Can superposition be attained for multiple states at once?

[Grinkle]

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?

 

[PeterDonis]

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.

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.

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.

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.

 

[Nugatory]

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##.

 

[Grinkle]

Thanks @PeterDonis @Nugatory .

Looks like I was confusing state variables with the state itself in my thinking about superposition.