Put a cat into a pure quantum state, without killing the cat?

[pervect]

Can we (in principle) put a cat into a pure quantum state, without killing the cat?

 

[dextercioby]

And how would you do that,namely putting the cat in the "pure" quantum state...?

Daniel.

 

[pervect]

And how would you do that,namely putting the cat in the "pure" quantum state...?

Daniel.

By measuring the cat (or it's component particles), the same way that one has a pure state of a particle after one measures it, rather than a mixed state.

(Why the scare quotes around "pure", anyway?)

A pure state can be represented by a ket |k>, a mixed state can only be represented by a statistical mixture of kets, or by a density matrix.

One of the problems is what specific observables we can measure about the cat without killing it. We can imagine, for instance, measuring the position of all the atoms (nucleii, actually), of the cat, at least up to the limits that quantum mechanics allows. While this would be a pure state (except in that it does not include the electronic part of the wavefunction), the process would kill the cat, as the momentum of the nucleii would be uncertain. Some other set of observables would have to be used.

 

[dextercioby]

I don't know,i've always though that asserting QM principles and rationales to a macroscopic (hence (very) many particle system) would lead nowhere...
In this line of logics,you might even suggest computing the "pure" quantum state of the cat at a moment "t" by applying the cat's evolution operator on the cat's ket at the moment "t_{0}"...

Daniel.

 

[selfAdjoint]

I don't know,i've always though that asserting QM principles and rationales to a macroscopic (hence (very) many particle system) would lead nowhere...
In this line of logics,you might even suggest computing the "pure" quantum state of the cat at a moment "t" by applying the cat's evolution operator on the cat's ket at the moment "t_{0}"...

Daniel.

But experimentalists have produced both macroscopic pure states and credible analogs of the cat.

Quantum principles extend as far as experimeters can make them extend. We can't put a notional cap on it.

 

[pervect]

I don't know,i've always though that asserting QM principles and rationales to a macroscopic (hence (very) many particle system) would lead nowhere...
In this line of logics,you might even suggest computing the "pure" quantum state of the cat at a moment "t" by applying the cat's evolution operator on the cat's ket at the moment "t_{0}"...

Daniel.

I wouldn't want to carry out the computation myself, but that's more or less what the universe does with a living cat, excpet that it evolves a density matrix rather than just a single ket.

As far as the number of particles being a limit - I can't see any point at which quantum mechanics should fail conceptually due to "too many particles", it's just that the computations become incredibly difficult.

I certainly don't mind sneaking up on the problem by talking about one particle, then two, then more than two.

If we represent the state of one particle by a ket |k1>, then the state of a two particle system is just the tensor product |k1> (tensor) |k2>, which is usually just written as |k1 k2>

For three particles, it's just |k1 k2 k3>, etc.

If we have a pair of particles, we can (and do) measure the total spin state of the pair of particles by measuring the spin of each particle separately. We usually pick one basis vector, |up> or |down>, then we find that the spin of one particle is either |up> or |down> (or a superposition), and that the spin state of a two particles is one of

|up up>, |up down>, |down up>, |down down> (or a superposition).

 

[dextercioby]

Of course,by that "nowhere" i meant it cannot make predictions with the same accuracy like in the case of very simple quantum systems...IIRC,not even the Helium atom would be totally predictable...
Anyway,apparently sA has given u some kind of answer you were looking after...

Daniel.

 

[pervect]

Of course,by that "nowhere" i meant it cannot make predictions with the same accuracy like in the case of very simple quantum systems...IIRC,not even the Helium atom would be totally predictable...
Anyway,apparently sA has given u some kind of answer you were looking after...

Daniel.

More or less - the question is still somewhat open. I tend to think that there must be a set of observables associated with a live cat because live cats exist in the real world. While quantum mechanics isn't the real world, I think QM should be in priincple a good enough theory to model chemistry, and hence cats. If it isn't good enough to do that by some mischance, it would be very good to know where it fails.

Unfortunately, it's still not very clear to me what these observables would be (observables compatible with live cats). An argument that they should probably exist because live cats exist is one thing, but it would be more convincing if we could write something down to describe what these alleged observables would actually be like.

At least a part of the problem is the issue of temperature. Basically there should be some sort of observable that wouldn't affect the statistical temperature of a group of particles. This wouldn't be a position observable, or a momentum observable, but some sort of hybrid.

This sort of observable would be a kind of bridge between the classical notion of statistical thermodynamics, and the world of quantum mechanics.

I've never read about anyone discussing this issue, but it seems like something that's so obvious that someone should have done it.

 

[vanesch]

Can we (in principle) put a cat into a pure quantum state, without killing the cat?

Assuming the cat can be described by a density matrix, and the density matrix being a symmetric matrix, it can be diagonalized in a certain basis of its hilbert state. At that point, we can say that the cat is in one of these states, but we lack knowledge of which one, which is described by the values on the diagonal of the matrix.
So as a matter of principle, it IS already solved: the cat is in one of these states, only we don't know which one. So we don't have to "put" it there, it is there. It is another matter to find out if we can learn which state without completely ALTERING the state (and hence killing the cat).
There might be a caveat to what I say if certain values on the diagonal are degenerate, in which this diagonalization procedure does NOT determine fully the set of states out of which we have to choose.

cheers,
patrick.

 

[slyboy]

Assuming the cat can be described by a density matrix, and the density matrix being a symmetric matrix, it can be diagonalized in a certain basis of its hilbert state. At that point, we can say that the cat is in one of these states, but we lack knowledge of which one, which is described by the values on the diagonal of the matrix.

The problem with that is that the density matrix can be composed into a probablistic mixture of (generally non-orthogonal) pure states in multiple different ways. What leads you to pick one particular decomposition as special?

There is also the problem that this is what some people call an improper mixture rather than a proper mixture. This means that it arises from tracing out part of a larger entangled state, including the cat's environment, rather than from knowing that the cat is definitely in one of an ensemble of pure states. There is quite a bit of debate about whether this actually matters, and for most practical purposes it is clear that it doesnt. However, if the cat is part of an entangled state, rather than being in a proper mixture, there could be non completely-positive dynamics of the cat's density matrix if she interacts with the same environmental degrees of freedom at a later time. This could not happen for a proper mixture, so I think it is clear that the two cases are not the same in principle.

 

[vanesch]

The problem with that is that the density matrix can be composed into a probablistic mixture of (generally non-orthogonal) pure states in multiple different ways. What leads you to pick one particular decomposition as special?

This is correct. I'm used to reason in the other way, in decoherence. But you're perfectly right that knowing the density matrix is not sufficient to know even which are a potential set of orthogonal states in which the system might be (and thus choose the observable that will not change the state).

thanks for correcting this!
Patrick.