Some context:
In the literature, there is different ways of using the word "state". A quick overview on this:
- Some say a "state" is a mathematical object, to be precise either
- a) a ray (set) of (normalized elements) of the usual hilbert space $\mathcal{H}$ (e.g. Marinescu (978-0-12-383874-2) on p.24) - again there seem to be 2 versions of this 1.a) but they only differ on whether they use normalized elements or not which we don't mind at this point
- b) an element of $\mathcal{H}$ (e.g. Nielsen & Chuang (978-1-107-00217-3) on p.13)
- Others use it not in a mathematical sense, but in a more general way used maybe in everyday language. This is the case when it is said that a "state" is formally represented by either "a ray of normalize elements of $\mathcal{H}$" or "an element of $\mathcal{H}$".
Why does the difference matter? Let me give 2 examples of situations that urge me to clarify the usage of "state". Adding to that, let's look at "states" of qubits specifically, since these are the simplest quantum mechanic systems. Let's not go into detail about whether one looks at the Hilbert space as abstract or whether one uses the isomorph Hilbert spaces. Instead, let's use rows of complex numbers (and therefore uses the coordinates of elements in the Hilbert space to represent them). From now on, let's call this Hilbert space $\mathcal{H}$.
Situation 1: We know that a result of the formal description of the measurement on a qubit is, that a global phase in $|\psi\rangle\in\mathcal{H}$ is not "observable" by a measurement. Therefore, one should consider $|\psi_1\rangle$ and $\psi_2=e^{i\gamma}|\psi_1\rangle$ (for $\gamma\in\mathbb{R}$ to be 1.a) both elements of the same ray 1.b) to be equal 2. to be formal descriptions of the same state.
From this situation, I conclude (at least that's my idea) that 1.b) is the wrong take, since clearly $|\psi_1\rangle\neq|\psi_2\rangle$. What's more, this observation kind of leads to using 1.a).
Moving on now, we have 1.a) and 2. as possibile ways of looking at the usage of "state". There is one more situation that creates problems:
Situation 2: We know that one usually uses expressions like "the state $|\psi\rangle$" and maybe later "$|\psi\rangle=(1,0)^T$" in a text. Meaning, one identifies the "state" with the formal object that is a element of a hilbert space.
This is no problem in 1.b) since a "state" is just that element of $\mathcal{H}$. There is a problem with 1.a), since one would identify the ray with an element in the ray (set).
There also is a problem with 2.): In terms of 2.) situation 2 identifies the label of a state (e.g. $|\psi\rangle$) with its formal representation. Because we can "forget about the global phase" (to put this rather complex observation into simple terms) there is some unease: If one has two formal descriptions of a state that only differ by a global phase - $|\psi_1\rangle$ and $|\psi_2\rangle$ - one would deduce that those two symbols are labels for the same state and therefore must be equal, say $|\psi_1\rangle=|\psi_2\rangle$.
My question: Obviously, the language in some (or many) books is very unprecise and leads to problems in understanding the basic concepts. What is the correct way of looking at it?
My try on giving an answer: Following the described "context", one is tempted to say that 1.a) should be the correct way of looking at it. But then, what should one call the state (referring to a everyday language use which is not referring to a mathematical object) - maybe "physical state"? Even worse, noone ever (not even Marinescu) calculate with rays but carries on and says "the state $|\psi\rangle\in\mathcal{H}$" (p. 26) and therefore abandons 1.a) immediately after using it for 1.b).
What should we make of all this?
