Determining the state from observation?

[pellman]

How does one determine the state of a system from observations?

Let us look at a simple example: the spin of a fermion. Any state of the system is a superposition of up and down eigenstates. Denote the coefficients in this superposition by a and b. The question here amounts to determining a and b up to an overall phase.By performing spin measurements on many identically prepared systems we can determine the ratio |a|^2 / |b|^2. But how do we get the relative phase?

 

[hilbert2]

If $\left|\right.\uparrow\left.\right>$ and $\left|\right.\downarrow\left.\right>$ are eigenstates of $\hat{S}_z$ and the state of the system is $a\left|\right.\uparrow\left.\right> + b\left|\right.\downarrow\left.\right>$, the relative phases of $a$ and $b$ affect what the probabilities of different eigenvalues of $\hat{S}_x$ and $\hat{S}_y$ are.

 

[Mentz114]

How does one determine the state of a system from observations?

Let us look at a simple example: the spin of a fermion. Any state of the system is a superposition of up and down eigenstates. Denote the coefficients in this superposition by a and b. The question here amounts to determining a and b up to an overall phase.By performing spin measurements on many identically prepared systems we can determine the ratio |a|^2 / |b|^2. But how do we get the relative phase?

I don't think you can because the only observables |a|², |b|²are independent of phase.

With light one can use tomography to get relative phase.

[Beaten by Hilbert2 with a better answer]

 

[A. Neumaier]

One can measure the spin in arbitrary directions, and this get information of the required form. In general, look up quantum tomography!

 

[A. Neumaier]

If $\left|\right.\uparrow\left.\right>$ and $\left|\right.\downarrow\left.\right>$ are eigenstates of $\hat{S}_z$ and the state of the system is $a\left|\right.\uparrow\left.\right> + b\left|\right.\downarrow\left.\right>$, the relative phases of $a$ and $b$ affect what the probabilities of different eigenvalues of $\hat{S}_x$ and $\hat{S}_y$ are.

But the probability is only one real parameter, and one needs to determine three. With one equation this is impossible.

 

[pellman]

Ok. Well the reason

If $\left|\right.\uparrow\left.\right>$ and $\left|\right.\downarrow\left.\right>$ are eigenstates of $\hat{S}_z$ and the state of the system is $a\left|\right.\uparrow\left.\right> + b\left|\right.\downarrow\left.\right>$, the relative phases of $a$ and $b$ affect what the probabilities of different eigenvalues of $\hat{S}_x$ and $\hat{S}_y$ are.

This is it. I worked it out on paper. The relative phase in the S_z representation can be expressed in terms of the absolute values of the coefficients in all three representations. But you need measurements with respect to all three axes. Thanks!

 

[pellman]

This question came to me because I often see statements like, given a system in state ψ, what is the probability of observing the system to be in state ϕ? And I finally wondered, what does it mean to "observe the system to be in state ϕ" when observations can only return limited information?

So, just for fun, how about the next simple example?
Determining the (time-independent) position wave function for a single particle in one dimension

In principle, from observations we can get the probability distributions in both the position representation and in the momentum representations. The wave functions in each representation are related by a Fourier transform.

So the theoretical question amounts to: if we know | f(x) |^2 and |F(k)|^2 for two complex-valued functions f(x) and F(k) which are Fourier transforms, can we recover f(x)?

This is probably a theorem out there somewhere. Anyone know?

 

[stevendaryl]

Knowing probabilities for three different axes is overkill.

Write a two-component spinor this way: $|\psi\rangle = e^{i\chi} \left( \begin{array} \\ cos(\frac{\theta}{2}) e^{-i \frac{\phi}{2}} \\ sin(\frac{\theta}{2}) e^{+i \frac{\phi}{2}} \end{array} \right)$. The overall phase $\chi$ is unobservable. So there are two unknowns: $\theta$ and $\phi$. So you shouldn't need more than 2 equations to determine them.

Let's let $P_x, P_y, P_z$ be the probabilities for measuring the particle to be spin-up in the three directions. Then, if I did the calculation correctly,

$P_x = \frac{1}{2} (1+sin(\theta) cos(\phi))$
$P_y = \frac{1}{2} (1+sin(\theta) sin(\phi))$ (I might have made a sign error---it might be $-sin(\theta) sin(\phi)$)
$P_z = \frac{1}{2} (1+cos(\theta))$

This implies a relationship between the three probabilities:

$\sum_j (2 P_j - 1)^2 = 1$

 

[pellman]

$P_x = \frac{1}{2} (1+sin(\theta) cos(\phi))$
$P_y = \frac{1}{2} (1+sin(\theta) sin(\phi))$ (I might have made a sign error---it might be $-sin(\theta) sin(\phi)$)
$P_z = \frac{1}{2} (1+cos(\theta))$

Right. But knowing $cos(\phi)$ alone is not enough because there are two values of $\phi$ with that value of $cos(\phi)$. You need both $cos(\phi)$ and $sin(\phi)$

 

[A. Neumaier]

So the theoretical question amounts to: if we know | f(x) |^2 and |F(k)|^2 for two complex-valued functions f(x) and F(k) which are Fourier transforms, can we recover f(x)?

This is probably a theorem out there somewhere. Anyone know?

This is a question you should ask at MathOverflow, where there are many professional mathematicians. You can then feed back the best answer from there to the present thread.

 

[pellman]

So the theoretical question amounts to: if we know | f(x) |^2 and |F(k)|^2 for two complex-valued functions f(x) and F(k) which are Fourier transforms, can we recover f(x)?

This is probably a theorem out there somewhere. Anyone know?

I need to modify this statement since it is easily shown that for a given f the function $g = e^{i \phi} f$ for constant $\phi$ has both $|g(x)|^2 = |f(x)|^2$ and $|G(k)|^2 = |F(k)|^2$. Which is fine from a quantum measurement standpoint. The overall phase is not observable.

So if we know | f(x) |^2 and |F(k)|^2 for two complex-valued functions f(x) and F(k) which are Fourier transforms, can we recover f(x) up to a constant factor of the form $e^{i \phi}$?

 

[pellman]

This is a question you should ask at MathOverflow, where there are many professional mathematicians. You can then feed back the best answer from there to the present thread.

Thanks. I will try that.