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Quantum indeterminacy 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. 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 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 mecahnics and quantum measurement continues to be an active research area in both theoretical and experimental physics. Possibly the first systematic attempt at a mathematical theory was developed by John von Neumann. The kind of measurements he investigated (discussed in the book Mathematical Foundations of Quantum Mechanics) 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 ψ, then immediately after measurement the system will occupy a state which is an eigenvector e of A and the observed value λ will be the corresponding eigenvalue A e = λ 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 ψ. The probaility is

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

where E(λ) is the projection onto the space of eigenvectors of A with eigenvalue λ.

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 some QM system. At the state ψ the values of σ₁ are +1 whereas the values of σ₃ 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 C, with each quantum state corresponding to a unit vector in C.

The Pauli spin matrices

${\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

are self-adjoint and correspond to spin-measurements along the coordinate axes.

The Pauli matrices all have the eigenvalues +1, − 1.

• For σ₁, these eigenvalues correspond to the eigenvectors

${\displaystyle {\frac {1}{\sqrt {2}}}(1,1),{\frac {1}{\sqrt {2}}}(1,-1)}$

• For σ₃, they correspond to the eigenvectors

${\displaystyle (1,0),(0,1)\quad }$

Thus in the state

${\displaystyle \psi ={\frac {1}{\sqrt {2}}}(1,1),}$

σ₁ has the determinate value +1, while measurement of σ₃ can produce either +1, −1 each with probability 1/2. In fact, there is no state in which measurement of both σ₁ and σ₃ have determinate values.

There are various questions that can be asked about the above indeterminacy assertion.

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?

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 his argument contained a flaw. The definitive negative answer to 1) has been established by experiment that Bell's inequalities are violated (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 σ₁ and (B) which measures only σ₃ of a spin system in the state ψ. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, −1 with probability 1/2.

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 J·s).

Incompleteness

Albert Einstein argued that if quantum mechanics is correct, then our view of "how the real world works" is no longer tenable. This view consists of two ideas: first that physical systems have a definite state which determines the values of all other measurable properties and second, that effects of local actions have a finite propagation speed. This 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. 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. Thus either spin in the x direction is not an element of physical reality or effects travel from Alice to Bob instantly.

See also

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

• Quantum mechanics
• Determinism