Quantum Teleportation
Quantum teleportation, is not about moving physical objects like humans, but it refers to the transfer of the quantum state of a single quantum object.
Unlike classical objects, the quantum state of a single object cannot be fully known through measurement due to projection and the no-cloning theorem, which prevents creating copies for repeated measurements.
Quantum teleportation is possible despite the inability to fully know or copy a quantum state. This is because teleportation involves transferring the quantum state, not creating a copy. The original quantum state is destroyed at the source as it is transferred, so it is not violating the no-cloning theorem.
To teleport the polarization state:
|\boldsymbol \varphi \left(\nu_0 \right) \rangle = \lambda | \mathbf x_0 \rangle + \mu | \mathbf y_0 \rangle
of photon \nu_0, an entangled pair of photons \nu_1, \nu_2 in the Bell state |\boldsymbol \Psi^-(\nu_1, \nu_2)\rangle is used:
|\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)
Photons \nu_0 and \nu_1 are combined using a beam splitter, and then jointly detected.
Initially, the state | \Upsilon (\nu_0, \nu_1, \nu_2)\rangle is:
| \Upsilon (\nu_0, \nu_1, \nu_2)\rangle = \frac{1}{\sqrt{2}} \left( \lambda | \mathbf x_0 \rangle + \mu | \mathbf y_0 \rangle \right) \otimes \left( |\mathbf x_1, \mathbf y_2\rangle - |\mathbf y_1, \mathbf x_2\rangle \right)
A partial Bell measurement is performed on photons \nu_0 and \nu_1 using a beam splitter and joint detection. A positive joint detection outcome, indicating a |\boldsymbol \Psi^-\rangle state for \nu_0 and \nu_1, triggers a signal to open shutter V, allowing photon \nu_2 to pass:
|\boldsymbol \Psi^-(\nu_0, \nu_1)\rangle = \frac{1}{\sqrt{2}} \left( |\mathbf x_0, \mathbf y_1\rangle - |\mathbf y_0, \mathbf x_1\rangle \right)
Determining the post-measurement state of \nu_2 requires projecting the initial three-photon state onto the eigenspace corresponding to the measured |\boldsymbol \Psi^-\rangle state of \nu_0, \nu_1, as required by quantum mechanics. The projection is spanned by the tensor product of |\boldsymbol \Psi^-(\nu_0, \nu_1)\rangle with | \mathbf x_2 \rangle and | \mathbf y_2 \rangle:
\left\{|\boldsymbol \Psi^-(\nu_0, \nu_1)\rangle \otimes | \mathbf x_2 \rangle \;, |\boldsymbol \Psi^-(\nu_0, \nu_1)\rangle \otimes | \mathbf y_2 \rangle \right\}
We can then project on each of these eigenvectors. Starting from the first one:
\begin{aligned} \left(\langle\boldsymbol \Psi^-(\nu_0, \nu_1)| \otimes \langle \mathbf x_2| \right) | \Upsilon (\nu_0, \nu_1, \nu_2)\rangle = & \frac{1}{\sqrt{2}} \left(\left( \langle \mathbf x_0, \mathbf y_1| - \langle \mathbf y_0, \mathbf x_1| \right) \otimes \langle \mathbf x_2| \right) | \frac{1}{\sqrt{2}} \left( \lambda | \mathbf x_0 \rangle + \mu | \mathbf y_0 \rangle \right) \otimes \left( |\mathbf x_1, \mathbf y_2\rangle - |\mathbf y_1, \mathbf x_2\rangle \right) \\ = & \frac{1}{2}\left( \langle \mathbf x_0,\mathbf y_1|\otimes\langle \mathbf x_2|-\langle \mathbf y_0,\mathbf x_1|\otimes\langle \mathbf x_2|\right)\left[\lambda|\mathbf x_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle-\lambda|\mathbf x_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle \right. \\ & \left. + \mu|\mathbf y_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle-\mu|\mathbf y_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right] \end{aligned}
Expanding term by term for the first bra factor \langle \mathbf x_0,\mathbf y_1|\otimes\langle \mathbf x_2|:
\begin{aligned} &\langle \mathbf x_0,\mathbf y_1|\otimes\langle \mathbf x_2|\;\left(\lambda|\mathbf x_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle\right) = \lambda\,\langle \mathbf x_0,\mathbf y_1|\mathbf x_0,\mathbf x_1\rangle\,\langle \mathbf x_2|\mathbf y_2\rangle = 0\\ &\langle \mathbf x_0,\mathbf y_1|\otimes\langle \mathbf x_2|\;\left(-\lambda|\mathbf x_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right) = -\lambda\,\langle \mathbf x_0,\mathbf y_1|\mathbf x_0,\mathbf y_1\rangle\,\langle \mathbf x_2|\mathbf x_2\rangle = -\lambda\\ &\langle \mathbf x_0,\mathbf y_1|\otimes\langle \mathbf x_2|\;\left(\mu|\mathbf y_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle\right) = \mu\,\langle \mathbf x_0,\mathbf y_1|\mathbf y_0,\mathbf x_1\rangle\,\langle \mathbf x_2|\mathbf y_2\rangle = 0\\ &\langle \mathbf x_0,\mathbf y_1|\otimes\langle \mathbf x_2|\;\left(-\mu|\mathbf y_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right) = -\mu\,\langle \mathbf x_0,\mathbf y_1|\mathbf y_0,\mathbf y_1\rangle\,\langle \mathbf x_2|\mathbf x_2\rangle = 0 \end{aligned}
For the second bra factor -\langle \mathbf y_0,\mathbf x_1|\otimes\langle \mathbf x_2|:
\begin{aligned} &-\langle \mathbf y_0,\mathbf x_1|\otimes\langle \mathbf x_2|\;\left(\lambda|\mathbf x_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle\right) = -\lambda\,\langle \mathbf y_0,\mathbf x_1|\mathbf x_0,\mathbf x_1\rangle\,\langle \mathbf x_2|\mathbf y_2\rangle = 0\\ &-\langle \mathbf y_0,\mathbf x_1|\otimes\langle \mathbf x_2|\;\left(-\lambda|\mathbf x_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right) = \lambda\,\langle \mathbf y_0,\mathbf x_1|\mathbf x_0,\mathbf y_1\rangle\,\langle \mathbf x_2|\mathbf x_2\rangle = 0\\ &-\langle \mathbf y_0,\mathbf x_1|\otimes\langle \mathbf x_2|\;\left(\mu|\mathbf y_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle\right) = -\mu\,\langle \mathbf y_0,\mathbf x_1|\mathbf y_0,\mathbf x_1\rangle\,\langle \mathbf x_2|\mathbf y_2\rangle = 0\\ &-\langle \mathbf y_0,\mathbf x_1|\otimes\langle \mathbf x_2|\;\left(-\mu|\mathbf y_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right) = \mu\,\langle \mathbf y_0,\mathbf x_1|\mathbf y_0,\mathbf y_1\rangle\,\langle \mathbf x_2|\mathbf x_2\rangle = 0 \end{aligned}
There is only a term that survives:
\left(\langle\boldsymbol \Psi^-(\nu_0, \nu_1)| \otimes \langle \mathbf x_2| \right) | \Upsilon (\nu_0, \nu_1, \nu_2)\rangle = \frac{1}{2}\,\left(-\lambda\right) = -\frac{\lambda}{2}
So the projection gives the state:
-\frac{\lambda}{2} |\boldsymbol \Psi^-(\nu_0, \nu_1)\rangle \otimes | \mathbf x_2 \rangle
Then the second eigenvector:
\begin{aligned} \left(\langle\boldsymbol \Psi^-(\nu_0, \nu_1)| \otimes \langle \mathbf y_2| \right) | \Upsilon (\nu_0, \nu_1, \nu_2)\rangle = & \frac{1}{\sqrt{2}} \left(\left( \langle \mathbf x_0, \mathbf y_1| - \langle \mathbf y_0, \mathbf x_1| \right) \otimes \langle \mathbf y_2| \right) | \frac{1}{\sqrt{2}} \left( \lambda | \mathbf x_0 \rangle + \mu | \mathbf y_0 \rangle \right) \otimes \left( |\mathbf x_1, \mathbf y_2\rangle - |\mathbf y_1, \mathbf x_2\rangle \right) \\ = & \frac{1}{2}\left( \langle \mathbf x_0,\mathbf y_1|\otimes\langle \mathbf y_2|-\langle \mathbf y_0,\mathbf x_1|\otimes\langle \mathbf y_2|\right)\left[\lambda|\mathbf x_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle-\lambda|\mathbf x_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle \right. \\ & \left. + \mu|\mathbf y_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle-\mu|\mathbf y_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right] \end{aligned}
Evaluating term by term:
\begin{aligned} \langle\mathbf x_0,\mathbf y_1|&\otimes\langle\mathbf y_2| \left(\lambda|\mathbf x_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle\right) =\lambda\,\langle\mathbf x_0,\mathbf y_1|\mathbf x_0,\mathbf x_1\rangle\,\langle\mathbf y_2|\mathbf y_2\rangle=0\\ \langle\mathbf x_0,\mathbf y_1|&\otimes\langle\mathbf y_2| \left(-\lambda|\mathbf x_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right) =-\lambda\,\langle\mathbf x_0,\mathbf y_1|\mathbf x_0,\mathbf y_1\rangle\,\langle\mathbf y_2|\mathbf x_2\rangle=0\\ \langle\mathbf x_0,\mathbf y_1|&\otimes\langle\mathbf y_2| \left(\mu|\mathbf y_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle\right) =\mu\,\langle\mathbf x_0,\mathbf y_1|\mathbf y_0,\mathbf x_1\rangle\,\langle\mathbf y_2|\mathbf y_2\rangle=0\\ \langle\mathbf x_0,\mathbf y_1|&\otimes\langle\mathbf y_2| \left(-\mu|\mathbf y_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right) =-\mu\,\langle\mathbf x_0,\mathbf y_1|\mathbf y_0,\mathbf y_1\rangle\,\langle\mathbf y_2|\mathbf x_2\rangle=0 \end{aligned}
For the second bra factor:
\begin{aligned} -\langle\mathbf y_0,\mathbf x_1|&\otimes\langle\mathbf y_2| \left(\lambda|\mathbf x_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle\right) =-\lambda\,\langle\mathbf y_0,\mathbf x_1|\mathbf x_0,\mathbf x_1\rangle\,\langle\mathbf y_2|\mathbf y_2\rangle=0\\ -\langle\mathbf y_0,\mathbf x_1|&\otimes\langle\mathbf y_2| \left(-\lambda|\mathbf x_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right) =\lambda\,\langle\mathbf y_0,\mathbf x_1|\mathbf x_0,\mathbf y_1\rangle\,\langle\mathbf y_2|\mathbf x_2\rangle=0\\ -\langle\mathbf y_0,\mathbf x_1|&\otimes\langle\mathbf y_2| \left(\mu|\mathbf y_0\rangle\otimes|\mathbf x_1,\mathbf y_2\rangle\right) =-\mu\,\langle\mathbf y_0,\mathbf x_1|\mathbf y_0,\mathbf x_1\rangle\,\langle\mathbf y_2|\mathbf y_2\rangle =-\mu\\ -\langle\mathbf y_0,\mathbf x_1|&\otimes\langle\mathbf y_2| \left(-\mu|\mathbf y_0\rangle\otimes|\mathbf y_1,\mathbf x_2\rangle\right) =\mu\,\langle\mathbf y_0,\mathbf x_1|\mathbf y_0,\mathbf y_1\rangle\,\langle\mathbf y_2|\mathbf x_2\rangle=0 \end{aligned}
Only the term -\mu survives; hence
\left(\langle\boldsymbol \Psi^-(\nu_0,\nu_1)|\otimes\langle\mathbf y_2|\right)|\Upsilon(\nu_0,\nu_1,\nu_2)\rangle =\frac{1}{2}(-\mu) =-\frac{\mu}{2}.
And the projection gives the state:
-\frac{\mu}{2}|\boldsymbol \Psi^-(\nu_0, \nu_1)\rangle \otimes | \mathbf y_2 \rangle
The total projection of | \Upsilon (\nu_0, \nu_1, \nu_2)\rangle is the sum of these two partial projections:
-\frac{1}{2}|\boldsymbol \Psi^-(\nu_0, \nu_1)\rangle \otimes \left(\lambda | \mathbf x_2 \rangle + \mu | \mathbf y_2 \rangle\right) = -\frac{1}{2}|\boldsymbol \Psi^-(\nu_0, \nu_1)\rangle \otimes | \boldsymbol \varphi (\nu_2) \rangle
So the quantum state of \nu_0 has been copied to \nu_2, and \nu_0 has disappeared in the process.
This teleportation method succeeds only when the Bell measurement on photons \nu_0 and \nu_1 yields the |\boldsymbol \Psi^-\rangle state, which occurs with a probability of 1/4, so the described quantum teleportation is probabilistic and successful in a quarter of attempts.
In principle, 100% efficient quantum teleportation is possible with a complete Bell state measurement. Current technology, which allows for distinguishing more Bell states like |\boldsymbol \Psi^+\rangle, can improve efficiency beyond 25%, although achieving 100% efficiency is not yet fully realized.
By also detecting the |\boldsymbol \Psi^+\rangle Bell state of \nu_0 and \nu_1, which also occurs with a 1/4 probability, the teleportation efficiency can be improved. This can achieved extended the apparatus has two additional polarized and four detectors in total.
In the case of a |\boldsymbol \Psi^+\rangle measurement, photon \nu_2, if we perform the same projection as above, it ends up in a state related to the initial state of \nu_0, but with a sign difference in the |\mathbf x \rangle component:
\frac{1}{2}|\boldsymbol \Psi^+(\nu_0, \nu_1)\rangle \otimes \left(-\lambda | \mathbf x_2 \rangle + \mu | \mathbf y_2 \rangle\right)
This difference can be corrected by applying a transformation, like using a half-wave plate or an electro-optic device, on photon \nu_2. By applying this correction when |\boldsymbol \Psi^+\rangle is detected and simply allowing photon \nu_2 to pass when |\boldsymbol \Psi^-\rangle is detected, the teleportation succeeds in half the cases.
By extending the Bell measurement to include |\boldsymbol \Phi^+\rangle and |\boldsymbol \Phi^-\rangle states and applying appropriate corrections at device V, teleportation can be generalized even further.
Classical communication, using two control bits, can instruct device V to perform the necessary transformations based on the four possible Bell measurement outcomes. Due to current limitations in achieving complete Bell measurements for single photons, this teleportation method is still limited to a 50\% success rate, corresponding to the detection of |\boldsymbol \Psi^-\rangle or |\boldsymbol \Psi^+\rangle.
Despite this, there is no fundamental barrier to realizing complete Bell measurements, suggesting the possibility of achieving 100\% efficient teleportation in the future.
Quantum teleportation utilizes two communication channels: a quantum channel and a classical channel. The quantum channel is formed by the entangled photon pair \nu_1, \nu_2. This entangled pair acts as a single quantum entity, enabling the projection of the state onto \nu_2. The classical channel is the communication line that transmits the results of the Bell measurement on \nu_0, \nu_1 to device V, instructing it to perform the necessary correction on \nu_2.
The description of quantum teleportation raises a timing issue. Photon \nu_2 arrives at device V around the same time as the measurement on \nu_0 and \nu_1 is performed. However, the setting of device V depends on the result of this measurement, which is communicated classically and cannot travel faster than light.
To resolve this, the arrival of photon \nu_2 at V needs to be delayed, allowing time for the classical information about the measurement outcome to reach and set device V. This delay can be implemented using an optical delay line, such as mirrors or an optical fiber, to lengthen the path of photon \nu_2.
A more sophisticated solution to the timing issue is to use a quantum memory to store the state of photon \nu_2. After a delay that allows for the classical communication of the measurement result to reach device V, the quantum memory would re-emit photon \nu_2 in its corrected state. Quantum memories are a crucial component for quantum communication and networks and are currently a subject of intense research. However, current quantum memory technology is not yet sufficiently advanced to efficiently store and re-emit single photons with a substantial storage duration.
Quantum teleportation, despite its quantum non-locality, does not enable faster-than-light communication of useful information because it relies on a classical communication channel for completion. However, quantum non-locality open new technological possibilities, and quantum teleportation is an example, essential for quantum networks and the future quantum internet.
It allows for long-distance quantum state transfer without signal attenuation, a significant advantage over direct transmission methods like sending qubits through optical fibers, provided entangled qubits are established at distant locations.