Secure and Efficient n-Qubit Entangled State Teleportation Using Partially Entangled GHZ Channels and Optimal POVM

TL;DR

This work tackles efficient teleportation of a restricted class of n-qubit entangled states by leveraging a partially entangled GHZ channel and an optimal POVM for unambiguous discrimination. The authors extend a prior two-qubit approach to an n-qubit setting, introducing (n+1)-qubit POVMs and a corresponding n-qubit receiver operation, achieving reduced classical communication costs relative to standard Bell-basis schemes. A key contribution is the generalization to broader entangled-state configurations and an explicit framework for incorporating reciprocal states via Eldar's semidefinite formulation, along with an efficient measurement scheme that scales linearly with n. The protocol enhances resource efficiency and offers resilience when teleporting encoded logical qubits, with potential benefits for secure, hop-by-hop quantum networks, particularly when alternating channel types hinders eavesdropping. Overall, the approach provides a principled, resource-conscious method for teleporting structured multipartite states while integrating with existing teleportation paradigms to bolster practical quantum networking.

Abstract

We introduce an efficient and versatile quantum teleportation protocol for specific types of n-qubit entangled states. By employing a partially entangled Greenberger-Horne-Zeilinger (GHZ) state as the quantum channel and an optimal Positive Operator-Valued Measure (POVM) based on an improved reciprocal state formulation, we achieve unambiguous state discrimination. The scheme has been generalized to support various entangled state configurations and demonstrates a notable reduction in classical communication costs for these states compared to standard Bell-basis teleportation. Its capacity for integration with conventional protocols is pivotal, enhancing quantum hop-by-hop communication security by allowing strategic choices in quantum channel and teleportation strategy

Figures (6)

• Figure 1: Schematic representation of the quantum teleportation protocol for an n-qubit GHZ-like entangled state. Alice holds the unknown n-qubit state $\ket{\chi_0}=\alpha \ket{0}^{\otimes n} + \beta \ket{1}^{\otimes n}$ and one qubit from the shared partially entangled GHZ state. She performs a Positive Operator-Valued Measurement (POVM) on her (n+1) qubits and transmits the classical outcomes $m_1$ and $m_2$ to Bob via a classical communication channel. Bob, who holds the remaining n qubits, applies appropriate unitary operations (X and Z gates) based on the received classical bits as per Eq. \ref{['gen unitary']} to reconstruct the original state $\ket{\chi_0}$.
• Figure 2: Quantum teleportation of a more generalized n-qubit entangled state. Alice holds an unknown n-qubit state $\ket{\chi_{st}} = \alpha \ket{t_1t_2...t_n} + (-1)^{s} \beta \ket{t_1^\prime t_2^\prime...t_n^\prime}$ and one qubit from a shared partially entangled GHZ channel. She performs a POVM on her (n+1) qubits and sends the classical outcomes $m_1$, $m_2$, and additional bits $t_1, t_2,\dots,t_n, s$ to Bob via a classical communication channel. Bob, holding the remaining n qubits, applies X and Z gates based on the received bits as per Eq. \ref{['t_prime']} to reconstruct $\ket{\chi_{st}}$.
• Figure 3: Quantum teleportation of a 3-qubit entangled state. Alice holds an unknown 3-qubit state $\ket{\chi_{st}^\prime} = \alpha \ket{101} - \beta \ket{010}$ and one qubit from a shared 4-qubit partially entangled GHZ state. After performing a POVM on these qubits, she sends the classical bits $m_1,m_2,t_1=1,t_2=0,t_3=1$ and $s=1$ to Bob. Bob, holding the remaining 3 qubits, applies X and Z gates based on the received bits as per Eq. \ref{['t_prime_expression']} to reconstruct $\ket{\chi_{st}^{\prime}}$.
• Figure 4: Graph showing the variation of success probability depending on the channel quality represented by the parameter b. According to the rule a>b for this specific protocol, the value of b ranges from 0 to $1/\sqrt{2}$. And the probability of successful teleportation $P=2b^2$ inherently implying that for maximally entangled channel i.e. when b=$1/\sqrt{2}$ the success probability i.e. fidelity becomes 1.
• Figure 5: Quantum hop-by-hop communication with enhanced security; Nodes communicate using either GHZ or Bell state entanglement, chosen randomly, when teleporting entangled states within the scope of the proposed scheme. Classical information ($m_1,m_2,s,t_1,t_2,\dots,t_n$) is shared over a public channel but does not reveal any details about the type of quantum channel used between any two immediate nodes. An eavesdropper (Eve) therefore has only a 50% chance of correctly identifying the entanglement type (GHZ or Bell) shared between any two nodes, such as Alice and Bob, enhancing the protocol’s security compared to using only Bell-type correlations.