Perfect Single Qubit Mirroring Effects on Two and Three Maximally Entangled Qubits

Abstract

Perfect quantum state mirroring in a chain of N spins is defined as the condition in which the state |i〉 of the chain is swapped into the state |N — i〉 within a time evolution interval τ. Such a phenomenon is an interesting way of transfering entanglement. An expressions for the perfect mirroring of a single qubit contained in a spin chain were proposed in the past. We exploit such an expressions for calculating the evolution times in chains of both two and three spins. In the case of a chain of two qubits, we derive conditions under which the associated four Bell states diagonalize the Hamiltonian. It is found that for the two Bell states |Φ+〉 and |Φ−〉, perfect mirroring does not occur (i.e. entanglement is not preserved under swapping). On the other hand, perfect single qubit mirror effect (entanglement preservation) indeed occurs for the other two Bell states |Ψ+〉 and |Ψ−〉 which are mapped into |Φ+〉 and |Φ−〉 respectively. For the case of a chain of three qubits, the effects of a perfect single qubit mirroring on a set of four maximally entangled three qubit states ψ1, ψ2, χ1, and χ2 are studied. Due to the fact that quantum mirroring preserves maximal entanglement, the states ψ1 and ψ2 are not altered. However, quantum mirroring changes the states χ1 and χ2 only if we apply perfect quantum state mirroring in the site a = 1 of the three qubits spin chain. The above constrains the preservation of maximal entanglement under qubit mirroring of such a state. Due to the fact that swapping has already been experimentally tested, a posible experimental implementations of single qubit mirroring is possible.