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Suppose Alice has two qubits $A'$ and $A$ and Bob has one qubit $B$. The system is in the state

$$\rho_{A'}\otimes \Phi^+_{AB} $$

where $\rho$ is an arbitrary state and $\Phi^+$ is the maximally entangled triplet state. Quantum teleportation consists of a protocol involving only LOCC that leaves the system in the state

$$\sigma_{A'A}\otimes\rho_B $$

where $\sigma$ is one of the four Bell states. This protocol maps a quantum state to a quantum state hence it should be possible to represent it as a quantum channel, but I'm not really sure how to represent this channel and how to model the classical communication, I tried the following:

After Alice measures $A'$ and $A$ in the Bell basis the post measurement state is

$$ \frac{1}{4}(\Phi^+_{AA'} \otimes \rho_B+\Phi^-_{AA'} \otimes Z\rho_BZ +\Psi^+_{AA'} \otimes X\rho_BX+\Psi^-_{AA'} \otimes XZ\rho_BZX)$$

and after the classical communication, it should be

$$ \frac{1}{4}(\Phi^+_{AA'}+\Phi^-_{AA'}+\Psi^+_{AA'}+\Psi^-_{AA'}) \otimes \rho_B=\frac{1}{4}\mathbb{1}_4\otimes \rho_B$$

Hence the teleportation channel is just a composition of the identity between $A'$ and $B$ and the completely depolarizing channel on $A'A$.

The trouble is that the channel acts this way only if the state on $AB$ is the correct one. What is the general channel that corresponds to teleportation when $AB$ is in the maximally entangled state?

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  • $\begingroup$ Measure AA' in the Bell basis and apply the corresponding Pauli to B? $\endgroup$ Nov 30, 2019 at 21:08

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What you're alluding to here is known as Channel Simulation, and is an extremely useful technique in the study of quantum channels, particularly in bounding channel capacities.

The basic idea revolves around what you said; That ideal quantum teleportation only occurs when the shared state between Alice and Bob $\rho$, (the resource state) is the ideal, maximally entangled state $\rho = \Phi_{AB}^+$. In this way, we say the ideal teleportation protocol of a state $\sigma_A$ from Alice to Bob simulates the identity channel, $\mathcal{I}(\sigma_A)$.

However, consider a teleportation protocol $\mathcal{T}$, which uses a resource state ${\rho_{AB}}$ that is not maximally entangled. The shared quantum correlation between Alice and Bob is now diminished. This diminished correlation manifests as "inaccuracy" on the local operation that Bob applies to his system in order to recreate Alice's state $\sigma_A$. This protocol $\mathcal{T}$ no longer simulates the non-decohering, zero noise identity channel, but instead some decohering, noisy channel $\mathcal{E}$.

More formally, we say that any quantum channel $\mathcal{E}$ admits a LOCC simulation that consists of a teleportation protocol $\mathcal{T}$ and a resource state $\rho$, such that for any input state $\sigma$ the channel output is $$ \mathcal{E}(\sigma) = \mathcal{T}(\sigma \otimes \rho). $$ It is the entanglement properties of the resource state that characterizes the channel being simulated. A channel that can be simulated using a resource state $\rho$ is called "$\rho$ - stretchable".

For more details see the following papers, Pirandola et al. '17 (for detailed theoretical definitions), Pirandola et al. '15, and Tserkis et al. '18 (for more descriptions, fascinating uses, and applications).

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