Wave functions for Coherence and Entanglement

[San K]

My understanding of wave-functions is close to zero, pardon me if the questions don't sound proper.

1. Do we have wave-function usage to describe a) Coherence and b) Entanglement?

2. Has a (mathematical/conceptual) way been developed to show the complementarity between both (a & b) via wave-functions?

3. is there some sort of inter-convertibility between two? (via wave-function treatment)

On a separate note:

Interestingly the wave function that emerges from a (single particle) double slit get stopped/blocked/terminated by the same types of obstacles that would effect a photon/light.

However entanglement is not effected.

In short: Coherence is effected by obstacles but entanglement is not and yet they are complementary.

Some of the other complementary "pairs" are position-momentum, time-energy, if we try to compare the pairs wonder if we can draw any insights/parallels

 

[tom.stoer]

It's conceptually simpler to use Bra-Ket notation in order to explain entanglement, but there is a wave-function notation as well.

Suppose we have two identical bosons n = 1, 2 with spin zero (b/c that's a very simple example). Suppose these two bosons are in states u and v with wave functions u(r) and v(r). Then symmetrization for bosons says that we cannot know that "particle 1 is in state u and particle 2 is in state v"; instead we have "one particle in state u and another particle in state v". Note the difference: we do not know which particle is in which state.

So instead of having a wave function

ψ(r₁, r₂) = u(r₁) v(₂)

where we know which particle (1, 2) is in which state (u, v) we have

ψ(r₁, r₂) = u(r₁) v(₂) + v(r₁) u(₂)

This is the mathematical expression for two particles in a state (u, v) w/o saying which particle is in which state.

This ψ describes an entangled pair.

 

[vanhees71]

...and for two (spin-0) bosons of the same kind the wave function MUST be in this state, which is symmetric under the operation of interchanging the particles, because there is nothing that can distinguish between the two individual particles (except they are different in at least one intrinsic quantum number like charge and/or mass). In this sense identical bosons are always entangled.

 

[San K]

Thanks Tom and Vanhees, great answers.

is there some equation/mathematical treatment that shows:

coherence must necessarily decrease when entanglement is increased...?
or vice versa

 

[tom.stoer]

You should read about decoherence

http://en.wikipedia.org/wiki/Quantum_decoherence
http://arxiv.org/abs/quant-ph/0312059
http://arxiv.org/abs/quant-ph/0105127
http://arxiv.org/abs/quant-ph/0306072