Quantum indeterminacy

Quantum indeterminacy (a terminology preferred over nondeterminism by Niels Bohr) is the apparent necessary incompleteness in the description of a physical system, that has become one the of characteristics of quantum physics. Prior to quantum physics, it was thought that a physical system had a determinate state which (a) uniquely determined all the values of its measurable properties and conversely (b) the values of its measurable properties uniquely determined the state. The subject matter of physics was discovering the laws that characterized the behavior of system state. 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 (see below).

Quantum indeterminacy can be quantitatively characterized by a probability distribution on the set of outcomes of measurements of an observable. For pairs of complementary observables, the dispersions of the corresponding distributions are related by the Heisenberg uncertainty principle.

Variability and errors

Variability of outcomes in the measurement process was not an innovation of quantum mechanics. However, by the latter half of the 1700s, measurement variability was thought to be well understood and it could be reduced by better equipment and accounted for by statistical error models. That is to say, the general supposition was that there was a "true" value to be measured and that the variation was due to "errors in measurement" explainable by some "error parameter."

Measurement requires intervention

Quantum indeterminacy is intrinsically involved with measurement which itself involves intervention. Measurement in quantum physics has proved to be unexpectedly subtle. How the might the object be affected by the measuring process? This question concerned Werner Heisenberg.

Measurement is theory laden

Many years after the fact, Heisenberg recalled how when he was developing the uncertainty principle that Albert Einstein had emphasized the importance of what later came to be called the "theory laden" character of measurement [Heisenberg 1971]:

It is quite wrong to try founding a theory on observable magnitudes alone. In reality the very opposite happens. It is the theory which decides what we can observe. You must appreciate that observation is a very complicated process. The phenomenon under observation produces certain events in our measuring apparatus. As a result, further processes take place in the apparatus, which eventually and by complicated paths produce sense impressions and help us to fix the effects in our consciousness. Along this whole path—from the phenomenon to its fixation in our consciousness—we must be able to tell how nature functions, must know the natural laws at least in practical terms, before we can claim to have observed anything at all. Only theory, that is, knowledge of natural laws, enables us to deduce the underlying phenomena from our sense impressions. When we claim that we can observe something new, we ought really to be saying th]at, although we are about to formulate new natural laws that do not agree with the old ones, we nevertheless assume that the existing laws—covering the whole path from the phenomenon to our consciousness—function in such a way that we can rely upon them and hence speak of “observation.”

Thomas Kuhn emphasized the "theory laden" character of observation in the development of quantum physics including further developing the thesis that it was Einstein who saw the revolutionary implications of the observations of black-body radiation whereas even Max Planck (who had pioneered in the work) initially resisted Einstein's views [Kuhn 1987].

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.

Einstein may have been first to make the argument

Chris Fuchs [2002] recounts that Einstein developed his argument for incompleteness over a long period of time:

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). His argument was simply that a quantum-state assignment for a system can be forced to go one way or the other by interacting with a part of the world that should have no causal connection with the system of interest. The paradigm here is of course the one well known through the Einstein, Podolsky, Rosen [1935] paper, but simpler versions of the train of thought had a long pre-history with Einstein himself (see [Fine 1986] and [Jammer 1985]).

A "real state of affairs" should not depend upon measurements that can be made on other causally unconnected systems

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 |ψ^AB> [Note not in original quotation: for an explanation of this notation see the Feynman Lectures on Physics Volume III, Claude Cohen-Tannoudji, Bernard Diu and Franck Laloë 1977 Volume I and Roger Penrose 2004]. 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 {|φ_i^B>} or another {|η_i^B>}, 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.

Quantum state cannot be a complete description

According to Fuchs [2002]:

Thus one must take it seriously that the new state (either a |φ_i^B> or |η_i^B>) 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

Einstein never fully accepted the necessary incompleteness of quantum physics. 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.

According to Steven Hawking in Does God Play Dice,

His [Einstein's] views were summed up in his famous phrase, 'God does not play dice'. He seemed to have felt that the uncertainty was only provisional: but that there was an underlying reality, in which particles would have well defined positions and speeds, and would evolve according to deterministic laws, in the spirit of Laplace. This reality might be known to God, but the quantum nature of light would prevent us seeing it, except through a glass darkly.

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.

Heisenberg later recalled his conversations with Einstein 1925-1927 in Heisenberg Recalls His Early Thoughts on the Uncertainty Principle as follows:

Namely, could it not be true that nature only allows for such situations which can be described with a mathematical scheme? Up to that moment, we had asked the opposite question. We had asked, given the situations in nature like the orbit in a cloud chamber, how can it be described with a mathematical scheme? But that wouldn't work, because by using such a word like "orbit", we of course assumed already that the electron had a position and had a velocity. But by turning it around, one could at once see that now it's possible, if I say nature only allows such situations as can be described with a mathematical scheme, then you can say, well, this orbit is really not a complete orbit. Actually, at every moment the electron has only an inaccurate position and an inaccurate velocity, and between these two inaccuracies there is this uncertainty relation. And only by this idea it was possible to say what such an orbit was. [emphasis as in original quotation]

Chris Fuchs [2002] concluded:

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.

Single particle indeterminacy

Quantum uncertainty can be illustrated in terms of a particle with a definitely measured momentum for which there must be a fundamental limit to how precisely its location can be specified. This quantum uncertainty principle can be expressed in terms of other variables, for example, a particle with a definitely measured energy has a fundamental limit to how precisely one can specify how long it will have that energy. The units involved in quantum uncertainty are on the order of Planck's constant (found experimentally to be 6.6 x 10^-34 J·s).

See also

• Quantum indeterminacy in computation
• Quantum mind
• just about any of the quantum mechanics articles

Reference

• A. Einstein, B. Podolsky, and N. Rosen,Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777–780 (1935).
• W. Pauli, letter to M. Fierz dated 10 August 1954, reprinted and translated in K. V. Laurikainen, Beyond the Atom: The Philosophical Thought of Wolfgang Pauli, (Springer-Verlag, Berlin, 1988), p. 226.
• David Bohm, Causality and Chance in Modern Physics, forward by Louis de Broglie, University of Pennsylvania Press, 1957. ISBN 0812210026
• Werner Heisenberg. Physics and Beyond: Encounters and Conversations translated by A. J. Pomerans (Harper & Row, New York, 1971), pp. 63–64.
• Claude Cohen-Tannoudji, Bernard Diu and Franck Laloë. Mecanique quantique (see also Quantum Mechanics translated from the French by Susan Hemley, Nicole Ostrowsky, and Dan Ostrowsky; John Wiley & Sons 1982) Hermann, Paris, France. 1977.
• M. Jammer, The EPR Problem in Its Historical Development in Symposium on the Foundations of Modern Physics: 50 years of the Einstein-Podolsky-Rosen Gedankenexperiment, edited by P. Lahti and P. Mittelstaedt (World Scientific, Singapore, 1985), pp. 129–149.
• A. Fine, The Shaky Game: Einstein Realism and the Quantum Theory, (University of Chicago Press, Chicago, 1986)
• Thomas Kuhn. Black-Body Theory and the Quantum Discontinuity, 1894-1912 Chicago University Press. 1987.
• A. Peres, Quantum Theory: Concepts and Methods, (Kluwer, Dordrecht, 1993).
• C. M. Caves and C. A. Fuchs,Quantum Information: How Much Information in a State Vector? in The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later, edited by A. Mann and M. Revzen, Ann. Israel Phys. Soc. 12, 226–257 (1996).
• Orly Alter and Yoshihisa Yamamoto. Quantum Measurement of a Single System Johh Wiley and Sons. 2001.
• Christopher Fuchs, Quantum mechanics as quantum information (and only a little more) in A. Khrenikov (ed.) Quantum Theory: Reconstruction of Foundations (Växjo: Växjo University Press, 2002).
• Asher Peres and Daniel Terno. Quantum Information and Relativity Theory Rev.Mod.Phys. 76 (2004) 93.
• Roger Penrose. The Road to Reality: A Complete Guide to the Laws of the Universe Alfred Knopf. 2004.