The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus.
Contents
- Measurement from a practical point of view
- Qualitative overview
- Quantitative details
- Measurable quantities observables as operators
- Measurement probabilities and wave function collapse
- Statistics of measurement
- Example
- Wave function collapse
- von Neumann measurement scheme
- Measurement of the first kind premeasurement without detection
- Measurement of the second kind with irreversible detection
- Decoherence in quantum measurement
- What physical interaction constitutes a measurement
- Does measurement actually determine the state
- Is the measurement process random or deterministic
- Does the measurement process violate locality
- Quotes
- References
Measurement from a practical point of view
Measurement plays an important role in quantum mechanics, and it is viewed in different ways among various interpretations of quantum mechanics. In spite of considerable philosophical differences, different views of measurement almost universally agree on the practical question of what results form a routine quantum-physics laboratory measurement. To understand this, the Copenhagen interpretation, which has been commonly used, is employed in this article.
Qualitative overview
In classical mechanics, a simple system consisting of only one single particle is fully described by the position
Once a quantum system has been prepared in laboratory, some measurable quantity such as position or energy is measured. For pedagogic reasons, the measurement is usually assumed to be ideally accurate. The state of a system after measurement is assumed to "collapse" into an eigenstate of the operator corresponding to the measurement. Repeating the same measurement without any evolution of the quantum state will lead to the same result. If the preparation is repeated, subsequent measurements will likely lead to different results.
The predicted values of the measurement are described by a probability distribution, or an "average" (or "expectation") of the measurement operator based on the quantum state of the prepared system. The probability distribution is either continuous (such as position and momentum) or discrete (such as spin), depending on the quantity being measured.
The measurement process is often considered as random and indeterministic. Nonetheless, there is considerable dispute over this issue. In some interpretations of quantum mechanics, the result merely appears random and indeterministic, whereas in other interpretations the indeterminism is core and irreducible. A significant element in this disagreement is the issue of "collapse of the wave function" associated with the change in state following measurement. There are many philosophical issues and stances (and some mathematical variations) taken—and near universal agreement that we do not yet fully understand quantum reality. In any case, our descriptions of dynamics involve probabilities, not certainties.
Quantitative details
The mathematical relationship between the quantum state and the probability distribution is, again, widely accepted among physicists, and has been experimentally confirmed countless times. This section summarizes this relationship, which is stated in terms of the mathematical formulation of quantum mechanics.
Measurable quantities ("observables") as operators
It is a postulate of quantum mechanics that all measurements have an associated operator (called an observable operator, or just an observable), with the following properties:
- The observable is a self-adjoint operator mapping a Hilbert space (namely, the state space, which consists of all possible quantum states) into itself.
- Thus, the observable's eigenvectors (called an eigenbasis) form an orthonormal basis that span the state space in which that observable exists. Any quantum state can be represented as a superposition of the eigenstates of an observable.
- Hermitian operators' eigenvalues are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable.
- For each eigenvalue there are one or more corresponding eigenvectors (eigenstates). A measurement results in the system being in the eigenstate corresponding to the eigenvalue result of the measurement. If the eigenvalue determined from the measurement corresponds to more than one eigenstate ("degeneracy"), instead of being in a definite state, the system is in a sub-space of the measurement operator corresponding to all the states having that eigenvalue.
Important examples of observables are:
Operators can be noncommuting. Two Hermitian operators commute if (and only if) there is at least one basis of vectors such that each of which is an eigenvector of both operators (this is sometimes called a simultaneous eigenbasis). Noncommuting observables are said to be incompatible and cannot in general be measured simultaneously. In fact, they are related by an uncertainty principle as discovered by Werner Heisenberg.
Measurement probabilities and wave function collapse
There are a few possible ways to mathematically describe the measurement process (both the probability distribution and the collapsed wave function). The most convenient description depends on the spectrum (i.e., set of eigenvalues) of the observable.
Discrete, nondegenerate spectrum
Let
Consider a system prepared in state
where
Usually
If the result of the measurement is
so any repeated measurement of
Continuous, nondegenerate spectrum
Let
Consider a system prepared in state
where
where
If the result of the measurement is
Alternatively, it is often possible and convenient to analyze a continuous-spectrum measurement by taking it to be the limit of a different measurement with a discrete spectrum. For example, an analysis of scattering involves a continuous spectrum of energies, but by adding a "box" potential (which bounds the volume in which the particle can be found), the spectrum becomes discrete. By considering larger and larger boxes, this approach need not involve any approximation, but rather can be regarded as an equally valid formalism in which this problem can be analyzed.
Degenerate spectra
If there are multiple eigenstates with the same eigenvalue (called degeneracies), the analysis is a bit less simple to state, but not essentially different. In the discrete case, for example, instead of finding a complete eigenbasis, it is a bit more convenient to write the Hilbert space as a direct sum of multiple eigenspaces. The probability of measuring a particular eigenvalue is the squared component of the state vector in the corresponding eigenspace, and the new state after measurement is the projection of the original state vector into the appropriate eigenspace.
Density matrix formulation
Instead of performing quantum-mechanics computations in terms of wave functions (kets), it is sometimes necessary to describe a quantum-mechanical system in terms of a density matrix. The analysis in this case is formally slightly different, but the physical content is the same, and indeed this case can be derived from the wave function formulation above. The result for the discrete, degenerate case, for example, is as follows:
Let
Assume the system is prepared in the state described by the density matrix ρ. Then measuring
where
Alternatively, one can say that the measurement process results in the new density matrix
where the difference is that
Statistics of measurement
As detailed above, the result of measuring a quantum-mechanical system is described by a probability distribution. Some properties of this distribution are as follows:
Suppose we take a measurement corresponding to observable
These are direct consequences of the above formulas for measurement probabilities.
Example
Suppose that we have a particle in a 1-dimensional box, set up initially in the ground state
Next suppose that the particle's position is measured. The position x will be measured with probability density
If the measurement result was x=S, then the wave function after measurement will be the position eigenstate
The new wave function
If we now leave this state alone, it will smoothly evolve in time according to the Schrödinger equation. But suppose instead that an energy measurement is immediately taken. Then the possible energy values
and moreover if the measurement result is
So in this example, due to the process of wave function collapse, a particle initially in the ground state can end up in any energy level, after just two subsequent non-commuting measurements are made.
Wave function collapse
The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured observable is called "collapse", or "wave function collapse". The final eigenstate appears randomly with a probability equal to the square of its overlap with the original state. The process of collapse has been studied in many experiments, most famously in the double-slit experiment. The wave function collapse raises serious questions regarding "the measurement problem", as well as questions of determinism and locality, as demonstrated in the EPR paradox and later in GHZ entanglement. (See below.)
In the last few decades, major advances have been made toward a theoretical understanding of the collapse process. This new theoretical framework, called quantum decoherence, supersedes previous notions of instantaneous collapse and provides an explanation for the absence of quantum coherence after measurement. Decoherence correctly predicts the form and probability distribution of the final eigenstates, and explains the apparent randomness of the choice of final state in terms of einselection.
von Neumann measurement scheme
The von Neumann measurement scheme, the ancestor of quantum decoherence theory, describes measurements by taking into account the measuring apparatus which is also treated as a quantum object.
"Measurement" of the first kind — premeasurement without detection
Let the quantum state be in the superposition
where
The transition
is often referred to as weak von Neumann projection, the wave function collapse or strong von Neumann projection
being thought to correspond to an additional selection of a subensemble by means of observation.
In case the measured observable has a degenerate spectrum, weak von Neumann projection is generalized to Lüders projection
in which the vectors
Measurement of the second kind — with irreversible detection
In a measurement of the second kind the unitary evolution during the interaction of object and measuring instrument is supposed to be given by
in which the states
Decoherence in quantum measurement
One can also introduce the interaction with the environment
which is related to the phenomenon of decoherence.
The above is completely described by the Schrödinger equation and there are not any interpretational problems with this. Now the problematic wave function collapse does not need to be understood as a process
represents a set of states that do not overlap in space, the appearance of collapse can be generated by either the Bohm interpretation or the Everett interpretation which both deny the reality of wave function collapse. Both of these are stated to predict the same probabilities for collapses to various states as the conventional interpretation by their supporters. The Bohm interpretation is held to be correct only by a small minority of physicists, since there are difficulties with the generalization for use with relativistic quantum field theory. However, there is no proof that the Bohm interpretation is inconsistent with quantum field theory, and work to reconcile the two is ongoing. The Everett interpretation easily accommodates relativistic quantum field theory.
What physical interaction constitutes a measurement?
Until the advent of quantum decoherence theory in the late 20th century, a major conceptual problem of quantum mechanics and especially the Copenhagen interpretation was the lack of a distinctive criterion for a given physical interaction to qualify as "a measurement" and cause a wave function to collapse. This is illustrated by the Schrödinger's cat paradox. Certain aspects of this question are now well understood in the framework of quantum decoherence theory, such as an understanding of weak measurements, and quantifying what measurements or interactions are sufficient to destroy quantum coherence. Nevertheless, there remains less than universal agreement among physicists on some aspects of the question of what constitutes a measurement.
Does measurement actually determine the state?
The question of whether (and in what sense) a measurement actually determines the state is one which differs among the different interpretations of quantum mechanics. (It is also closely related to the understanding of wave function collapse.) For example, in most versions of the Copenhagen interpretation, the measurement determines the state, and after measurement the state is definitely what was measured. But according to the many-worlds interpretation, measurement determines the state in a more restricted sense: In other "worlds", other measurement results were obtained, and the other possible states still exist.
Is the measurement process random or deterministic?
As described above, there is universal agreement that quantum mechanics appears random, in the sense that all experimental results yet uncovered can be predicted and understood in the framework of quantum mechanics measurements being fundamentally random. Nevertheless, it is not settled whether this is true, fundamental randomness, or merely "emergent" randomness resulting from underlying hidden variables which deterministically cause measurement results to happen a certain way each time. This continues to be an area of active research.
If there are hidden variables, Bell's theorem suggests they would have to be "nonlocal".
Does the measurement process violate locality?
In physics, the Principle of locality is the concept that information cannot travel faster than the speed of light (also see special relativity). It is known experimentally (see Bell's theorem, which is related to the EPR paradox) that if quantum mechanics is deterministic (due to hidden variables, as described above), and counterfactual definiteness is true, then it is nonlocal (i.e. violates the principle of locality). Nevertheless, there is not universal agreement among physicists on whether quantum mechanics is nondeterministic, nonlocal, or both.
Quotes
A measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement.