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.
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
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:
Let us describe the simplest case of quantum correlated systems: the
case n=m=2, incarnated as the spins of two electrons.
- 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).
- 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.
- 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.
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
(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
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
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
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
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).
The EPR paradox - The double-slit experiment - Quantum decoherence
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