Effective Dynamics of Open Quantum Systems: Stochastic vs Unitary Models

[A. Neumaier]

Are you saying that quantum dynamics cannot describe this "jump", but that it necessarily have to be described by classical physics or something outside of any model/theory?

Not quite. But it necessarily has to be described by a different quantum model than unitary dynamics if it is an open system and the rest of the universe is not explicitly modeled.

For convenience, physicists often want to describe a small quantum system in terms of only its Hilbert space, when it is in reality not isolated but coupled to a detector (and hence should be described by a unitary deterministic dynamics in a much bigger Hilbert space). This necessarily leads to an effective description of the dynamics of the state of a a small quantum system alone.

Even if the full dynamics of the state of system+detector is deterministic and unitary, the effective dynamics of the state of the system alone is stochastic and nonunitary (dissipative). It can be given by a classical stochastic process for the state vector of the small system. The form of this stochastic process can be derived by refinements of traditional techniques in quantum statistical mechanics. Depending on the kind of coupling to the detector, the effective dynamics is in certain cases a classical jump process described by a master equation, and in other cases a classical diffusion process described by a Fokker-Planck equation, and in general by a combination of both and a deterministic drift term. Here classical refers to the form of the stochastic description - it is still quantum in the sense that the dynamics of a state vector (ray in Hilbert space) is described.

This means that at any fixed time the system is described by a state vector which changes stochastically with time. In a jump process, the state vector changes at random times to another state vector (generalizing the classical dynamics of a Markov chain), and the trajectories are formed by piecewise constant state vectors. In a diffusion process it satisfies instead a Fokker Planck equation (generalizing the classical dynamics of Brownian motion), and the trajectories form Hoelder continuous paths of state vectors, with exponent 1/2.

If a drift term is present, the diffusion process changes to a process described by a stochastic differential equation with a noise term, and the jump process changes to the von Neumann picture of quantum dynamics - namely continuous unitary dynamics interrupted by discontinuous jumps. The jumps are in general governed by POVM probabilities and states.

In special cases the jumps are governed by Born probabilities and eigenstates. Thus von Neumann's dynamics with unitary dynamics interrupted by jumps defined by nonunitary collapse is the correct effective description of the dynamics of certain open systems.

In particular, the stochastic process assigns a trajectory of state vectors to each particular realization of the process, and hence to each single system.

See Section 7 of http://arxiv.org/abs/1511.01069 for a summary how the state of single atoms can be continuously monitored and shows jump of diffusion properties depending on the kind of measurement it is subjected to.

A formal description of the technical side of the reduction process that produces the reduced quantum jump description from the unitary dynamics is given in the references discussed in post #28 below.

These references justify the collapse as an instantaneous approximation on the system-only level to what happens in an interaction with an appropriate measurement device on the system+detector level.

 

[kith]

This means that at any fixed time the system is described by a state vector which changes stochastically with time.

I'm only familiar with the Lindblad equation / quantum optical master equations. There, an initial pure state gets mixed over time. Is what you write about stochastically changing state vectors supposed to be a process which is underlying this or is it a process which is incompatible with it?

In a jump process, the state vector changes at random times to another state vector (generalizing the classical dynamics of a Markov chain)

I didn't get from your text whether these jump processes involve an approximation or not. In the usual open quantum systems approach, the Markov property is the result of an approximation (which is justified by the nature of the full system including the environment).

 

[vanhees71]

Not quite. But it necessarily has to be described by a different quantum model than unitary dynamics if it is an open system and the rest of the universe is not explicitly modeled.

For convenience, physicists often want to describe a small quantum system in terms of only its Hilbert space, when it is in reality not isolated but coupled to a detector (and hence should be described by a unitary deterministic dynamics in a much bigger Hilbert space). This necessarily leads to an effective description of the dynamics of the state of a a small quantum system alone.

Even if the full dynamics of the state of system+detector is deterministic and unitary, the effective dynamics of the state of the system alone is stochastic and nonunitary (dissipative). It can be given by a classical stochstic process for the state vector of the small system. The form of this stochastic process can be derived by refinements of traditional techniques in quantum statistical mechanics. Depending on the kind of coupling to the detector, the effective dynamics is in certain cases a classical jump process described by a master equation, and in other cases a classical diffusion process described by a Fokker-Planck equation, and in general by a combination of both and a deterministic drift term. Here classical refers to the form of the stochastic description - it is still quantum in the sense that the dynamics of a state vector (ray in Hilbert space) is described.

This means that at any fixed time the system is described by a state vector which changes stochastically with time. In a jump process, the state vector changes at random times to another state vector (generalizing the classical dynamics of a Markov chain), and the trajectories are formed by piecewise constant state vectors. In a diffusion process it satisfies instead a Fokker Planck equation (generalizing the classical dynamics of Brownian motion), and the trajectories form H"older continuous paths of state vectors, with exponent 1/2.

If a drift term is present, the diffusion process changes to a process described by a stochastic differential equation with a noise term, and the jump process changes to the von Neumann picture of quantum dynamics - namely continuous unitary dynamics interrupted by discontinuous jumps. The jumps are in general governed by POVM probabilities and states.

In special cases the jumps are governed by Born probabilities and eigenstates. Thus von Neumann's dynamics with unitary dynamics interrupted by jumps defined by nonunitary collapse is the correct effective description of the dynamics of certain open systems.

In particular, the stochastic process assigns a trajectory of state vectors to each particular realization of the process, and hence to each single system.

See Section 7 of http://arxiv.org/abs/1511.01069 here it is shown how the state of single atoms can be continuously monitored and shows jump of diffusion properties depending on the kind of measurement it is subjected to.

This justifies the collapse as an instantaneous approximation on the system-only level to what happens in an interaction with an appropriate measurement device on the system+detector level.

I like all this, but that's not collapse! That's the effective description of quantum dynamics of open systems. It's the opposite of introducing a collapse sotosay, i.e., the derivation why the shutup-and-calculate description of the Born probabilities work with real-world measurements by taking the interaction/coupling of the object with/to the measurement apparatus into account (and providing in addition a formalism for more general types of "weak measurements" in terms of the POVMs).

The title of this very interesting and nice paper is thus misleading: It's a very good description of how to get rid of the inconsistencies of the flavors of the Copenhagen interpretation invoking a collapse as a hokus-pokus mechanism outside of (effective) quantum dynamics! It's also clearly shown that "jump" has an effective meaning. In reality nothing jumps, but only on some macroscopic scale it can be a good practical approximation to talk about a "jump".

 

[A. Neumaier]

I'm only familiar with the Lindblad equation / quantum optical master equations. There, an initial pure state gets mixed over time. Is what you write about stochastically changing state vectors supposed to be a process which is underlying this or is it a process which is incompatible with it?

The stochastic process is related to the Lindblad equation roughly the way as a classical stochastic differential equation is related to the Fokker-Planck equation. Roughly only as in the latter case, the descriptions are equivalent for single-time statements (and are the classical analogue of Heisenberg vs. Schroedinger representation). Whreas in the former case, some process information is lost by going to the lattter though the resulting Lindblad equation is an exact consequence. But it ignores the (in principle measurable) classical information that escapes into the environment.

I didn't get from your text whether these jump processes involve an approximation or not. In the usual open quantum systems approach, the Markov property is the result of an approximation (which is justified by the nature of the full system including the environment).

The derivation I talked of is not exact but also assumes the Markov approximation. Otherwise one doesn't get pure differential equations but has additional memory terms. But the derivation of the Lindblad equations - which are heavily used in practice and are usually fully adequate - also needs the Markov approximation.

 

[A. Neumaier]

but that's not collapse! That's the effective description of quantum dynamics of open systems.

Collapse is an effective description of quantum dynamics of open systems! In von Neumann's book it is what happens during the [idealized as infinitely] brief moment where a short flash of polarized light passes the polarizer and changes its polarisation plane, thereby losing part of its intensity.
For low intensity laser light (where only the 0- and 1-particle sector needs to be accounted for) this is a nonunitary change of a superposition $\sin\alpha$|0>+$\cos\alpha|1,\psi$> into one of the two states of definite particle number |0> or |1,$\phi$>, the latter with the Born probability $|\alpha|\cdot|\phi^*\psi|^2$, where $\phi$ is the polarizer plane. Similarly for all other cases where you prefer to talk about subensemble selection within the ensemble interpretation. The collapse is the way Nature achieves the automatic subensemble selection!

And the description in post #1 applies not only to an ensemble but to each individual quantum system - in particular to the single atom etc. in the experiments described in the papers I had referred to in the other post. The ensemble interpretation is no longer the most complete description of what one can say about a quantum system! This is why very distinguished people such as Plenio and Knight wrote their article about quantum jumps mentioned in another post. See post #28 for further details.

 

[atyy]

See Section 7 of http://arxiv.org/abs/1511.01069 here it is shown how the state of single atoms can be continuously monitored and shows jump of diffusion properties depending on the kind of measurement it is subjected to.

Is this just another exposition of his earlier work like http://arxiv.org/abs/1411.2025? In the paper you linked, he suggests the cut is objective, so quantum mechanics will fail at some level. His earlier work certainly does not support the view that one can have a deterministic unitarily evolving wave function of the universe, and nothing else.

 

[A. Neumaier]

In the paper you linked, he suggests the cut is objective, so quantum mechanics will fail at some level.

I was only referring to Section7, where the only reference to a cut (after (52)) is immediately rejected. But he gives no detail anyway; I gave the paper only as a very recent reference to a short summary.

The appropriate reference for the technical part is the paper by Plenio and Knight mentioned in post #5. I don't think that they use a cut. What is used instead is a correlation assumption that introduces a dissipative arrow of time. This is done everywhere in statistical mechanics, even classically, as otherwise it would be impossible to get dynamical information from the statistical approach.

 

[vanhees71]

Collapse is an effective description of quantum dynamics of open systems! In von Neumann's book it is what happens during the [idealized as infinitely] brief moment where a short flash of polarized light passes the polarizer and changes its polarisation plane, thereby losing part of its intensity.
For low intensity laser light (where only the 0- and 1-particle sector needs to be accounted for) this is a nonunitary change of a superposition $\sin\alpha$|0>+$\cos\alpha|1,\psi$> into one of the two states of definite particle number |0> or |1,$\phi$>, the latter with the Born probability $|\alpha|\cdot|\phi^*\psi|^2$, where $\phi$ is the polarizer plane. Similarly for all other cases where you prefer to talk about subensemble selection within the ensemble interpretation. The collapse is the way Nature achieves the automatic subensemble selection!

And the description in post #1 applies not only to an ensemble but to each individual quantum system - in particular to the single atom etc. in the experiments described in the papers I had referred to in the other post. The ensemble interpretation is no longer the most complete description of what one can say about a quantum system! This is why very distinguished people such as Plenio and Knight wrote their article about quantum jumps mentioned in another post.

But that's then quantum dynamics. I'm confused now. I always thought that, if somebody talks about a collapse, he means some (rather vague) process that is not describable by quantum dynamics and leads to the projection of the original pure or mixed state to the corresponding pure eigenstate of the measured observable, i.e., if you measure $A$ and find eigenvalue $a$ of the corresponding $\hat{A}$ then "the quantum state collapses" (instantaneously) to
$$\hat{\rho}'=\frac{1}{Z} \sum_{\alpha} |a,\alpha \rangle \langle a,\alpha|\hat{\rho}|a,\alpha \rangle \langle a,\alpha|,$$
where $\hat{\rho}$ is the state before the measurment, and $|a,\alpha \rangle$ a orthonormal basis of the subspace $\mathrm{Eig}(\hat{A},a)$ and
$$Z=\mathrm{Tr} \sum_{\alpha} |a,\alpha \rangle \langle a,\alpha|\hat{\rho}|a,\alpha \rangle \langle a,\alpha=\sum_{\alpha} \langle a,\alpha|\hat{\rho}|a,\alpha \rangle$$
is the probability that the measurement gives $a$ as the result.

 

[A. Neumaier]

But that's then quantum dynamics.

Of course. It is the quantum dynamics according to von Neumann's 1932 book, where as long as the system is isolated the state evolves according to the unitary dynamics, while when it interacts with an unmodeled instrument measuring X, the state evolves by instantaneous projection to an eigenstate of X. The latter has later been called the collapse (in 1951 by Bohm, according to Wikipedia).

Much later [apparently in the 1970s; cf. C.W. Helstrom, Quantum Detection and Estimation Theory. Academic Press 1976] it was realized that the collapse to an eigenstate happens in very special circumstances only, and that in the instantaneous approximation, collapse of $\psi$ to a normalized multiple of one of $P_k\psi$ (with probability $|P_k\psi|^2$), where $\sum_kP_k^*P_k=1$ guarantees that the probbilities sum to 1, is the generic discrete measurement situation, with the $P_k$ determined by the instrument.

 

[vanhees71]

In your cited paper is no instantaneous collapse but only a (assumed rapid) decay of the initially prepared excited state. Of course (50) is an idealization. There's nothing in QED that makes this rigorously true.

 

[A. Neumaier]

In your cited paper is no instantaneous collapse but only a (assumed rapid) decay of the initially prepared excited state. Of course (50) is an idealization. There's nothing in QED that makes this rigorously true.

Both the collapse to the ground state and the excitation to the excited state are very fast, and the time needed for them is negligible compared to the time the atom stays in one of these states. Thus treating them as instantaneous is justified on the level of an effective description valid for not too high time resolution. Fact is that highly regarded practitioners in the field model it in this way.

Instantaneous collapse is an idealization inherent in von Neumann's treatment (together with other idealizations) of the measurement process. For a light pulse, passing a polarizer takes an extremely short time, so that the instantaneous approximation is often justified. It is no different from assuming in the usual textbook derivation of the Boltzmann equation that collisions are instantaneous.

 

[atyy]

The appropriate reference for the technical part is the paper by Plenio and Knight mentioned in post #5. I don't think that they use a cut. What is used instead is a correlation assumption that introduces a dissipative arrow of time. This is done everywhere in statistical mechanics, even classically, as otherwise it would be impossible to get dynamical information from the statistical approach.

The Plenio and Knight paper http://arxiv.org/abs/quant-ph/9702007 assumes collapse throughout.

They simply postulate that "The jumps that occur in this description can be considered as due to the increase of our knowledge about the system which is represented by the wave-function (or the density operator) describing the system."

There is no derivation of that assumption from unitary evolution alone. Every time the knowledge changes, they collapse the wave function.

 

[atyy]

Here are Plenio's QM lectures http://www3.imperial.ac.uk/pls/portallive/docs/1/613904.PDF

Here is Knight's quantum optics book https://books.google.com.sg/books?id=CgByyoBJJwgC&source=gbs_navlinks_s

Both teach standard QM with collapse as a postulate.

Just to make it clear, I do believe collapse can be derived from unitarity under some conditions. For example, Bohmian Mechanics gives a derivation that is rigrourous enough at the physics level. I also respect approaches like MWI or Allahverdyan, Balian and Nieuwenhuizen, even if I am not sure they are technically correct. I also do respect approaches like Bohr's or Heisenberg's which implicitly recognize the problem, except that they would say it's a feature and not a problem. What I object to is simply removing the postulate, and trivializing the measurement problem by handwaving claims that the ensemble interpretation solves it.

 

[A. Neumaier]

There is no derivation of that assumption from unitary evolution alone. Every time the knowledge changes, they collapse the wave function.

I had read the paper by Plenio and Knight a long time ago, and obviously didn't recall its precise content.

I still think the picture I had painted in post #1 is the correct one. But I need to do a more thorough literature search to find out te precise status of the theory of quantum jump processes. For example, there is lots of related rigorous work by Accardi and collegues on quantum stochastic processes in the Heisenberg picture, but I don't recall the details. There is also old work by Jadczyk on a purely phenomenological basis.

Rechecking the literature may take quite some time, though, since during the term I have much less time for physics.

 

[A. Neumaier]

Section 1 of http://arxiv.org/abs/quant-ph/0204056 contains a summary and references to Jadczyk's work. His work describes the piecewise continuous quantum jump process in some detail but gives no derivation from microscopic theory.

 

[stevendaryl]

Just to make it clear, I do believe collapse can be derived from unitarity under some conditions. For example, Bohmian Mechanics gives a derivation that is rigrourous enough at the physics level.

I don't think it's that clear with Bohmian mechanics. In a certain sense, Bohmian mechanics is "pre-collapsed", since particles have definite positions at all times. But that's not the full story of collapse. In Bohmian mechanics, you have two "state variables": The actual position, $x$, and the wave function $\psi$. The wave function provides a nonlocal force acting on the particle. But the collapse hypothesis is this:

After measuring an observable $\hat{O}$ and getting result $\lambda$, then immediately afterward, the appropriate wave function to use is the projection of $\psi$ onto the subspace of eigenstates of $\hat{O}$ with eigenvalue $\lambda$.​

There might be an argument that that is the appropriate thing to do in Bohmian mechanics, but it isn't obvious that such an argument is any easier to do in Bohmian mechanics than in any other interpretation.

 

[atyy]

I don't think it's that clear with Bohmian mechanics. In a certain sense, Bohmian mechanics is "pre-collapsed", since particles have definite positions at all times. But that's not the full story of collapse. In Bohmian mechanics, you have two "state variables": The actual position, $x$, and the wave function $\psi$. The wave function provides a nonlocal force acting on the particle. But the collapse hypothesis is this:

After measuring an observable $\hat{O}$ and getting result $\lambda$, then immediately afterward, the appropriate wave function to use is the projection of $\psi$ onto the subspace of eigenstates of $\hat{O}$ with eigenvalue $\lambda$.​

There might be an argument that that is the appropriate thing to do in Bohmian mechanics, but it isn't obvious that such an argument is any easier to do in Bohmian mechanics than in any other interpretation.

Yes, it is pre-collapsed. And one of the triumphs of BM is that it derives collapse. This is one of the reasons why BM is the interpretation that has the greatest claim to solving the measurement problem for some realm of QM.

 

[stevendaryl]

Yes, it is pre-collapsed. And one of the triumphs of BM is that it derives collapse.

I don't see how it derives collapse of the wave function. I thought that in Bohmian mechanics, the wave function always evolves unitarily, which would mean no collapse, I would think. The POSITION is always definite in Bohmian mechanics, but in Bohmian mechanics, the position being definite does not imply that the wave function is localized. So how do you derive in Bohmian mechanics that the wave function is localized after a position measurement has been made?

 

[atyy]

I don't see how it derives collapse of the wave function. I thought that in Bohmian mechanics, the wave function always evolves unitarily, which would mean no collapse, I would think. The POSITION is always definite in Bohmian mechanics, but in Bohmian mechanics, the position being definite does not imply that the wave function is localized. So how do you derive in Bohmian mechanics that the wave function is localized after a position measurement has been made?

Yes, that's the point. There is no collapse, and the wave function evolves unitarily. However, Bohmian mechanics reproduces the predictions of quantum mechanics without hidden variables and with collapse. So that is a way of deriving collapse from unitarity.

 

[A. Neumaier]

Bohmian mechanics reproduces the predictions of quantum mechanics without hidden variables and with collapse.

Really? How does it do that, in case not position but spin or momentum is measured? How does the effective collapse appear?

 

[StevieTNZ]

Would it be fair to say that if Bohmian Mechanics only allows for a definite position of quantum systems, other observables are still subject to the measurement problem?

 

[atyy]

Really? How does it do that, in case not position but spin or momentum is measured? How does the effective collapse appear?

The key idea is that all measurements ultimately are position measurements, eg. of the pointer. Spin is a bit complicated (but it can be done), but an example of a momentum measurement by position measurement is that if you have a slit, the far field Fraunhofer limit of position is the (Fourier transform of position = momentum) just after the slit.

The effective collapse occurs by decoherence. Essentially, in decoherence, collapse appears as the transformation of the reduced density matrix from an improper to a proper ensemble (this is true whether one uses Copenhagen or Ensemble language, ie. whether one talks about pure states of individual system or the assignment of sub-ensembles). In BM, the assumption that there is a trajectory means the ensemble is always proper.

 

[atyy]

Would it be fair to say that if Bohmian Mechanics only allows for a definite position of quantum systems, other observables are still subject to the measurement problem?

The idea is that all measurements are ultimately position measurements (eg. position of a pointer).

So there is no measurement problem in the sense that an observer is not needed - we return to classical reality, but nonlocal. However, the treatment of spin is very interesting, and some will say spin in BM is still not real - I find spin in BM tricky, so can't answer off the top of my head - I'm sure Demystifier will be able to give more insight into this.

 

[A. Neumaier]

The key idea is that all measurements ultimately are position measurements, eg. of the pointer.

My question is precisely how an arbitrary measurement M is reduced to a position measurement P in such a way that Born's rule for M follows.

an example of a momentum measurement by position measurement is that if you have a slit, the far field Fraunhofer limit of position is the (Fourier transform of position = momentum) just after the slit.

A slit is fixed in space, and measurement takes time. Therefore passing a slit never measures the 3D position of a fast moving particle. It can only measure the transversal components of position and the momentum component in the direction of motion. These commute.
And it measures these only approximately. Thus there is something to be explained already for a spinless particle. The position in the longitudinal direction must be inferred from the time the measurement takes and the value of the longitudinal particle momentum, and hence is always very uncertain at the typical speeds of elementary particles.

Moreover, since position measurements are always approximate, how is it explained that spin measurement can be exact? And how that the measurement of angular momentum (which also has a discrete spectrum) can be exact?

In general, Bohr's rule states that for the measurement of a set of commuting orthogonal projectors, the result will be 1 for one of these and zero for the others, and the state collapses to an eigenstate of the corresponding projector. This property, at least in the case of one projector for each electron in a crystal or fluid, is essential for explaining how photodetection works for single photons. A claim of Bohmian mechanics to solve the measurement problem must derive this property! Else it has no explanation for the photoeffect!

 

[atyy]

My question is precisely how an arbitrary measurement M is reduced to a position measurement P in such a way that Born's rule for M follows.

Everything is done as in the usual quantum mechanics with decoherence and position as a pointer observable.

A slit is fixed in space, and measurement takes time. Therefore passing a slit never measures the 3D position of a fast moving particle. It can only measure the transversal components of position and the momentum component in the direction of motion. These commute.
And it measures these only approximately. Thus there is something to be explained already for a spinless particle. The position in the longitudinal direction must be inferred from the time the measurement takes and the value of the longitudinal particle momentum, and hence is always very uncertain at the typical speeds of elementary particles.

Yes, so in general one should use POVMS for the real measurements we do. The orthogonal sharp measurements are the same sort of thing as imagining perfect decoherence which never happens, but we believe is good enough at the heuristic level. One can find a derivation of POVMs from BM in http://arxiv.org/abs/quant-ph/0308039 and http://arxiv.org/abs/quant-ph/0308038

Moreover, since position measurements are always approximate, how is it explained that spin measurement can be exact? And how that the measurement of angular momentum (which also has a discrete spectrum) can be exact?

In general, Bohr's rule states that for the measurement of a set of commuting orthogonal projectors, the result will be 1 for one of these and zero for the others, and the state collapses to an eigenstate of the corresponding projector. This property, at least in the case of one projector for each electron in a crystal or fluid, is essential for explaining how photodetection works for single photons. A claim of Bohmian mechanics to solve the measurement problem must derive this property! Else it has no explanation for the photoeffect!

Spin is tricky, I don't know it well enough to answer off the top of my head. I refer you to http://arxiv.org/abs/1206.1084 or maybe Demystifier can help out.

 

[A. Neumaier]

My question is precisely how an arbitrary measurement M is reduced to a position measurement P in such a way that Born's rule for M follows.

Everything is done as in the usual quantum mechanics with decoherence and position as a pointer observable.

I don't understand how this is done in the usual quantum mechanics - there Born's rule is assumed, not obtained.

Please point to a paper where usual quantum mechanics with decoherence and position as a pointer observable leads to a derivation of the Born rule for an arbitrary measurement.

 

[A. Neumaier]

Spin is tricky, I don't know it well enough to answer off the top of my head. I refer you to http://arxiv.org/abs/1206.1084 or maybe Demystifier can help out.

The paper doesn't consider spin measurement. After (114), it is assumed ''for simplicity'' that the operator to be measured has no degeneracy in the big Hilbert space of particle plus detector. This is not the case for spin. Thus the later discussion doesn't apply to spin measurements.

It is also not the case for the measurement of the projection operators to the bound spectrum of the electrons in a photodetector, needed for analyzing photodetection.

In fact, when measuring a system whose Hilbert space is a tensor product, most operators one typically measures are highly degenerate. Thus treating their measurement well is essentially for coping with the measurement problem.

Let us see what @Demystifier can say about this.

 

[A. Neumaier]

The Plenio and Knight paper http://arxiv.org/abs/quant-ph/9702007 assumes collapse throughout.

No, not throughout. The paper is a survey paper and describes many approaches, including approaches freely using collapse.

But Plenio and Knight also describe a derivation by Gardiner (1988) that starts from the unitary evolution and does not use collapse: The description of this derivation begins on p.31. Formula (78) contains the Hamiltonian of the complete system. The collapse is avoided by the following technical trick:

The idea is now to perform the Markovian limit directly in the Schroedinger equation instead of performing this limit on the results

which is then carried out using the quantum Ito calculus.

An equivalent but far less technical derivation was later given in the paper
H. P. Breuer, F. Petruccione, Stochastic dynamics of reduced wave functions and continuous measurement in quantum optics, Fortschritte der Physik 45, 39-78 (1997).
In particular, pp.53-58 of this paper describe a fairly elementary derivation of a quantum jump process responsible for photodetection, starting with the unitary dynamics and involving no collapse but only standard approximations from statistical mechanics.

The quantum jump processes for general measurement situations are derived from unitarity in the more technical papers [30-32] by Breuer and Petruccione cited in the paper mentioned above. All four papers can be downloaded from http://omnibus.uni-freiburg.de/~breuer/

 

[atyy]

I don't understand how this is done in the usual quantum mechanics - there Born's rule is assumed, not obtained.

Please point to a paper where usual quantum mechanics with decoherence and position as a pointer observable leads to a derivation of the Born rule for an arbitrary measurement.

Right. What I mean is in the usual quantum mechanics, with no hidden variables, the Born rule is assumed. In this regular QM, we can (at the physics level) couple spin to a pointer, and read spin off as a position.

In BM, the Born rule is derived. However, additional assumptions are made that enable this derivation - the hidden variables and the condition of "quantum equilibrium". This leads to a derivation of the Born rule for position. Consequently, if we use a final readout that is position, then we get the same results in QM without hidden variables and in BM with the hidden variables and the condition of "quantum equilibrium".

After you look at BM, it is ugly and "obvious". In mathematical terms, what BM solves is asking whether the non-simplex state space of QM can be embedded into a state space that is a simplex. Once the state space is a simplex, we recover regular classical probability, and the measurement problem goes away (of course we still have all the problem associated with regular probability).

In the typical presentation of BM, the Born rule is not fully derived from determinism, since the "quantum equilibrium" condition puts the stochasticity into the initial conditions. However, one can show that there are deterministic dynamics that establish the "quantum equilibrium" quickly - the Valentini H-theorem https://en.wikipedia.org/wiki/Quantum_non-equilibrium.

 

[Demystifier]

I am tired of explaining how Bohmian mechanics cannot violate the Born rule for this or that specific type of measurement.

This is like explaining how classical mechanics explains that perpetuum mobile cannot work for this or that specific mechanism
https://www.google.hr/search?q=perp...X&ved=0ahUKEwi19P6Ao67LAhWDfRoKHcWOBZoQsAQIGw
It is not so easy to explain in detail why each particular perpetuum mobile cannot work. But unless you are a crackpot, you don't need a detailed explanation for all those examples. It is sufficient to derive the general theorem of energy conservation. Since it is general, it works for any case, so you don't need to know all the details specific for this or that particular example.

The same is with Bohmian mechanics and the Born rule. Unless your understanding of Bohmian mechanics is at the level of a crackpot, all you really need to understand is the general theorem that measurement of any observable in non-relativistic Bohmian mechanics leads to probabilities given by the Born rule. The theorem is given in most books and reviews on Bohmian mechanics, yet some people don't want to read it. Please, read the general theorem! If you have any objections about the general theorem, I will be glad to answer. But don't ask about particular examples without understanding the general theorem.

Of course, any theorem has its assumptions, and this one is not an exception. You are free to doubt about the validity of these assumptions. But please phrase your doubts in terms of the assumptions of the general theorem, not in terms of this or that particular example.

One place where the general theorem is presented is my own paper
http://arxiv.org/abs/1112.2034
Secs. 2.1-2.2. The word "theorem" is not explicitly used, nevertheless the proof ends with the proof of the Born rule in Eq. (13).

 

[stevendaryl]

Yes, that's the point. There is no collapse, and the wave function evolves unitarily. However, Bohmian mechanics reproduces the predictions of quantum mechanics without hidden variables and with collapse. So that is a way of deriving collapse from unitarity.

Somehow, I'm not being clear. I understand that that's the claim, but I'm questioning whether it's true. Or rather, I'm question how it's true--how Bohmian mechanics makes predictions equivalent to collapse.

I understand the basic Bohmian idea:

1. Assume that particle positions are initially distributed according to $|\psi(x)|^2$
2. Use the wave function $\psi(x)$ to compute the "quantum potential" that influences particle motion.
3. Prove that particle motion, together with the quantum potential, insures that probability distribution remains $|\psi(x)|^2$

So without collapse, Bohmian mechanics has the same probabilistic predictions as the standard interpretation. But now, if we introduce collapse for the standard interpretation, but NOT for the Bohmian interpretation, then the two interpretations will be using different functions $\psi(x)$. The Bohmian analysis will be using $\psi_{uncollapsed}$ and the standard analysis will be using $\psi_{collapsed}$. So it's not immediately clear that the two interpretations give the same result: Bohmian analysis will be using a probability distribution $|\psi_{uncollapsed}|^2$, while the standard interpretation will be using a probability distribution $|\psi_{collapsed}|^2$. So they predict different probabilities for future position measurements (or at least, seem to). It would seem to me that for Bohmian mechanics to be equivalent to the standard interpretation, they would have to use $\psi_{collapsed}$, rather than $\psi_{uncollapsed}$ in computing the quantum potential.

Now, I think that the answer might be something like entanglement. When you measure a particle, the particle becomes entangled with the measuring device. So the actual "quantum potential" that should be used afterward is not derived from the wave function of the particle alone, but from the wave function of the composite particle + measuring device. This more sophisticated analysis may reproduce the same predictions as if they used $\psi_{collapsed}$, but it certainly isn't at all obvious, and the equivalence (if they are equivalent) is not particular easy to see.

 

[A. Neumaier]

This leads to a derivation of the Born rule for position. Consequently, if we use a final readout that is position

Everything is obviously built in for position. But I was asking for the Born rule for operators with a discrete spectrum.

In particular, I was asking for the projection operators that project the wave function of the $i$th electron is a macroscopic array of independent, distinguishable electrons to its bound part. There is no macroscopic pointer for measuring this projection operator. Only one for measuring an unknown one of them. But the Born rule for each of it is needed in the conventional arguments used to predict the correct multiphoto counting statistics. It is also needed for applications to quantum computing - where one actually measures only once at the end, but invokes the generalized Born rule for designing the computing equipment.

 

[atyy]

Yeah, I'll have to hope Demystifier answers that.

 

[stevendaryl]

I am tired of explaining how Bohmian mechanics cannot violate the Born rule for this or that specific type of measurement.

I'm not sure what this quote is in response to, but the issue that I have with Bohmian mechanics is not really about Bohmian mechanics (except indirectly), but is about the equivalence of assuming and not assuming the collapse hypothesis.

We can ask the question purely in terms of the standard interpretation of QM, rather than the Bohmian interpretation. Ostensibly, the collapse hypothesis seems to have empirical content:

Initially, the wave function is $\psi_{uncollapsed}(x)$. After a measurement of an observable $O$ is found to give value $\lambda$, the collapse hypothesis claims that the wave function is now given by $\psi_{collapsed}(x)$, which is obtained from $\psi_{collapsed}$ by projecting onto the subspace of eigenstates of $O$ with eigenvalue $\lambda$. In subsequent experiments, you use $\psi_{collapsed}$.

Now, surely there is a difference between using $\psi_{collapsed}$ for prediction of the probabilities of results of future experiments and using $\psi_{uncollapsed}$. So the collapse hypothesis seems to have testable consequences. So shouldn't we be able to decide, once and for all, whether collapse happens, or not?

Well, I think the answer is not so simple, and the reason is that once you've performed a measurement, from that point on, the particle being measured has become entangled with the system doing the measurement. So a noncollapse interpretation (whether Bohmian, or Many-Worlds, or minimalist interpretation) has to consider the total system of particle + detector, rather than the particle alone. So the key theorem making collapse and non-collapse interpretations equivalent is that the effect of using the uncollapsed wave function for the total system is equivalent to using the collapsed wave function for the particle alone. My point is not that such a theorem is not particularly helped by the Bohmian interpretation, since it's a theorem about wave functions, not about particles.

 

[Demystifier]

But I was asking for the Born rule for operators with a discrete spectrum.

If you ask about operators such as spin, take my analysis in the paper linked in post #30, and in all equations replace $^*$ with $^\dagger$.