Measurement in quantum mechanics

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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.

Measurement from a practical point of view

Measurement is viewed in different ways in the many interpretations of quantum mechanics; however, despite the considerable philosophical differences, they almost universally agree on the practical question of what results from a routine quantum-physics laboratory measurement. To describe this, a simple framework to use is the Copenhagen interpretation, and it will be implicitly used in this section; the utility of this approach has been verified countless times^{[citation needed]}, and all other interpretations are necessarily constructed so as to give the same quantitative predictions as this in almost every case.

Qualitative overview

The quantum state of a system is a mathematical object that fully describes the quantum system with respect to the attributes being modeled. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical representation of the quantum state then reflects that preparation. Once the quantum state has been prepared, some dynamical aspect of it is measured (for example, its position or energy). Typically, there is an assumption of ideally accurate measurements (for pedagogical convenience: real measuring devices are not 100% accurate), in which case the dynamic state of the system after measurement is assumed to be an eigenstate of the mathematical operator used to represent that measurement, with the eigenvalue that corresponds to the result of measurement. Thus, repeated measurement of the same dynamic variable, without any significant evolution of the quantum state, will produce the same result. If the preparation is repeated (and it does not put the system into an eigenstate of the measurement operator), subsequent measurements will likely produce different values.

The expected result of the measurement is in general described by a probability distribution of measurement results. The probability distribution is determined by an "average" or "expectation" of the measurement operator based on the quantum state of the prepared system. (This distribution can be either discrete or continuous, depending on what is being measured.)

The measurement process is often said to be random and indeterministic. (However, there is considerable dispute over this issue; in some interpretations of quantum mechanics, the result merely appears random and indeterministic, in other interpretations the indeterminism is core and irreducible.) A significant element in this disagreement is the issue of "collapse of the wavefunction" 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 and/or averages.

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

Main article: Observable

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:

1. The observable is a Hermitian (self-adjoint) operator mapping a Hilbert space (namely, the state space, which consists of all possible quantum states) into itself.
2. 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.
3. Hermitian operators' eigenvalues are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable.
4. 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:

• The Hamiltonian operator, representing the total energy of the system; with the special case of the nonrelativistic Hamiltonian operator: ${\hat H} = {\hat p^2 \over 2m} + V( \hat x )$.
• The momentum operator: ${\hat p} = -i\hbar {\partial \over \partial x}$ (in the position basis).
• The position operator: ${\hat x}$, where ${\hat x} = x$ (in the position basis), or ${\hat x}= i\hbar {\partial \over \partial p}$ (in the momentum basis).

Operators can be noncommuting. Two Hermitian operators commute if (and only if) there is at least one basis of vectors, 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 a direct consequence of the wave-like nature of the quantum postulate, and are associated with disturbance-due-to measurement due to the fundamental contributions of Werner Heisenberg.

Measurement probabilities and wavefunction collapse

There are a few possible ways to mathematically describe the measurement process (both the probability distribution and the collapsed wavefunction). The most convenient description depends on the spectrum (i.e., set of eigenvalues) of the observable.

Discrete, nondegenerate spectrum

Let ${\hat O}$ be an observable, and suppose that it has discrete eigenstates $|n \rang$ (in bra-ket notation) for $n = 1, 2, 3, \ldots$ and corresponding eigenvalues $O_1, O_2, O_3, \ldots$, no two of which are equal.

Assume the system is prepared in state $|\psi \rang$. Since the eigenstates of an observable form a basis (the eigenbasis), it follows that $|\psi\rang$ can be written in terms of the eigenstates as

$|\psi\rang = c_1 | 1 \rang + c_2 | 2 \rang + c_3 | 3 \rang + \cdots$

(where $c_1,c_2,\ldots$ are complex numbers). Then measuring ${\hat O}$ can yield any of the results $O_1, O_2, O_3, \ldots$, with corresponding probabilities given by

$\Pr( O_n ) = \frac{ | c_n |^2 }{\sum_k | c_k |^2} = \frac{ |\lang n | \psi \rang|^2}{\lang \psi | \psi\rang}$

Usually $|\psi\rang$ is assumed to be normalized, in which case this expression reduces to

$\Pr( O_n ) = | c_n |^2 = |\lang n | \psi \rang|^2$

If the result of the measurement is $O_n$, then the system's quantum state after the measurement is

$| \psi' \rang = | n \rang$

so any repeated measurement of ${\hat O}$ will yield the same result $O_n$. When there is a discontinuous change in state due to a measurement that involves discrete eigenvalues, that is called wavefunction collapse. For some, this is simply a description of a reasonably accurate discontinuous change in a mathematical representation of physical reality; for others, depending on philosophical orientation, this is a fundamentally serious problem with quantum theory.

Continuous, nondegenerate spectrum

Let ${\hat O}$ be an observable, and suppose that it has a continuous spectrum of eigenvalues filling the interval (a,b). Assume further that each eigenvalue x in this range is associated with a unique eigenstate $|x\rang$.

Assume the system is prepared in state $|\psi\rang$, which can be written in terms of the eigenbasis as

$|\psi\rang = \int_a^b c(x) | x \rang \, dx$

(where $c(x)$ is a complex-valued function). Then measuring ${\hat O}$ can yield a result anywhere in the interval (a,b), with probability density function $|c(x)|^2$; i.e., detecting the system between y and z will occur with probability

$\Pr( y<x<z ) = \frac{ \int_y^z | c(x) |^2 \, dx }{\int_a^b | c(x) |^2 \, dx}$

Again, $|\psi\rang$ is often assumed to be normalized, in which case this expression reduces to

$\Pr( y<x<z ) = \int_y^z | c(x) |^2 \, dx$

If the result of the measurement is x, then the new wave function will be

$|\psi'\rang = |x\rang.$

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 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

Main article: Density matrix

Instead of performing quantum-mechanics computations in terms of wavefunctions (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 wavefunction formulation above. The result for the discrete, degenerate case, for example, is as follows:

Let ${\hat O}$ be an observable, and suppose that it has discrete eigenvalues $O_1,O_2,O_3,\ldots$, associated with eigenspaces $V_1,V_2,\ldots$ respectively. Let $P_n$ be the projection operator into the space $V_n$.

Assume the system is prepared in the state described by the density matrix ρ. Then measuring ${\hat O}$ can yield any of the results $O_1, O_2, O_3, \ldots$, with corresponding probabilities given by

$\Pr( O_n ) = \mathrm{Tr}(P_n \rho)$

where $\mathrm{Tr}$ denotes trace. If the result of the measurement is n, then the new density matrix will be

$\rho' = \frac{P_n \rho P_n}{\mathrm{Tr}(P_n \rho)}$

Alternatively, one can say that the measurement process results in the new density matrix

$\rho'' = \sum_n P_n \rho P_n$

where the difference is that $\rho''$ is the density matrix describing the entire ensemble, whereas $\rho'$ is the density matrix describing the sub-ensemble whose measurement result was $n$.

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 $\hat O$, on a state whose quantum state is $|\psi\rang$.

• The mean (average) value of the measurement is (see Expectation value (quantum mechanics))

$\lang \psi | \hat O | \psi \rang$.

• The variance of the measurement is

$\lang \psi | \hat O^2 | \psi \rang - (\lang \psi | \hat O | \psi \rang)^2$

• The standard deviation of the measurement is

$\sqrt{\lang \psi | \hat O^2 | \psi \rang - (\lang \psi | \hat O | \psi \rang)^2}$

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 $|\psi_1\rang$. As can be computed from the time-independent Schrödinger equation, the energy of this state is $E_1=\frac{\pi^2\hbar^2}{2mL^2}$ (where m is the particle's mass and L is the box length), and the spatial wavefunction is $\lang x|\psi_1\rang = \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{\pi x}{L}\right)$. If the energy is now measured, the result will always certainly be $E_1$, and this measurement will not affect the wavefunction.

Next suppose that the particle's position is measured. The position x will be measured with probability density

$\Pr(S<x<S+dS) = \frac{2}{L}~{\rm sin}^2\left(\frac{\pi S}{L}\right)dS.$

If the measurement result was x=S, then the wavefunction after measurement will be the position eigenstate $|x=S\rang$. If the particle's position is immediately measured again, the same position will be obtained.

The new wavefunction $|x=S\rang$ can, like any wavefunction, be written as a superposition of eigenstates of any observable. In particular, using energy eigenstates, $| \psi_n\rang$, we have

$|x=S\rang = \sum_n | \psi_n \rangle \left\langle \psi_n | x=S \right\rangle = \sum_n | \psi_n \rangle \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{n \pi S}{L}\right)$

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 $E_n$ will be measured with relative probabilities:

$\Pr(E_n) = |\lang \psi_n | S \rang|^2 = \frac{2}{L}~{\rm sin}^2\left(\frac{n \pi S}{L}\right)$

and moreover if the measurement result is $E_n$, then the new state will be the energy eigenstate $|\psi_n\rang$.

So in this example, due to the process of wavefunction collapse, a particle initially in the ground state can end up in any energy level, after just two subsequent non-commuting measurements are made.

Wavefunction collapse

Main article: Wave function collapse

[This and the following sections need major overhaul. The author(s)did not have sufficient understanding to engage in this. (1) The double-slit experiment is primarily about wave-like interference, and sometimes a major player in discussions involving wave vs. particle like behavior (especially of photons). The issue of collapse applies to all measurements. (2) The language in description of von Neumann's measurement scheme needs a bit more clarity. ".. the operator that needs to be measured" needs better language {that has been fixed.} (3) In von Neumann's analysis of measurement, you show 2 different equations (A -> B) where a pure state becomes an entangled state. There is no clue how that happens, either physically or mathematically. (4) There is a system upon which a measurement operator represents a measurement, then there is the quantum representation of a classical device that produces a classical measurement such as a pointer reading—and that needs a different name. Etc. Most likely some symbols (e.g., M1 & M2 or CMD [classical measurement device], whatever) will need to be used in order to keep things clear. Whoever is doing this, please get someone who is knowledgeable to proof-read what you are writing. von Neumann's analysis of measurement is a serious challenge; it would be really worthwhile to have a real expert contribute to this!]

The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured observable is called "collapse", or "wavefunction 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 wavefunction 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. Let the quantum state be in the superposition $\scriptstyle |\psi\rang = \sum_n c_n |\psi_n\rang$, where $\scriptstyle |\psi_n\rang$ are eigenstates of the operator for the measurement prior to von Neumann's second apparatus. In order to make the measurement, the system described by $\scriptstyle |\psi\rang$ needs to interact with the measuring apparatus described by the quantum state $\scriptstyle |\phi\rang$, so that the total wave function before the measurement and interaction with the second apparatus is $\scriptstyle |\psi\rang |\phi\rang$. During the interaction of object and measuring instrument the unitary evolution is supposed to realize the following transition from the initial to the final total wave function: [This is confusing |psi> is the state prior to interaction with the original measuring device and with von Neumann's apparatus. The next equation shows the transition from the original unmeasured state to an entangled state involving both pieces of apparatus. It is not clear how that happens or how one can, by a unitary transformation, go from an original pure state to an entangled state. Without that, this is all smoke & mirrors.]

$|\psi\rang |\phi\rang \rightarrow \sum_n c_n |\psi_n\rang |\phi_n\rang \quad \text{(measurement of the first kind),}$

where $\scriptstyle |\phi_n\rang$ are orthonormal states of the measuring apparatus. The unitary evolution above is referred to as premeasurement. The relation with wave function collapse is established by calculating the final density operator of the object $\scriptstyle \sum_n |c_n|^2 |\psi_n\rang\lang \psi_n|$ from the final total wave function. This density operator is interpreted by von Neumann as describing an ensemble of objects being after the measurement with probability $\scriptstyle |c_n|^2$ in the state $\scriptstyle |\psi_n\rang.$

The transition

$|\psi\rang \rightarrow \sum_n |c_n|^2 |\psi_n\rang \lang \psi_n|$

is often referred to as weak von Neumann projection, the wave function collapse or strong von Neumann projection

$|\psi\rang \rightarrow \sum_n |c_n|^2 |\psi_n\rang \lang \psi_n| \rightarrow |\psi_n\rang$

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

$|\psi\rang \rightarrow \sum_n |c_n|^2 P_n,\; P_n = \sum_i |\psi_{ni}\rang \lang \psi_{ni}|,$

in which the vectors $\scriptstyle |\psi_{ni}\rang$ for fixed n are the degenerate eigenvectors of the measured observable. For an arbitrary state described by a density operator $\scriptstyle \rho$ Lüders projection is given by

$\rho \rightarrow \sum_n P_n \rho P_n.$

Measurements of the second kind

In a measurement of the second kind the unitary evolution during the interaction of object and measuring instrument is supposed to be given by

$|\psi\rang |\phi\rang \rightarrow \sum_n c_n |\chi_n\rang |\phi_n\rang,$

in which the states $\scriptstyle |\chi_n\rang$ of the object are determined by specific properties of the interaction between object and measuring instrument. They are normalized but not necessarily mutually orthogonal. The relation with wave function collapse is analogous to that obtained for measurements of the first kind, the final state of the object now being $\scriptstyle |\chi_n\rang$ with probability $\scriptstyle |c_n|^2.$ Note that many present-day measurement procedures are measurements of the second kind, some even functioning correctly only as a consequence of being of the second kind. For instance, a photon counter, detecting a photon by absorbing and hence annihilating it, thus ideally leaving the electromagnetic field in the vacuum state rather than in the state corresponding to the number of detected photons; also the Stern–Gerlach experiment would not function at all if it really were a measurement of the first kind.

Decoherence in quantum measurement

One can also introduce the interaction with the environment $\scriptstyle |e\rang$, so that, in a measurement of the first kind, after the interaction the total wave function takes a form

$\sum_n c_n |\psi_n\rang |\phi_n\rang |e_n \rang,$

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 wavefunction collapse does not need to be understood as a process $\scriptstyle |\psi\rangle \rightarrow |\psi_n\rang$ on the level of the measured system, but can also be understood as a process $\scriptstyle |\phi\rangle \rightarrow |\phi_n\rang$ on the level of the measuring apparatus, or as a process $\scriptstyle |e\rangle \rightarrow |e_n\rang$ on the level of the environment. Studying these processes provides considerable insight into the measurement problem by avoiding the arbitrary boundary between the quantum and classical worlds, though it does not explain the presence of randomness in the choice of final eigenstate. If the set of states

$\{ |\psi_n\rang\}$, $\{ |\phi_n\rang\}$, or $\{ |e_n\rang\}$

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 wavefunction 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.

Philosophical problems of quantum measurements

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 wavefunction to collapse. This is best 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 wavefunction 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, they would have to be "nonlocal".

Does the measurement process violate locality?

Main articles: Quantum nonlocality and Principle of 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), 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.

See also

• Measurement related problems and paradoxes
  • Afshar experiment
  • Measurement problem
  • Wavefunction collapse
  • Quantum Zeno effect
  • EPR paradox
  • Quantum pseudo-telepathy
  • Renninger negative-result experiment
  • Elitzur–Vaidman bomb-testing problem
  • Schrödinger's cat
  • Popper's experiment
• Interpretations of quantum mechanics
  • Transactional interpretation
  • Copenhagen interpretation
  • Many-worlds interpretation
  • Hidden variables theory
• Quantum mechanics formalism
  • Quantum mechanics
  • Mathematical formulation of quantum mechanics
  • Schrödinger equation
  • Bra-ket notation
  • Generalized measurement (POVM, Positive operator valued measure)

Further reading

• John A. Wheeler and Wojciech Hubert Zurek, eds. (1983). Quantum Theory and Measurement. Princeton University Press. ISBN 0-691-08316-9. 
• Vladimir B. Braginsky and Farid Ya. Khalili (1992). Quantum Measurement. Cambridge University Press. ISBN 0-521-41928-X. 
• George S. Greenstein and Arthur G. Zajonc (2006). The Quantum Challenge: Modern Research On The Foundations Of Quantum Mechanics (2nd ed.). ISBN 076372470X. 

External links

• "The Double Slit Experiment". (physicsweb.org)
• "Measurement in Quantum Mechanics" Henry Krips in the Stanford Encyclopedia of Philosophy
• Decoherence, the measurement problem, and interpretations of quantum mechanics
• Measurements and Decoherence
• The conditions for discrimination between quantum states with minimum error
• Quantum behavior of measurement apparatus