About creating a product state

[KFC]

Hi all,
I am reading a book about fundamental quantum mechanics, in which there mentioned many time about the product state $|a\rangle|b\rangle$ of two states $|a\rangle$ and $|b\rangle$ . To my understanding, product state means to combine two small systems to get a bigger one. So I am thinking if there is any experimental way that I can combine two separate states into product?

My second question, let's say I already have $|a\rangle$ and $|b\rangle$, is that any way in experiment I can tell the overlap between them?

 

[DrClaude]

Prepare one system in state $|a \rangle$ and the other in state $|b \rangle$, and voilà, the combined system is in state $|a \rangle \otimes |b \rangle$

 

[KFC]

Prepare one system in state $|a \rangle$ and the other in state $|b \rangle$, and voilà, the combined system is in state $|a \rangle \otimes |b \rangle$

Thanks for that information. I understand that in math. But how actually people do that in experiment?

 

[kith]

Preparing a system in a state isn't math. You go to the lab, define what your system of interest is and use an appropriate experimental apparatus to prepare it in a certain state. For example, you can use an oven which emits silver atoms and use a Stern-Gerlach apparatus to prepare them in state $|+_z \rangle$.

You can write down a product state whenever you have another system which is well-prepared. For example you could use a double slit to prepare a photon in the state $|\text{photon went through slit A} \rangle + |\text{photon went through slit B}\rangle$.

Then your product state would be $|+_z \rangle \otimes (|\text{photon went through slit A} \rangle + |\text{photon went through slit B}\rangle)$

 

[KFC]

Thanks. I think I understand that now. But I am still looking for the answer for the second question. Let say I have two states $|a\rangle$ and $|b\rangle$, what is the significance of this $\langle a | b \rangle$, by reading some examples found in the text, can I say that it stands for the probability to find state a when b is given? In experiment, how do we measure the $|\langle a | b \rangle|^2$? Do we measure state b and state a? If state a and b are independent, can conclude that

$|\langle a | b \rangle|^2 = |\langle a|a\rangle|^2 + |\langle b|b\rangle|^2$

Thanks

 

[Zafa Pi]

Preparing a system in a state isn't math. You go to the lab, define what your system of interest is and use an appropriate experimental apparatus to prepare it in a certain state. For example, you can use an oven which emits silver atoms and use a Stern-Gerlach apparatus to prepare them in state $|+_z \rangle$.

You can write down a product state whenever you have another system which is well-prepared. For example you could use a double slit to prepare a photon in the state $|\text{photon went through slit A} \rangle + |\text{photon went through slit B}\rangle$.

Then your product state would be $|+_z \rangle \otimes (|\text{photon went through slit A} \rangle + |\text{photon went through slit B}\rangle)$

I'm glad this topic was brought up, because I'm confused.
Let's say we want a horizontally polarized photon (state = |0º⟩). Well we prepare some photons one at a time from a filament and filter them with a polarized lens with a horizontal axis. But how do you know we have one? Maybe we have two and thus |0º⟩|0º⟩, or maybe none. We could see if one hits a detector/screen, but then it no longer exists. I see how to run tests on a sequence of them, but I don't see how we can know when we "have" one.

 

[PeterDonis]

Let say I have two states $|a\rangle$ and $|b\rangle$, what is the significance of this $\langle a | b \rangle$

You seem to be confusing two different things. This $\langle a | b \rangle$ is not a product state, in fact it's not a state at all, it's a number that represents the probability of measuring a system to be in state $a$ if it is prepared in state $b$. A product state where we prepare one system in state $a$ and another in state $b$ is written $|a\rangle |b\rangle$; note how this is a product of two kets, rather than a bra and a ket.

 

[Zafa Pi]

Thanks. I think I understand that now. But I am still looking for the answer for the second question. Let say I have two states $|a\rangle$ and $|b\rangle$, what is the significance of this $\langle a | b \rangle$, by reading some examples found in the text, can I say that it stands for the probability to find state a when b is given? In experiment, how do we measure the $|\langle a | b \rangle|^2$? Do we measure state b and state a? If state a and b are independent, can conclude that

$|\langle a | b \rangle|^2 = |\langle a|a\rangle|^2 + |\langle b|b\rangle|^2$

Thanks

States are unit vectors (olden times) ⟨a|b⟩ is their inner product so |⟨a|b⟩| ≤ 1, and |⟨a|a⟩| = 1, thus your formula is never valid.

If you have a source of |a⟩s and a source of |b⟩s, say polarized photons. find a polarized lens that let's all the |a⟩s pass (try rotating the lens until the intensity of the source = intensity of the filtered). Now if any of the |b⟩s get thru that lens then you know there's overlap (i.e. they are not orthogonal).

 

[KFC]

Preparing a system in a state isn't math. You go to the lab, define what your system of interest is and use an appropriate experimental apparatus to prepare it in a certain state. For example, you can use an oven which emits silver atoms and use a Stern-Gerlach apparatus to prepare them in state $|+_z \rangle$.

You can write down a product state whenever you have another system which is well-prepared. For example you could use a double slit to prepare a photon in the state $|\text{photon went through slit A} \rangle + |\text{photon went through slit B}\rangle$.

Then your product state would be $|+_z \rangle \otimes (|\text{photon went through slit A} \rangle + |\text{photon went through slit B}\rangle)$

You seem to be confusing two different things. This $\langle a | b \rangle$ is not a product state, in fact it's not a state at all, it's a number that represents the probability of measuring a system to be in state $a$ if it is prepared in state $b$. A product state where we prepare one system in state $a$ and another in state $b$ is written $|a\rangle |b\rangle$; note how this is a product of two kets, rather than a bra and a ket.

Thanks for the reply. I didn't confuse those two things. I though I clarify that at the beginning by saying that they are two questions but obviously I didn't make it clear enough. Anyway, my last question is about how to measure $\langle a|b\rangle$ (nothing to do with the product state here). I understand your explanation that this is a number represents the probability of measuring a system to be in state $a$ if it is prepared in state $b$. But what happens if $|a\rangle$ and $|b\rangle$ are state from two independent system, does it has any significance to compute $\langle a|b\rangle$ ?

 

[PeterDonis]

what happens if $|a\rangle$ and $|b\rangle$ are state from two independent system, does it has any significance to compute $\langle a|b\rangle$ ?

No; in fact it can't even be done, because the inner product is only defined for two states in the same Hilbert space, and if the two systems are independent, their states are in different Hilbert spaces.