0|100⟩ + 훼0훽1|111⟩ − 훼1훽0|000⟩ − 훼1훽1|011⟩ − − 훼0훽0|111⟩ − 훼0훽1|100⟩ − 훼1훽0|011⟩ + 훼 훽 |000⟩ |휙⟩(푎0)(푏0)(푎1)(푐)(푏1)퐴퐵 = 1 1 1 [훼 훽 (|00000⟩ + |00111⟩ + |11000⟩ + |11111⟩)|00⟩ + 2 0 0 TABLE V 훼0훽1(|00001⟩ + |00110⟩ + |11001⟩ + |11110⟩)|01⟩ + APPLYING Z UNITARY OPERATION Alice’s Result Bob’s Result Unitary Operation on (푏0)(푎1) 훼1훽0(|10000⟩ + |10111⟩ + |01000⟩ + |01111⟩)|10⟩ + + + 퐼⨂퐼 훼1훽1(|10001⟩ + |10110⟩ + |01001⟩ + |01110⟩)|11⟩] (4) + − 퐼⨂푍 − + 푍⨂퐼 Step 3: Alice and Bob perform a single-qubit measurement in − − 푍⨂푍 Z-basis on qubits 푎 , 푏 respectively. The other qubits collapse 0 1 Step 7: Charlie notify the distribution status of his qubit to Alice to one of four possible states with the same probability (see and Bob with two classical bits (see Table I).Then, he measures Table II). his qubit in X-basis and informs Alice and Bob of his result. If TABLE II Charlie’s measured result is |+⟩ (|−⟩), then, the states of other MEASUREMENT RESULTS BASED ON Z AND COLLAPSED STATES. qubits are equal to either Eq. (7) or (8), respectively. The Alice’s Bob’s measurement results on Charlie’s qubit are shown in Table VI The collapsed state of qubits (푏0)(푎1)(푐)퐴퐵 Result Result for the different channels of Table I. 훼0훽0|00000⟩ + 훼0훽1|01101⟩ + 훼1훽0|10010⟩ 0 0 (훼0훽0|00⟩ + 훼0훽1|01⟩ + 훼1훽0|10⟩ + 훼훽1|11⟩)(푏0)(푎1) (7) + 훼1훽1|11111⟩ 훼 훽 |01100⟩ + 훼 훽 |00001⟩ + 훼 훽 |11110⟩ 0 1 0 0 0 1 1 0 (훼0훽0|00⟩ − 훼0훽1|01⟩ + 훼훽0|10⟩ − 훼1훽1|11⟩)(푏0)(푎1) (8) + 훼1훽1|10011⟩ 훼 훽 |10000⟩ + 훼 훽 |11101⟩ + 훼 훽 |00010⟩ Step 8: According to Charlie’s results, Alice and Bob apply 푍 1 0 0 0 0 1 1 0 + 훼1훽1|01111⟩ unitary operation (see Table VI). Finally, Alice and Bob can 훼 훽 |11100⟩ + 훼 훽 |10001⟩ + 훼 훽 |01110⟩ 1 1 0 0 0 1 1 0 reconstruct the transmitted states again and the BQCT is + 훼1훽1|00011⟩ successfully completed (see Eq. (9) and (10)). |휓⟩퐴 = 푏0|0⟩ + 푏1|1⟩ (9) Step 4: First, Alice and Bob notify the Z-basis measurement |휓⟩퐵 = 푎0|0⟩ + 푎1|1⟩ (10) TABLE VI APPLYING Z UNITARY OPERATION FOR THE DIFFERENT CHANNELS SHOWED IN TABLE I. Coding bits showing different Charlie’s Results The collapsed state of qubits Unitary Operation on (푏0)(푎1) channels (푏0)(푎1) |+⟩ |00⟩ + |01⟩ + |10⟩ + |11⟩ 퐼⨂퐼 00 |−⟩ |00⟩ + |01⟩ + |10⟩ + |11⟩ 퐼⨂퐼 |+⟩ |00⟩ + |01⟩ + |10⟩ + |11⟩ 퐼⨂퐼 01 |−⟩ |00⟩ − |01⟩ + |10⟩ − |11⟩ 퐼⨂푍 |+⟩ |00⟩ + |01⟩ + |10⟩ + |11⟩ 퐼⨂퐼 10 |−⟩ |00⟩ + |01⟩ − |10⟩ − |11⟩ 푍⨂퐼 |+⟩ |00⟩ + |01⟩ + |10⟩ + |11⟩ 퐼⨂퐼 11 |−⟩ |00⟩ − |01⟩ − |10⟩ + |11⟩ 푍⨂푍 Afterward, Hadamard gates are applied to qubits 푏0 and 푎1 A.

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