Why do the off-diagonal terms not matter in density matrices?

[cube137]

can all quantum state be entangled without any exception even if their phases don't coincide? is the term to call this mixed state entanglement accurate? does it have to do with Fourier addition?

this is related to environmental entanglement...

when you are shaking hands with another person.. the atoms in the hands have interaction.. or say the thermal photons from your hands interact with the electrons in the hands of another person... can you call this entanglement?

 

[mfb]

What do you mean by "if their phases don't coincide"? Entanglement would lead to a known relation between phases.

when you are shaking hands with another person.. the atoms in the hands have interaction.. or say the thermal photons from your hands interact with the electrons in the hands of another person... can you call this entanglement?

Only if you somehow avoid decoherence, which you cannot in systems like this.

 

[cube137]

What do you mean by "if their phases don't coincide"? Entanglement would lead to a known relation between phases.Only if you somehow avoid decoherence, which you cannot in systems like this.

I meant.. if the phases won't have interference.. in pure state, the phases will have interferences.. so entanglement can also work even if there was no inteferences but only known relation between phases? Or in other worlds.. all things can entangle as long as they have waveforms? but all matter have wavelength.. so 100% of matter entangle? Is it related to Fourier addition of waveform?

 

[mfb]

in pure state, the phases will have interferences

Which phases where?
In the general context that statement does not make sense.

so entanglement can also work even if there was no inteferences but only known relation between phases?

Yes.

so 100% of matter entangle?

Every particle can be entangled in some properties.

 

[cube137]

Which phases where?
In the general context that statement does not make sense.Yes.Every particle can be entangled in some properties.

I meant, the off diagonal term of the density matrix has none positive values and has interference. If the value goes from large to tiny, do you also consider it as being a pure state? whats' the threshold for the off diagonal term values (% from minimum and maximum) being considered a pure state?

 

[atyy]

I meant, the off diagonal term of the density matrix has none positive values and has interference. If the value goes from large to tiny, do you also consider it as being a pure state? whats' the threshold for the off diagonal term values (% from minimum and maximum) being considered a pure state?

The off-diagonal terms are not that important, because even a pure state can have off-diagonal terms.

The density matrix is pure if squaring it produces the same matrix, or if the trace of its square is 1.
[/PLAIN]
http://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf (see #30)

 

[cube137]

The off-diagonal terms are not that important, because even a pure state can have off-diagonal terms.

The density matrix is pure if squaring it produces the same matrix, or if the trace of its square is 1.
[/PLAIN]
http://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf (see #30)

Yes I've read it and tried to understand it.
Why is the off-diagonal terms not that important when you can tell from it whether it's pure or mixed state? And for a pure state, can you say the off-diagonal term has value of 100%?

In an electron entangling with the nucleus in an atom.. what are the mixed states.. is it the position eigenvalues? I don't think it is pure state.

 

[Nugatory]

Why is the off-diagonal terms not that important when you can tell from it whether it's pure or mixed state? And for a pure state, can you say the off-diagonal term has value of 100%?

If you're going to dig as deeply into the formalism as you want to, you're going to have to learn the math - there is no other way to get to where you want to be. Atyy's link is very good, but it is written for people who have already been through a no-kidding college-level introduction to quantum mechanics, where the basic notion of states as vectors in a Hilbert space is taught. Only after you've been through that will you be ready to take on the density matrix formalism.

But a quick answer to why the off-diagonal terms don't matter is that you can make them disappear just by changing the basis. As an exercise, you might try writing the density matrices for the following states, using the spin-up/spin-down and spin-left/spin-right bases so you write the density matrix in two different forms for each case:
1) A spin-1/2 particle has been prepared in the spin-up state by selecting it from the upwards-deflected beam of a vertically oriented Stern-Gerlach device.
2) A spin-1/2 particle has been prepared in the spin-left state by selecting it from the leftwards-deflected beam of a horizontally oriented Stern-Gerlach device.
3) A spin-1/2 particle has been randomly selected from one of the two beams coming out of a vertically-oriented Stern-Gerlach device.
4) A spin-1/2 particle has been randomly selected from one of the two beams coming out of horizontally-oriented Stern-Gerlach device.
#1 and #2 are pure states. #3 and #4 are mixed states. #1 is a superposition when written in the left/right basis but not when written in the up/down basis; #2 is the other way around. All four of these states will have off-diagonal elements in one basis or the other.