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Please be careful with your notation; don't get confused with the number of qubits that are in $|0_{Eve}\rangle$ (I'm not necessarily saying that you are - just mentioning this as a precaution). Since the state $|\psi\rangle = \frac{1}{\sqrt{N}}\sum_{j \in 0,1,2....N-1}|j\rangle$ that Alice wants to send is a $n$-qubit state with $N=2^{n}$, Alice needs to ...

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It does exist. It's basically the controlled-not gate generalised to higher dimensional systems. The important thing to realise is that means that the bit that Bob ends up with will be highly entangled with what Eve has, and that will have significant observable consequences. As part of a cryptographic protocol, for exampe, Bob could detect that Eve is ...

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The basic answer is to just write, for a given bipartite state $X$, $\Phi(X)\equiv\operatorname{Tr}_B(X)$. This defines a quantum channel which acts on the input as a partial trace. You can check that this does indeed qualify as a quantum channel etc. A misconception that might have led you to ask this question can be found in your saying that "I have no ...

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The first and foremost thing to realize is that the partial trace over a density matrix is indeed a linear CPTP map $\Lambda$, but it is not a map from any $\mathcal{C}^{n\times n} \rightarrow \mathcal{C}^{n\times n}$ (i.e. to `itself' - the same dimension), but rather to a density operator space with a lower dimension: \$\mathcal{C}^{n\times n} \rightarrow \...

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