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The question says it all. Explain any difference in terminology that can be encountered across Physics, Quantum Computing and Mathematics. Thanks!

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You are correct that "basis states" and "basis vectors" essentially mean the same thing, but the terms do have a bit of a different connotation.

The eigenvectors that span a Hamiltonian matrix are "states" in which a quantum mechanical system can exist. The ones with the smallest eigenvalue are "ground state wavefunctions", the ones with the next smallest eigenvalue are the "first excited state wavefunctions", etc. The eigenvectors of a Hamiltonian matrix are also basis vectors/states, because Hamiltonians are Hermitian.

However the eigenvectors of a Hermitian matrix arising in some finance application that has nothing to do with quantum mechanics, will be called "basis vectors" but not "basis states" because they don't necessarily have the meaning of being "states" of anything.

A bigger distinction comes from the fact that not all quantum systems are discrete, for example the position $x$ of a particle is often treated to be a continuous variable. Here we are reminded that the Hamiltonian is not just a "matrix" but it's something more general called an "operator". There is still an eigensystem associated with this operator, but instead of calling the wavefunctions "eigenvectors", we call them "eigenfunctions". Eigenfunctions are functions like $\psi(x) = e^{-x^2}$ and you would not usually call this a "vector" (although you certainly can if you want to).

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    $\begingroup$ Thank you for the detailed explanation! $\endgroup$ – M. Al Jumaily Apr 12 at 8:23
  • $\begingroup$ No problem, anytime :) $\endgroup$ – user1271772 Apr 12 at 15:14
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Any quantum state is described by a complex vector of dimension $n$. When you take $n$ linearly independent vectors, they form a basis. The same is true for quantum states as they are described by vectors, hence these states are also called basis.

So, answer is: yes, basis states are composed of a vector space basis.

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  • $\begingroup$ Note: this is true for quantum computing mathematical description only, answer by user127177 is more general. $\endgroup$ – Martin Vesely Mar 23 at 20:39
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    $\begingroup$ There's continuous variable quantum computing: In that case we can have "eigenfunctions" instead of "eigenvectors", but as you said, an eigenfunction can still be considered a "vector" in the mathematical vector space sense if you want :) $\endgroup$ – user1271772 Mar 23 at 23:44
  • $\begingroup$ @user1271772: Thanks for pointing that out. Could you please give an example where continuous variable is used? I thought that quantum computer function is described by matrices and hence it is inherently discrete. $\endgroup$ – Martin Vesely Mar 24 at 5:08
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    $\begingroup$ I haven't been active on this stack exchange in the last 18 months or so, therefore I don't remember the best examples. However there was a lot of discussion about CV quantum computing in the early days: quantumcomputing.stackexchange.com/questions/tagged/… (almost all of these questions are from 2018). $\endgroup$ – user1271772 Mar 24 at 5:31
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    $\begingroup$ Yes! And an example of a "continuous variable" is the position variable, x, like in here: en.wikipedia.org/wiki/Quantum_harmonic_oscillator $\endgroup$ – user1271772 Mar 24 at 6:22

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