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I'm reading the book "Quantum Computing 4 real IT people" by Chris Bernhardt and I have a question about the following phrase in chapter 3 which says that

An ordered orthonormal basis corresponds to measuring the spin in the vertical direction and is specified by a pair of vectors (1; 0) and (0; 1). The first vector in the basis corresponds to an electron with spin N (North) in the direction of 0°, and the second vector - to an electron with spin S (South) in the direction of 0°.

But I don't understand how that could be possible... Because the 1st basis vector (1; 0) has horizontal direction and the 2nd basis vector (0; 1) has a vertical direction (in usual Cartesian coordinate system):

enter image description here

glS
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Ksenia
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    That's the worst sentence on spins I've ever read. You're rightfully confused. I'm not sure what the author of that book wants to achieve but there's no connection between "orthonormal basis" (computational basis?) and "spins", on the contrary. It just happens that there are two spin states of an electron and they can be described by two orthonormal vectors. The latter statement can be taken as an axiom. Those two vectors can be identified with the standard basis $(1,0)$ and $(0,1)$ (but there's no need to). – Markus Heinrich Oct 27 '21 at 14:59
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    Welcome to QCSE, Ksenia! This appears to be the (not uncommon) mix-up of the physical space with the Hilbert space. Perhaps this post answers your question? – Adam Zalcman Oct 27 '21 at 16:34
  • @MarkusHeinrich that's probably good enough to be an answer! – glS Nov 02 '21 at 11:18

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The qubit space is continuous and is expressed on the Bloch sphere rather than the standard coordinate system.

https://en.wikipedia.org/wiki/Bloch_sphere

The north pole of the Bloch sphere is (1,0) state and the south pole of the Bloch sphere is (0,1) state. The use of the Bloch sphere is analog with the spin of the electron, where (1,0) state signifies the +h/2 spin of an electron and (0,1) signifies the -h/2 spin of an electron. You can try the visualization given below.

https://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/blochsphere/blochsphere.html

user18876
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