User:TLee115/sandbox/Quantum repeater

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In quantum communication, a quantum repeater is a device which allows the retransmission of quantum information (such as qubits encoded on photons) in order to overcome the effects of loss, decoherence, and channel noise over long distances and extend the range of transmission. Quantum repeaters are so named by analogy with repeaters used in classical telecommunications.^[1]

Unlike a classical repeater, which can retransmit an arbitrary classical signal, a quantum repeater cannot retransmit an arbitrary quantum state by directly amplifying or reconstructing it due to the no-cloning theorem. Instead, the quantum information is transferred via quantum teleportation. Quantum teleportation can be used either to extend the transmission range of a single qubit or to generate entanglement between distant locations via entanglement swapping.^[1][2]

Context

Repeaters in classical telecommunication

In classical fibre-optic telecommunications, information is encoded by transmitting pulses of light through an optical fibre. For example, a binary digit ("0" or "1") may be encoded onto the optical intensity measured at a given time, where a "0" represents zero intensity (or intensity below a threshold) and "1" represents intensity above a threshold.

Although silica fibres are highly transparent to light, optical signals will inevitably be degraded or distorted over long distances due to loss (e.g. due to material absorption or Rayleigh scattering), chromatic dispersion, or (at sufficiently high powers) parasitic noise generated by nonlinear optical effects. For example, an optical pulse transmitted into a fibre with a loss of 0.2 dB/km will be attenuated to 1% of its initial power after a distance of 100 km. Therefore, to perform high-bandwidth long-distance communication it is necessary to introduce optical repeaters at nodes periodically spaced along the fibre network, separated by tens of kilometres. Each of these repeaters amplify and reshape the distorted optical signal, effectively producing a copy of the original transmitted signal.^[3]

Quantum communication

Qubits

In quantum communication, information consists of qubits which have a quantum state that can be expressed as:$${\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,}$$ where ${\displaystyle \alpha ,\beta \in \mathbb {C} }$  are complex-valued coefficients satisfying ${\displaystyle |\alpha |^{2}+|\beta |^{2}=1}$  and the set of orthonormal states ${\displaystyle \{|0\rangle ,|1\rangle \}}$  is referred to as the computational basis. The ${\displaystyle |0\rangle }$  and ${\displaystyle |1\rangle }$  states can be encoded using degrees of freedom of a photon, such as polarization, spatial mode, or time bin. For example, the above state ${\displaystyle |\psi \rangle }$  could be encoded using polarization states, with the horizonal state ${\displaystyle |H\rangle }$  representing the computational state ${\displaystyle |0\rangle }$ ; and the vertical polarization ${\displaystyle |V\rangle }$  representing the computational state ${\displaystyle |1\rangle }$ , so that a photon has the overall state:$${\displaystyle |\psi \rangle =\alpha |H\rangle +\beta |V\rangle .}$$ A similar approach can be taken using other degrees of freedom; for example, time bin encoding might employ photons in states corresponding to "early" and "late" arrival times, relative to a reference timing.^[4]

Limits to long-distance quantum communication

Long-distance fibre-based quantum communication also requires a method of relaying a signal between nodes because optical losses may prevent the qubit from reaching the intended recipient. For instance, for the same 100-km length of fibre described above, only one photon/qubit is expected to be received for every 100 transmitted, thus imposing a severe limit on achievable bit rates. Moreover, decoherence of quantum states can severely degrade quantum information processes or protocols, such as quantum key distribution.

A classical repeater cannot be used for this purpose by attempting to directly amplify the photonic signal because the no-cloning theorem states that it is not possible to create a copy of an arbitrary or unknown quantum state between two systems, i.e. there is no physical process which can enable the operation:^[1] $${\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}\longmapsto |\psi \rangle _{A}\otimes |\psi \rangle _{B},}$$ where the subscripts A and B correspond to two distinct quantum systems or Hilbert spaces, ${\displaystyle |\psi \rangle }$  is the desired state to be copied, ${\displaystyle |\phi \rangle }$  is the initial state of system B to be transformed into ${\displaystyle |\psi \rangle }$ , and ${\displaystyle \otimes }$  denotes the tensor product.

This limitation cannot be overcome by first measuring the state and attempting to reproduce it, as a measurement causes the state to reduce to one eigenstate (${\displaystyle |0\rangle }$  or ${\displaystyle |1\rangle }$  in the computational basis), rather than the original unreduced pure state ${\displaystyle |\psi \rangle }$ .^[5]

Instead of directly amplifying the photonic signal, a quantum repeater retransmits a qubit by transferring the quantum state itself between two systems via quantum teleportation, as proposed by C. Bennett et al. in 1993.^[6]

Quantum teleportation of a single qubit

 

Illustration of quantum teleportation of an unknown quantum state. Alice's unknown state is first contained in system a. An EPR pair (or Bell state) in systems A and B is generated by Charlie and sent to Alice and Bob respectively. The result of Alice's Bell state measurement (BSM) on systems a and A is communicated to Bob via a classical channel; Bob subsequently applies the appropriate operation (${\displaystyle {\hat {\sigma }}_{x}}$ , ${\displaystyle {\hat {\sigma }}_{y}}$ , ${\displaystyle {\hat {\sigma }}_{z}}$ , or ${\displaystyle {\hat {I}}}$ , represented by the blue circle) on his system B. After applying the appropriate operator, the system B is now in the unknown state that was originally contained in system a.

Overview

Suppose the sender, Alice (A), wishes to transmit the pure state ${\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle }$  to her intended recipient, Bob (B), using a repeater node Charlie (C). The basic idea is as follows:

• Alice and Bob must first establish a classical channel between each other, and a quantum channel each between themselves and Charlie.
• Charlie is to prepare a pair of entangled photons, known as a Bell state or EPR pair, and send one photon each to Alice and Bob so that they share a maximally entangled state.
• Alice will then perform a joint measurement between the state ${\displaystyle |\psi \rangle }$  and her portion of the entangled pair generated by Charlie.
• Alice then communicates the result of her joint measurement to Bob via the classical channel.
• Based on the result communicated by Alice, Bob performs a transformation on his portion of the entangled photon pair, which causes that photon to now carry the pure state ${\displaystyle |\psi \rangle }$ .

Protocol

Establishing a maximally-entangled state between Alice and Bob

Charlie prepares the entangled photon pair in any of the four Bell states given by:

${\displaystyle |\Phi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B}{\big )}}$ 

${\displaystyle |\Phi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B}{\big )}}$ 

${\displaystyle |\Psi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B}{\big )}}$ 

${\displaystyle |\Psi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}{\big )}}$ 

where the subscripts A and B correspond to the qubit sent to Alice and Bob respectively. Although any of the four states can be used, for illustrative purposes we consider only the case of using ${\displaystyle |\Psi ^{-}\rangle }$ .

This Bell state pair is then entangled with the photon prepared in state ${\displaystyle |\psi \rangle }$  by Alice. The resulting three-photon state is given by:$${\displaystyle |\psi \rangle _{a}\otimes |\Psi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}{\big (}|\psi \rangle _{a}\otimes |0\rangle _{A}\otimes |1\rangle _{B}-|\psi \rangle _{a}\otimes |1\rangle _{A}\otimes |0\rangle _{B}{\big )}.}$$ where the subscript a denotes the system in which Alice initially prepares the state ${\displaystyle |\psi \rangle }$ . Substituting the above expression for ${\displaystyle |\psi \rangle }$ :$${\displaystyle |\psi \rangle _{a}\otimes |\Psi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}{\big (}\alpha |0\rangle _{a}\otimes |0\rangle _{A}\otimes |1\rangle _{B}-\alpha |0\rangle _{a}\otimes |1\rangle _{A}\otimes |0\rangle _{B}+\beta |1\rangle _{a}\otimes |0\rangle _{A}\otimes |1\rangle _{B}-\beta |1\rangle _{a}\otimes |1\rangle _{A}\otimes |0\rangle _{B}{\big )}.}$$ For a tensor product of two qubits, we note that the following identities apply:

${\displaystyle |0\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle +|\Phi ^{-}\rangle )}$ 

${\displaystyle |0\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle +|\Psi ^{-}\rangle )}$ 

${\displaystyle |1\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle -|\Psi ^{-}\rangle )}$ 

${\displaystyle |1\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle -|\Phi ^{-}\rangle )}$ 

Applying these identities to the systems a and A, the three-photon system can be written as: $${\displaystyle |\psi \rangle _{a}\otimes |\Psi ^{-}\rangle _{AB}={\frac {1}{2}}\left\{|\Psi ^{-}\rangle _{aA}\otimes {\big (}-\alpha |0\rangle _{B}-\beta |1\rangle _{B}{\big )}+|\Psi ^{+}\rangle _{aA}\otimes {\big (}-\alpha |0\rangle _{B}+\beta |1\rangle _{B}{\big )}+|\Phi ^{-}\rangle _{aA}\otimes {\big (}\alpha |1\rangle _{B}+\beta |0\rangle _{B}{\big )}+|\Phi ^{+}\rangle _{aA}\otimes {\big (}\alpha |1\rangle _{B}-\beta |0\rangle _{B}{\big )}\right\}.}$$ 

Performing a joint measurement on Alice's photons

Alice now performs a Bell state measurement on the systems a and A. Upon performing the measurement, the three-photon system will be reduced into one of four states with equal probability:

${\displaystyle |\Psi ^{-}\rangle _{aA}\otimes -{\big (}\alpha |0\rangle _{B}+\beta |1\rangle _{B}{\big )}}$ 

${\displaystyle |\Psi ^{+}\rangle _{aA}\otimes {\big (}-\alpha |0\rangle _{B}+\beta |1\rangle _{B}{\big )}}$ 

${\displaystyle |\Phi ^{-}\rangle _{aA}\otimes {\big (}\alpha |1\rangle _{B}+\beta |0\rangle _{B}{\big )}}$ 

${\displaystyle |\Phi ^{+}\rangle _{aA}\otimes {\big (}\alpha |1\rangle _{B}-\beta |0\rangle _{B}{\big )}.}$ 

Applying a transformation on Bob's qubit

Alice informs Bob of the result of the Bell state measurement. In response, Bob will apply one of four operations on the qubit B: the identity operator ${\displaystyle {\hat {I}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}=|0\rangle \langle 0|+|1\rangle \langle 1|}$ , or any of the three Pauli operators

${\displaystyle {\hat {\sigma }}_{x}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}=|0\rangle \langle 1|+|1\rangle \langle 0|}$ ,

${\displaystyle {\hat {\sigma }}_{y}={\begin{bmatrix}0&-i\\i&0\end{bmatrix}}=-i|0\rangle \langle 1|+i|1\rangle \langle 0|}$ ,

${\displaystyle {\hat {\sigma }}_{z}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}=|0\rangle \langle 0|-|1\rangle \langle 1|}$ .

Depending on the state of qubit B which corresponds to Alice's result, these four operators can be used to transform the qubit B into the state ${\displaystyle |\psi \rangle _{B}}$  (up to a global phase) as follows:

${\displaystyle {\hat {I}}{\big (}-\alpha |0\rangle _{B}-\beta |1\rangle _{B}{\big )}=-{\big (}\alpha |0\rangle _{B}+\beta |1\rangle _{B}{\big )}=-|\psi \rangle _{B}}$ 

${\displaystyle {\hat {\sigma }}_{z}{\big (}-\alpha |0\rangle _{B}+\beta |1\rangle _{B}{\big )}=-{\big (}\alpha |0\rangle _{B}+\beta |1\rangle _{B}{\big )}=-|\psi \rangle _{B}}$ 

${\displaystyle {\hat {\sigma }}_{x}{\big (}\alpha |1\rangle _{B}+\beta |0\rangle _{B}{\big )}={\big (}\alpha |0\rangle _{B}+\beta |1\rangle _{B}{\big )}=|\psi \rangle _{B}}$ 

${\displaystyle {\hat {\sigma }}_{y}{\big (}\alpha |1\rangle _{B}-\beta |0\rangle _{B}{\big )}=-i{\big (}\alpha |0\rangle _{B}+\beta |1\rangle _{B}{\big )}=-i|\psi \rangle _{B}.}$ 

Thus, the state ${\displaystyle |\psi \rangle }$  has been transferred from Alice's original system a onto the system B sent to Bob, provided that Bob performs the correct operation on system B. Upon making her Bell state measurement, Alice can inform Bob of which operator to apply in order to obtain the original state ${\displaystyle |\psi \rangle }$ . The correct choice of Bob's operator is determined by Alice, who sends to Bob a two-bit code corresponding to the Bell state she measured, for example according to the table below.^[5][7]

Alice measures:	Alice sends:	Bob applies:
${\displaystyle |\Psi ^{-}\rangle _{aA}}$	00	${\displaystyle {\hat {I}}}$
${\displaystyle |\Psi ^{+}\rangle _{aA}}$	01	${\displaystyle {\hat {\sigma }}_{z}}$
${\displaystyle |\Phi ^{-}\rangle _{aA}}$	10	${\displaystyle {\hat {\sigma }}_{x}}$
${\displaystyle |\Phi ^{+}\rangle _{aA}}$	11	${\displaystyle {\hat {\sigma }}_{y}}$

Using this protocol, the repeater has been used to "teleport" the quantum state between Alice and Bob, with the cost that a classical communication channel must be used in order for Alice to indicate the outcome of her measurement. The photon carrying the original state ${\displaystyle |\psi \rangle }$  has not physically transported from Alice to Bob; in fact, the photons corresponding to systems a and B never directly interact throughout this procedure.^[8] Instead, the information about the quantum state has been transferred to system B without being copied. This process avoids violating the no-cloning theorem because no measurement has been performed on the state ${\displaystyle |\psi \rangle }$ .

The above process describes the process for one repeater between Alice and Bob. If multiple repeaters are to be used, this is equivalent to performing the above quantum teleportation process sequentially across several repeaters. A new Bell state measurement would be performed every time the state ${\displaystyle |\psi \rangle }$  is teleported.^[1][2]

Equivalent protocols can be employed for any of the other three Bell states.^[5]

Practical considerations

Because quantum teleportation requires Alice and Bob to communicate classical information, Bob's system B must function as a quantum memory which stores the quantum state long enough for the classical communication to occur. This implies that quantum teleportation of individual qubits requires successful implementation of quantum memories.^[9]

Entanglement swapping

 

Illustration of entanglement swapping. Alice and Bob first share entangled systems A and C1 (yellow) and B and C2 (green) with Charlie. After Charlie performs a BSM between C1 and C2, the result is classically communicated to both Alice and Bob, who apply the appropriate transformations on systems A and B. Systems A and B are entangled (red) as a result of this protocol.

Entanglement swapping extends the idea of quantum teleportation described above by allowing Alice and Bob to generate a maximally entangled state between them, despite not creating a direct interaction between their qubits.^[8] Such a procedure was first proposed by M. Żukowski et al. in 1993,^[8] and experimentally demonstrated by J. Pan et al. in 1998.^[10]

In this protocol, Alice and Bob each generate a Bell state which are both shared with Charlie (rather than Charlie creating and sending a Bell state pair to Alice and Bob). Alice's Bell state is created in systems A and C1, and Bob's Bell state is in systems B and C2. Charlie now performs a Bell state measurement on systems C1 and C2, and simultaneously communicates the outcome (via a classical channel) to Alice and Bob. Alice and Bob then apply appropriate operators (${\displaystyle {\hat {\sigma }}_{x}}$ , ${\displaystyle {\hat {\sigma }}_{y}}$ , ${\displaystyle {\hat {\sigma }}_{z}}$ , ${\displaystyle {\hat {I}}}$ , or possibly a combination of them) on systems A and B. After performing their operations, systems A and B will now be in a maximally entangled state.^[1][8]

For example, say Alice and Bob wish to generate a Bell state ${\displaystyle |\Phi ^{+}\rangle _{AB}}$  between their two systems. They would create Bell states ${\displaystyle |\Phi ^{+}\rangle _{AC1}}$  and ${\displaystyle |\Phi ^{+}\rangle _{C2B}}$  and share the appropriate systems with Charlie. Proceeding similarly to the example of a single-qubit teleportation above, Charlie would measure C1 and C2 to be in one of the four Bell states with equal probability, and inform Alice and Bob of the outcome. Alice and Bob would then apply the appropriate operation on their qubits according to the table below^[1]:

Charlie measures:	Alice apples on A:	Bob applies on B:
${\displaystyle |\Psi ^{-}\rangle _{C1C2}}$	${\displaystyle {\hat {I}}}$	${\displaystyle {\hat {\sigma }}_{z}}$ ${\displaystyle {\hat {\sigma }}_{x}}$
${\displaystyle |\Psi ^{+}\rangle _{C1C2}}$	${\displaystyle {\hat {I}}}$	${\displaystyle {\hat {\sigma }}_{x}}$
${\displaystyle |\Phi ^{-}\rangle _{C1C2}}$	${\displaystyle {\hat {I}}}$	${\displaystyle {\hat {\sigma }}_{z}}$
${\displaystyle |\Phi ^{+}\rangle _{C1C2}}$	${\displaystyle {\hat {I}}}$	${\displaystyle {\hat {I}}}$

(Note that in the ${\displaystyle |\Psi ^{-}\rangle _{C1C2}}$  case, Bob applies the ${\displaystyle {\hat {\sigma }}_{x}}$  operator first, before applying the ${\displaystyle {\hat {\sigma }}_{z}}$  operator.)

After applying the appropriate operations, systems A and B will now be in the state ${\displaystyle |\Phi ^{+}\rangle _{AB}}$ .

See also

• Quantum key distribution
• Qubit
• Quantum memory
• Quantum entanglement
• Bell's theorem
• No-cloning theorem
• Quantum entanglement swapping
• Quantum teleportation

Further reading

• Barnett, S. M. (2009). Quantum information. Oxford master series in physics. Atomic, optical and laser physics. Oxford: Oxford University Press. ISBN 978-0-19-852762-6. OCLC 316430129.
• Kok, Pieter; Lovett, Brendon W. (2010). Introduction to optical quantum information processing. Cambridge New York: Cambridge University Press. ISBN 978-0-511-77618-2.
• Bergou, János A.; Hillery, Mark Stephen; Saffman, Mark (2021). Quantum information processing: theory and implementation. Graduate texts in physics (2nd ed.). Cham: Springer. ISBN 978-3-030-75435-8.
• Nielsen, Michael A.; Chuang, Isaac (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0521632358. OCLC 43641333.

References

1. Azuma, K.; Economou, S. E.; Elkouss, D.; Hilaire, P.; Jiang, L.; Lo, H.-K.; Tzitrin, I. (2003). "Quantum repeaters: From quantum networks to the quantum internet". Reviews of Modern Physics. 95 (4). doi:10.1103/RevModPhys.95.045006. ISSN 0034-6861.
2. Kok, P.; Lovett, B. W. (2010). Introduction to optical quantum information processing. Cambridge New York: Cambridge University Press. ISBN 978-0-511-77618-2.
3. Agrawal, G. P. (2022). Fiber-optic communication systems. Wiley series in microwave and optical engineering (5th ed.). Hoboken, NJ: Wiley. ISBN 978-1-119-73738-4.
4. Marcikic, I. (2002). "Time-bin entangled qubits for quantum communication created by femtosecond pulses". Physical Review A. 66 (6). doi:10.1103/PhysRevA.66.062308.
5. Barnett, S. M. (2009). Quantum information. Oxford master series in physics. Atomic, optical and laser physics. Oxford: Oxford University Press. ISBN 978-0-19-852762-6. OCLC 316430129.
6. Bennett, C. H.; Brassard, G.; Crépeau, C.; Jozsa, R.; Peres, A.; Wootters, W. K. (1993). "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels". Physical Review Letters. 70 (13): 1895–1899. doi:10.1103/PhysRevLett.70.1895.
7. Bergou, J. A.; Hillery, M. S.; Saffman, M. (2021). Quantum information processing: theory and implementation. Graduate texts in physics (2nd ed.). Cham: Springer. ISBN 978-3-030-75435-8.
8. Pan, J.-W.; Bouwmeester, D.; Weinfurter, H.; Zeilinger, A. (1998). "Experimental Entanglement Swapping: Entangling Photons That Never Interacted". Physical Review Letters. 80 (18): 3891–3894. doi:10.1103/PhysRevLett.80.3891.
9. Munro, William J.; Azuma, K.; Tamaki, K.; Nemoto, K. (2015). "Inside Quantum Repeaters". IEEE Journal of Selected Topics in Quantum Electronics. 21 (3): 78–90. doi:10.1109/JSTQE.2015.2392076.
10. Żukowski, M.; Zeilinger, A.; Horne, M. A.; Ekert, A. K. (1993). "Event-ready-detectors Bell experiment via entanglement swapping". Physical Review Letters. 71 (26): 4287–4290. doi:10.1103/PhysRevLett.71.4287.