Probabilistic quantum teleportation via thermal entanglement

Abstract

We study the probabilistic (conditional) teleportation protocol when the entanglement needed for its implementation is given by thermal entanglement, i.e., when the entangled resource connecting Alice and Bob is an entangled mixed state described by the canonical ensemble density matrix. Specifically, the entangled resource we employ here is given by two interacting spin-$\frac{1}{2}$ systems (two qubits) in equilibrium with a thermal reservoir at temperature $T$. The interaction between the qubits is described by a Heisenberg-like Hamiltonian, encompassing the Ising, the $XX$, the $XY$, the $XXX$, and $XXZ$ models, with or without external fields. For all those models, we show analytically that the probabilistic protocol is exactly equal to the deterministic one whenever we have no external field. However, when we turn on the field, the probabilistic protocol outperforms the deterministic one in several interesting ways. Under certain scenarios, for example, the efficiency (average fidelity) of the probabilistic protocol is greater than the deterministic one and increases with increasing temperature, a counterintuitive behavior. We also show regimes in which the probabilistic protocol operates with relatively high success rates and, at the same time, with efficiency greater than the classical limit $2/3$, a threshold that cannot be surpassed by any protocol using only classical resources (no entanglement shared between Alice and Bob). The deterministic protocol's efficiency under the same conditions is below $2/3$, highlighting that the probabilistic protocol is the only one yielding a genuine quantum teleportation. We also show that near the quantum critical points for almost all those models the qualitative and quantitative behaviors of the efficiency change considerably, even at finite $T$.