What is the difference between entanglement and coherence

[sciencejournalist00]

It is said that a coherent light beam is described by a single wave because all the individual waves add up in phase to produce a single big wave.

It is also said that entangled light is described by a single wavefunction.
https://qph.is.quoracdn.net/main-qimg-74a1e026576eb278294f9e2ac41996d2?convert_to_webp=true

These two descriptions are similar and have been a source of confusion for years for me, even for the teachers at my university. They asked me to find out what the difference between light "described by a single wave" and light "described by a single wavefunction" is.

From the pictures above, I see no difference between coherence and entanglement, so the Internet won't do for explaining the difference to me. Please do so yourself.

 

[f95toli]

First of all, light does not have a wavefunction (there are some ways to formulate the math that will give a you something a bit similar, but it is not a QM wavefunction)
Hence. "Light described by a single wavefunction" has no physical meaning.

Secondly, I am not even sure how the two terms could be confused since they are used in different contexts. "Coherence" is a term that is used for many different systems, not only light, the most obvious example would be a single two-level system which can be coherent or incoherent depending on its dynamics .
Entanglement would be a property of (at least) two connected systems (or perhaps two parameters of a single system).

 

[sciencejournalist00]

First of all, light does not have a wavefunction (there are some ways to formulate the math that will give a you something a bit similar, but it is not a QM wavefunction)
Hence. "Light described by a single wavefunction" has no physical meaning.

Secondly, I am not even sure how the two terms could be confused since they are used in different contexts. "Coherence" is a term that is used for many different systems, not only light, the most obvious example would be a single two-level system which can be coherent or incoherent depending on its dynamics .
Entanglement would be a property of (at least) two connected systems (or perhaps two parameters of a single system).

But coherence means constant phase difference. Adding waves with constant phase difference gives a single resultant wave, with higher or lower amplitude.

Therefore, coherent light is described by a single wave. A wave which is a displacement measured in micrometers from the equilibrium position of the electric force.

And entangled photons have the linked quantum state, which means their probabilities of having a state are connected. Wavefunction refers to probability, not to displacement from the equilibrium position of a force.

So entangled photons have linked probabilities and coherent photons have linked displacement vectors.

My question is, why can't coherent photons also considered to have linked quantum states? Doesn't linking displacement vectors also link probability amplitudies?

 

[kith]

Just a few general remarks: coherence is a property of a single quantum system while entanglement is a property of multiple (sub)systems. So if you can't split your system somehow, it doesn't make sense to talk about entanglement.

An entangled state is a superposition of states of the combined system and therefor coherent. The individual systems however can't be assigned individual states but only probabilities for certain states. So they are in what is called an incoherent mixture of states.

It is quite difficult to convey this with words only and different people will use different words. In the maths, the terminology is very clear. You seem to be willing to put in quite some effort into reading about experiments and their pop-science descriptions. Why don't you learn some basic QM? Off the top of my head, I would recommend Leonard Susskind's Theoretical Minimum book on QM which is aimed at laypersons who want the real thing. This will greatly improve the communication between you and people who know QM.