Quantum Information splitting using a pair of GHZ states

We describe a protocol for quantum information splitting (QIS) of a restricted class of three-qubit states among three parties Alice, Bob and Charlie, using a pair of GHZ states as the quantum channel. There are two different forms of this three-qubit state that is used for QIS depending on the distribution of the particles among the three parties. There is also a special type of four-qubit state that can be used for QIS using the above channel. We explicitly construct the quantum channel, Alice’s measurement basis and the analytic form of the unitary operations required by the receiver for such a purpose.

💡 Research Summary

The paper proposes a protocol for quantum information splitting (QIS) of a restricted class of multi‑qubit states using only a pair of three‑qubit GHZ states as the entangled resource. The authors first review the landscape of quantum teleportation and QIS, noting that while single‑qubit QIS can be achieved with a single GHZ state and two‑qubit QIS with two Bell pairs, extending these ideas to three‑qubit or higher‑dimensional states has remained challenging. Their central contribution is to demonstrate that two GHZ states, when appropriately distributed among three parties (Alice, Bob, and Charlie), can serve as a quantum channel for splitting certain three‑qubit and one special four‑qubit states.

Two distinct distributions of the GHZ pair are considered. In the first (A‑A‑B‑B‑B‑C) configuration, the combined channel is a five‑particle state |Ψ_G⟩ = ½(|000000⟩+|010110⟩+|101001⟩+|111111⟩). The authors show that a three‑qubit state of the form |ξ⟩ = α|000⟩+β|011⟩+γ|100⟩+δ|111⟩ can be split. Alice holds five qubits (the first five of the combined system) and performs a five‑qubit measurement in a specially constructed orthonormal basis {η_i}, i = 0,…,15. Each η_i is generated by applying tensor products of Pauli‑X operators on selected qubits according to the binary representation of i. After measurement, Alice sends the 4‑bit outcome (i) to Bob, while Charlie measures his single qubit in the Hadamard basis and sends a single classical bit (±). Using the 5‑bit classical information, Bob applies a predetermined tensor product of Pauli operators (I, σ_x, σ_y, σ_z) on his three qubits, as listed in Table 1, thereby reconstructing the original three‑qubit state.

A second distribution (A‑A‑B‑B‑B‑C) leads to a different channel |˜Ψ_G⟩ = ½(|000000⟩+|010011⟩+|101100⟩+|111111⟩). Here the splittable three‑qubit state is |˜ξ⟩ = α|000⟩+β|001⟩+γ|110⟩+δ|111⟩. The measurement basis {˜η_i} is defined analogously, and the correction operations for Bob are given in Table 2. Both tables illustrate a systematic mapping from the classical bits to Pauli corrections, demonstrating that the protocol is deterministic and requires only local operations and classical communication (LOCC).

The authors further extend the scheme to a special four‑qubit state |ζ⟩ = α(|0000⟩+|0011⟩)+β(|1100⟩+|1111⟩). Using the same GHZ pair (now in an A‑A‑B‑B‑B‑C layout with six qubits total), Alice performs a five‑qubit measurement in the basis {ν_i}, i = 0,…,3, each constructed from superpositions of computational basis states with appropriate signs. After Alice’s measurement and Charlie’s Hadamard measurement, Bob receives five classical bits and applies the unitary operations listed in Table 3 to recover the original four‑qubit state.

Key technical insights include:

1. The construction of measurement bases using tensor products of σ_x and σ_z conditioned on the binary representation of the outcome index, which exploits the symmetry of GHZ states.
2. The reduction of required classical communication to five bits (four from Alice, one from Charlie) regardless of whether a three‑ or four‑qubit state is being split.
3. The deterministic nature of the protocol: every possible measurement outcome has a corresponding correction, eliminating the need for probabilistic post‑selection.
4. The restriction to specific state families; arbitrary three‑qubit states cannot be split with this channel, highlighting the trade‑off between resource simplicity (only two GHZ states) and universality.

The paper concludes by emphasizing that while the protocol does not achieve universal QIS for arbitrary multi‑qubit states, it establishes that a minimal entanglement resource—two GHZ states—suffices for a non‑trivial class of three‑ and four‑qubit states. The authors suggest that future work could explore extensions to more general states, alternative multipartite entangled resources, and experimental implementations, noting that precise five‑qubit joint measurements and low‑error Pauli corrections are the primary technical challenges.