Quantum teleportation: transferring quantum states
Quantum teleportation is a process in quantum mechanics and it isn’t about moving physical objects from one place to another. Instead, quantum teleportation is the transfer of the quantum state of a quantum system to another location, where the original state is destroyed.
Unlike classical objects, we cannot fully determine the quantum state of a single quantum object by measurement. This is due to the projection postulate of quantum mechanics and the no-cloning theorem, which states that an unknown quantum state cannot be perfectly copied. Therefore, teleportation achieves transferring a quantum state without fully knowing or copying it.
Let’s consider the teleportation of the polarization state of a photon, say photon \nu_0. We want to teleport its state:
|\boldsymbol \varphi \left(\nu_0 \right) \rangle = \lambda | \mathbf x_0 \rangle + \mu | \mathbf y_0 \rangle
To do this, we use an entangled pair of photons, \nu_1 and \nu_2, prepared in a Bell state, for example |\boldsymbol \Psi^-(\nu_1, \nu_2)\rangle:
|\boldsymbol \Psi^-(\nu_1, \nu_2)\rangle = \frac{1}{\sqrt{2}} \left( |\mathbf x_1, \mathbf y_2\rangle - |\mathbf y_1, \mathbf x_2\rangle \right)
The teleportation protocol involves performing a Bell measurement on photons \nu_0 and \nu_1. This measurement projects the combined state of \nu_0 and \nu_1 onto one of the Bell states. If the measurement outcome corresponds to the |\boldsymbol \Psi^-\rangle state, a signal is sent to a device, say device V, to let photon \nu_2 pass. Through quantum mechanical projection, it turns out that photon \nu_2 will then acquire a state related to the initial state of photon \nu_0. In fact, in this specific case, photon \nu_2 directly inherits the state |\boldsymbol \varphi \left(\nu_2 \right) \rangle = \lambda | \mathbf x_2 \rangle + \mu | \mathbf y_2 \rangle.
This method is probabilistic. The successful teleportation described above happens only when the Bell measurement on \nu_0 and \nu_1 yields the |\boldsymbol \Psi^-\rangle state, which occurs with a probability of 1/4. However, by performing a complete Bell state measurement, which distinguishes between all four Bell states (|\boldsymbol \Psi^-\rangle, |\boldsymbol \Psi^+\rangle, |\boldsymbol \Phi^+\rangle, |\boldsymbol \Phi^-\rangle), and applying appropriate corrections to photon \nu_2 based on the measurement outcome, it’s in principle possible to achieve 100% efficient quantum teleportation. Current technology allows us to distinguish more Bell states, improving efficiency, but full 100% efficiency is still a technological challenge.
Quantum teleportation utilizes both a quantum channel and a classical channel. The entangled photon pair (\nu_1, \nu_2) forms the quantum channel. The classical channel is used to communicate the result of the Bell measurement from the measurement apparatus to device V, which then applies a correction if needed.
A timing issue arises because photon \nu_2 arrives at device V almost simultaneously with the measurement on \nu_0 and \nu_1. However, the operation of device V depends on the outcome of this measurement, which is communicated classically and thus cannot travel faster than light. To address this, we need to delay photon \nu_2’s arrival at V. This delay can be implemented using an optical delay line. A more advanced approach involves using a quantum memory to store the state of photon \nu_2 until the classical information about the measurement outcome arrives and the necessary correction can be applied before releasing photon \nu_2. Quantum memories are important components for future quantum networks and are an active area of research.
Despite its quantum non-locality, quantum teleportation does not allow for faster-than-light communication of classical information because the classical channel is essential to interpret the teleported quantum state. Nonetheless, quantum teleportation is a key protocol for quantum technologies, especially for quantum networks and the quantum internet. It enables long-distance quantum state transfer without signal loss, offering a significant advantage over direct transmission once entanglement can be distributed over long distances.
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