I have read about the no-cloning theorem and read that if the states are known then it is a different situation, I was wondering about this transportation of qubits, can the $U$ be for example an $X$-gate then we say this is possible? $$𝑈|1⟩_𝐴|0⟩_𝐵=|0⟩_𝐴|1⟩_𝐵$$
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1$\begingroup$ Yes, it is called the SWAP gate en.wikipedia.org/wiki/Quantum_logic_gate#Swap_gate $\endgroup$– CondoCommented Apr 5, 2022 at 17:24
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The operation $𝑈|\psi⟩_𝐴|\phi⟩_𝐵=|\phi⟩_𝐴|\psi⟩_𝐵$ (swapping the state of the two qubits) is a unitary operation. So, it is a valid quantum operation. It is known in quantum computing as the $\text{SWAP}$-gate and is represented by the matrix $$\text{SWAP} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
On the other hand, cloning means to create an independent and identical copy of an arbitrary unknown quantum state. That is:
$$𝑈|\psi⟩_𝐴|0⟩_𝐵=|\psi⟩_𝐴|\psi⟩_𝐵$$
No-cloning theorem states that such unitary operator $U$ does not exist.
answered Apr 5, 2022 at 18:13
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3$\begingroup$ Just to be clear, the unitary $U$ as specified by the OP is not uniquely defined. SWAP is one possible option. Even easier (from some perspective) would be $X\otimes X$ $\endgroup$ Commented Apr 6, 2022 at 6:31
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$\begingroup$ Indeed. But since he mentioned this as an example, I just focused on what he called "transportation of qubits". I updated my answer for more clarity $\endgroup$ Commented Apr 6, 2022 at 9:48