Aspects of Quantum Measurement

Posted on 28 November 2008

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In the mathematical formalism of quantum theory, there are two levels that are important to keep clearly distinct: 1) observables/operators and 2) outcomes/eigenstates. At the first level, one observable is selected by the experimenter from among various observables. At the second level, the measurement of a selected observable is actually performed and an outcome is randomly selected by nature from among the eigenstates of that observable’s operator.

For example, consider observable such as spin with an associated operator, S. The eigenvectors of this operator S correspond to the possible outcomes of a measurement of that observable, e.g., |up> and |down>. These eigenvectors have corresponding eigenvalues which are the probability amplitudes of each outcome. The probabilities of all the outcomes of the single observable are normalized. For a classical analog to the spin, we can consider an observable S for a coin that corresponds to the measurement of its heads/tails position. This single observable S has two outcomes, |heads> and |tails>, each with probability 50%. We’re still just talking about one observable, though.

We can envision another observable T of the coin, e.g., the measurement of its angular momentum. The operator for this observable will have its own set of orthogonal eigenvectors corresponding to the different possible outcomes of measuring this momentum observable. The corresponding probabilities of these momentum outcomes will be normalized amongst themselves.

The two different operators S and T may be non-commuting, i.e., the two observables may be incompatible, which means that they can not be simultaneously measured with arbitrary precision. What is important to note here is that the different orthogonal eigenvectors |heads> and |tails> of the single observable S are very different from different non-commuting operators T and S. In other words, the two different outcomes of a single observable S are not the same as incompatible observables S and T. For example, commutation relations and Heisenberg’s uncertainty principle apply to two incompatible observables, not to two possible outcomes of one observable. Normalization and the collapse of the state vector apply to the possible outcomes of a single observable, not to incompatible observables.

The selection of an observable (e.g, position or momentum) is a free choice of the experimenter. In effect, it is a choice of a reference frame or coordinate system. Mathematically, it is a choice of the basis according to which the state vector of the system is represented. The choice of an observable is not part of the dynamics of quantum theory. It is not something the theory predicts, but something we must define, just as in relativity we are free to select any particular reference frame for observation. We are, in effect, talking about different ways of describing the world, not different underlying physics. A unique feature of quantum physics, however, is the incompatibility of different observables. Multiple complementary perspectives are needed to see the “whole” of reality; but we can not simultaneously view reality from these different perspectives. This is one unique and mysterious aspect of quantum theory.

Once an observable is selected, the selection of a particular outcome of a measurement of that observable is not a choice of the experimenter. The result of a measurement appears to be a free choice of nature, subject to the probability distributions dictated by quantum theory. If the same observable is measured, each individual outcome is unpredictable, but the ensemble of outcomes will conform to the theoretically predicted probability distribution.

Thus, there are two quite distinct aspects of quantum measurement. The first is the free selection by the experimenter of the observable to be measured. This selection defines the set of possible measurement results and their respective probabilities, but does not select a particular result. Even if the measurement interaction takes place, and the system decoheres, there is still not a selection of one result. All that has taken place is that the coherence between the different possible results is no longer measurable. The selection of one result does not take place until a result has been actualized by the actualization of a unique measurement result. This selection of one result does not obey a physical law (in fact, it violates Schrodinger’s equation). Nor is it, like the selection of the observable, something that the experimenter is free to choose. It is a unique and mysterious feature of quantum theory.

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Posted in: Science