Is there an objective asymmetry between a collapsed and un-collapsed wave function?

Question

In a quantum double slit experiment, one starts at t0 with a wave function that propagates through two slits, interferes, and probabilities for various positions at the final stage at t1 can be computed. This is a physical representation that can be endorsed by an observer at the beginning of the experiment and allows her to predict possible outcomes. However at the end of the experiment, an observer has observed a definite outcome corresponding to a precise position (in Copenhagen parlance, there's a "collapse"). So in principle, this final observer could represent what has just happened differently: have a "collapsed" wave function with well defined position (up to a certain precision maybe) at t1 and apply the schrödinger equation backward in time to see what the wave function was at t0. Then the result would be different: let's call psi0(t) the wave function of the initial observer and psi1(t) that of the final observer.

Whether this kind of backward in time calculation is legitimate, meaningful, useful or anything is not my concern (it is feasible so let's assume it is legitimate for the sake of the question). My question is: how would the two wave functions psi0 and psi1 be related? It seems that the final observer has "more information" than the initial observer: does it translate in one way or another? Is there an asymmetry between psi0 and psi1, an objective way to tell that psi1 contains more information about the world?

What I find puzzling is that any wave function can be represented as a vector in Hilbert space, and I can't see how a vector could be said to carry more "information" than another. They're just vectors. On the other hand, it seems obvious that the final observer has more information than the initial one, since she has one more measurement record.

Answer 1

What you say is correct. Indeed the final state vector carries much less information when compared to the initial state vector. The wave-function $|\psi(t)\rangle$ by definition carries the entire information that the system has to offer. Imagine $|\psi(t)\rangle$ to be a vector in an n-dimensional Hilbert space. Any vector can be decomposed into a linear composition of basis vectors. Let's say that: \begin{equation} |\psi(t)\rangle=\sum_i^{\infty}c_n|\phi_n(t)\rangle \end{equation} Where $|\phi_n(t)\rangle$ are the basis vectors. What any measurement does is that it takes the projection of the state vector along a particular basis vector (let's say along $|\phi_k(t)\rangle$). Then what we have is; \begin{equation} \langle\phi_k(t)|\psi(t)\rangle=\sum_i^{\infty}c_n\langle \phi_k(t)|\phi_n(t)\rangle=c_k \end{equation} The above equation works out because of the orthonormality of the basis vectors. What the state vector is now reduced to $|\phi_k(t)\rangle$ and the measurement value is the scaling factor along that basis vector which is $c_k$. This change of the state vector of the system from $|\psi(t)\rangle$ to $|\phi_k(t)\rangle$ is essentially known as the collapse of wave function. What you are now left with after the measurement is $|\phi_k(t)\rangle$ and there is no operation possible that can revert it back to $|\psi(t)\rangle$ since we don't know every $c_n$.

It has to be mentioned that if you have a number of same state vectors, then you can perform measurements along all possible basis to reconstruct the initial state. This process is known as state tomography.

Answer 2

What I find puzzling is that any wave function can be represented by a vector in Hilbert space and I can't see how a vector can be said to carry more information than another.

If we model collapse stochastically then a filtration on the sample space that specifies the degree of information available at time $t$:

the filtration can be interpreted as all historical information available about a stochastic process, with the algebraic object ... gaining in complexity. A stochastic process that is adapted to a filtration is called non-anticipating, ie one that cannot see into the future.

However, I haven't seen this mathematical technology used in QM ...

Answer 3

This is an experimentalist's answer:

Here is the double slit experiment one electron at a time.

Each random looking spot in frame a) is the footprint of an electron in the (x,y) plane of the detector. This electron before it hit the screen was described by a wavefunction and by the boundary conditions picking up the specific experimental setup: incoming from a beam direction, scattering off two slits with specific distances and width. These boundary conditions give a specific wavefunction for the single electron whose modulus square gives the probability distribution for finding an electron at a specific (x,y).

To get experimentally this distribution a third boundary condition enters, the screen at z and the atoms/molecules that the electron hits so that it is seen. That is a different wavefunction . Once the electron hits the screen and is detected as a point the original wavefunction no longer holds, i.e. it has collapsed because an instance of the probability distribution it predicted was picked.

It is the same as with throwing dice. Once the dice falls it has no meaning to speak of the other probable values that could have come up too. One has to throw the dice many times to get a probability distribution , which as you must know is flat. Throwing many electrons at the screen shows the probability distribution for finding the electron at (x,y,z) and lo, it shows the interference pattern inherent in the wavefunction describing the specific system.

What I find puzzling is that any wave function can be represented as a vector in Hilbert space,

It is simpler to think in terms of wavefunction solutions of the Schrodinger or Dirac equations. Thinking about Hilbert spaces you are ignoring boundary conditions that differentiate one experimental setup from another.

and I can't see how a vector could be said to carry more "information" than another. They're just vectors.

Boundary conditions pick up a subset of the Hilbert vector space.

On the other hand, it seems obvious that the final observer has more information than the initial one, since she has one more measurement record.

Please note that there cannot be initial and final observers without introducing interactions, i.e. changing boundary conditions: observation means interaction. Each electron is part of a beam and is uniquely observed at the screen where it interacts and its wavefunction gives one of the values according to the probability distribution describing it, this is called collapse. At the screen the electron loses its energy in interacting to give the point seen, and it is a completely different wavefunction and boundary conditions that would describe it from then on.