Let say we have a product state $$|\phi^+\rangle_{12}|\phi^+\rangle_{34}$$ with particles (1,4) belong to $A$ and (2,3) belong to $B$. Where $A$ and $B$ are say at distance of $L$ from each other. When particle $A$ measures his qubits $(1,4)$ in the BSM (Bell state measurement), there is correlation to the BSM of the $(2,3)$ qubits of $B$. Suppose this was part of some secret sharing scheme, what cost does it incur, starting from the initial creation of the state, to the transmission of data (quantum) which includes communication the classical information.

Does the distance between these two parties $A$ and $B$ have some role to play? My feeling is that it should have a role since the quantum data when transmitted over the channel may be exposed to noise or any other errors, so the longer the distance the higher is the probability of the error in transmission.

Is this cost some function of $L$ and the number of qubits (in this case $4$). And if there is some transmission cost how does one reduce this?

Can somebody atleast give some references?


I'm unsure of this exact protocol you're referring to, but if you're interested in the "cost" or reliability of teleportation , then I have some information pertaining to the original setup.

Teleportation is a protocol that uses local operations and classical communications to teleport a state $\rho_{\text{in}}$ from Alice (A) to Bob (B) who share an entangled resource state $\sigma_{AB}$ (typically a Bell state $|{\Phi}\rangle$). Alice performs a Bell measurement on her input state $\rho_{\text{in}}$ and her half of the resource state, the result of which is classically communicated to Bob. They then exploit their shared correlations to deduce a corrective unitary in order to transform Bob's half of the resource into $\rho_{\text{out}} \approx \rho_{\text{in}}$.

During this protocol, quantum data is not transmitted - only classical information about Alice's Bell measurement. Therefore the distance between parties only impacts the quality of the classical communication between the two parties, which we can typically assume is ideal (see here for the interesting impact of imperfect classical communications on teleportation).

What crucially impacts the performance of teleportation is the quality of the entanglement between Alice and Bob. The ideal resource state is maximally entangled in order to optimise the accuracy of Bob's corrective unitary. In this ideal scenario one can think of teleportation as a simulation of transmission of an input state through an identity channel $ \rho_{\text{out}} = \mathcal{I}(\rho_{\text{in}}) = \rho_{\text{in}}. $ But if the resource state entanglement is sub-maximal, it introduces error on this operation. Instead, we simulate some non-identity, decohering channel $ \rho_{\text{out}} = \mathcal{E}(\rho_{\text{in}}) \neq \rho_{\text{in}}. $ Therefore one can quantify the performance of a teleportation protocol based on the entanglement properties of the resource state.

Similar theory could be derived for multi-qudit teleportation schemes, since these are based on the original two qudit scheme. Then indeed, in a multi-qudit scenario, one can imagine that it is indeed harder to guarantee a high degree of entanglement across larger entangled resource states, subsequently reducing the protocol's reliability.

These ideas have some really cool applications in quantum channel capacities, metrology and channel discrimination, and see this paper for some better insight.

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