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Introduction to quantum physics: states, correlations and measurements

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Entropy in information theory
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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.

Other people also reacted to this article:

Here is a copy of my reply to his articles (replacing "you" by "him"), which I consider interesting to publish for pedagogical reasons, as these remarks may help other students learning about decoherence and the measurement problem, who may be tempted with the same misunderstandings.

He wrote, after equation (4):

"His misunderstanding here is his assumption that a clear "fundamental" separation exists between what is "fundamental" and what is only "for current technologies". By definition, a measurement is a measurement if the measuring device is macroscopic, that is, it is made of a very large number of atoms. But, roughly speaking, the difficulty to "reverse" the measurement that was made so as to measure the interference, is (something like) exponentially difficult with respect to the size of the device, and also as time passes and decoherence happens.For current technologies it would be very difficult indeed to observe interference with a (near) macroscopic device. Fundamentally speaking, however, equation (3) does permit the observation of interference. Hence we cannot disregard the superposition in a fundamental description, as future technologies may bring it within experimental grasp."

Formally, the "measurement of the interference" is a measurement that does not commute with the initial measurement, so that it is a logical contradiction to claim that both happened. More precisely, this means that his hypothetical "future technologies to measure the interference" are actually undoing the first measurement, which means that this first measurement has to be understood as never having taken place at all.

However, the thing is that, while this "undoing" of the measurement does not seem "fundamentally impossible" when looking at the fundamental equations, it quickly turns out to be completely impossible by any conceivable technologies when he puts the problem in context. And the obstacle is the same that explains the time arrow of thermodynamics, that is the irreversibility of entropy creation.

A naive idea in trying to reverse the measurement process (or the entropy creation process) at an elementary level, would be to "bounce it back", by a sort of "mirror". This is strictly impossible because, depending on how "try to make it":

- Reversing the state to let further natural evolution bring it backwards ? This reversion would have to be an antilinear transformation

http://en.wikipedia.org/wiki/Antilinear_map

which is fundamentally impossible since all evolution has to be linear

- Keeping the state as such but changing the sign of the Hamiltonian so as to reverse the evolution ? We can change the potential energy function in different ways but we cannot change the sign of the kinetic energy function.

The only remaining "solution" is to completely isolate the system and "wait a long time" until its natural evolution comes back to a configuration making the wanted new measurement practically possible again. But this is impossible in practice for the following reasons:

One is that we need to use huge supercomputers in order to predict exactly how long we need to make this very long wait before making the new measurement that can successfully measure the thing we wanted to measure.

The other reason is that the isolation of the system has to remain perfect during this very long wait. And the average needed time of wait is an exponential function of the size (number of atoms...) of the isolated system. But in practice, totally isolating a big system for a very long time is completely impossible because there are too many interactions with the environment, which makes bigger and bigger the size of the total system (including the affected environment) where the information is dispersed and that we would have to isolate again for an even much longer time.

These interactions can be : any thermic contact ; any exchange of a photon by thermic radiation ; and finally, any exchange of a graviton, which we obviously cannot bounce back to remain confined inside the limits of the system, by whatever "mirror for gravitons" we may wish to think about.

This irreversibility can be compared to the event of falling into a black hole : the horizon after which nothing can come back is not locally well-defined, however from a global viewpoint the fall is irreversible.

After this he wrote

"To be short : yes we can know it, and for this we do not need to appeal to another measuring apparatus. We know it because the laws of physics are such that the risks of perturbations are insignificant when applied to the shape of the first measuring apparatus, which has been precisely designed as to be robust in this way, otherwise it would not be considered a measuring apparatus at all.Can we know if the observed pointer reading accurately represents the “system value” or was it perturbed (perhaps even strongly) by the environment?Clearly, one cannot know without an appeal to another measuring apparatus, for which the same difficulty may arise"

"Phase shifts do not matter here.It is obvious to every experimentalist that there is a great difficulty in building a device which acts on a system to produce perfect (or even near perfect) correlation between states without introducing uncontrollable phase shifts"

(7)What matters is that |

With his analogy of the boat and the wave, the point is that the boat is much bigger than the waves: the correct analogy is that we have a 100-meters long boat reflecting centimeter-size waves. Even if we have only one boat and a thousand waves, the waves will be reflected by the boat in a way that depends on the position and orientation of the boat, but these position and orientation will not be significantly affected by the waves, but can be reliably measured again by new waves, to be clearly distinguished from completely different positions and orientations of the boat.

After this I directly skipped to his section about the Bohm interpretation. There again he has a misunderstanding : while the "position of the electron" appears fixed in an hydrogen atom, this does not constitute a dipole because in the Bohm interpretation, the diplole is not defined by the "position" of the charge but by quantum properties of its guiding wave, so that the effective predictions remain identical to those of orthodox quantum mechanics.

Now in the section on "consistent histories". Again his criticism is wrong. Because he refers maybe to an expression of "consistent histories" that did not make use of decoherence.

"the set of histories such that the particle has a well defined positionNo, they are not valid realms because they do not satisfy the condition of decoherence. To satisfy the condition of decoherence, a realm has to correspond to a physical evolution that includes a measurement process of the considered observables, in other words, happens in a context where decoherence makes the observable accessible in a classical rather than quantum manner.x_{0}att_{0}forms a realm. In a like manner, one may show that the set of histories such that the particle has a well defined momentump_{0}att_{0}also forms a realm."

Any two decoherent realms will then be compatible in the sense that their observables commute, unlike the examples he gave.

"There are 2 possibilities: either the system is closed and the experimenter does not measure anything in between. Then the realm is only a choice of interpretation of what happens, no matter what happens.Let us demonstrate that the probability for a given realm is necessary for consistency and argue that it is an obvious quantity to be deduced from experiment"

Or the experimenter makes some measurements in between, but these very acts of physical measurements are themself disturbing and changing the course of evolution, forcing the inclusion of his observables in the realm of evolution of what happens (now in the presence of these measurements), thus cannot constitute any measure of adequacy of a realm describing what would happen in the absence of these measurements.

"No. He imagines a problem where there is absolutely no problem. Because the situation is exactly the same as in the following sentence:Firstly, the consistent histories formalism deems that it must follow a particular history (in some realm). Secondly, there is nothing outside the system to enact realm selection. It is straightforward to see that there are only two ways to guarantee the consistency of these two statements"

"This vector has particular values of its coordinates (in some coordinates system). But there is nothing outside the vector to decide the selection of the coordinates system".

There is no a paradox here. At least not mathematically. Hum ok there is a metaphysical dimension here that makes things different, but empirically if we don't include a conscious observer in the system but only look at the end result or in the many-worlds interpretation, there would be no problem.

"(8)A good reason why the state of the environment cannot be measured is that it is very quickly lost as entropy.

The answer most often cited is that of environmental ignorance. Namely, the environment is not measured or the experimenter lacks control over the environment.

The mention of “measurement” or “ignorance” in an argument for the trace procedure is unacceptable."

"The adjective "unitary", and the question whether an evolution is unitary or not, does not make sense in the absolute but only relatively to the Hilbert space of the system that he is considering. The problem, what makes the evolution non-unitary, is that we are not considering the same system along the way, so that the evolution is not taking place inside a fixed Hilbert space. First we consider a system (or measurement apparatus), then we introduce the environment, then the system interacts with the environment, then the environment leaves and we consider the state of the system separately from the environment.the assumption of unitary evolution of closed systems is incompatible with any nonunitary evolution whatever"

"Precisely not : by his oversimplification he loses the necessary concepts to understand how things happen, and this is one of the causes creating the troubles he complains of.It is evident that a realistic environment has a Hilbert space with a large number of dimensions; however, insight can be had by considering much simplified models."

"The initial state of the environment is not a pure state but a very mixed state made of an astronomical number of components (according to its entropy). Among these components he can find theIn particular, we have introduced an environment whose approximatebase may be taken to be..."

"Not even : (6) remains true (with different final states of the environment) when replacingOf particular note is that the initial environment state cannot involve a superposition of (...), because such an initial superposition immediately spoils the interpretation of (6),"

Then I looked at his other article : "Structural Instability and Quantum Lying in the Many-worlds (Relative State) Interpretation" I only had a very short look at his article, but my guess (unless he has anything deeper that how it seems) is that this is simply a naive mistake he is making: while "theoretically possible", these "quantum liar states" are effectively impossible for thermodynamic reasons, that is, for the exact same reason that it is impossible for thermodynamically irreversible processes to be reversed.

More related pages

Quantum Entropy

The interpretation problem of quantum physics

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