Problem of quantum measurement

The problem of quantum measurement was formalized for the first time by John von Neumann in 1932 in its book “the mathematical bases of quantum mechanics” (chapter VI). It, in 1935, was popularized thereafter by Erwin Schrödinger with its famous “paradox of the cat”.

Since, this problem was the subject of many debates and remains still today the object of polemics, even if solutions were established and are accepted by a majority (but not totality) of the physicists.

Exposed problem

The problem of measurement consists in fact in a whole of problems, which highlight difficulties of correlation between the Postulats of quantum mechanics and the macroscopic world such as it appears to us or such as it is measured.

These problems are:

• PMQ1 : Is the evolution of the function of causal and deterministic wave being (postulate 6), and representing all recognizable information on a system (postulate 1), why the result of a quantum measurement basically indeterminist (postulate 4 and postulate 5)?

• PMQ2 : Can the evolution of the function of wave being linear and unit (postulate 6), how the quantum superpositions disappear (postulate 5), whereas the linearity/unitarity leads naturally to a safeguarding of the superimposed states?

Even if these two problems are dependant, it is important to distinguish them because certain solutions as the Décohérence bring a response to PMQ2, but not to PMQ1.

Development and explanation of the problems

Postulate 5 can be seen like mathematically and logically soft with postulate 6. Indeed, according to postulates 1 and 6, the physical status and its evolution are entirely and completely describes by a vector and its evolution by the equation of Schrödinger. The postulate 5 (which also describes a certain “evolution” of the function of wave) thus does not have logically a raison d'être and it should normally be “contained” in a way hidden in postulate 6 if postulate 1 is correct.

However one sees badly, a priori, how postulate 5 could result from postulate 6 (and it is well the reason for which there exist two different postulates):

• According to John von Neumann, the unit evolution of the function of wave described by postulate 6 is thermodynamically reversible , whereas the collapse of the function of wave described by postulate 5 is thermodynamically irreversible . One can reformulate this remark by saying that the unit evolution is causal, or deterministic, whereas the collapse of the function of wave is acausale and indeterminist (PMQ1).

• Another problem, major, involved in PMQ1 is that the result of a measurement is single whereas the function of wave describes a multiple reality, and does not lead - mathematically and physically - to a single reality. Here still, there are conflict and inconsistency between postulate 5 and postulate 6. It is the “problem of unicity” (Roland Omnès) or the problem of “or-and” (John Bell).

• One can as notice as the evolution of the function of wave is continuous basically , whereas the collapse of the function of wave is discontinuous (related to PMQ2).

• Enfin, and especially, the evolution described in postulate 6 is unit , i.e. it preserves the standard and thus the scalar product. However, the process describes by postulate 5 is basically not-unit, and does not preserve the scalar product, since there is projection . At a certain time, the probability (which - postulate 4 - is given by a scalar product) of being in a certain state is with a certain value 0 < P < 1, and at the next moment (according to postulate 5) this probability becomes either 1, or 0 (PMQ2).

It raises also the question, compared to this problem, of knowing when (or on which criteria) to employ postulate 5 rather than postulate 6 (or conversely) to treat evolution of a system. There does not exist mathematical formal criterion to know if it is necessary, vis-a-vis a certain quantum system, to rather employ one or the other to deal with its evolution.

This last question is of an major importance with regard to the quantum Informatique since the latter rests on the control of the evolution of a quantum system, and application of postulate 5 (to obtain the result of a quantum algorithm) compared to the postulate 6 (which controls the mechanism of the quantum algorithm itself). There is thus, compared to this problem, of the new brief replies to await development of this discipline.

Solutions

These considerations led many physicists to call into question either postulate 1 (theories to nonlocal hidden variables, intervention of the conscience), or postulate 5 (multiple universes, décohérence). The only solution not calling into question no postulate is the Interprétation of Copenhagen.

There still does not exist, nowadays, of solution unanimously recognized by the community of the physicists, even if some are accepted than of others.

Positivist approaches: quantum mechanics is not supposed to describe reality

This approach rests on the observation that there is true problem only if it is considered that the Postulats of quantum mechanics have some ontology, and describe (at least partially) reality. If one takes the position which the postulates do not describe reality in itself , but what we can pragmatically know on it, then these problems becomes deprived of direction because these problems then do not relate to any more reality in itself, but axiomatic which is “such as it is” and which does not have to justify its inconsistencies as long as it gives results who are, for whatever purpose it may serve, correct. Even less has it to justify lack of coherence between its formalism and our preconceived ideas on it what the world should resemble. Like W.H. Zurek writes it: " The only failure of quantum mechanics is not to have been able to agree with our préjugés".

Besides this pragmatic approach and positivist constitute the gasoline of the Interprétation of Copenhagen of the quantum physics.

Interpretation of Copenhagen

See also: Interpretation of Copenhagen

This approach rests on the conviction that quantum mechanics is only one description of all that we can know of reality, but does not describe reality in itself.

Moreover, being of obedience positivist, this approach denies that the concept of “reality” have a scientific direction, and any reasoning or scientific problem of nature compared to a “reality” is excluded.

Stephen Hawking summarizes this approach well, with its direction of the formula: “ I do not ask that a theory correspond to reality, because I do not know what is reality. It is not something which one can test with paper pH. All that is essential to me is that the theory correctly envisages the result of an experiment. ”.

This approach constitutes, still nowadays, the “official solution” of the problem of measurement, and remains a majority opinion among the physicists, even if a growing number of those grant interest to the theory of the Décohérence.

Physicians representative of this approche

Niels Bohr (at the origin of the interpretation of Copenhagen), Stephen Hawking

Other approaches positivists

Since the interpretation of Copenhagen, the current positivist continued to comment on the problem of measurement, by taking account of modern epistemology. Apart from the physicists, the epistemologists and the philosophers it are seized this problem.

Running wittgensteinien

This current tries to show that the problem of measurement is a problem due to the inaccuracy of the language in which on the one hand the theoretical quantum one is formulated and on the other hand the paradox itself.

• Example 1 Low van Fraassen (Mechanics Quantum, Year Empiricist View):

1 phrases: “After the preparation of Schrödinger, the cat is in the state : $\{(|living\; \backslash \; rangle+|died\; \backslash \; rangle)\}/\backslash \; sqrt\; \{2\}$”

Phrases 2: “It is found in experiments that the cat is either in the “alive” state or in the “dead” state ”.

Without the common word state the two sentences would not be contradicted.

In sentence 1, it is about a state dynamique= “How the system would evolve/move if he were insulated”

In sentence 2, it is about a state of valeur= “Which observable has a value, and which is this value”

• Example 2:

Is “Which the solution of the problem of measurement? I say that it is this one: when one measures X with clean states $|X\_i\; \backslash \; rangle$, the $x\_i$ result is observed with the probability: $|\backslash langle\backslash psi|X\_i\; \backslash \; rangle|^2$, where $|\backslash \; psi\; \backslash \; rangle$ is the initial state. It is it with what return we from there, and that will also be appropriate like starting point ” (S. Saunders, 1994)

Negation of postulate 5

Certain positivists follow the same track as the theory of the décohérence (see low) to solve PMQ2, by denying the postulate 5. This position is in any case that of the most consequent positivists and more “hard” (the standard positivists only prescribe to use the postulate 5 like a useful “receipt”). Bas van Fraassen writes as follows: “Neither the postulate of projection nor no other principle of interpretation are necessary to explain the repeatability”. Useless “to reduce” the vector of state to take account of the information acquired at the time of a sequence of former experiments. Useless “to reduce it in particular” to explain why the probability of immediately repeating the same result in the event of the second identical measurement after is equal to 1. The “dynamic state” does not have to be modified brutally; it evolves/moves, it intricate, but it forever does not have to be “reduced”. Roland Omnès, which is however far from being a positivist, showed that one can perfectly do without the postulate 5. According to its result, it is not essential to follow the usual sequence: (A) preparation, (b) definition of a vector of state, (c) the first measurement, (d) reduction of the vector of state, then (E) calculus the probability for the result of a new measurement starting from the vector of reduced state. In the place of that, one can calculate, directly starting from the vector of initial state, the conditional probability to obtain a result during the second measurement if such result were obtained for the first measurement. This result comes in support of the sentence from Saunders which suggests that all which one needs “to interpret” the quantum mechanics in the most economic possible way is to apply the postulate 4 extensively of evaluation of the probabilities.

This approach, just as the theory of the décohérence, does not answer entirely PMQ1.

Éléments in favor of the approach positiviste

• Corresponds to the latent epistemology of part of the community of the physicists (see Hawking): a scientific theory must (almost by definition, or at least in good practice) be of nature positivist: to limit themselves to the predictions and to avoid the ontological aspects .

• cannot be contradicted, as long as the results of measurement are in agreement with the theory.

Éléments in discredit of the approach positiviste

• This approach does not answer the problem of measurement, but declares that this problem does not have a scientific direction. That can be regarded as unsatisfactory.

• “ It acts of a dissolution and not of a solution of the problem of quantum measurement ” (Michel Bitbol)

The question of knowing if this is an element unfavourable or favorable to interpretation positivist remains however open. After all, it can be useful, with certain stages of the development of science, to show that a problem is badly posed or does not exist. It is enough to think of Galileo (Galileo Galilei), which rightly considered that it is not necessary to wonder, between two mobiles in uniform translation one compared to the other, which is “really” at motionless rest and which “really”. At the time, the aristotelicians reproached him for not wanting to answer “a true” scientific question; but Galileo stuck to his position, according to which its principle of relativity removes any direction with the question of the “real” movement.

• the majority of the approaches positivists do not approach a face, or overlook, the problem of the mathematical inconsistency between postulate 6 and postulate 5, whereas it is about a purely mathematical problem, and does not depend on any ontology or terminology.
• the others try to solve the problem of the inconsistency by denying postulate 5, but leave PMQ1 not entirely solved.

Realistic approaches

Contrary to the approach positivist, a certain number of physicists think that the postulates of quantum mechanics say something in connection with physical reality to us and thus seek the coherence and the direction of the postulates, and their adequacy with reality itself.

Quantum mechanics describes reality completely

These approaches rest on the conviction that postulates 1 and 6 are exact, i.e. reality is entirely determined by a vector of state, whose evolution is governed by the equation of Schrödinger.

Postulate 5 then either is denied, or deduced from postulate 6.

Multiple worlds or “theory of the relative states”

See also: Theory of Everett

This approach, initiated by Hugh Everett in 1957, takes the party to consider that all reality is described by postulate 6, and stipulates that postulate 5 is only a illusion .

That means that, when a quantum measurement can give several different results, the whole of the superpositions of all the possible values of measurement coexist in a Multivers, but we would be aware only one of only one possibility because our conscience (which is by assumption, in this theory, a purely physical phenomenon) is found quantiquement intricate with one and only one result of measurement.

In other words, $is\; a\; quantum\; state|\backslash \; psi\; \backslash \; rangle$ to be measured by a Observable, broke up into a together complete of orthogonal projectors $E\_i$.

According to postulate 5, the quantum state $|\backslash \; psi\; \backslash \; rangle$ evolves/moves in a state (by chance given) $E\_i|\backslash \; psi\; \backslash \; rangle$ after a measurement by this observable.

According to the theory of the multiple worlds, the quantum state $|\backslash \; psi\; \backslash \; rangle$, after the very measurement observable one, evolves/moves in:

$E\_1|\backslash psi\backslash rangle|State\; \backslash \; of\; \backslash \; conscience\_1\; \backslash \; rangle\; +\; E\_2|\backslash psi\backslash rangle|State\; \backslash \; of\; \backslash \; conscience\_2\; \backslash \; rangle\; +\dots \; +\; E\_n|\backslash psi\backslash rangle|State\; \backslash \; of\; \backslash \; conscience\_n\; \backslash \; rangle$

The observer (and the intricate part of the universe with him) “are thus divided”, to each time a quantum measurement can give several different results.

It is thus impossible, for a given state of consciousness, to perceive the whole of the states superimposed, however real according to this theory. That gives a direction to the postulate 5 which then does not describe reality, but an illusion due to our conscience.

PMQ1 and PMQ2 are thus explained: the random and discontinuous aspect (PMQ1) of the evolution of the function of wave, as well as the rupture of linearity and unitarity (PMQ2) is only one misleading appearance and does not exist on the level of the multivers.

NB: It is important to note that, although utilizing the concept of conscience, this approach is to be distinguished clearly from the other approaches utilizing the conscience. Indeed, in the theory of the multiple worlds, the conscience is a physical phenomenon which enters entirely within the framework of postulate 6 . In the other approaches utilizing the conscience, this one is apart from the quantum laws, or is described by modified quantum physics. One could just as easily speak “about state of the measuring device” rather than “of state of consciousness”.

Éléments in favor of this approche

• This approach has the merit to as well bring a clear answer to PMQ 1 qu ' to PMQ2.

• Moreover, it agrees well (and a ontology brings) to the Intégrale of way of Richard Feynman.
• also Brings an answer to the problem of the contrafactuality.

Éléments in discredit of this approche

• does not bring a clear answer on the process of measurement in itself:

• From which moment, and on which criteria, the projectors in work and the universes (and consciences) are they subdivided-T enter?
• does not bring an explanation to postulate 4 (rule of probability theory)
• the number inconceivably large (even, according to of Espagnat an infinity continues) parallel universe which this theory implies. Is the universe truly at this “expensive” point?
• That to think of the multitude “of alter ego” which in this theory each conscious individual has?

Physicians representative of this approche

Hugh Everett, David Deutsch, John Wheeler, DeWitt & Graham

Décohérence

See also: Décohérence

This approach aims at showing that postulate 5 is a consequence of postulate 6, although these two postulates seem a priori incompatible. The idea of this approach is that, if a quantum system cannot be perfectly insulated, then its interaction with its environment involves necessarily the disappearance of the quantum superpositions.

In other words, according to this approach, reality indeed is described only and entirely by postulate 6, but the fact that a system quantum can never be “pure” and décorélé of its environment involves the need for postulate 5.

The décohérence is modelled by using the formalism of the Matrice density. One can show whereas the matrix density corresponding to a quantum system tends very quickly to becoming diagonal when this quantum system is put in interaction with a “environment”. The fact that the matrix density becomes “diagonal” means that the superimposed states (particle in two states at the same time for example) tend to disappear. One found oneself then in a situation very close to that stipulated by postulate 5.

It is important to note that postulate 5 strictly and rigorously is not shown and is not deduced by this theory. The differences compared to postulate 5 are:

• the not-diagonal elements tend towards zero but never become strictly null (what is logical, because all that is deduced from the postulate 6, which mathematically does not allow the total disappearance of the superimposed states, because of the linearity and the unitarity).
• collapse does not occur instantaneously as postulate 5 stipulates it, but in an extremely short time.
• and especially, the state which one obtains after a measurement is not a state chosen among several possible states as postulate 5 stipulates it, but rather a coexistence several states without correlation (problem of “or-and” John Bell)

Consequently, postulate 5 is seen by this theory only as one (excellent) approximation from what actually occurs during a process of measurement quantum.

This approach thus brings an answer to PMQ2: the superimposed states tend to disappear (become undetectable), and one can detect in practice only the quantum states having a defined value, from where postulate 5.

On the other hand, this approach leaves unexplained PMQ1. Here a quotation of Erich Joos on this subject: “ does the décohérence solve the problem of measurement? Certainly not. With a stage or with another we will have to still observe the usual probabilistic rules of the quantum theory. They for example are hidden in the matrices density ”

Éléments in favor of this approche

• Theory sufficiently formalized to be able to make testable predictions by the experiment.

• a growing number of experiments which show an good agreement between the theory and the results of measurement.

Éléments in discredit of this approche

• does not say anything on PMQ1

• does not reproduce postulate 5 completely
• Problème of “or-and” John Bell

Physicians representative of this approche

Hans Dieter Zeh, Wojciech Hubert Zurek, Erich Joos, Serge Haroche

Consistent stories

This approach was proposed by Robert B. Griffith in 1984, and then was taken again and developed by Roland Omnès 1987 and Murray Freezing-Mann in 1990.

It consists in modelling the evolution of a quantum system by a “consistent history”. A history is a sequence of vectorial subspaces $F\_1.\; ,\; F\_n$ (which, point out it, according to postulate 1, represent each one a quantum state of the system), at times $t\_1.\; ,\; t\_n$.

Times $t\_1.\; ,\; t\_n$ is not unspecified, but is characterized by a particular event, or changes of properties of the system , according to the experiment carried out and of the system described. At each time $t\_i$ a Observable $A\_i$ is associated which itself breaks up into a together complete of orthogonal projectors $E\_j$.

At each time $t\_i$, observable associated the $A\_i$ subdivides the history in progress out of N different stories, N being the number of orthogonal projectors of the observable one. For example, starting from a state (a vectorial subspace) $F\_1$ at time $t\_1$, one has N subspaces F_2.1, F_2.2. , F_2.n at time $t\_2$ etc One thus has then a tree of history which ramifies at each time T.

A history thus consists in following a way in this tree, by selecting at each time T a subspace among all those possible.

Among all these stories, all these ways, some are described as consistent , if they satisfy certain conditions. These conditions express primarily that, whatever under spaces $(F\_i,\; F\_j)$ taken in a history, the states corresponding are without quantum interferences, i.e. are excluded mutually. They are only the stories retained in calculations, the others are regarded as “unreal”.

This model makes it possible to find the rules of probability calculus described by postulate 4, and to make certain checked experimental forecasts. That makes it possible to justify that the soft stories are indeed unreal. Under these conditions, that makes it possible to bring an answer to PMQ2: the superimposed states are unreal, and like one does not leave postulate 6 to arrive at this conclusion (but a model), it does not have there contradiction with postulate 6, which also makes it possible to answer to PMQ1.

Physicians representative of this approche

Robert B. Griffith, Roland Omnès, Murray Freezing-Mann, Jim Hartle

Quantum mechanics does not describe reality completely and must be modified (or replaced)

The solutions seen up to now are based, entirely or partially, on the postulates of quantum mechanics which are regarded as exact. The solutions of this chapter, on the contrary, consider that true solutions with the problem of measurement cannot be brought that while calling more or less basically into question these postulates.

Pilot wave De Broglie/Bohm

This approach was imagined initially by the French physicist Louis de Broglie to solve the problem of the duality wave/corpuscle. The basic idea is that quantum reality consists of two basic components: a wave, known as Pilot wave (without material substrate), and corpuscles themselves. The wave is governed according to the equation of Schrödinger. The corpuscles are judicious being “guided” by this wave and would all the more be likely to follow a certain direction in the space which the wave has a module raised in this area. The physical nature of the pilot wave is not clarified: she is regarded as the demonstration of hidden variables, nonlocal.

This approach, given up by Broglie initially, was improved by David Bohm in 1952 until being able to reproduce, qualitatively and quantitatively, all predictions of standard quantum mechanics.

In this formalism, there does not exist quantum superposition: a particle is, at every moment, with a given position of space and, overall, in one and only one given state.

This theory thus explains very well the problem of measurement because, for PMQ1, the indeterminism is generated by the hidden variables, and in this theory there does not exist quantum superposition, therefore PMQ2 is without object and is a non-problem presented by a false formalism.

On the other hand, this theory could not be reconciled completely with relativity and is not completely covariant (i.e. its laws are not completely expressed same manner in all the reference frames).

Éléments in favor of this approche

• simple and clear Explanation of PMQ1 and PMQ2.

• absolutely Reproduces all the predictions of standard quantum mechanics.

Éléments in discredit of this approche

• No additional predictions compared to standard quantum mechanics.

• Natural physics of the not very clear pilot wave.
• Problems with restricted relativity.

Physicians representative of this approche

Louis de Broglie, David Joseph Bohm, John Stewart Beautiful

Intervention of the conscience

This interpretation leaves the report which the problem of measurement does not exist that if there exist conscious individuals to take note of the result of a measurement. Indeed, as long as there is not conscience result of a measurement on a system (for example the opening of the box containing the Chat of Schrödinger), there is absolutely nothing which brings to think that the system is not, actually, in a superimposed state that postulate 6 implies. From where the affinity which seems to exist, with the eyes of holding of this theory, between the conscience and postulate 5.

For those, the conscience is a phenomenon apart from the physics and which escapes description by quantum mechanics , and it is it which causes the collapse of the function of wave describes by postulate 5. Consequently, the mathematical inconsistency and logic between postulates 5 and 6 are included/understood, because it is only the reflection of the opposition between a physical world and a nonphysical world. It is thus a direct answer to PMQ2, as with PMQ1 because it is the conscience which brings to the ingredient “indeterminist” of quantum measurement.

But if she answers the problem of measurement, this approach opens other questions which are perhaps not easier to solve. Which physics is not” of the universe this component “? At which time the Earth it did pass from a state superimposed to a defined state, with the appearance of the first conscious being? What is it experiments which show that the Earth had a well defined state (composition of atmosphere etc) before the appearance of the first Man? etc

Éléments in favor of this approche

• Answers PMQ1 and PMQ2

• Explique the affinity which seems to exist between the conscience and the problem of measurement.

Éléments in discredit of this approche

• Introduces a component “not physics”, which opens the door with scientistic or mystical drifts. was

• Which the state of the universe when it did not have nobody there to observe it before the appearance of the life?

Physicians representative of this approche

John Von Neumann, Eugene Wigner, Fritz London & Edmond Bauer

Reduction of the package of objective wave

This approach tackles the problem in the most frontal way and most direct: as postulate 5 is (according to the partisans of this approach) irremediably incompatible with postulate 6, one must deduce from it that there exist still unknown physical phenomena which objectively cause (i.e. without intervention of a conscience which would be except physics) the collapse of the function of wave. This approach thus utilizes of the “hidden variables”, nonlocal, which would be responsible for postulate 5.

This approach thus adds terms additional, nonlinear, with the equation of Schrödinger in order to find the results of postulate 5. These terms represent variable physical phenomena according to the authors:

• Ghirardi/Rimini/Weber (GRW) proposes that the equation of Schrödinger is disturbed by chance, all the 100 million years approximately (for a given particle), by a phenomenon which causes to multiply the function of wave by Gaussian located, causing the collapse of the function of wave.
• Penrose thinks that the Gravitation can influence the evolution of the equation of Schrödinger.

These approaches share a certain number of disadvantages.

Initially, as the approach of Bohm, the non-linear terms associated with hidden variables have evil to be relativistement covariants. However recent work (e.g. has Relativistic Version off the Model Ghirardi-Rimini-Weber) tends to overcome these difficulties.

Then, one can reproach, for these theories, an approach ad hoc (i.e. the theory is invented according to the awaited results, and nonfounded on an existing theory which would lead naturally to these results).

Up to now, one reproached these theories for being not-refutable, but there still of recent progress proposals for experiments bring (e.g. Towards quantum superpositions off has mirror).

On the other hand, with these approaches, PMQ1 and PMQ2 obtain clear and direct answers. The ingredient “not-determinist” is brought by the hidden variables. And the non-linearity of postulate 5 comes from the non-linearity of the additional terms.

Éléments in favor of this approche

• Explains in a simple way and direct PMQ1 and PMQ2.

• Approach among more “reasonable”, in the direction where it do not suppose extraordinary elements like parallel universes or a conscience except physics.
• Refutable for the most advanced versions.

Éléments in discredit of this approche

• Problems with restricted relativity.

• Problems with the conservation of energy.
• Rests on not yet shown assumptions ad hoc or physical assumptions.

Physicians representative of this approche

Roger Penrose, Ghirardi/Rimini/Weber

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