Quantum indeterminacy

Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics. Prior to quantum physics, it was thought that (a) a physical system had a determinate state which uniquely determined all the values of its measurable properties, and conversely (b) the values of its measurable properties uniquely determined the 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.

Quantum indeterminacy can be quantitatively characterized by a probability distribution on the set of outcomes of measurements of an observable. The distribution is uniquely determined by the system state, and moreover quantum mechanics provides a recipe for calculating this probability distribution.

Indeterminacy in measurement was not an innovation of quantum mechanics, since it had established early on by experimentalists that errors in measurement may lead to indeterminate outcomes. However, by the latter half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.

Measurement
An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of quantum mechanics and quantum measurement continues to be an active research area in both theoretical and experimental physics (Braginski and Khalili 1992.) Possibly the first systematic attempt at a mathematical theory was developed by John von Neumann. The kind of measurements he investigated are now called projective measurements. That theory was based in turn on the theory of projection-valued measures for self-adjoint operators which had been recently developed (by von Neumann and independently by Marshall Stone) and the Hilbert space formulation of quantum mechanics (attributed by von Neumann to Paul Dirac).

In this formulation, the state of a physical system corresponds to a vector of length 1 in a Hilbert space H over the complex numbers. An observable is represented by a self-adjoint operator A on H. If H is finite dimensional, by the spectral theorem, A has an orthonormal basis of eigenvectors. If the system is in state &psi;, then immediately after measurement the system will occupy a state which is an eigenvector e of A and the observed value &lambda; will be the corresponding eigenvalue of the equation A e = &lambda; e. It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is &psi;. The probability is


 * $$ \operatorname{Pr}(\lambda)= \langle \operatorname{E}(\lambda) \psi \mid \psi \rangle $$

where E(&lambda;) is the projection onto the space of eigenvectors of A with eigenvalue &lambda;.

Example
Bloch sphere showing eigenvectors for Pauli Spin matrices. The Bloch sphere is a two-dimensional surface the points of which correspond to the state space of a spin 1/2 particle. At the state &psi; the values of &sigma;1 are +1 whereas the values of &sigma;2 and &sigma;3 take the values +1, -1 with probability 1/2. In this example, we consider a single spin 1/2 particle (such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional Hilbert space C2, with each quantum state corresponding to a unit vector in C2 (unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right.

The Pauli spin matrices

\sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} $$ are self-adjoint and correspond to spin-measurements along the 3 coordinate axes.

The Pauli matrices all have the eigenvalues +1, &minus;1.
 * For &sigma;1, these eigenvalues correspond to the eigenvectors
 * $$ \frac{1}{\sqrt{2}} (1,1), \frac{1}{\sqrt{2}} (1,-1) $$


 * For &sigma;3, they correspond to the eigenvectors
 * $$ (1, 0), (0,1) \quad $$

Thus in the state
 * $$ \psi=\frac{1}{\sqrt{2}} (1,1), $$

&sigma;1 has the determinate value +1, while measurement of &sigma;3 can produce either +1, &minus;1 each with probability 1/2. In fact, there is no state in which measurement of both &sigma;1 and &sigma;3 have determinate values.

There are various questions that can be asked about the above indeterminacy assertion. Von Neumann formulated the question 1) and provided an argument why the answer had to be no, if one accepted the formalism he was proposing, although according to Bell, von Neumann's formal proof did not justify his informal conclusion (Bell: 2004, pg. 5). The definitive negative answer to 1) has been established by experiment that Bell's inequalities are violated, on the assumption that hidden variables must be local (see Bell test experiments). The answer to 2) depends on how disturbance is understood (particularly since measurement is disturbance), but in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) which measures exclusively  &sigma;1 and (B) which measures only &sigma;3 of a spin system in the state &psi;. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, &minus;1 with probability 1/2.
 * 1) Can the indeterminacy be understood as similar to an error in measurement explainable by an error parameter? More precisely, is there a hidden parameter that could account for the statistical indeterminacy in a completely classical way?
 * 2) Can the indeterminacy be understood as a disturbance of the system being measured?

Other examples of indeterminacy
Quantum indeterminacy can also 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&middot;s).

Indeterminacy and incompleteness
Quantum indeterminacy is the assertion that the state of a system does not determine a unique collection of values for all its measurable properties. Indeed in the quantum mechanical formalism, for a given quantum state, each one of these measurable values will be obtained non-deterministically in accordance with a probability distribution which is uniquely determined by the system state. Note that the state is destroyed by measurement, so when we refer to a collection of values, each measured value in this collection must be obtained using a freshly prepared state.

This indeterminacy might be regarded as a kind of essential incompleteness in our description of a physical system. Notice however, that the indeterminacy as stated above only applies to values of measurements not to the quantum state. For example, in the spin 1/2 example discussed above, the system can be prepared in the state &psi; by using measurement of &sigma;1 as a filter which retains only those particles such that &sigma;1 yields +1. By the von Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state &psi;.

However, Einstein did believe that quantum state cannot be a complete description of a physical system and, it is commonly thought, never came to terms with quantum mechanics. In fact, Einstein, Boris Podolsky and Nathan Rosen did show that if quantum mechanics is correct, then the classical view of how the real world works (at least after special relativity) is no longer tenable. This view included the following two ideas: This failure of the classical view was one of the conclusions of the EPR thought experiment in which two remotely located observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It was a conclusion of EPR, using the formal apparatus of quantum theory, that once Alice measured spin in the x direction, Bob's measurement in the x direction was determined with certainty, whereas immediately before Alice's measurement Bob's outcome was only statistically determined. From this it follows that either value of spin in the x direction is not an element of reality or that the effect of Alice's measurement has infinite speed of propagation.
 * 1) A measurable property of a physical system whose value can be predicted with certainty is actually an element of reality (this was the terminology used by EPR).
 * 2) Effects of local actions have a finite propagation speed.

Indeterminacy for mixed states
We have described indeterminacy for a quantum system which is in a pure state. Mixed states are a more general kind of state obtained by a statistical mixture of pure states. For mixed states the "quantum recipe" for determining the probability distribution of a measurement is determined as follows:

Let A be an observable of a quantum mechanical system. A is given by a densely defined self-adjoint operator on H. The spectral measure of A is a projection-valued measure defined by the condition
 * $$ \operatorname{E}_A(U) = \int_U \lambda d \operatorname{E}(\lambda), $$

for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:
 * $$ \operatorname{D}_A(U) =

\operatorname{Tr}(\operatorname{E}_A(U) S). $$ This is a probability measure defined on the Borel subsets of R which is the probability distribution obtained by measuring A in S.