# A Collapse from unitarity

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1. Mar 5, 2016

### A. Neumaier

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.

Last edited: Mar 7, 2016
2. Mar 5, 2016

### kith

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?

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).

3. Mar 5, 2016

### vanhees71

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".

4. Mar 5, 2016

### A. Neumaier

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.
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.

5. Mar 5, 2016

### A. Neumaier

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.

Last edited: Mar 7, 2016
6. Mar 5, 2016

### atyy

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.

7. Mar 5, 2016

### A. Neumaier

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.

Last edited: Mar 5, 2016
8. Mar 5, 2016

### vanhees71

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.

9. Mar 5, 2016

### A. Neumaier

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.

Last edited: Mar 5, 2016
10. Mar 5, 2016

### 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.

11. Mar 5, 2016

### A. Neumaier

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.

Last edited: Mar 5, 2016
12. Mar 5, 2016

### atyy

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.

13. Mar 5, 2016

### 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.

Last edited: Mar 5, 2016
14. Mar 6, 2016

### A. Neumaier

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.

15. Mar 6, 2016

### 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.

16. Mar 6, 2016

### stevendaryl

Staff Emeritus
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.

17. Mar 6, 2016

### atyy

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.

18. Mar 6, 2016

### stevendaryl

Staff Emeritus
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?

19. Mar 6, 2016

### atyy

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.

20. Mar 6, 2016

### A. Neumaier

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