Introduction to quantum theory

(the theory of quantum states and measurements)

"Hold on to the brush, I remove the ladder !"

Introduction and preliminaries

The laws of quantum physics are fundamentally probabilistic. Thus, to understand them, we need to express probabilistic laws of evolution. The geometric expression of Markov processes (the general case of classical probabilistic law of evolution for material systems) is a prerequisite here, as it provides the language of this presentation.
We are going to express the concepts of quantum states and measurements in a simplified but still mathematically accurate manner (compared to usual courses of quantum physics) as a variant of that mathematical description of Markov processes (in other words, in terms of how quantum probability differs from classical probability). This will describe and show the coherence of some famous "paradoxes" of quantum physics, making them almost seem natural and intuitive. It includes the concepts of indetermination, the role of the observer, and the description of the quantum system involved in the double slit experiment.
We shall also explain the details of the measurement process as a particular case of quantum evolution, by the concept of decoherence ; but that only works "for all practical purposes", not completely in ontological terms (the question of what is real). The remaining discrepancy between the role of measurement in the principles of the theory, and its explanation as a physical process, is the subject of the debate on the interpretations of quantum physics.

See also:

Axioms and fundamental properties of quantum states

We can obtain the basic principles of quantum theory, from the classical probabilistic concepts we just developed, by slight modifications and specifications, as follows (you may ask "Why are things this way ?" well, we just know that they are this way because experience has confirmed it countless times) :
Indeed consider the simplest example, n=2 : any point inside the sphere can be obtained as a barycenter of 2 "pure states", that is, points one the sphere. These are the 2 intersections of the sphere with any line going through the point. You can see that any point of the sphere fits with one of these possible decompositions.

Let us focus on this case n=2 to examine how things work there in more details.

The spin 1/2

The most natural case of a physical system with "2 possible states", is the spin 1/2 of a particle. The simplest and most common example of a particle with 2 states due to its spin 1/2 is the electron, so that we will fix the discussion on it, but some other particles such as the proton (hydrogen kernel), the neutron, and some other atoms, kernels or ions, have this property too (it does not matter whether a particle is elementary or not).

What is a spin ? The first idea for describing a spin, would be that of a rotating ball that must keep rotating because of the conservation of angular momentum. However a rotating ball has too much details that an electron does not have: we can draw a mark on the ball and see it moving around; the ball may stop spinning and become at rest, or spin at different speeds.
The electron, on the other hand, has no such details: it cannot stop spinning, and has no mark on its face that can be seen moving around. Its spin state only consists in the data of its angular momentum, and thus remains constant in time as long as it is not modified by interaction with the environment (namely, by the magnetic field). For any system, the momentum can only vary by integers (to multiply by the Planck constant). The electron has only 2 possible values of the momentum, 1/2, thus with a difference of 1.
In order to measure the spin of an electron and getting one of both possibilities (clockwise vs. counterclockwise), we first need to choose the direction of the axis around which this spin will be measured. And the probabilities of results will of course depend on the axis chosen (as a continuous change of possible choice ending up in exchanging both ends, will of course exchange both probabilities).

Before choosing an axis, any electron's spin is naturally in some state. Like any angular momentum, it is a pseudo-vector. This means it belongs to a 3-dimensional vector space, but its representation as a vector in our space depends on a convention of orientation of space, and is reversed when we change this convention. For example, the angular momentum of the Earth can be represented by a vector towards the North pole, but a representation by a vector towards the South pole would be an equally possible convention. We just have to fix the convention once for all.

So, once this space orientation convention is fixed, the ball B of all spin states of an electron, whose surface is the sphere S of pure states, is figured by a ball in space.

Spin measurement

Let us describe measurements of this spin.
As before, each possible measurement result goes with a probability calculated as an affine function from B to real numbers, and more precisely into [0,1]. It can be any such function. So, it can be represented geometrically by the data of both parallel planes P0 and P1 where this affine function, extended to the whole space, would take the values 0 and 1 (so, outside B, and having B between them).
In the case of a binary (yes/no) measurement, the other possible measurement has the complementary probability (so that the sum is 1), represented with P0 and P1 exchanging their roles.

Now that we have specified "what is to be measured" (the probabilities of measurement results as depending on the initial state), what can be the state of the system after the measurement ?

Contrary to classical physics, quantum physics cannot admit a measurement being done on an elementary system without a physical interaction with it, that disturbs it (there is no more such a thing as a non-disturbing measurement). We need a measuring apparatus to interact with a system, and let the result of the measurement appear in a macroscopic way, where its description can be summed up (approximated) in the form of classical probabilities that we first presented, thanks to the process of decoherence.
We shall precisely describe the form of this necessary disturbance happening during measurement (effect of the physical processes making the measurement).

Instead of a non-disturbing measurement, we have the concept of a least-disturbing measurement. Let us describe its effects geometrically, for the spin of the electron.

The simplest case is the case of a complete measurement, that is where the probability 0 and 1 planes are tangent to S at two opposite points. This measurement collapses the spin onto the point of tangency which is the only pure state having the probability 1 of giving the observed result. As the two possible measurement results collapse the spin onto 2 opposite points, this is why we say that the "number of possible states" of the spin is 2.

This collapsing effect works more generally for any case when P0 is tangent to the sphere, disregarding whether P1 is also tangent or not, and so collapses the spin onto the maximum probability point (opposite to the 0 probability one).

Indeed, we already explained with classical probabilities, that the meaning of a measurement, and thus its effect on the state of the system, does not change if the function that gave the probability of reaching it, was multiplied by a constant.

In this sense, just like in the classical case, the set of all measurements has the same geometrical shape (a ball) as the set of all states (and this correspondence also works for any other "number of states"). To see this, you just need to divide the probability function of a measurement, by its value at the center of S, which will thus become 1 (and divide again the result by 2 if you want it to give a meaningful probability, with values in [0,1] over the sphere). In a Cartesian coordinates system (for the 3-dimensional space containing the sphere), you just need to reinterpret the coefficients (a,b,c) of this function (x,y,z) -> ax+by+cz+1, as the coordinates of the measurement in a space of measurements.

In other words, a measurement, as specified by its zero-probability plane outside the sphere, will be represented by the point inside the sphere, on the line from the center and orthogonal to the plane, and at a distance from the center which is the inverse of the distance of the center to the plane (if the sphere has radius 1), and on the opposite side.
This way, each measurement is represented by the point where it sends the center of the sphere (the totally undetermined state) according to its least-disturbing effect.
Each pure measurement is figured as the element of the sphere where it has its maximal probability, while others are figured inside it.

So, there are many other possible sorts of least-disturbing measurements: binary measurements where one possible result collapses the spin while the other doesn't; or where none does; measurements with arbitrary numbers of possible results, with arbitrary respective probability functions on B, provided that they are positive, affine, and that their sum is 1 all over B.

Weak measurements

Now let us describe other cases, when the measurement result does not happen to provide certainty on the state of the system,. i.e. where P0 is not tangent, but away from S. Then the effect is that of a projective transformation of the space that sends P0 to infinity, and globally preserves B and S : each pure state becomes another pure state.
Only two pure states remain fixed (in the least-disturbing case): those that were nearest and furthest to P0

(These projective transformations of the 3-dimensional space that preserve a sphere, are also those acting on the set of speeds considered as relatively to different observers according to Special Relativity theory: the elements of the sphere define the speed vectors whose length correspond to the speed of light, thus expressing the fact that going at the speed of light, is a property that does not depend on the movement of the observer that measures this speed.)

More remarks

We can see here that the concept of non-perturbing measurement cannot make sense in general: not all pure states (points of the sphere) can be preserved in such a projective transformation. Only two can, and so must be specified to make sense of the "non-perturbing" claim.

Popular accounts of quantum physics mention the Heisenberg inequalities. One of these inequalities say that the position and the momentum of a particle cannot be both determined, and the more precisely one of these quantities is known, the less the other is.
What we just explained about the spin, already presents such an indetermination: it is neither possible to measure nor predict the spin of the electron along several axis at the same time.

Energy and evolution

The evolution of a physical system is determined by the energy differences between its possible states.
We will describe the situation in the case of the spin of the electron, but the same law applies to any other system as well. The explanation will be based on some concepts of classical mechanics. Many concepts of classical mechanics are no more valid in quantum theory, however some properties like those we will mention here, still apply somehow and can help to understand the situation intuitively.

The electron has a magnetic moment associated to its spin. This means that it behaves like a little magnet with the same orientation as its spin. Like any magnet, its interaction with an external magnetic field gives it a potential energy that is minimal when the magnetic moment is aligned with the magnetic field, and maximal when they are opposite. When the magnetic moment is not aligned with the magnetic field, the magnetic field exerts a torque on the magnet, which in the case of ordinary magnets pushes them towards the minimum energy configuration, aligned with the field. But the axis of the electron's spin is not like a fixed object that is turned in the way forces push to turn it. Instead, as it is defined by the angular momentum, the torque exerted by the magnetic field produce a gyroscopic precession of this spin around the direction of the magnetic field.

Now let us express the situation in the terms of quantum physics.

One of the Heisenberg inequalities says that the energy and the time cannot be both determined. Thus, whenever the energy of a system has an exact well-defined value, nothing can happen to it along time.

The spin has two possible states, and thus two possible values of the energy (when the environment is classically fixed). Each of both pure states of the spin along the direction of the magnetic field, has a well-defined value of the energy. For any other state of the spin, the energy in undetermined.
The measurement of the energy of the electron, coincides with the measurement of its spin along the direction of the magnetic field.
These two pure states of well-defined energy remain fixed in time, and give the axis of the rotation of the set B of all spin states along time.
The frequency of this rotation is proportional to the difference of energy between both possible values of the energy. This rotational movement of the spin, being also a rotation of the magnetic momentum of the electron which affects the surrounding magnetic field, generates an electromagnetic wave. This is the frequency of the photon emitted by the electron, by which it will lose its energy in the long term, and reach its state of lowest energy.
But to say this, means that we don't consider the spin of the electron as an isolated system anymore.

Other comments on energy in quantum physics

The photon

The quantum theory of electromagnetism is very complex with strange properties, but here we will focus on the simple case of a single photon with a well-defined frequency and propagating in a unique direction,

Like the electron, the photon has a spin, also called polarization, whose number of possible states is 2, even though the two values of its angular momentum are no more 1/2 but 1. Unlike the electron whose spin could be mesured along any axis in space, the spin of the photon is only defined with respect to the axis which is the direction of propagation. Still, it is possible to measure this spin along any other direction of its abstract sphere of states, but the  (below described) correspondence between these abstract directions and our usual space-time differs from the spin 1/2 case; while the angular momentum that a photon may carry with respect to other directions, takes the form of the spatial configuration of the wave (position and direction of propagation) and will not be discussed here.

We can first understand the polarization in the case of a classical electromagnetic wave: this is a transverse wave, which means that the oscillation of the electric field is perpendicular to the direction of propagation (and the magnetic field too, which at every point of space-time, coincides with the electric field turned 90 around the direction of propagation).

On the abstract sphere of states of the photon's polarization, let us mark 6 points, configured like the centers of faces of a cube containing this abstract sphere; as a cube defines a coordinates system, so these points are expressed by their 3 coordinates.
Imagine that the photon propagates horizontally, so that the oscillation of the field happens in a vertical plane.

Let us also represent in the last column of the following table, another case of a 2-states system: the two possible states of the electromagnetic field that correspond to the undetermined presence of a given circularly polarized photon.

Abstract Position Coordinates Polarization type for a photon Possibly absent circular photon
Left (-1,0,0) Linear, horizontal Electric field to the left
Right (1,0,0) Linear, vertical Electric field to the right
Front (0,-1,0) Linear, diagonal Electric field to the top
Back (0,1,0) Linear, other diagonal Electric field to the bottom
Top (0,0,1) Circular clockwise One circular photon
Bottom (0,0,-1) Circular counter-clockwise Zero photon

(The situation would be the same for the presence/absence of an electron as here with a photon, except that there is no direct measurement possible for this system in any other direction of that sphere than the presence/absence direction, in contrast with the case of the photon where such a measurement can be done in terms of the electric field. In other words, unlike the photon, it is not possible to "see" any oscillation in the electron, despite the fact that such an oscillation somehow exists relatively to some contexts such as the double-slit experiment, see below)

Note that in the case of the possibly absent photon, the electric field oscillates circularly at the frequency usually said to be the frequency of the photon, because each of both poles of the sphere (one photon/zero photon) has a different well-defined energy, which makes the sphere of states rotate around this axis at the frequency defined by the energy difference, which is the energy of the photon.

Also note that we have a nice correspondence between the sphere of spin states of the electron precessing in the magnetic field, and the sphere of states for the undetermined presence of a photon: this is the way the electron comes down to its minimum energy level by emitting a photon and thus transferring its state to it.
We described the case of the circularly polarized photon. It is what would be emitted by the spin of the electron in the direction of the magnetic field, in the case the photon would be detected in this direction, as the rotation of the electric field follows the rotation of the spin.

But the photon is emitted in all directions, so that if we only try to detect it in one direction, we may not get it as it may be going to another direction instead. In other words, the detection of the photon in a direction is correlated to its non-detection in another direction.
So, let us consider a photon detector all around the electron, with a way out in some angular area around the direction of the magnetic field.
The fact that no photon is detected around, defines a partial measurement with respect to the initial spin state of the electron: it is the sure outcome if the electron was already in its minimal energy level, but it also has a chance to be so if it was in the maximal energy level, as the photon can go by the exit (circularly polarized). Thus this case of absence of any photon emitted in other directions, makes a physical evolution defined by a projective transformation from the initial spin state of the electron to the final state of presence/absence of the circular photon emitted in the direction of the magnetic field; this transformation maps pure states into pure states.

Or, if we don't wait enough time to let the electron come down to its minimum energy level for sure, then the presence of an emitted photon will be correlated with the remaining spin state of the electron.

Let us now examine the concept of correlation in quantum theory.

Quantum Correlations

Let us recall the description of correlation in the classical probabilistic theory:

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.

Now with quantum theory, the situation is very similar:
An n-states system aside an m-states system, together form an mn-states system, as it is possible to distinguish there mn 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 n-1 and m-1. The set AB of all (correlated) states of the global system, has dimension nm-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 nm-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 nm-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 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 ks 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).
(The state -s/3 can be obtained as a classical correlation too... 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)

The double-slit experiment

Let us now explain this famous experiment that expresses the strangeness of quantum physics with its wave-particle duality.

Consider a photon going towards a plate with two slits.
It may be stopped by the plate, or go through the slits.
To simplify, let us assume that the case when the photon is stopped by the plate, is detected and eliminated from consideration.

The state of the electromagnetic field in each slit, is undetermined, in between the presence and the absence of the photon. The global system of both slits, is in a pure state of correlation of the slits.
And it is located "in the middle" between both states defined by (1 photon, no photon) and (no photon, 1 photon). Let us denote C the circle of points of the sphere "in the middle" (the equator) between them (taken as poles).

We have explained what are such "equatorial" points, elements of C: they are correlated states defined by an indirect isometry between both abstract spheres of presence/absence of the photon, exchanging the presence of the photon in the one, with the absence of the photon in the other.
Therefore, both equators are just following each other. Points of these equators correspond to different orientations of the electric field. Different elements of C give different correspondances between the electric fields of both slits: they say how much delay separates their oscillations (how late is one's oscillation with respect to the other). One whole way around C, makes this time difference change up to one whole period, coming back to the initial correspondence.

The photon has been sent to the slits from one specific direction, thus specifying the element of C involved (the phase difference between the slits).

After the slits, we have a screen with many points that are sensitive to the photon. So we have a measurement with many possible results (which point of the screen will detect the photon).
Each of these results consists in the pure measurement of a point of C. (The barycenter of these many points of C has to be the center of that sphere, for this to be a possible measurement.)

If we could detect which slit the photon was in (distinguish between (1 photon, no photon) and (no photon, 1 photon)), this would be a measurement along the axis of C (orthogonal in space), and would collapse the state of the system onto either (1 photon, no photon) or (no photon, 1 photon). It would no more be at its position on C. its probability of being detected by this measurement, thus as a pure measurement characterized as coming from some point of C (=whose probability cancels at the opposite point), is half the probability for this point by this measurement (=its maximum probability).

Further remarks on the double slit experiment

In principle, this experiment can be proceeded with any particle, however it becomes more and more difficult (sensitive to disturbance) as its mass increases. The biggest "particle" so far on which it has been made and interferences have been observed, is the fullerene (C60) molecule.

Instead of having a continuous range of possible positions of detections of the particle, corresponding to all points of C, it is possible to redesign the experience to have only two possible results, corresponding to diametrically opposite positions in C. This is an experience with photons going through mirrors and semi-reflecting mirrors (I forgot the reference).
So, a photon has a probability 1/2 of going through either path; if from only one path it would have a probability 1/2 of ending in either of possible final locations; but with this way of keeping both possibilities of paths, it can only reach one destination; but if a modification is done on one of these paths to inverse its phase, this operates a 180 turn of C so that the photon can only end up to the other destination. Strange conclusion: by only affecting the path of statistically half of the photons, the destination of all photons is changed !
See also my further explanations of quantum physics in this discussion, some explanations about the wave/particle duality and the classical approximations of quantum physics by classical mechanics.

Decoherence, how are measurements physically proceeded (in principle)

There is no physical law that say whether and when the state of systems happen to be measured (observed), but for all practical terms of how things appears, there is a condition specifying the time when the hypothesis that measurement happened starts to be "possible" (but not necessary). This condition is decoherence.
Decoherence is NOT an interpretation, but it is an effective (though only emergent) physical property that can be deduced from quantum theory disregarding the choice of interpretation.

Its precise definition is :
A system S is said to have decohered with respect to a possible measurement M, if there will be no more difference on the probabilities of any future possible measurement of S, whether or not the wave-function of S is assumed to be now already collapsed with respect to M. Therefore in all practical terms, the indetermination of the state of the system has been collapsed from a quantum indetermination to a classical one, to be represented by an element of the (n-1)-simplex (rather than the quantum n-states shape) whose ends have irreversibly become clearly distinguishable by measurements, at the expense of any other direction of measurement.

In other words, a decoherence is NOT a spontaneous collapse, but it is the description of the circumstances where the question whether a collapse happened or not, becomes unverifiable, so that the "already collapsed" hypothesis becomes compatible with the predictions of quantum theory on future measurement results (while a collapse before decoherence would violate the predictions of quantum theory on future measurements).

However, this property of decoherence is an emergent property that only makes sense as a limit property of large systems instead of elementary ones, because it depends on which future measurements can remain possible or not in practice, and this is a fuzzy condition. It is not exactly an internal change, but an external irreversible loss of future opportunities to make measurements capable of deducing the past characters of the system expressed by components of the wave-function that an hypothetical present collapse would destroy.

In practice, decoherence happens as soon as (but not only if) a measurement has been "physically processed", in the sense that we have a macroscopic delivery of the measured result, that is, when the information of the result is "out of the box" with many copies of this information escaping in the environment, for example by radiation or gravitation, so that it cannot be anymore securely hidden by any further operation.

Let us now explain how the laws of quantum physics allow measurements to be proceeded, finally reducing quantum states into classical probabilistic states at a macroscopic level.

The interaction of a 2-states system we want to observe with a measurement apparatus, will end up to produce correlated states of the system with the apparatus. 

We described above the sphere of correlated states between (↑,↓ ) and (↓,↑).
By just rotating one spin, we get a similar sphere of correlated states between (↓,↓ ) and (↑,↑), (and similarly to adapt to the chosen direction of measurement).

Assuming that the measurement apparatus was initially in a known pure state, the sphere of initial states of the system, will evolve into such a sphere of correlations, with ↓ evolving into (↓,↓ ), and ↑ into (↑,↑) (where one component represents the state of the measurement apparatus, and the other represents the state of the system after measurement).

This would operate the complete observation of the system that would collapse it into the perceived state... if we could observe the state of the measurement apparatus.
So, how to do it ? The advantage of the measurement apparatus, will be that it will let its state appear macroscopically, which means that it will will make many copies of its state in the same way. Such copies are faithful for copying the wanted classical bit of information: whatever the state we can have in the sphere of states between (↓,↓ ) and (↑,↑), if ever the first component is measured in the intended direction (↑ or ↓), then the possible result ↑ for one copy will collapse the other copy to ↑ too, while the other possible result ↓ on one copy will collapse the other copy to ↓ too. And their respective probabilities properly reflect the wanted observation.

Moreover, the mere fact of losing one of the copies away in the environment, suffices to collapse the sphere of possible initial states, by projecting it (orthogonally) onto its diameter, which represents the segment of classical probabilistic states between the 2 possible results we wanted to measure.

The cases of weak measurements, can be obtained by some other ways of mapping the sphere of the object's initial state, into a correlated state with the measurement apparatus. This is completed by the same exact copying procedure for the obtained bit of information as in the exact case.

Next :
Interpretation of quantum physics.
The nature of entropy and how it is created.


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