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According to the knowledge I have, there are routers, switches etc. Therefore, packets would have to be "measured" before continuing on. (If not, how will anyone know the damn IP address?)

But this effectively ends any advantage of quantum channel..

So how is quantum network working theoretically?

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You would transmit the packet from one end to another by quantum teleportation, which only requires classical communication and entangled resources. If you do make measurements on the qubits, you only do this on an error-correcting encoding of the original data, with the goal of reversing any potential decoherence. You would need a huge amount of entangled bits at distant locations to transmit large amounts of quantum information, but you wouldn't measure any quantum information at any transit station.

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No one know how will a quantum Internet work as it is still in its infancy of the idea. The main difficulty is that quantum information are tight to a particular physical system. In classical communication, you can copy any signal, store and retransmit later easily. However, in quantum communication, once you have done measurement on the whole system, everything is gone. Also, the coupling with the environments are serious problem for communication.

The purpose of quantum Internet is how to move quantum information around, which would allow the task like distributing quantum key and collaborating between different quantum computers (unlike classical computing, a quantum server can work on a job but it never know what it is working on). There are two main different ways for quantum communication.

The first scheme is the use of the quantum teleportation. We assume two parties are sharing unlimited entangled pair. To transmit quantum information, they only need to preform local operations and classical communication (LOCC). Transmission of one qubit requires a measurement on one particle of an entangled pair and sending two classical bits. The model requires a reliable source that generates large amount of entangled pairs and transmit to both parties. Also the operations of two qubits gates need to be precise.

The second scheme is the direct transmission of qubits. Using this method, the qubits is usually carried by the spin-1/2 photon. The generation of the photon likely involve the transfer of quantum information from the internal state of a quantum computer and then convert back in the receiving party. This depends on how a quantum computer might actually work and we dont know yet.

Both schemes require "quantum router" to store and redirect the qubits to the correct destination. One method is to slow the light to nearly stop, so that it can be stored in the quantum router for a while. Another approach would be to transfer qubit from the photon to an atomic state and then convert back later, which could be stored for a longer time.

As said before, it is unclear how the final quantum Internet would work. The decoherent should be the main issue in the deciding the practical schemes. Probably, we may use the neutrino for interplanetary communication one day as it has very weak interaction with normal particles and so the quantum state can be perserved over lightyear.

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There is nothing avoiding you to transmit routing information by classical channels, then the quantum packets would be routed according to the classical information and shared by the receiver and sender in those headers

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