Definition of quantum spin networks and something else I recently started to have an interest in quantum computing (I am a student in Mathematics). I have various questions for terminologies, and I wonder if I am correctly understanding concepts.

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*As a definition of a quantum spin network, is it okay to say 'a system of quantum states together with their couplings.'


*In the paper 'Perfect Transfer of Arbitrary States in Quantum Spin Networks' (https://arxiv.org/abs/quant-ph/0411020), I wonder why they keep mentioning 'couplings' instead of `entanglements.' I read the difference between them in Wikipedia. Based on the difference described in (https://en.wikipedia.org/wiki/Quantum_coupling), I am just guessing that since entanglements are not likely to happen in empirical settings, they use the term 'couplings.'


*As I read some articles related to 'perfect state transfer,' I can see this sentence quite a lot 'in quantum computing, it is important to transfer a quantum state from one location to another.' I am not clear about what stuff is transferred from where to where. Here is what I understand. The word 'quantum state' is an abstract concept that represents a thing satisfying some properties such as superposition and so on in quantum mechanics. So, a quantum state can be a photon, a qubit, etc. In the context of quantum computing, a quantum state is regarded as a qubit, and in empirical settings, it is some atom. Then, that atom has the information about various spins. Here is what I am confused as I read several articles.
(i) A state transfer means a transfer of the information regarding spins in an atom from one (literally) place to another. That is, it is transferring the spin information, not some substance (we can see or touch) to one place to another.
(ii) A qubit is a quantum state; so, quantum state transfer is `transfer of a qubit (an atom) from one place to another.'
As I read several articles, it seems authors assume that a qubit and the related information are identical despite qubits are some substances. Could you let me know where I started to misunderstand?


*I would appreciate if you recommend some book for a beginner for me to polish my thoughts up.

 A: Short answers to your questions:

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*This is not quite enough. This misses the important tensor product structure of the Hilbert space, together with the physical geometry of the system. These properties are given by Eqs. 2 and 1 respectively in the paper you linked (explained by the surrounding text).


*No, entanglement is a property of a quantum state (eg. Eq 5 in the paper you linked), whereas a coupling is a parameter (denoted $J_{ij}$ in the paper you linked) from the Hamiltonian (the generator of dynamics, Eq 3. in the paper you linked).


*If two subsystems $\mathcal{H}_A, \mathcal{H}_B$ of the global Hilbert space have the same structure (in quantum computing it is sufficient that they consist of the same number of qubits) one can imagine transferring a state from one to the other. Specifically, given (1) a state $| \psi \rangle$ with initial reduced density matrix $\rho_A = \mathrm{tr}_\bar{A}(| \psi \rangle \langle \psi |)$ (here $\bar{A}$ denotes the complement of $A$), (2) a series of operations $U = U_N \ldots U_3 U_2 U_1$ and (3) an the output state on the second subsystem $\rho_B = \mathrm{tr}_\bar{B}( U | \psi \rangle \langle \psi | U^\dagger)$, the operations $U$ have transferred the state from $A$ to $B$ if $\rho_A = \rho_B$.
A comment: If you are of a mathematics background, and interested in this topic, I recommend Nielsen and Chuang's classic pedagogical text. Reading this may significantly aid your efforts to engage with research in the field.
