Operator (quantum physics)

Incompleteness of quantum physics

Incompleteness of quantum physics is the assertion that the state of a physical system, as formulated by quantum mechanics, does not give a complete description for the system, assuming the usual philosophical requirements ("reality", "nonlocality", etc.). Einstein, Podolsky, and Rosen had proposed their definition of a "complete" description as one which uniquely determines the values of all its measurable properties. The existence of indeterminacy for some measurements is a characteristic of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle.

Incompleteness can be understood in two fundamentally different ways:

  1. QM is incomplete because it is not the "right" theory; the right theory would provide descriptive categories to account for all observable behavior and not leave "anything to chance".
  2. QM is incomplete, but is a faithful picture of nature.

Incompleteness understood as 1) would motivate search for a hidden variables theory featuring nonlocality, owing to results of Bell test experiments. There are many variants of 2) which is widely considered to be the more orthodox view of quantum mechanics.

Einstein's argument for the incompleteness of quantum physics

Albert Einstein may have been the first person to carefully point out the radical effect the new quantum physics would have on our notion of physical state. For a historical background of Einstein's thinking in regard to QM, see Jankiw and Kleppner [2000], although his best known critique was formulated in the EPR thought experiment. See Bell [1964].

According to Fuchs [2002], Einstein developed a very good argument for incompleteness:

The best [argument of Einstein] was in essence this. Take two spatially separated systems A and B prepared in some entangled quantum state |ψAB>. By performing the measurement of one or another of two observables on system A alone, one can immediately write down a new state for system B. Either the state will be drawn from one set of states {|φiB>} or another {|ηiB>}, depending upon which observable is measured. The key point is that it does not matter how distant the two systems are from each other, what sort of medium they might be immersed in, or any of the other fine details of the world. Einstein concluded that whatever these things called quantum states be, they cannot be “real states of affairs” for system B alone. For, whatever the real, objective state of affairs at B is, it should not depend upon the measurements one can make on a causally unconnected system A.

Einstein's argument shows that quantum state is not a complete description of a physical system, according to Fuchs [2002]:

Thus one must take it seriously that the new state (either a |φiB> or |ηiB>) represents information about system B. In making a measurement on A, one learns something about B, but that is where the story ends. The state change cannot be construed to be something more physical than that. More particularly, the final state itself for B cannot be viewed as more than a reflection of some tricky combination of one’s initial information and the knowledge gained through the measurement. Expressed in the language of Einstein, the quantum state cannot be a “complete” description of the quantum system.

Reality of incompleteness

Although Einstein was one of the first to formulate the necessary incompleteness of quantum physics, he never fully accepted it. In a 1926 letter to Max Born, he made a remark that is now famous:
Quantum mechanics is certainly imposing. But an inner voice tells me it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the Old One. I, at any rate, am convinced that He does not throw dice.

Einstein was mistaken according to Stephen Hawking in Does God Play Dice,

Einstein's view was what would now be called, a hidden variable theory. Hidden variable theories might seem to be the most obvious way to incorporate the Uncertainty Principle into physics. They form the basis of the mental picture of the universe, held by many scientists, and almost all philosophers of science. But these hidden variable theories are wrong. The British physicist, John Bell, who died recently, devised an experimental test that would distinguish hidden variable theories. When the experiment was carried out carefully, the results were inconsistent with hidden variables. Thus it seems that even God is bound by the Uncertainty Principle, and can not know both the position, and the speed, of a particle. So God does play dice with the universe. All the evidence points to him being an inveterate gambler, who throws the dice on every possible occasion.

Chris Fuchs [2002] summed up the reality of the necessary incompleteness of information in quantum physics as follows, attributing this idea to Einstein "He [Einstein] was the first person to say in absolutely unambiguous terms why the quantum state should be viewed as information (or, to say the same thing, as a representation of one’s beliefs and gambling commitments, credible or otherwise).

Fuchs adds:

Incompleteness, it seems, is here to stay: The theory prescribes that no matter how much we know about a quantum system—even when we have maximal information about it—there will always be a statistical residue. There will always be questions that we can ask of a system for which we cannot predict the outcomes. In quantum theory, maximal information is simply not complete information [Caves and Fuchs 1996]. But neither can it be completed.

The kind of information about the physical world that is available to us according to Fuchs [2002] is “the potential consequences of our experimental interventions into nature” which is the subject matter of quantum physics.

The Copenhagen Interpretation

It should however be noted that according to the generally accepted Copenhagen Interpretation of quantum mechanics (Niels Bohr) the philosophical requirements assumed by Einstein are not true: according to this interpretation quantum mechanics is neither "real", since a quantum mechanical measurement does not simply state, but instead prepare the physics of a system. Quantum mechanics is also not "local", essentially because the state of a system is described by the Hilbert vector |psirangle, which includes the value at every site, |psirangle to psi (x,y,z).

So in this respect Einstein was simply wrong, although he "pinpointed" the formalism of quantum mechanics exceptionally sharp.

Relational Quantum Physics

According to Relational Quantum Physics [Laudisa and Rovelli 2005], the way distinct physical systems affect each other when they interact (and not of the way physical systems "are") exhausts all that can be said about the physical world. The physical world is thus seen as a net of interacting components, where there is no meaning to the state of an isolated system. A physical system (or, more precisely, its contingent state) is described by the net of relations it entertains with the surrounding systems, and the physical structure of the world is identified as this net of relationships. In other words, “Quantum physics is the theoretical formalization of the experimental discovery that the descriptions that different observers give of the same events are not universal.”

The concept that quantum mechanics forces us to give up the concept of a description of a system independent from the observer providing such a description; that is the concept of the absolute state of a system. I.e., there is no observer independent data at all. According to Zurek [1982], “Properties of quantum systems have no absolute meaning. Rather they must be always characterized with respect to other physical systems.”

Does this mean that there is no relation whatsoever between views of different observers? Certainly not. According to Rovelli [1996] “It is possible to compare different views, but the process of comparison is always a physical interaction (and all physical interactions are quantum mechanical in nature).”


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  • Federico Laudisa and Carlo Rovelli. "Relational Quantum Mechanics" The Stanford Encyclopedia of Philosophy (Fall 2005 Edition).
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