Measurement in quantum mechanics

The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications.

Measurable quantities ("observables") as operators
An observable quantity is represented mathematically by an Hermitian or self adjoint operator. The set of the operator's eigenvalues represent the set of possible outcomes of the measurement. For each eigenvalue there is a corresponding eigenstate (or "eigenvector"), which will be the state of the system after the measurement. Some properties of this representation are

Important examples are:
 * 1) The eigenvalues of Hermitian matrices are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable.
 * 2) A Hermitian matrix can be unitarily diagonalized (See Spectral theorem), generating an orthonormal basis of eigenvectors which spans the state space of the system. In general, the state of a system can be represented as a linear combination of eigenvectors of any Hermitian operator. Physically, this is to say that any state can be expressed as a superposition of the eigenstates of an observable.
 * 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: $$ {\hat p} = {\hbar \over i}{\partial \over \partial x} $$ (in the position basis).
 * The position operator: $$ {\hat x} $$, where $$ {\hat x} = {-\hbar \over i}{\partial \over \partial p} $$ (in the momentum basis).

Operators can be noncommuting. In the finite dimensional case, two Hermitian operators commute if they have the same set of {normalized} eigenvectors. Noncommuting observables are said to be incompatible and can not be measured simultaneously. This can be seen via the uncertainty principle.

Eigenstates and projection
Assume the system is prepared in state $$|\psi\rang$$. Let $$ {\hat O} $$ be a measurement operator, an observable, with eigenstates $$|n\rang $$ for $$ n = 1, 2, 3, ... $$ and corresponding eigenvalues $$O_1, O_2, O_3, ...$$. If the measurement outcome is $$ O_N$$, the system will then "collapse"  to the state $$ |N\rang $$ after measurement.

The case of a continuous spectrum is more involved, since, physically speaking, the basis has uncountably many eigenstates, but the general concept is the same. In the position representation, for instance, the eigenstates can be represented by the set of delta functions, indexed by all possible positions of the particle. In the experimental setting, the resolution of any given measurement is finite, and therefore the continuous space may be divided into discrete segments. Another solution is to approximate any lab experiments by a "box" potential (which bounds the volume in which the particle can be found, and thus ensures a discrete spectrum).

Wavefunction collapse

 * Given any quantum state which is a superposition of eigenstates at time t
 * $$ | \psi \rang = c_1 e^{-i E_1 t} | 1 \rang + c_2 e^{-i E_2 t} | 2 \rang + c_3 e^{-i E_3 t} | 3 \rang + \cdots \ \, $$
 * if we measure, for example, the energy of the system and receive E2
 * (this result is chosen randomly according to probability given by
 * $$ \Pr( E_n ) = \frac{ | c_n |^2 }{\sum_k | c_k |^2} $$),
 * then the system's quantum state after the measurement is
 * $$ | \psi \rang = e^{-i E_2 t} | 2 \rang $$
 * so any repeated measurement of energy will  yield E2.

''Figure 1. The process of wavefunction collapse illustrated.''

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 of determinism and locality, as demonstrated in the EPR paradox and later in GHZ entanglement.

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. While this theory correctly predicts the form and probability distribution of the final eigenstates, it does not explain the randomness inherent in the choice of final state.

There are two major approaches toward the "wavefunction collapse":
 * 1) Accept it as it is. This approach was supported by Niels Bohr and his Copenhagen interpretation which accepts the collapse as one of the elementary properties of nature (at least, for small enough systems). According to this, there is an inherent randomness embedded in nature, and physical observables exist only after they are measured (for example: as long as a particle's speed isn't measured it doesn't have any defined speed).
 * 2) Reject it as a physical process and relate to it only as an illusion. This approach says that there is no collapse at all, and we only think there is. Those who support this approach usually offer another interpretation of quantum mechanics, which avoids the wavefunction collapse.

von Neumann measurement scheme
The von Neumann measurement scheme, an 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 $$ |\psi\rang = \sum_n c_n |\psi_n\rang $$, where $$ |\psi_n\rang $$ are eigenstates of the operator that needs to be measured. In order to make the measurement, the measured system described by $$ |\psi\rang $$ needs to interact with the measuring apparatus described by the quantum state $$ |\phi\rang $$, so that the total wave function before the interaction is $$ |\psi\rang |\phi\rang $$. After the interaction, the total wave function exhibits the unitary evolution $$ |\psi\rang |\phi\rang \rightarrow \sum_n c_n |\psi_n\rang |\phi_n\rang $$, where $$ |\phi_n\rang$$ are orthonormal states of the measuring apparatus. The unitary evolution above is referred to as premeasurement. One can also introduce the interaction with the environment $$ |e\rang $$, so that, 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 $$ |\psi\rangle \rightarrow |\psi_n\rang $$ on the level of the measured system, but can also be understood as a process $$ |\phi\rangle \rightarrow |\phi_n\rang $$ on the level of the measuring apparatus, or as a process $$ |e\rangle \rightarrow |e_n\rang $$ 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; they both, though, predict the same probabilities for collapses to various states as does the conventional interpretation. 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.

Example
Suppose that we have a particle in a box. If the energy of the particle is measured to be $$ E_N = \frac{N^2\pi^2\hbar^2}{2mL^2} $$ then the corresponding state of the system is $$|\psi_N\rang = \int | x\rang \lang x|\psi_N\rang dx$$ where $$\lang x|\psi_N\rang = \lang x|N\rang = \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi x}{L}\right)$$, which is determined by solving the Time-Independent Schrödinger equation for the given potential.

Alternatively, if instead of knowing the energy of the particle the particle's position is determined to be a distance $$ S $$ from the left wall of the box, the corresponding system state is $$|\psi_S\rang = \int | x\rang \lang x|\psi_S\rang dx$$ where $$\lang x|\psi_S\rang = \lang x|S\rang = \delta( S - x ) $$.

These two state functions $$|\psi_N\rang$$ and $$|\psi_S\rang$$ are distinct functions (of the position $$ x $$ after we left multiply by the bra state $$\lang x|$$), but they are in general not orthogonal to each other:

$$ \lang \psi_S | \psi_N\rang = \lang S | N\rang = \int \lang S | x \rang \lang x | N\rang dx =\int_0^L ~\delta( S - x)~\sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi x}{L}\right) dx = \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi S}{L}\right) $$.

The two systems are therefore distinct; a position measurement is instantaneous whereas a definite value of energy $$ E_N $$ is established only in the limit of an infinitely long observation period.

Completeness of eigenvectors of Hermitian operators guarantees that either system state, being the eigenvector to one measurement operator, can be expressed as a linear combination of eigenvectors of the other measurement operator:

$$ |S\rang = \sum_n | n \rangle \left\langle n | S \right\rangle = \int \sum_n | x \rangle \langle x | n \rangle \left\langle n | S \right\rangle dx = \int | x \rangle \frac{2}{L}~\sum_n {\rm sin}\left(\frac{n \pi x}{L}\right)~{\rm sin}\left(\frac{n \pi S}{L}\right) dx = \int | x \rangle \delta( S - x ) dx $$, i.e. $$ |S\rang = \int | x \rangle \delta( S - x ) dx $$

and

$$ |N\rang = \int ~|s\rang \left\langle s | N \right\rangle ds = \int ~|s\rang \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi s}{L}\right) ds$$.

The time dependence of the system states is determined by the Time Dependent Schrödinger equation. In the preceding example, with energy eigenvalues $$E_n$$, it follows that the time dependent solution is

$$|\psi( t )\rang = \sum_n |n\rang \lang n|\psi_S\rang ~e^{-i t E_n/\hbar} $$,

where $$ t $$ represents the time since the particle's location in space was measured. Consequently

$$ \lang n|\psi( t )\rang = \lang n|\psi_S\rang ~e^{-i t E_n/\hbar} = \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{n \pi S}{L}\right)~e^{-i t E_n/\hbar} ~{\not =}~ 0 $$

at least for several distinct energy eigenstates $$|n\rang $$, for all values $$ t $$, and for all $$ 0 < S < L $$.

The particle state $$ |\psi_S \rang$$ therefore can not have evolved (in the above technical sense) into state $$ |\psi_N \rang $$ (which is orthogonal to all energy eigenstates, except itself), for any duration $$ t $$. While this conclusion may be characterized accordingly instead as "the wave function of the particle having been projected, or having collapsed into" the energy eigenstate $$ |\psi_N\rang $$, it is perhaps worth emphasizing that any definite value of energy $$ E_N $$ can be established only in the limit of a long-lasting trial and never for any finite value of time.

Optimal quantum measurement
What is the optimal quantum measurement to distinguish mixed states from a given ensemble? This is a natural question of which the solution is well understood, and is given by a semidefinite programming.

More specifically, suppose a mixed state $$\rho_i$$ is drawn from the ensemble with probability $$p_i$$, we wish to find a POVM measurement $$\{ \Pi_i \}$$ so that $$ \sum_i\ p_i \mathrm{tr}(\Pi_i\ \rho_i)$$ is maximized. This is clearly a semidefinit programming:

$$\mathrm{max}\ \sum_i\ p_i \mathrm{tr}(\Pi_i\ \rho_i)\quad\textrm{s.t.} \ \Pi_i\ge0, \ \sum_i \Pi_i=I.$$

Interestingly, the dual problem has a nice description:

$$ \mathrm{min}\ tr(X) \quad \textrm{s.t.}\ X-p_i\rho_i \ge0. $$

Let $$ \hat\Pi_i$$ and $$\hat X$$ be the solutions of the primal and the dual, we have

$$ \hat\Pi_i\cdot(\hat X - p_i\rho_i) = 0.$$

From this one can conclude that if all $$\rho_i$$ are pure states, then $$\hat\Pi_i$$ must also be of rank $$1$$. Furthermore, if $$\rho_i$$'s are in addition independent, then the optimal measure is a von Neumann measurement.

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 best illustrated by the Schrödinger's cat paradox.

Major philosophical and metaphysical questions surround this issue:


 * The concept of weak measurements.
 * Macroscopic systems (such as chairs or cats) do not exhibit counterintuitive quantum properties, which can only be observed in microscopic particles such as electrons or photons. This invites the question of when a system is "big enough" to behave classically and not quantum mechanically?

Quantum decoherence theory has successfully addressed other questions that previously haunted quantum measurement theory:


 * Does a measurement depend on the existence of a self-aware observer?


 * Answer: No. Coupling an isolated quantum system to another quantum system with many degrees of freedom generically transfers the coherence of the first system into mutual coherence of the two systems. The initially isolated quantum system then appears to "collapse." Interpreting the second system as a measurement apparatus, as in the von Neumann scheme, shows that no consciousness or self-awareness is necessary for collapse of the first system.


 * What interactions are strong enough to constitute a measurement?


 * This question is quantitatively answered by decoherence theory, given a model for the measurement apparatus. The scaling of the measurement effects with the system/apparatus interaction strength usually only weakly depends on the choice of a model for the apparatus, so one can give a generic description of the strength of a measurement induced by a given interaction.

Does measurement actually determine the state?
The question of whether a measurement actually determines the state, is deeply related to the Wavefunction collapse.

Most versions of the Copenhagen interpretation answer this question with an unqualified "yes".

See also:
 * Philosophies: Copenhagen interpretation.
 * People (actualist philosophers): Henri Poincaré, Niels Bohr.

The quantum entanglement problem
See EPR paradox.