Measurement in quantum mechanics

Measurement in quantum mechanics

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The framework of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics
Interpretation of quantum mechanics
An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations...

, 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
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

, and it will be implicitly used in this section; the utility of this approach has been verified countless times, 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. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Once the quantum state has been prepared, some aspect of it is measured (for example, its position or energy). If the experiment is repeated, so as to measure the same aspect of the same quantum state prepared in the same way, the result of the measurement will often be different.

The expected result of the measurement is in general described by a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

that specifies the likelihoods that the various possible results will be obtained. (This distribution can be either discrete or continuous, depending on what is being measured.)

The measurement process is often said to be random
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

and indeterministic
Indeterminism
Indeterminism is the concept that events are not caused, or not caused deterministically by prior events. It is the opposite of determinism and related to chance...

. (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.) This is because an important aspect of measurement is wavefunction collapse
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...

, the nature of which varies according to the interpretation adopted.

What is universally agreed, however, is that if the measurement is repeated, without re-preparing the state, one finds the same result as the first measurement. As a result, after measuring some aspect of the quantum state, we normally update the quantum state to reflect the result of the measurement; it is this updating that ensures that if an immediate re-measurement is repeated without re-preparing the state, one finds the same result as the first measurement. The updating of the quantum state model is called wavefunction collapse.

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
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

.

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:
1. The observable is a Hermitian (self-adjoint
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

) operator mapping a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

(namely, the state space
State space (physics)
In physics, a state space is a complex Hilbert space within which the possible instantaneous states of the system may be described by a unit vector. These state vectors, using Dirac's bra-ket notation, can often be treated as vectors and operated on using the rules of linear algebra...

, which consists of all possible quantum states) into itself.
2. The observable's eigenvalues are real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

. The possible outcomes of the measurement are precisely the eigenvalues of the given observable.
3. For each eigenvalue there are one or more corresponding eigenvectors (which in this context are called eigenstates), which will make up the state of the system after the measurement.
4. The observable has a set of eigenvectors which span
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...

the state space. It follows that each observable generates an orthonormal basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

of eigenvectors (called an eigenbasis). Physically, this is the statement that any quantum state can always be represented as a superposition
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

of the eigenstates of an observable.

Important examples of observables are:
• The Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

operator, representing the total energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

of the system; with the special case of the nonrelativistic Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

operator: .
• The momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

operator: (in the position basis).
• The position operator
Position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L2, the Hilbert space of complex-valued and square-integrable ...

: , where (in the momentum basis).

Operators can be noncommuting
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

. 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
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

, as a consequence of the Robertson-Schrödinger relation.

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 be an observable, and suppose that it has discrete eigenstates (in bra-ket notation
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

) for and corresponding eigenvalues , no two of which are equal.

Assume the system is prepared in state . Since the eigenstates of an observable form a basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

(the eigenbasis), it follows that can be written in terms of the eigenstates as
(where are complex numbers). Then measuring can yield any of the results , with corresponding probabilities given by
Usually is assumed to be normalized, in which case this expression reduces to

If the result of the measurement is , then the system's quantum state after the measurement is
so any repeated measurement of will yield the same result . (This phenomenon is called wavefunction collapse
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...

.)
Continuous, nondegenerate spectrum

Let be an observable, and suppose that it has a continuous spectrum
Continuous spectrum
The spectrum of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum.If H is a topological vector space and A:H \to H is a linear map, the spectrum of A is the set of complex numbers \lambda such that A - \lambda I : H \to H is not...

of eigenvalues filling the interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

(a,b). Assume further that each eigenvalue x in this range is associated with a unique eigenstate .

Assume the system is prepared in state , which can be written in terms of the eigenbasis as
(where is a complex-valued function). Then measuring can yield a result anywhere in the interval (a,b), with probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

; i.e., a result between y and z will occur with probability
Again, is often assumed to be normalized, in which case this expression reduces to
If the result of the measurement is x, then the new wave function will be

Alternatively, it is often possible and convenient to analyze a continuous-spectrum measurement by taking it to be the limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

of a different measurement with a discrete spectrum. For example, an analysis of scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

involves a continuous spectrum of energies, but by adding a "box" potential
Particle in a box
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...

(which bounds the volume in which the particle can be found), the spectrum becomes discrete
Discrete spectrum
In physics, an elementary explanation of a discrete spectrum is that it is an emission spectrum or absorption spectrum for which there is only an integer number of intensities. Atomic electronic absorption and emission spectrum are discrete, as contrasted with, for example, the emission spectrum...

. 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
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....

of the original state vector into the appropriate eigenspace.
Density matrix formulation

Instead of performing quantum-mechanics computations in terms of wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

s (kets
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

), it is sometimes necessary to describe a quantum-mechanical system in terms of a density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

. 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 be an observable, and suppose that it has discrete eigenvalues , associated with eigenspaces respectively. Let be the projection operator
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....

into the space .

Assume the system is prepared in the state described by the density matrix ρ. Then measuring can yield any of the results , with corresponding probabilities given by
where Tr denotes trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

. If the result of the measurement is n, then the new density matrix will be
Alternatively, one can say that the measurement process results in the new density matrix
where the difference is that ρ ' ' is the density matrix describing the entire ensemble, whereas ρ ' 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 , on a state whose quantum state is .
• The mean
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

(average) value of the measurement is (see Expectation value (quantum mechanics)
Expectation value (quantum mechanics)
In quantum mechanics, the expectation value is the predicted mean value of the result of an experiment. Despite the name, it is not the most probable value of a measurement...

).
• The variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

of the measurement is
• The standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

of the measurement is

These are direct consequences of the above formulas for measurement probabilities.

Example

Suppose that we have a particle in a 1-dimensional box
Particle in a box
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...

, set up initially in the ground state . As can be computed from the time-independent Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

, the energy of this state is (where m is the particle's mass and L is the box length), and the spatial wavefunction is . If the energy is now measured, the result will always certainly be , 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
If the measurement result was x=S, then the wavefunction after measurement will be the position eigenstate . If the particle's position is immediately measured again, the same position will be obtained.

The new wavefunction can, like any wavefunction, be written as a superposition of eigenstates of any observable. In particular, using energy eigenstates, , we have
If we now leave this state alone, it will smoothly evolve in time according to the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

. But suppose instead that an energy measurement is immediately taken. Then the possible energy values will be measured with relative probabilities:
and moreover if the measurement result is , then the new state will be the energy eigenstate .

So in this example, due to the process of wavefunction collapse
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...

, a particle initially in the ground state can end up in any energy level, after just two subsequent non-commuting
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

Wavefunction collapse

The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...

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
Double-slit experiment
The double-slit experiment, sometimes called Young's experiment, is a demonstration that matter and energy can display characteristics of both waves and particles...

. The wavefunction collapse raises serious questions regarding "the measurement problem", as well as, questions of determinism
Determinism
Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...

and locality
Principle of locality
In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...

, as demonstrated in the EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

and later in GHZ entanglement
Greenberger-Horne-Zeilinger state
In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state which involves at least three subsystems . It was first studied by D. Greenberger, M.A. Horne and Anton Zeilinger in 1989...

. (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
Quantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...

, 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.

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 , where are eigenstates of the operator that needs to be measured. In order to make the measurement, the measured system described by needs to interact with the measuring apparatus described by the quantum state , so that the total wave function before the interaction is . During the interaction of object and measuring instrument the unitary
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...

evolution is supposed to realize the following transition from the initial to the final total wave function:
where 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 from the final total wave function the final density operator of the object as This density operator is interpreted by von Neumann as describing an ensemble of objects being after the measurement with probability in the state

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 for fixed n are the degenerate eigenvectors of the measured observable. For an arbitrary state described by a density operator
Lüders projection is given by

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

in which the states 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 with probability 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).
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Decoherence in quantum measurement

One can also introduce the interaction with the environment , so that, in a measurement of the first kind, after the interaction the total wave function takes a form
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 on the level of the measured system, but can also be understood as a process on the level of the measuring apparatus, or as a process on the level of the environment. Studying these processes provides considerable insight into the measurement problem
Measurement problem
The measurement problem in quantum mechanics is the unresolved problem of how wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer...

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
, , or
represents a set of states that do not overlap in space, the appearance of collapse can be generated by either the Bohm interpretation
Bohm interpretation
The de Broglie–Bohm theory, also called the pilot-wave theory, Bohmian mechanics, and the causal interpretation, is an interpretation of quantum theory. In addition to a wavefunction on the space of all possible configurations, it also includes an actual configuration, even in situations where...

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
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

. 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
Quantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...

theory in the late 20th century, a major conceptual problem of quantum mechanics and especially the Copenhagen interpretation
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

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
Schrödinger's cat
Schrödinger's cat is a thought experiment, usually described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The scenario presents a cat that might be...

paradox. Certain aspects of this question are now well understood in the framework of quantum decoherence theory, such as an understanding of weak measurement
Weak measurement
Weak measurements are a type of quantum measurement, where the measured system is very weakly coupled to the measuring device. After the measurement the measuring device pointer is shifted by what is called the "weak value". So that a pointer initially pointing at zero before the measurement would...

s, 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
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

, the measurement determines the state, and after measurement the state is definitely what was measured. But according to the Many-worlds interpretation
Many-worlds interpretation
The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction, but denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternative histories and futures are real, each representing an...

, 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
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...

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
Principle of locality
In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...

", see below.)

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
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

(also see special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

). It is known experimentally (see Bell's theorem
Bell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...

, which is related to the EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

) 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.

• Measurement related problems and paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...

es
• Afshar experiment
Afshar experiment
The Afshar experiment is an optical experiment, devised and carried out by Shahriar Afshar in 2001, which investigates the principle of complementarity in quantum mechanics...

• Measurement problem
Measurement problem
The measurement problem in quantum mechanics is the unresolved problem of how wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer...

• Wavefunction collapse
• Quantum Zeno effect
Quantum Zeno effect
The quantum Zeno effect is a name coined by George Sudarshan and Baidyanath Misra of the University of Texas in 1977 in their analysis of the situation in which an unstable particle, if observed continuously, will never decay. One can nearly "freeze" the evolution of the system by measuring it...

The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

• Quantum pseudo-telepathy
Quantum pseudo-telepathy
Quantum pseudo-telepathy is a phenomenon in quantum game theory resulting in anomalously high success rates in coordination games between separated players. These high success rates would require communication between the players in a purely classical world; however, the game is set up such that...

• Renninger negative-result experiment
Renninger negative-result experiment
In quantum mechanics, the Renninger negative-result experiment is a thought experiment that illustrates some of the difficulties of understanding the nature of wave function collapse and measurement in quantum mechanics...

• Elitzur–Vaidman bomb-testing problem
• Schrödinger's cat
Schrödinger's cat
Schrödinger's cat is a thought experiment, usually described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The scenario presents a cat that might be...

• Popper's experiment
Popper's experiment
Popper's experiment is an experiment proposed by the 20th century philosopher of science Karl Popper, an advocate of an objective interpretation of quantum mechanics. He wanted to test the Copenhagen interpretation, a popular subjectivist interpretation of quantum mechanics...

• Interpretations of quantum mechanics
Interpretation of quantum mechanics
An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations...

• Transactional interpretation
Transactional interpretation
The transactional interpretation of quantum mechanics describes quantum interactions in terms of a standing wave formed by retarded and advanced waves. It was first proposed in 1986 by John G...

• Copenhagen interpretation
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

• Many-worlds interpretation
Many-worlds interpretation
The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction, but denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternative histories and futures are real, each representing an...

• Hidden variables theory
• Quantum mechanics formalism
• Quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

• Mathematical formulation of quantum mechanics
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

• Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

• Bra-ket notation
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

• Generalized measurement
POVM
In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...

(POVM, Positive operator valued measure)

• John A. Wheeler and Wojciech Hubert Zurek  (eds), Quantum Theory and Measurement, Princeton University Press
Princeton University Press