# Quantum Correlations (entanglement)

Let us recall the geometric description of correlation in classical probabilistic processes:

Consider a classical n-states system, whose states space A has dimension n-1, correlated with an m-states system, whose states space B has dimension m-1.
Each correlated state is expressed by a projective transformation from the (n-1)-dimensional set A* (dual set to A) of all possible measurements of the first system, into B; which can be equivalently expressed by a projective transformation from B* (measurements of the second system) into A.
The set of all such correlated states had dimension nm-1, as the global system is an nm-states system.

With quantum theory, the situation is very similar:
An n-states system aside an m-states system together form an nm-states system, as it is possible to distinguish there nm distinct pure states by measurements with certainty (which means that any two from such a list are clearly distinct, being 2 opposite points of the sphere of states they are forming). And distinguishing n states on the one and m states on the other, is a way to make such a distinction of nm states on the global system.

Now, the states sets A and B of these systems have respective dimensions n²-1 and m²-1. The set AB of all (correlated) states of the global system, has dimension n²m²-1. Each one is represented by a projective transformation from A* into B, or equivalently from B* into A.

Let us call classically correlated state, any state (element of AB) which can be obtained as a barycenter with positive coefficients, of a list of uncorrelated states, where an uncorrelated state is defined by a pair (a,b), of states in A and B (corresponding to the limit sort of "transformation" that collapses A* onto b and collapses B* onto a).

There are 3 differences between classical and quantum correlations:
1. The set of projective transformations from A* to B has the same dimension n²m²-1 as AB, and AB is included there, but some of its elements (projective transformations mapping A* inside B), do not belong to AB (they do not express physically possible states of the system).
2. The set of classically correlated states has the same dimension n²m²-1, and is included in AB, but is not all AB: some physically possible correlations cannot be obtained as a classical correlation. Bell inequalities are inequalities satisfied by all classically correlated states, but not always by other elements of AB.
3. In particular, the set of pure states in AB has dimension 2mn-2, while its subset of uncorrelated pure states (a,b) where a is pure in A and b is pure in B, has a lower dimension, sum of the dimensions of variations of a and b: (2m-2)+(2n-2)=2m+2n-4. Thus, most pure states of AB are correlated but can't be classically correlated, because classically correlated states can't be pure.
Let us describe the simplest case of quantum correlated systems: the case n=m=2, incarnated as the spins of two electrons.

A natural process that can produced entangled particles, is to take a pair of electrons and separate them passively in a spatial manner without using a magnetic field. Then their spins are entangled (opposite to each other in any direction of measurement). Still it is not used for experiments at long distance because it is not practical to transport electrons and they can lose their spin by interaction with photons that are usually present in the form of thermic radiation (does anyone know at which speed for given temperatures ?). Also it may be hard to produce, as pairs of electrons are present in atoms but don't easily get out. Shooting away the kernel of an helium atom is possible but, well, not easy. Experiments were mainly done with entangled photons instead.
A pure correlated state naturally appears in the form of an electron pair. Indeed an electron pair is a 1-state system, thus pure. But both electrons there are together. In order to obtain a system made of two subsystems (electrons), we first need to separate both electrons from the pair. This is done by spatially introducing a separation (a wall or the like), and checking that exactly one electron is present on each side, without disturbing the system any further.

So, after the separation, we have a system made of 2 subsystems, which is in a pure correlated state. Both spins are opposite, no matter the common direction in which they will be measured.
The corresponding projective transformation from A* to B is very simple: it is the central symmetry of the sphere.

As this central symmetry maps the center into the center, the first measurement of any of these spins has probability 1/2 for each of its both possible results. And whatever is the result, the knowledge of this result collapses the state of the other electron's spin onto the opposite point.

Then, what other pure correlated spin states are there ?
An easy way is to take the one we got, and modify it by simply rotating one of the spins (by a magnetic field). This way, the possible relations we will get between the spins, will be anyone defined by the composition of a central symmetry with a rotation, so, any indirect isometry (around the center of the sphere).

More generally, all pure correlated states are represented by all projective transformations that map the sphere of pure measurements of one spin, to the sphere of pure states of the other spin, and reversing the orientation.

We can describe their whole set as follows: the pure correlated states of spins, are those mapping the sphere of pure measurements of the one, onto the sphere of pure states of the other, and reversing the orientation.

This reversing of the orientation is required: the projective transformations preserving the sphere but also preserving its orientation, do not define any physically possible state of spin correlation.

To understand what these transformations look like, we can study the orientation-preserving transformations instead (as both cases are exchanged by central symmetry).
These are conformal transformations of the sphere: they map circles on the sphere to other circles, because circles are the intersections of the sphere with planes in space, and projective transformations of the space map planes into planes. Those who are not isometries, are expanding some side of the sphere and shrinking the opposite side.
It is possible to understand these transformations of the sphere, by considering a sphere taken on picture in perspective, and reinterpreting the same picture of the sphere as if it was viewed from different distances to the sphere, or viewed from infinity.

Now, inside the quantum 4-states shape of all states of a system of 2 electrons'spins, let us consider its the particular 2-states sphere, of the states between (↑,↓ ) and (↓,↑). That is, all the states that have probability 1 of being "either (↑,↓ ) or (↓,↑)".
We can represent these two states (↑,↓ ) and (↓,↑) as the poles (north and south) of this sphere.  Then, what is its equator made of ?
It is made of all negative isometries of the sphere that exchange ↑ with ↓.
The central symmetry is one of them. To get all others, you just need to apply a rotation around the vertical axis. In particular, the one opposite to the central symmetry, is the reflection with respect to the horizontal plane.

Generally, any two correlated states defined by (negative) isometries of the sphere, can be completely separated by a measurement after some interaction between both spins, if an only if these two isometries differ by composition by an axial symmetry.
So, while the 4 states  (↑,↓ ), (↓,↑), (↑, ↑) and (↓,↓) are a possible set of completely distinct 4 pure states generating the whole quantum 4-states shape of all states of correlated spins, another possible list of completely distinct pure states, is made by 4 isometric correlated states differing from each other by compositions by axial symmetries around all 3 axis of a Cartesian coordinates system.

Whatever the choice of such a list, the isobarycenter of these 4 states, or global center of this quantum 4-states shape, is the uncorrelated state given by the centers of each sphere of spin.
When applied to this center, any pure state has a probability 1/4 to be obtained.
In particular, the measure of presence of a pure correlated state, such as the one of central symmetry s (from a pair of electrons), has this probability 1/4.
When applied to the state defined by the transformation k.s (i.e. homothety with ratio -k) for any real number k, its probability is the affine function of k. that gives 1 for k=1 and gives 1/4 for k=0. Thus it cancels for k = -1/3 (the state -s/3, the dilation with ratio 1/3), thus a mere combination of "the 3 remaining states"), and k cannot go lower (dilations with ratios higher than 1/3 can't be reached).
Unlike classical physics, quantum physics discriminates between correlations that preserve the orientiation of the space of possible spin states, and those which reverse it : both have the same limit for correlation preserving orientation (the state -s/3, i.e. homothety with ratio 1/3, is the limit as well for classical correlations, as it can be obtained by taking at random one of the 3 space dimensions in which both spins will be correlated) but for those reversing it, classical correlations have the same limit but quantum ones do better : s instead of s/3. but as the possibilities of classical correlations don't depend on the sign, they can't make a k higher than +1/3 either, so that the state s, that exists in quantum systems, is quite far from reach by classical correlations)

So, it is useless to discuss partial entanglements as quantum theory does not distinguish a blurred entanglement from a classical correlation : a entangled pair of electrons produced as above that has 1/3 probability to be preserved and 2/3 probability to be blurred (replaced by uncorrelated random spins) is the same quantum system as a classical correlated pair produced by a 1/3 probability of having been produced by a classical correlation in each of the 3 dimensions. In other words, by throwing a dice to decide if the spins are (up,down), (down,up), (right,left), (left,right), (face,back) or (back,face).
Next: The EPR paradox - The double-slit experiment - Quantum decoherence

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