Is there really such a thing as a definite state?

[Albert V]

In quantum mechanics we have the projection postulate saying that during a measurement the wave function of an observable collapse into a definite state. However, a friend of mine is convinced that there is no collapse, only linearity. After a measurement the wave function of an observable is still present, but is (for unresolved reasons) shifted towards a definite value. Hence, other outcomes become insignificant.

Are there any findings that suggest that it really is such a thing as a definite state?

 

[vanhees71]

That's a pretty delicate issue, and the answer depends on the personal view of the physicst you ask. I'm following the "minimal statistical interpretation", which just takes the minimal assumptions, i.e., the Born rule (together with the other mathematical postulates of QT, i.e., the projective Hilbert space for the (pure) states, self-adjoint operators to represent observables, unitary time evolution, etc.) to make the connection between the abstract formalism of QT with observations in the real world.

This implies that within the framework of quantum theory, after preparing an object in a definite state, which in the real world is given by an equivalence class of manipulations on this object to bring it into this state (e.g., preparing a beam of particles in an accelerator like the LHC), we only know probabilities about the outcome of measurements on the system. This implies that an observable has a determined value if and only if the system is in an eigenstate of the representing operator of this observable, and the corresponding eigenvalue is the value of this observable. Any other observables have no determined value, and the probability assignment according to Born's rule tells us with which frequency we can expect to find a possible value of this observable, when we repeat the experiment on an ensemble of many equally but independently from each other prepared systems.

There is no need for a collapse in this interpretation. The collapse hypothesis leads to a lot of inconsistencies within the framework of quantum theory, which cannot be justified by any observation today. First of all, the collapse hypothesis implies that quantum theory does not provide the complete description of the dynamics of the system, because with unitary time evolution there won't occur a collapse. This implies that there must be another theory describing the interaction of the measured system with the measurement apparatus. According to Bohr that theory is classical physics. However, there is no hint that on a fundamental level there is a "cut" between classical and quantum behavior. For macroscopic systems usually it is difficult to isolate them strictly enough from the environment to prevent decoherence. Through decoherence the behavior of the system becomes classical with high accuracy, but there is no principle reason to believe that quantum theory is invalid and classical theory must take over on a fundamental level. Nowadays one can prepare pretty large macroscopic systems in away that they show quantum behavior, including fascinating properties like entanglement, i.e., non-local correlations (not interactions!) between distant objects.

Another problem with the collapse hypothesis is related to entanglement, as was famously pointed out by Einstein, Podolsky, and Rosen: When the wave function of entangled far-distant objects would really collapse by measurement on one object, and this measurement implies the local interaction of the object with the measurement apparatus, has instantaneous causal implications for the other far-away object. This contradicts the causality structure of relativistic spacetime according to which no causal signal propagation can occur that is transmitted faster than the speed of light.

For all these reasons I think it is far more simple to abandon the unnecessary collapse hypothesis and stick to the minimal statistical interpretation!

 

[Albert V]

Thanks Vanhees, Since you hold this view it would be interesting to have your comments on the following:

I had an impression that it was a fundamental problem with the idea of "unitary evolution only", and at the same time that this theory is universally valid. I thought that the projection postulate was introduced (by Von Neumann) because he ran into an infinite regress when trying to perform measurements from within. Thomas Breuer seems also to have put much effort into arguing against universal valid theories and self-measurability. eg: https://homepages.fhv.at/tb/cms/?download=tbDISS.pdf‎

From the paper:
“Is quantum theory consistent with our experience of the macroscopic world? Can we regard macroscopic bodies as consisting of a large number of quantum systems?
There are famous historical arguments why this should be impossible. Such an argument is for example Schrödinger’s cat paradox. These arguments intend to show that
(1) at least some properties of macroscopic bodies (such as a cat being alive or dead) are definite in all states; (I will call such properties classical; and properties for which there are states in which they are not definite, quantum)
(2) quantum theory does not allow for classical properties.
Assuming that quantum theory is universally valid, (1) and (2) contradict each other. If one wants to avoid this contradiction, and a the same time stick to the universal validity of quantum theory, then clearly one has to drop (1) or (2).”


It is not easy to deny (1). One can drop (2), but then it follows that quantum states can be definite in the strong sence.

 

[meBigGuy]

Is the minimal statistical interpretation saying that things are only as definite as their surroundings (interactions) require?

 

[vanhees71]

@meBigGuy: The minimal statistical interpretation, just takes Born's rule as the interpretation of the physical content of the state and nothing more, i.e., the quantum mechanical (pure or mixed) state describes the probabilities for the outcome of measurements under the condition that the measured quantum system is prepared in that state. These probabilities you can only find by preparing many systems in that state and doing the appropriate statistical evaluation of the results (which is what many experimental colleagues in fact mainly do working, e.g., at the LHC, RHIC, and other accelerators around the world). Thus the minimal statistical interpretation refers to an ensemble of identically (but independently from each other) prepared quantum systems. The reference to the single system is only given by the preparation procedure. In this sense you can shortly say the quantum system's state is linked to reality to (an equivalence class) of preparation procedures; see also Mermin's very concise book on these issues:

A. Peres, Quantum Theory: Concepts and Methods, Kluver Academic Publ. (2002)

@Albert V: Unfortunately I could not open the link to Breuer's work. I can only give my personal opinion concerning the cited text in your posting. I'm of the opposite opinion: The macroscopic laws are only consistent with the observed "atomistic structure" of matter within quantum mechanics. The macroscopic laws are emergent phenomena derivable from quantum mechanics. The reason for the classical behavior of macroscopic objects, including cats ;-)), is decoherence and the coarse-grained description of macroscopic observables as averages over many microstates in the sense of quantum-many-body theory. Since without quantum theory, given the atomistic structure of matter, macroscopic stable matter couldn't exist, in precise contradiction to statement (2) I'd rather say "only quantum theory allows for classical properties" (particularly the stability of matter).

 

[Albert V]

@vanhees71
Is the minimal statistical interpretation compatible witht this work?

http://arxiv.org/abs/1111.3328

"Quantum states are the key mathematical objects in quantum theory. It is therefore surprising
that physicists have been unable to agree on what a quantum state truly represents. One possibility is that a pure quantum state corresponds directly to reality. However, there is a long history of suggestions that a quantum state (even a pure state) represents only knowledge or information about some aspect of reality. Here we show that any model in which a quantum state represents mere information about an underlying physical state of the system, and in which systems that are prepared independently have independent physical states, must make predictions which contradict those of quantum theory."