State vector behaviour during coupling

[James MC]

The Schrödinger equation rotates the state vector in Hilbert space continuously (i.e. without jumps). This makes sense for individual systems, but I'm finding this hard to reconcile with coupling or entanglement. For example, consider how Schrödinger's cat paradox is typically presented (in Dirac notation). We have a cat initially in the state |Alive> and we have a vile of poisonous gas in the state |contained> and we have some radioactive material in the superposition a|decay> + b|~decay>.
The overall system is set up so that if the material decays (i.e. is in state |decay>) then the vile of gas enters the state |released> and the cat evolves to |Dead>.
The linearity of the Schrödinger equation then entails the entire system should evolve to:
|Alive>(a|decay>|released> + b|~decay>|contained>)
And then to:
a|decay>|released>|Dead> + b|~decay>|contained>|Alive>

Here's what I don't understand: If the Schrödinger equation always rotates the state vector continuously (without sudden jumps) then how could the state vector in the composite state space go from this:
|Alive>(a|decay>|released> + b|~decay>|contained>)
to this:
a|decay>|released>|Dead> + b|~decay>|contained>|Alive>

These two vectors (in the composite vector space) are surely not an infinitesimal rotation apart.
But then how could amplitudes a and b just pass immediately from the radioactive material, to the vile, to the cat?

I'm clearly missing something here. Any help would be appreciated.

 

[Fredrik]

You seem to have confused yourself by using exactly the same notation for different states of the same object. For example, if the cat is |alive> at one point, then it's not going to be |alive> later. It may be alive, but you'd have to write it as |alive'> or something, because it's not the same state.

 

[Jilang]

What you are asking about is the wave function collapse and is always problematic if it is equated with the real state of a system. It can help to think of the wave function as the information we have about a system. There are different views on this, but this one works for me.

 

[James MC]

What you are asking about is the wave function collapse and is always problematic if it is equated with the real state of a system. It can help to think of the wave function as the information we have about a system. There are different views on this, but this one works for me.

No I am not asking about wave function collapse. In fact the whole point of Schrodinger's thought experiment is to show what happens (a) when you correlate the state of a macro-system to the quantum state of a particle and (b) when collapse NEVER occurs, i.e. when the dynamics is given completely by the linear Schrodinger equation.

Your epistemic/informational interpretation of the wave function is in my eyes refuted by the PBR theorem. But this has nothing at all to do with my original question and so is not something I want to enter into in this thread. My question is purely formal, and simply concerns the mathematical representation of statevector evolution during entanglement, at least as it's standardly used to present Schrodinger's paradox.

 

[James MC]

You seem to have confused yourself by using exactly the same notation for different states of the same object. For example, if the cat is |alive> at one point, then it's not going to be |alive> later. It may be alive, but you'd have to write it as |alive'> or something, because it's not the same state.

You're absolutely right that a more precise statement of Schrodinger's paradox would account for the fact that e.g. the cat is shedding fur etc. But how does that help? It seems my question still remains even when we account for this by stating the evolution as:

[t1] |Alive>|contained>(a|decay> + b|~decay>)
[t2] |Alive'>(a|decay>|released'> + b|~decay>|contained'>
[t3] a|decay>|released''>|Dead''> + b|~decay>|contained''>|Alive''>

It seems the question still remains:
The transition from [t1] to [t2] (or [t2] to [t3]) does not appear to be an infinitesimal rotation of the state vector. But then does this (standard!) presentation of Schrodinger's paradox violate the principle that the Schrodinger equation only ever rotates the statevector continuously?

 

[James MC]

You seem to have confused yourself by using exactly the same notation for different states of the same object. For example, if the cat is |alive> at one point, then it's not going to be |alive> later. It may be alive, but you'd have to write it as |alive'> or something, because it's not the same state.

Are you trying to say that going from:
a|Alive>|decay>+b|Alive>|~decay>
to:
a|Dead>|decay>+b|Alive'>|~decay>
involves a mere infinitesimal rotation of the state vector, within the vector space with basis vectors that include: |Alive>|decay> and |Alive>|~decay> and |Dead>|decay> and |Alive'>|~decay>?

 

[Fredrik]

The notation is still a bit confusing. Instead of just using primes to distinguish between different states, I will now also use the same number of primes on states at the same time.

The general idea here is that the experiment is set up so that the time evolution operator U(t) for some time t will take |Alive>|decay> to |Dead'>|decay'>, and |Alive>|~decay> to |Alive'>|~decay'>. The linearity of U(t) then implies that

U(t)|Alive>(a|decay> + b|~decay>) = a|Dead'>|decay'> + b|Alive'>|~decay'>​

U(t) is a unitary operator. I guess that's what you mean by a "rotation" (since unitary operators preserve the norm), but it's only appropriate to use the term "infinitesimal" when you expand in a series U(t)=1+iHt+... and throw away second and higher order terms.

Your [t2] to [t3] is just a change from

a|decay'>|released'>|Alive'> + b|~decay'>|contained'>|Alive'>​

to

a|decay''>|released''>|Dead''> + b|~decay''>|contained''>|Alive''>​

I don't see why you think this can't be a unitary transformation. Note that this differs from what you wrote initially (without primes) because the primes make it clear that U(t) affects both terms. In post #1, your notation made it look like only the first term was affected, and therefore like the time evolution operator would be non-linear. I assumed that this was the root of the problem.

 

[James MC]

Thanks. This is getting to the issue...

The notation is still a bit confusing. Instead of just using primes to distinguish between different states, I will now also use the same number of primes on states at the same time.

The general idea here is that the experiment is set up so that the time evolution operator U(t) for some time t will take |Alive>|decay> to |Dead'>|decay'>, and |Alive>|~decay> to |Alive'>|~decay'>. The linearity of U(t) then implies that

U(t)|Alive>(a|decay> + b|~decay>) = a|Dead'>|decay'> + b|Alive'>|~decay'>​

U(t) is a unitary operator. I guess that's what you mean by a "rotation" (since unitary operators preserve the norm), but it's only appropriate to use the term "infinitesimal" when you expand in a series U(t)=1+iHt+... and throw away second and higher order terms.

It's slightly more than the fact that U(t) preserves the norm i.e. only changes the direction but never the unit-length of the state vector. I'm also assuming that U(t) never induces sudden jumps regarding the direction it points to, so there are only continuous rotations of the state vector. Is that right? I take it that there's a sense in which U(t) does induce sudden jumps: in the sense that it is an operator that can in principle do all sorts of things e.g. reflect the state vector about an axis. But I'm assuming that this can only be interpreted as a continuous process that takes time, in particular, that U(t)'s effect on the state vector is to be interpreted as a continuous rotation undertaken over the time period t. This is what I'm assuming. If this is right then we must interpret this:

U(t)|Alive>(a|decay> + b|~decay>) = a|Dead'>|decay'> + b|Alive'>|~decay'>​

as a continuous rotation of the state vector that takes time t. Then the question is what states the system possesses during t, i.e. in between possessing |Alive>(a|decay> + b|~decay>) and a|Dead'>|decay'> + b|Alive'>|~decay'>. I'm assuming that there must be some weird states possessed by the system in between since the following are all basis vectors and therefore orthogonal:

(i) |Alive>|decay>
(ii) |Alive>|~decay>
(iii) |Dead'>|decay'>
(iv) |Alive'>|~decay>

So if the state vector is to move from a weighted sum of (i) and (ii) to a weighted sum of (iii) and (iv) then that vector should continuously move through all sorts of other states.

My problem, then, is that surely the system does not evolve through all those other states. For example, it does not evolve through a weighted sum of (i), (ii) and (iii). That's why it can't be a continuous rotation of the state vector.

Perhaps the simplest way to put my worry is this: you say that

the time evolution operator U(t) for some time t will take |Alive>|decay> to |Dead'>|decay'>

But since |Alive>|decay> and |Dead'>|decay'> are eigenstates of the overall system, they are orthogonal. But then if U(t) continuously rotates the state vector, with no eigenstate-to-eigenstate jumps, then even this evolution will go through superpositions that the system does not actually exhibit.

I hope this makes more sense of why I'm still quite confused!

 

[f95toli]

Your question seems to be related to so-called quantum jump problems(?), although that is usually encountered in an environment with dissipation.
If so, this is a common problem in e.g. quantum optics and there ways of handling it.

See e.g.
http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.70.101


(also on the arXiv)
http://arxiv.org/abs/quant-ph/9702007

edit: Also, this should be exactly analogous to a system with a spin 1/2 system coupled to an excitation (photon) in a resonator; if so it is standard Jaynes-Cummings physics.

 

[James MC]

Your question seems to be related to so-called quantum jump problems(?),

Not if by "quantum jump" you mean wave-function collapse or state reduction. I'm here only considering purely unitary linear Schrodinger dynamics.

although that is usually encountered in an environment with dissipation.

That makes it sound like you're considering the dynamics of an open microsystem in a heat bath or something. But I'm considering the dynamics of closed composite systems. For this reason I didn't see the relevance of the article.

My question is how it can be possible for the Schrodinger dynamics to spread specific amplitudes during entanglement in the manner described above, yet still rotate the state vector continuously. These seem like inconsistent principles.

 

[kith]

But then if U(t) continuously rotates the state vector, with no eigenstate-to-eigenstate jumps, then even this evolution will go through superpositions that the system does not actually exhibit.

Why shouldn't it? For simplicity, let's look at only one term, e.g.

U(t)|Alive>|decay> = |Dead'>|decay'>​

Possible intermediate states would be

U(t')|Alive>|decay> = a(t')|Alive>|decay> + b(t')|Dead'>|decay'>​

where |a(t')| and |b(t')| vary from 1 to 0 (resp. from 0 to 1) continuously with t'.

 

[James MC]

Why shouldn't it? For simplicity, let's look at only one term, e.g.

U(t)|Alive>|decay> = |Dead'>|decay'>​

Possible intermediate states would be

U(t')|Alive>|decay> = a(t')|Alive>|decay> + b(t')|Dead'>|decay'>​

where |a(t')| and |b(t')| vary from 1 to 0 (resp. from 0 to 1) continuously with t'.

Thanks for your clearly stated question, I think you've located the thing that's troubling me.

I've been trying to suggest that two propositions are inconsistent:
(i) Schrodinger evolution rotates the state vector continuously.
(ii) It is possible for Schrodinger's cat to simply die (without first going through life/death superpositions), even when the cat/vile/particle system as a whole is governed entirely by the Schrodinger equation.

You have asserted (i) and you have given a (helpful!) formal presentation of how (i) applies to the cat/vile/particle system.

But do you agree that (i) is inconsistent with (ii)? Regarding (ii) I'm thinking that if the particle is in a definite state, a non-superposed state, the eigenstate of decaying with certainty; then because the system has been set up to release the vile and kill the cat if the particle decays, then it must be that the cat simply dies.

Here's why the (i)/(ii) conflict is important. It's important because (ii), or some variant of it, is used to define the measurement problem; indeed (ii) or some variant of it, is required to state all of the primary "interpretive" difficulties of quantum mechanics that have led to the endless "interpretations". This last point is non-trivial, and requires further explanation, but I leave this last point here as it's important that the (i)/(ii) conflict is understood first.

Please do let me know your thoughts!

 

[Jilang]

I would suggest that assertion ii) is the problem as death by radiation poisoning is not instantaneous. All physical processes being reversible to some degree.

Similarly with poison gas and thermonuclear devices.

 

[f95toli]

Not if by "quantum jump" you mean wave-function collapse or state reduction. I'm here only considering purely unitary linear Schrodinger dynamics.

No, "quantum jumps" refers to techniques used to describe e.g. V system (e.g. systems with 3 levels and shelving); it has nothing to do with collapse as such.

That makes it sound like you're considering the dynamics of an open microsystem in a heat bath or something. But I'm considering the dynamics of closed composite systems. For this reason I didn't see the relevance of the article.

But I suspect part of the problem here is that you system is NOT closed.
You can't stricktly speaking have decay in a closed system; i.e. without some sort of dissipation the system will never "relax" into a "dead cat" that stays permanently dead.

Also, I also believe talking about cats etc is just confusing things.

If I understood you correctly we could re-label as

¦e> = Live cat
¦g> = Dead cat
¦0> = No decay
¦1> = particle has decayed

I.e. a system comprising a standard 1/2 spin system and another two state system. If e.g. the other system is a photon (which in this case seems appropriate) you end up with a standard Jaynes-Cummings Hamilonian (note that I realize this is not neccesarily the best description, but it is easy, see below) If we now initialize the system as ¦e>¦0> and let it evolve we will find that we get Rabi oscillations where the "cat" oscillates between being alive and dead out of phase with the excitation of the vacuum (zero or one photons).

You could of course instead describe the decay using a collapse operator in a Lindbladian, and then the "cat" would just decay (presumably exponentially, but that will depend a bit on the form of the collapse operator) to its ground state ("dead") and stay there. But, again, this would not be a closed system.

 

[kith]

But do you agree that (i) is inconsistent with (ii)?

Yes and I don't think that (ii) is compatible with QM.

I'm thinking that if the particle is in a definite state, a non-superposed state, the eigenstate of decaying with certainty, [...] the cat simply dies.

Unlike (ii), your statement here conflicts with (i) only if a decayed particle eigenstate is reached before a dead cat eigenstate. How do you think this happens?

 

[kith]

If we now initialize the system as ¦e>¦0> [...]

The initial state of the OP translates to a|e,1> + b|e,0>. There, the Rabi oscillations would involve a third state, |g,2>, wouldn't they? This is fine within the JCM but doesn't represent two two state systems.

I agree with what you write about dissipation. Sooner or later, it will have to enter the picture.