Simple Explanation for Quantum Entanglement?

[CramBoom]

Can anyone explain simply what quantum entanglement is and how it works (with equations). I understand the basic principles, but I found the in depth information a little bit confusing. Thank you.

[Sensei]

f.e. when two gamma photons with E=510,999 eV will collide they will produce positron and electron in pair production process.

When we will learn that spin of electron is up, we know in advance that positron will have spin down. Or reverse. These particles are entangled.

[CramBoom]

f.e. when two gamma photons with E=510,999 eV will collide they will produce positron and electron in pair production process.

When we will learn that spin of electron is up, we know in advance that positron will have spin down. Or reverse. These particles are entangled.

Thanks, I know all the stuff about wave function, wave collapse, non-local interaction, and the basics. I just wanted to know the formulae involved and the more complex reasoning for why it is happening (theoretical reasoning).

[swansont]

Thanks, I know all the stuff about wave function, wave collapse, non-local interaction, and the basics. I just wanted to know the formulae involved and the more complex reasoning for why it is happening (theoretical reasoning).

 

 

http://www.lecture-notes.co.uk/susskind/quantum-entanglements/lecture-5/example-states/

[imatfaal]

f.e. when two gamma photons with E=510,999 eV will collide they will produce positron and electron in pair production process.

When we will learn that spin of electron is up, we know in advance that positron will have spin down. Or reverse. These particles are entangled.

 

gamma-gamma interactions - two photon physics - is exceedingly rare (presently be studied indirectly at the LEP at Cern) and does not proceed through a simple e+ e- pair production. photons do not really interact with each other - but at high enough energies a single photon can produce a fermion/antifermion pair, the other photon can now interact with this photon. gamma - gamma interaction is used to study the structure of the photon (which I must admit fries my brain a little). A much simpler way of getting entanglement is to send a proton beam through a Beta Barium Borate crystal which causes spontaneous parametric down-conversion - you get divergent beams of signal and idler photons

[Enthalpy]

As far as I understand (serious warning, not rhetoric):

- Entanglement is observed when the wavefunction collapses.

- The measure of each particle is uncertain as if the particle were alone.

- Only the correlation among the measures reveals an entanglement.

- The entanglement itself bears uncertainty, consistent with Heisenberg's law.

- Several particles are described by one single wavefunction that gives an observation probability for "particle A in this state AND particle B in that". When you can deduce a probability for A that depends much of B, you call it entanglement. But if the probabilities are independent enough, you reasonably take one wavefunction per particle.

- A wavefunction collapse can mean the choice of a weighed sum of the function basis chosen by humans to describe the particles.

 

----------

 

Take two electrons on a 1s orbital. Each one has a distribution with spherical symmetry, but as they repel an other, if you observe one near a position, the other is unlikely there. The proper way to describe this is one single wavefunction for both: psi(x1, y1, z1, x2, y2, z2). From that, you may deduce a probability density to find any electron near a position, but this distribution does not contain all the information: it lacks the correlation.

 

Then the spin: if you observe one electron up, the other is down - but both can have any orientation before detection.

 

----------

 

Or take two transitions: one electron drops 3s -> 2p -> 1s. 3s and 1s are spherical, but 2p isn't: the eigenfunction is a peacock along x, y, z or a doughnut around x, y, z - or a weighed sum of them.

http://winter.group.shef.ac.uk/orbitron/AOs/2p/index.html

The Orbitron shows only the peacock functions; add two with 90° geometric angle and 90° phase shift to have a doughnut function.

 

If the intermediate 2p is a peacock along z, the first photon (emitted by the 3s+2p not-eigen function) is linear, with a higher probability (a cosine) to move near the xy plane and with z polarization. The the second photon (emitted by the 2p+1s not-eigen function) has the same caracteristics, approximately, because the 2p+1s function has the same orientation as 3s+2p. Provided that the 2p state was untouched meanwhile, which relates to the possibility of observation.

 

The intermediate 2p can also be a doughnut around y for instance, and then both photons are circularly polarized and propagate more probably near the y direction. The two directions are linked, but with an uncertainty. [For those who like radiocomms, it's the directivity of a short dipole antenna or loop].

 

Two sets of eigenfunctions (peacock or doughnut) are equally good to describe the 2p orbitals. They are equally "eigen"; the doughnuts are weighed sums of peacocks; the peacocks are weighed sums of doughnuts (peacock along z is a sum of counterrotative doughnuts around y for instance). The corresponding photons are equally "eigen" as circular (a sum of linear) or linear (a sum of circular).

 

What's fun is that you get entanglement with both types of detectors, linear or circular. This is what decided among several interpretation of quantum mechanics. The photon pair does not "decide" when emitted whether its vertical, horizontal or biassed, because this wouldn't explain the correlation among circular detectors. The pair doesn't decide neither to be right, left or elliptic, because one would see no correlation using linear detectors. There is no "hidden parameter": the pair is undetermined until detection.

 

I wrote that the collapse can be a weighed sum: if you write one linear EM wave for both functions, then the (complex) weighed sum can be circular if needed, and bear the correlation.

 

More puzzling: to my understanding (take with caution), one (particle pair) wavefunction does not suffice to describe the situation. It takes two waves (like horizontal and vertical polarizations); not even a weighed sum describes the particles before detection. That is, the uncertainty results not only from the property of a wave, which already bears x versus p, E versus t... uncertainty.

[DParlevliet]

.... I just wanted to know the formulae involved and the more complex reasoning for why it is happening ....

 

I think nobody knows why it is happening. It is measured and what is observed is written in a formula.

[Enthalpy]

 

I think nobody knows why it is happening.

Your opinion.

[Enthalpy]

"With equations", asked the first message...

 

The one I know is quite disappointing. It consists in writing one single wavefunction for both particles A and B:

psi (position A, position B, state A, state B, time)

where state includes the angular momentum. Psi can then give a probability density, serve to compute time evolution and eigenstates...

 

The difference with

PsiA (position A, state A, time) * PsiB (position B, state B, time)

is the entanglement. That is, the chances to observe the particle A in state A or position A depend on particle B being in state B or at position B - plus, if particles A and B are of the same kind the chances to observe BA instead of AB, computed properly depnding on bosons or fermions.

 

This single wavefunction can formulate nearly anything, hence I wrote "disappointing": as is, it gives very little hints about what is independent or linked in the particles' behaviour. It's more a general frame where on can model, by writing details of Psi, how he believes the particles are linked.

 

What is happening: the link between the particle results from the process that emitted them. For instance the intermediate state of an electron that defines the polarisations and directions of two photons.

 

Though, what makes entanglement special is that particles can "decide" their state late, upon detection for instance. Ancient comprehension would have wanted the electron's intermediate state, or whatever decides the states of the emitted particles, to be certain - this is disproven. One interpretation of QM wanted the states be already decided upon particle emission, but not observed by us - and this as well is disproven by EPR-type experiments. That is, the entangled particles are in a superposition of states until their state is measured.

 

Disproven "by experiments" means also "nobody knows why it is happening", yes. No justification by theory, but "superposition until measured" is a formulation that fits experiments.

Edited December 25, 2013 by Enthalpy

[Enthalpy]

My opinion about the "wavefunction collapse", measurements, and particles "deciding" has evolved.

The collapse was introduced long ago as an ad hoc addition to QM, with nor firm basis nor formulation. More recent experiments, like "quantum erasers", tell that interactions don't reduce the possible states of a particle. A measurement being an interaction (followed by many interactions), it shouldn't reduce neither the possible states.

It's only that apparatus are built to indicate a reduced set of possibilities - give certainty. This results from the decorrelation among the other possible states, which is extremely difficult to avoid in macroscopic states (making quantum computers difficult), and the decorrelation makes the other possible outcomes unobservable from a state that observes one outcome.

So all possible stories do happen, they interfere and this can sometimes be observed, but decorrelation gives the observer the states that observed A the impression that he doesn't exist in the states that observed B. No collapse at all. Only an illusion.

Some funny experiments tend to rule out that the observer's spirit makes the collapse. For instance with light emitted by stars long before the observer was born.

I don't know in detail what the specialists' opinion is, but the kind of experiments they make suggests that they have already taken this step. The idea needs only time to percolate the science community.

What relation with entanglement and the EPR paradox?

Both photons (or whatever you want) are emitted with any polarization: all the possible stories happen. But the only possible stories link the polarization of both photons.

Both detectors see a photon or not, consistently with the possible pairs of states. No collapse at the detectors and the measurement: the detectors, the apparatus, the observer exist in all the possible states that result from the possible photon pair stories.

The photons, the detectors... don't need to transmit any information, neither slowly nor quickly. In very possible state of the experiment and the observer, the photons are correlated because all possible stories want it.

Some state of the photon pair at the detectors, say "seen and seen", results in many possible states of the experiment and observer, depending on every collision with an air molecule for instance, and the sum of these many states is small as a mean and non-repeatable, so this set of many states has no effect on any state of the experiment and observer that resulted from a "missed and missed" state of the photon pair at the detectors. This gives the illusion of a wavefunction collapse.

[Enthalpy]

Wavefunction collapse, again. Changed my mind since 12/18/18.

In an EPR experiment, the photon pair can't be emitted in all states. The argument is the same as against states decided at the emission:

• States with linear polarization would sometimes be observed by circular detectors, with no correlation between the detections.
• States with circular polarization would sometimes be observed by linear detectors, with no correlation between the detections.
• Even if all possible states, linear and circular, were emitted, the resulting correlation would be smaller than observed.

In simpler cases where the states are exclusive, like the up and down magnetic moment for an electron, absorbed or not for a photon... a story like "both options coexist" is possible. Here with photon polarizations, we have linear combinations, and such stories don't work as is.

I wonder how the parallel universes interpretation copes with that.

EPR experiments are famous for supraluminal or simultaneous thingies, but there is worse. The particle that emitted two entangled photons was in a state that is decided when the photons reach the detectors, not before. Imagine a pair of transitions 3s -> 2p -> 1s: same momentum in 3s and 1s, the sum of the emitted photons' momenta is zero. But whether the photons were observed by a linear or a circular detector tells, back in time, that the 2p intermediate state of the electron too was linear or circular.

This is possible because the intermediate state of the electron was not observed, so the backward propagation does not transport chosen information nor action.

I had read vague allusions to backward "causality" in QM, it must relate with that. The conservation of momentum, angular momentum, energy... was already problematic with a reduction of the wavefunction, so backward propagation doesn't even need entangled photons.

[joigus]

On 11/25/2013 at 2:58 AM, CramBoom said:

Can anyone explain simply what quantum entanglement is and how it works (with equations). I understand the basic principles, but I found the in depth information a little bit confusing. Thank you.

(my emphasis)

This is a very old post, I realize. But I think it may be worth keeping the fire burning.

I'd love to have a go at getting into some details. But I don't think anybody can make it simple. The mathematics is not very hard, compared with, e.g., GR, but the connection with the physical intuition is bizarre at best. It would be nice to be able to dispel some common misconceptions.

 

On 4/26/2020 at 10:11 PM, Enthalpy said:

Wavefunction collapse, again. Changed my mind since 12/18/18.

Very interesting. One of my old teachers --now retired-- changed his mind twice in 48 hours!

I changed mine circa 1999 --old geezer-- and never went back. Suddenly realized that you don't need to collapse anything if you keep track of mixed states instead of pure states. Nobody would listen back then, though the real savvies knew about it. Life went on and I licked my wounds. I learnt about Ballentine's approach very similar to my view, although something was wanting: What happens to pure states? Are they a figment of the physicist's imagination?

Hopefully, someone might catch me in any possible mistakes, imprecision, ambiguity or digression, misquotation.

 

I've compiled a basic dictionary of everything that might be needed:

 

1) Dirac bra/ket notation. If not, the row/column complex vector would suffice.

2) Spin eigenstates and eigenvalues, Pauli matrices. For example, does this ring a bell? --no pun intended,

\[\left(\begin{array}{cc} n_{z} & n_{x}-in_{y}\\ n_{x}+in_{y} & -n_{z} \end{array}\right)\]

3) The multi-particle formalism of quantum mechanics, otherwise known as "tensor products of states". 2-particle will do.

4) The postulates of quantum mechanics (states, observables, eigenvalues and eigenstates, probability amplitudes as Hermitian products, commutators & incompatibility, Hamiltonian or linear+isometric = unitary evolution, projection postulate.)

5) Yes/No observables, otherwise known as projectors (P linear & P²=P). That is, observables of simple bi-valued spectrum {0,1}. Plus useful lemma:

 

\[A^{2}=I\Rightarrow P_{\pm}\stackrel{{\scriptstyle \textrm{def}}}{=}\frac{1}{2}\left(I\pm A\right)\:\textrm{are mutually orthogonal projectors}\]

6) Completeness relation or resolution of the identity for orthogonal projectors,

 

In a nutshell: as much of the basics of the formalism as possible.

 

To really understand the fundamentals, I would also strongly advise anybody to learn about:

 

-The position/momentum representations of the wave function

-The concept of a complete set of commuting observables (CSCO)

-Superselection observables (those for which superposition cannot be applied)

-The density matrix (distincion between pure and mixed states)

-Decoherence

-Local conservation principles, in particular, local conservation of probability densities

 

Just for hairy details of measurement, QM of open systems, how states actually evolve in space and time, etc.

 

I do have a feeling of impending doom, though. This topic has brought me unbearable pain in the past. I can no longer feel pain, so I'm thinking what the hell.

 

My intellectual appetizer would be the statement of this common misunderstanding:

Quantum mechanics is an out and out local theory. The conundrum is more to do with: systems that look to my rational mind as pairs of things (triplets: GHZ-M) are really one thing in some strange, uniquely quantum sense, because they're internally connected. It's non-separability that's at the root of all this, not non-locality.

Edited May 9, 2020 by joigus
mistyped

[Markus Hanke]

16 hours ago, joigus said:

The mathematics is not very hard, compared with, e.g., GR

GR is hard but not complicated; QM is complicated, but not hard  

16 hours ago, joigus said:

It's non-separability that's at the root of all this, not non-locality.

Well said. While these two concepts sometimes coincide (depending on what available information about a quantum system you are looking at), non-separability is more fundamental and more general.

[joigus]

3 hours ago, Markus Hanke said:

GR is hard but not complicated; QM is complicated, but not hard  

🤓 ... 🤔 ... 🤭 

"But... what happens at zero?," asked the GR disciple.

"Zero is not zero," the GR master replied.

"How could that be?," said the disciple.

"Coordinates are meaningless," the great master said.

"I don't understand," the student declared, utterly puzzled.

"Every answer has a question in it; every question departs from an answer, they both build on each other," was the master's reply, and hit him with a tensor on his head.

The student was immediately enlightened. 

 

Edited May 10, 2020 by joigus
wrong use of diacritics

[joigus]

On 5/9/2020 at 4:53 PM, joigus said:

 

I've compiled a basic dictionary of everything that might be needed:

 

[...]

 

In a nutshell: as much of the basics of the formalism as possible.

 

To really understand the fundamentals, I would also strongly advise anybody to learn about:

 

-The position/momentum representations of the wave function

-The concept of a complete set of commuting observables (CSCO)

-Superselection observables (those for which superposition cannot be applied)

-The density matrix (distincion between pure and mixed states)

-Decoherence

-Local conservation principles, in particular, local conservation of probability densities

 

[...]

Silly me, forgot:

-Entropy: \[s\left(\rho\right)=-\textrm{tr}\rho\ln\rho\]

When most physicists talk about entangled states, what they really mean is maximum-entropy non-factorizable pure n-particle states. They are something more than just entangled --non-separable.

Edited May 11, 2020 by joigus
abridging of self-quotation

[joigus]

2 hours ago, joigus said:

When most physicists talk about entangled states, what they really mean is maximum-entropy non-factorizable pure n-particle states. They are something more than just entangled --non-separable.

Again, imprecise.

Non-factorizable pure n-particle states that, once an ideal measurement has been performed on them, become strict mixtures of maximum entropy.

I think I got it right now. It is much shorter to say 'entangled,' but dangerously vague. I think I've read 'maximally entangled' somewhere...

Otherwise, for pure states, you pick your state as making up the first vector of a basis, and then, the entropy is clearly identically zero. For the GHZ experiment, e.g., if you pick the GHZ state as the first vector of your basis,

\[\rho=\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)\]

so that,

\[s\left(\rho\right)=-1\ln1=0\]

Instead, if you perform and ideal measurement, coherences are erased and,

\[\rho=\left(\begin{array}{ccc} \frac{1}{3} & 0 & 0\\ 0 & \frac{1}{3} & 0\\ 0 & 0 & \frac{1}{3} \end{array}\right)\]

so that,

\[s\left(\rho\right)=-3\textrm{tr}\left(\frac{1}{3}\ln\frac{1}{3}\right)=\ln3\]

 

 

Edited May 11, 2020 by joigus
mistyped

[studiot]

53 minutes ago, joigus said:

I think I've read 'maximally entangled' somewhere..

Entanglement is complicated by the fact that there can be degrees of entanglement.
Full entanglement is called maximal entanglement,
Partial entanglement results in various special states, for instance with photons the colour may be entangled for 3 photons, resulting in what is called the W state.

https://en.wikipedia.org/wiki/W_state

[joigus]

34 minutes ago, studiot said:

Entanglement is complicated by the fact that there can be degrees of entanglement.
Full entanglement is called maximal entanglement,
Partial entanglement results in various special states, for instance with photons the colour may be entangled for 3 photons, resulting in what is called the W state.

https://en.wikipedia.org/wiki/W_state

Colour or polarisation state? Sorry, I'm not quite up-to-date in quantum information technology.

[studiot]

On 5/11/2020 at 1:19 PM, joigus said:

Colour or polarisation state? Sorry, I'm not quite up-to-date in quantum information technology.

https://phys.org/news/2019-09-three-photon-color-entangled-state.html

[joigus]

2 minutes ago, studiot said:

https://phys.org/news/2019-09-three-photon-color-entangled-state.html

Thank you so much.