cianfa72
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- About the representation of Qubit state space set by mean of projective space built over the Hilbert space of dimension 2
I'd like to discuss some aspects of quantum systems' state space from a mathematical perspective.
Take for a instance a qubit, e.g. a two-state quantum system and consider the set of its pure states. This set as such is a "concrete" set, namely the "bag" containing all the qubit's pure states. Such a set, however, doesn't carry a vector space structure much less an Hilbert space structure.
Nevertheless, quantum physics employs the abstract Hilbert space ##\mathcal H## of dimension 2 to model such a concrete set. Namely we take the set of rays of the abstract space ##\mathcal H## and map/define a one-to-one/onto correspondence (bijective map) between the set of qubit's pure states and set of ##\mathcal H## rays (namely the elements of the projective Hilbert space ##P(\mathcal H)## built over ##\mathcal H##).
Up to this point no "concrete" or specific "instances" of the abstract Hilbert space of dimension 2 are involved (note that such abstract Hilbert space of dimension 2 is unique by its very definition).
Next fixing a basis in ##\mathcal H## one can define a straightforward isomorphism with ##\mathbb C^2## which is a concrete instance/realization of the abstract Hilbert space ##\mathcal H##. Let's call ##\ket{\uparrow}## and ##\ket{\downarrow}## such basis elements in ##\mathcal H##.
From the above it follows that ##\ket{\uparrow}## or ##\ket{\downarrow}## are not qubit states themselves, any of them actually represents a qubit state. Formally what is really going on is taking the ray the vector ##\ket{\uparrow}## or ##\ket{\downarrow}## belongs to in order to select/pick the unique corresponding element within ##P(\mathcal H)##. Such an element maps back (bijectively) to an element of the qubit state space's set.
What do you think, does it actually make sense ?
Take for a instance a qubit, e.g. a two-state quantum system and consider the set of its pure states. This set as such is a "concrete" set, namely the "bag" containing all the qubit's pure states. Such a set, however, doesn't carry a vector space structure much less an Hilbert space structure.
Nevertheless, quantum physics employs the abstract Hilbert space ##\mathcal H## of dimension 2 to model such a concrete set. Namely we take the set of rays of the abstract space ##\mathcal H## and map/define a one-to-one/onto correspondence (bijective map) between the set of qubit's pure states and set of ##\mathcal H## rays (namely the elements of the projective Hilbert space ##P(\mathcal H)## built over ##\mathcal H##).
Up to this point no "concrete" or specific "instances" of the abstract Hilbert space of dimension 2 are involved (note that such abstract Hilbert space of dimension 2 is unique by its very definition).
Next fixing a basis in ##\mathcal H## one can define a straightforward isomorphism with ##\mathbb C^2## which is a concrete instance/realization of the abstract Hilbert space ##\mathcal H##. Let's call ##\ket{\uparrow}## and ##\ket{\downarrow}## such basis elements in ##\mathcal H##.
From the above it follows that ##\ket{\uparrow}## or ##\ket{\downarrow}## are not qubit states themselves, any of them actually represents a qubit state. Formally what is really going on is taking the ray the vector ##\ket{\uparrow}## or ##\ket{\downarrow}## belongs to in order to select/pick the unique corresponding element within ##P(\mathcal H)##. Such an element maps back (bijectively) to an element of the qubit state space's set.
What do you think, does it actually make sense ?
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