Can superposition be attained for multiple states at once?

In summary: The state after the measurement will be either up or down, and neither of those is an eigenstate of left/right. So you could view the state after measurement as a superposititon of left and right. But as above, that's just a mathematical choice and doesn't affect the physics.
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Grinkle
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I hope my articulation makes sense.

Can I prepare a particle so that it has >1 states in superposition and resolve them at different times? I will make up states to try and illustrate my question better.

Prepare a particle so that spin up and spin down are in a state of superposition. Also, the states of spin left and spin right are in a state of superposition. I mean these states (up/down and left/right) to be independent of each other.

Then I measure up/down. Have I also fixed left/right even if don't observe it or can left/right still be said to be in a state of superposition for this particle?
 
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Grinkle said:
Prepare a particle so that spin up and spin down are in a state of superposition. Also, the states of spin left and spin right are in a state of superposition.

This is easily done, just prepare the particle in a state that is not an eigenstate of up/down or left/right. However, the particle still only has one state; it doesn't have multiple states. Viewing the state as being a "superposition" of up/down and left/right is a mathematical choice and doesn't affect the physics.

Grinkle said:
I mean these states (up/down and left/right) to be independent of each other.

I don't know what you mean by this. The particle only has one state.

Grinkle said:
Then I measure up/down. Have I also fixed left/right even if don't observe it

You can't measure up/down and left/right in the same measurement; those observables don't commute.

Grinkle said:
can left/right still be said to be in a state of superposition for this particle?

The state after the measurement will be either up or down, and neither of those is an eigenstate of left/right. So you could view the state after measurement as a superposititon of left and right. But as above, that's just a mathematical choice and doesn't affect the physics. The particle still has only one state.
 
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Grinkle said:
Can I prepare a particle so that it has >1 states in superposition and resolve them at different times?
A quantum system always has only one state. Superposition just means that it is mathematically possible to write the state in different ways. For example any state ##|\psi\rangle## can be written in the form ##|\psi\rangle=\alpha|up\rangle+\beta|down=\gamma|left\rangle+\delta|right\rangle=\epsilon|45\rangle+\zeta|225\rangle## - a superposition of up/down and also a superposition of left/right and of the two diagonal axes, and we could choose any other angle as well if we wanted. So yes, a state can be a superposition of more than one thing; in fact, it always is.

Prepare a particle so that spin up and spin down are in a state of superposition. Also, the states of spin left and spin right are in a state of superposition. I mean these states (up/down and left/right) to be independent of each other.
You can’t make them independent of one another, because there’s only one way of writing any particular state as (for example) a sum of up/down and as a sum of left/right. So once you’ve chosen your state ##|\psi\rangle## to be a particular superposition of up and down (that is, you've chosen particular values for ##\alpha## and ##\beta##) you've determined the values of ##\gamma## and ##\delta## so in that sense they're not independent. However, all four of them can be non-zero so the state still a superposition of up/down and left/right and you cannot predict with certainty a measurement on either axis. So say you have prepared the particle in such a state and then...
Then I measure up/down. Have I also fixed left/right even if don't observe it or can left/right still be said to be in a state of superposition for this particle?
The measurement of up/down will collapse the wave function into either ##|up\rangle## or ##|down\rangle##. Both of these states are superpositions of left and right; for example ##|up\rangle=|left\rangle+|right\rangle##.
 
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Thanks @PeterDonis @Nugatory .

Looks like I was confusing state variables with the state itself in my thinking about superposition.
 
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FAQ: Can superposition be attained for multiple states at once?

1. Can superposition be attained for multiple states at once?

Yes, superposition is a quantum mechanical phenomenon in which a particle can exist in multiple states at the same time. This means that the particle can have multiple values for its properties, such as position and momentum, simultaneously.

2. How is superposition achieved?

Superposition is achieved through the process of quantum superposition, in which a particle is put into a state of quantum uncertainty by being exposed to a superposition of different energy states. This allows the particle to exist in multiple states at once.

3. What is the significance of superposition?

The significance of superposition is that it allows for the exploration of different states of a particle simultaneously, which can lead to a better understanding of quantum mechanics and the behavior of particles at a subatomic level. It also has practical applications in quantum computing and cryptography.

4. Can superposition be observed in everyday life?

No, superposition is a phenomenon that occurs at a subatomic level and is not observable in everyday life. It is only observable through experiments conducted in a controlled environment, such as a laboratory.

5. What is the relationship between superposition and entanglement?

Superposition and entanglement are both quantum mechanical phenomena that occur at the subatomic level. Superposition refers to the ability of a particle to exist in multiple states at once, while entanglement refers to the correlation between the states of two or more particles. Superposition is often used in the process of creating entangled particles.

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