Pain and pleasure

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Pain_and_pleasure

Some philosophers, such as Jeremy Bentham, Baruch Spinoza, and Descartes, have hypothesized that the feelings of pain (or suffering) and pleasure are part of a continuum.

Perception of pain

Sensory input system

From a stimulus-response perspective, the perception of physical pain starts with the nociceptors, a type of physiological receptor that transmits neural signals to the brain when activated. These receptors are commonly found in the skin, membranes, deep fascias, mucosa, connective tissues of visceral organs, ligaments and articular capsules, muscles, tendons, periosteum, and arterial vessels. Once stimuli are received, the various afferent action potentials are triggered and pass along various fibers and axons of these nociceptive nerve cells into the dorsal horn of the spinal cord through the dorsal roots. A neuroanatomical review of the pain pathway, "Afferent pain pathways" by Almeida, describes various specific nociceptive pathways of the spinal cord: spinothalamic tract, spinoreticular tract, spinomesencephalic tract, spinoparabrachial tract, spinohypothalamic tract, spinocervical tract, postsynaptic pathway of the spinal column.

Neural coding and modulation

Activity in many parts of the brain is associated with pain perception. Some of the known parts for the ascending pathway include the thalamus, hypothalamus, midbrain, lentiform nucleus, somatosensory cortices, insular, prefrontal, anterior and parietal cingulum.

Perception of pleasure

Pleasure can be considered from many different perspectives, from physiological (such as the hedonic hotspots that are activated during the experience) to psychological (such as the study of behavioral responses towards reward). Pleasure has also often been compared to, or even defined by many neuroscientists as, a form of alleviation of pain.

Neural coding and modulation

Pleasure has been studied in the systems of taste, olfaction, auditory (musical), visual (art), and sexual activity. Neural hotspots involved in the processing of pleasure include the nucleus accumbens, posterior ventral pallidum, amygdala, other cortical and subcortical regions. The prefrontal and limbic regions of the neocortex, particularly the orbitofrontal region of the prefrontal cortex, anterior cingulate cortex, and the insular cortex have all been suggested to be pleasure causing substrates in the brain.

Psychology of pain and pleasure (reward-punishment system)

One approach to evaluating the relationship between pain and pleasure is to consider these two systems as a reward-punishment based system. When pleasure is perceived, one associates it with reward. When pain is perceived, one associates with punishment. Evolutionarily, this makes sense, because often, actions that result in pleasure or chemicals that induce pleasure work towards restoring homeostasis in the body. For example, when the body is hungry, the pleasure of rewarding food to one-self restores the body back to a balanced state of replenished energy. Like so, this can also be applied to pain, because the ability to perceive pain enhances both avoidance and defensive mechanisms that were, and still are, necessary for survival.

Opioid and dopamine systems in pain and pleasure

The neural systems to be explored when trying to look for a neurochemical relationship between pain and pleasure are the opioid and dopamine systems. The opioid system is responsible for the actual experience of the sensation, whereas the dopamine system is responsible for the anticipation or expectation of the experience. Opioids work in the modulation of pleasure or pain relief by either blocking neurotransmitter release or by hyperpolarizing neurons by opening up a potassium channel which effectively temporarily blocks the neuron.

Pain and pleasure on a continuum

Arguments for pain and pleasure on a continuum

It has been suggested as early as 4th century BC that pain and pleasure occurs on a continuum. Aristotle claims this antagonistic relationship in his Rhetoric:

"We may lay it down that Pleasure is a movement, a movement by which the soul as a whole is consciously brought into its normal state of being; and that Pain is the opposite."^[7]

Common neuroanatomy

On an anatomical level, it can be shown the source for the modulation of both pain and pleasure originates from neurons in the same locations, including the amygdala, the pallidum, and the nucleus accumbens. Not only have Siri Leknes and Irene Tracey, two neuroscientists who study pain and pleasure, concluded that pain and reward processing involve many of the same regions of the brain, but also that the functional relationship lies in that pain decreases pleasure and rewards increase analgesia, which is the relief from pain.

Arguments against pain and pleasure on a continuum

Asymmetry between pain and pleasure

Thomas Szasz notes that although we often refer to pain and pleasure as opposites in such a way, that this is incorrect; we have receptors for pain, but none in the same way for pleasure; and so it makes sense to ask "where is the pain?" but not "where is the pleasure?". With this vantage point established, the author delves into the topics of metaphorical pain and of legitimacy, of power relations, and of communications, and of myriad others.

Evolutionary hypotheses for the relationship between pain and pleasure

South African neuroscientists presented evidence that there was a physiological link on a continuum between pain and pleasure in 1980. First, the neuroscientists, Mark Gillman and Fred Lichtigfeld demonstrated that there were two endogenous endorphin systems, one pain producing and the other pain relieving. A short time later they showed that these two systems might also be involved in addiction, which is initially pursued, presumably for the pleasure generating or pain relieving actions of the addictive substance. Soon after they provided evidence that the endorphins system was involved in sexual pleasure.

Opponent process theory

The opponent-process theory is a model that views two components as being pairs that are opposite to each other, such that if one component is experienced, the other component will be repressed. Therefore, an increase in pain should bring about a decrease in pleasure, and a decrease in pain should bring about an increase in pleasure or pain relief. This simple model serves the purpose of explaining the evolutionarily significant role of homeostasis in this relationship. This is evident since both seeking pleasure and avoiding pain are important for survival. Leknes and Tracey provide an example:

"In the face of a large food reward, which can only be obtained at the cost of a small amount of pain, for instance, it would be beneficial if the pleasurable food reduced pain unpleasantness."

They then suggest that perhaps a common currency for which human beings determine the importance of the motivation for each perception can allow them to be weighed against each other in order to make a decision best for survival.

Motivation-decision model

The Motivation-Decision Model, suggested by Howard L. Fields, is centered around the concept that decision processes are driven by motivations of highest priority. The model predicts that in the case that there is anything more important than pain for survival will cause the human body to mediate pain by activating the descending pain modulation system described earlier.

Clinical applications

Related diseases

The following neurological and/or mental diseases have been linked to forms of pain or anhedonia: schizophrenia, depression, addiction, cluster headache, chronic pain.

Animal trials

A great deal of what is known about pain and pleasure today primarily comes from studies conducted with rats and primates.

Insertion of electrode during deep brain stimulation surgery using a stereotactic frame

Deep brain stimulation

Deep brain stimulation involves the electrical stimulation of deep brain structures by electrodes implanted into the brain. The effects of this neurosurgery has been studied in patients with Parkinson's disease, tremors, dystonia, epilepsy, depression, obsessive-compulsive disorder, Tourette's syndrome, cluster headache and chronic pain. A fine electrode is inserted into the targeted area of the brain and secured to the skull. This is attached to a pulse generator which is implanted elsewhere on the body under the skin. The surgeon then turns the frequency of the electrode to the voltage and frequency desired. Deep brain stimulation has been shown in several studies to both induce pleasure or even addiction as well as ameliorate pain. For chronic pain, lower frequencies (about 5–50 Hz) have produced analgesic effects, whereas higher frequencies (about 120–180 Hz) have alleviated or stopped pyramidal tremors in Parkinson's patients.

There is still further research necessary into how and why exactly DBS works. However, by understanding the relationship between pleasure and pain, procedures like these can be used to treat patients suffering from a high intensity or longevity of pain. So far, DBS has been recognized as a treatment for Parkinson's disease, tremors, and dystonia by the Food and Drug Administration (FDA).

Phenomenology

Valence is an inferred criterion from instinctively generated emotions; it is the property specifying whether feelings/affects are positive, negative or neutral. The existence of at least temporarily unspecified valence is an issue for psychological researchers who reject the existence of neutral emotions (e.g. surprise, sublimation). However, other psychological researchers assume that neutral emotions exist. Two contrasting views in the phenomenology of valence are that of a constrained valence psychology, where the most intense experiences are generally no more than 10 times more intense than the mildest, and the Heavy-Tailed Valence hypothesis, which states that the range of possible degrees of valence is far more extreme.

Some philosophers question whether the structure of affective experience supports a strict positive-negative valence binary. For example, it has been argued that while suffering is clearly negatively valenced, introspective attempts to identify a phenomenologically opposite state—such as “anti-suffering”—fail to reveal a distinct experiential counterpart. This suggests that valence may not always correspond to simple oppositional categories. Rather than a linear scale, emotional valence might reflect a more complex and asymmetrical space of affective states, where the absence of suffering is not necessarily equivalent to the presence of pleasure.

Transhumanism

Transhumanist philosophers such as David Pearce and Mark Alan Walker have argued that future technologies will eventually make it feasible to eradicate suffering entirely and artificially induce states of perpetual bliss. Walker coined the term "biohappiness" to describe the idea of directly manipulating the biological roots of happiness in order to increase it. Pearce argues that suffering could eventually be eradicated entirely, stating that: "It is predicted that the world's last unpleasant experience will be a precisely dateable event." Proposed technological methods of overcoming the hedonic treadmill include wireheading (direct brain stimulation for uniform bliss), which undermines motivation and evolutionary fitness; designer drugs, offering sustainable well-being without side effects, though impractical for lifelong reliance; and genetic engineering, the most promising approach. Pearce argues that physical pain could be replaced with "gradients of bliss" that provide the same functionality of pain, e.g. avoiding injury, but without the suffering. Genetic recalibration through hyperthymia-promoting genes could raise hedonic set-points, fostering adaptive well-being, creativity, and productivity while maintaining responsiveness to stimuli. While scientifically achievable, this transformation requires careful ethical and societal considerations to navigate its profound implications

Measurement in quantum mechanics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics

In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic.

The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about local hidden variables within quantum systems.

Measuring a quantum system generally changes the quantum state that describes that system. This is a central feature of quantum mechanics, one that is both mathematically intricate and conceptually subtle. The mathematical tools for making predictions about what measurement outcomes may occur, and how quantum states can change, were developed during the 20th century and make use of linear algebra and functional analysis. Quantum physics has proven to be an empirical success and to have wide-ranging applicability.

On a more philosophical level, debates continue about the meaning of the measurement concept. The different interpretations of quantum mechanics, concern of solving what is known as the measurement problem.

Mathematical formalism

"Observables" as self-adjoint operators

For broader coverage of this topic, see Observable (quantum mechanics).

Main article: Canonical quantization

Further information: Dirac–von Neumann axioms

In quantum mechanics, each physical system is associated with a Hilbert space, each element of which represents a possible state of the physical system. The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable". These observables play the role of measurable quantities familiar from classical physics: position, momentum, energy, angular momentum and so on. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom. Many treatments of the theory focus on the finite-dimensional case, as the mathematics involved is somewhat less demanding. Indeed, introductory physics texts on quantum mechanics often gloss over mathematical technicalities that arise for continuous-valued observables and infinite-dimensional Hilbert spaces, such as the distinction between bounded and unbounded operators; questions of convergence (whether the limit of a sequence of Hilbert-space elements also belongs to the Hilbert space), exotic possibilities for sets of eigenvalues, like Cantor sets; and so forth. These issues can be satisfactorily resolved using spectral theory; the present article will avoid them whenever possible.

Projective measurement

See also: Projection-valued measure

The eigenvectors of a von Neumann observable form an orthonormal basis for the Hilbert space, and each possible outcome of that measurement corresponds to one of the vectors comprising the basis. A density operator is a positive-semidefinite operator on the Hilbert space whose trace is equal to 1. For each measurement that can be defined, the probability distribution over the outcomes of that measurement can be computed from the density operator. The procedure for doing so is the Born rule, which states that

${\displaystyle P(x_i)=\mathrm{tr}({\mathrm{\Pi }}_i\rho ),}$

where ${\displaystyle \rho }$ is the density operator, and ${\displaystyle {\mathrm{\Pi }}_i}$ is the projection operator onto the basis vector corresponding to the measurement outcome ${\displaystyle x_i}$. The average of the eigenvalues of a von Neumann observable, weighted by the Born rule probabilities, is the expectation value of that observable. For an observable ${\displaystyle A}$, the expectation value given a quantum state ${\displaystyle \rho }$ is

${\displaystyle ⟨A⟩=\mathrm{tr}(A\rho ).}$

A density operator that is a rank-1 projection is known as a pure quantum state, and all quantum states that are not pure are designated mixed. Pure states are also known as wavefunctions. Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system (i.e., ${\displaystyle P(x)=1}$ for some outcome ${\displaystyle x}$). Any mixed state can be written as a convex combination of pure states, though not in a unique way. The state space of a quantum system is the set of all states, pure and mixed, that can be assigned to it.

The Born rule associates a probability with each unit vector in the Hilbert space, in such a way that these probabilities sum to 1 for any set of unit vectors comprising an orthonormal basis. Moreover, the probability associated with a unit vector is a function of the density operator and the unit vector, and not of additional information like a choice of basis for that vector to be embedded in. Gleason's theorem establishes the converse: all assignments of probabilities to unit vectors (or, equivalently, to the operators that project onto them) that satisfy these conditions take the form of applying the Born rule to some density operator.

Generalized measurement (POVM)

Main article: POVM

In functional analysis and quantum measurement theory, a positive-operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVMs) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs. In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see Schrödinger–HJW theorem); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics, and can also be used in quantum field theory. They are extensively used in the field of quantum information.

In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices ${\displaystyle \{F_i\}}$ on a Hilbert space ${\displaystyle \mathcal{H}}$ that sum to the identity matrix,

${\displaystyle \sum _{i=1}^n{F}_i=\mathrm{I}.}$

In quantum mechanics, the POVM element ${\displaystyle F_i}$ is associated with the measurement outcome ${\displaystyle i}$, such that the probability of obtaining it when making a measurement on the quantum state ${\displaystyle \rho }$ is given by

${\displaystyle \text{Prob}(i)=\mathrm{tr}(\rho F_i)}$,

where ${\displaystyle \mathrm{tr}}$ is the trace operator. When the quantum state being measured is a pure state ${\displaystyle |\psi ⟩}$ this formula reduces to

${\displaystyle \text{Prob}(i)=\mathrm{tr}(|\psi ⟩⟨\psi |F_i)=⟨\psi |F_i|\psi ⟩}$.

State change due to measurement

Main article: Quantum operation

A measurement upon a quantum system will generally bring about a change of the quantum state of that system. Writing a POVM does not provide the complete information necessary to describe this state-change process. To remedy this, further information is specified by decomposing each POVM element into a product:

${\displaystyle E_i=A_i^{†}{A}_i.}$

The Kraus operators ${\displaystyle A_i}$, named for Karl Kraus, provide a specification of the state-change process. They are not necessarily self-adjoint, but the products ${\displaystyle A_i^{†}{A}_i}$ are. If upon performing the measurement the outcome ${\displaystyle E_i}$ is obtained, then the initial state ${\displaystyle \rho }$ is updated to

${\displaystyle \rho \to {\rho }^{\prime }=\frac{A_i\rho A_i^{†}}{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}(i)}=\frac{A_i\rho A_i^{†}}{\mathrm{tr}(\rho E_i)}.}$

An important special case is the Lüders rule, named for Gerhart Lüders. If the POVM is itself a PVM, then the Kraus operators can be taken to be the projectors onto the eigenspaces of the von Neumann observable:

${\displaystyle \rho \to {\rho }^{\prime }=\frac{{\mathrm{\Pi }}_i\rho {\mathrm{\Pi }}_i}{\mathrm{tr}(\rho {\mathrm{\Pi }}_i)}.}$

If the initial state ${\displaystyle \rho }$ is pure, and the projectors ${\displaystyle {\mathrm{\Pi }}_i}$ have rank 1, they can be written as projectors onto the vectors ${\displaystyle |\psi ⟩}$ and ${\displaystyle |i⟩}$, respectively. The formula simplifies thus to

${\displaystyle \rho =|\psi ⟩⟨\psi |\to {\rho }^{\prime }=\frac{|i⟩⟨i|\psi ⟩⟨\psi |i⟩⟨i|}{|⟨i|\psi ⟩{|}^2}=|i⟩⟨i|.}$

Lüders rule has historically been known as the "reduction of the wave packet" or the "collapse of the wavefunction". The pure state ${\displaystyle |i⟩}$ implies a probability-one prediction for any von Neumann observable that has ${\displaystyle |i⟩}$ as an eigenvector. Introductory texts on quantum theory often express this by saying that if a quantum measurement is repeated in quick succession, the same outcome will occur both times. This is an oversimplification, since the physical implementation of a quantum measurement may involve a process like the absorption of a photon; after the measurement, the photon does not exist to be measured again.

We can define a linear, trace-preserving, completely positive map, by summing over all the possible post-measurement states of a POVM without the normalisation:

${\displaystyle \rho \to \sum _i{A}_i\rho A_i^{†}.}$

It is an example of a quantum channel, and can be interpreted as expressing how a quantum state changes if a measurement is performed but the result of that measurement is lost.

Examples

Bloch sphere representation of states (in blue) and optimal POVM (in red) for unambiguous quantum state discrimination on the states ${\displaystyle |\psi ⟩=|0⟩}$ and ${\displaystyle |\phi ⟩=(|0⟩+|1⟩)/\sqrt{2}}$. Note that on the Bloch sphere orthogonal states are antiparallel.

The prototypical example of a finite-dimensional Hilbert space is a qubit, a quantum system whose Hilbert space is 2-dimensional. A pure state for a qubit can be written as a linear combination of two orthogonal basis states ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$ with complex coefficients:

${\displaystyle |\psi ⟩=\alpha |0⟩+\beta |1⟩}$

A measurement in the ${\displaystyle (|0⟩,|1⟩)}$ basis will yield outcome ${\displaystyle |0⟩}$ with probability ${\displaystyle |\alpha {|}^2}$ and outcome ${\displaystyle |1⟩}$ with probability ${\displaystyle |\beta {|}^2}$, so by normalization,

${\displaystyle |\alpha {|}^2+|\beta {|}^2=1.}$

An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for ${\displaystyle 2\times 2}$ self-adjoint matrices:

${\displaystyle \rho =\frac{1}{2}\left(I+r_x{\sigma }_x+r_y{\sigma }_y+r_z{\sigma }_z\right),}$

where the real numbers ${\displaystyle (r_x,r_y,r_z)}$ are the coordinates of a point within the unit ball and

${\displaystyle {\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).}$

POVM elements can be represented likewise, though the trace of a POVM element is not fixed to equal 1. The Pauli matrices are traceless and orthogonal to one another with respect to the Hilbert–Schmidt inner product, and so the coordinates ${\displaystyle (r_x,r_y,r_z)}$ of the state ${\displaystyle \rho }$ are the expectation values of the three von Neumann measurements defined by the Pauli matrices. If such a measurement is applied to a qubit, then by the Lüders rule, the state will update to the eigenvector of that Pauli matrix corresponding to the measurement outcome. The eigenvectors of ${\displaystyle {\sigma }_z}$ are the basis states ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$, and a measurement of ${\displaystyle {\sigma }_z}$ is often called a measurement in the "computational basis." After a measurement in the computational basis, the outcome of a ${\displaystyle {\sigma }_x}$ or ${\displaystyle {\sigma }_y}$ measurement is maximally uncertain.

A pair of qubits together form a system whose Hilbert space is 4-dimensional. One significant von Neumann measurement on this system is that defined by the Bell basis, a set of four maximally entangled states:

${\displaystyle \begin{array}{rl}|{\mathrm{\Phi }}^{+}⟩& =\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |0{⟩}_B+|1{⟩}_A\otimes |1{⟩}_B)\\ |{\mathrm{\Phi }}^{-}⟩& =\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |0{⟩}_B-|1{⟩}_A\otimes |1{⟩}_B)\\ |{\mathrm{\Psi }}^{+}⟩& =\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |1{⟩}_B+|1{⟩}_A\otimes |0{⟩}_B)\\ |{\mathrm{\Psi }}^{-}⟩& =\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |1{⟩}_B-|1{⟩}_A\otimes |0{⟩}_B)\end{array}}$

Probability density ${\displaystyle P_n(x)}$ for the outcome of a position measurement given the energy eigenstate ${\displaystyle |n⟩}$ of a 1D harmonic oscillator

A common and useful example of quantum mechanics applied to a continuous degree of freedom is the quantum harmonic oscillator. This system is defined by the Hamiltonian

${\displaystyle H=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{x}^2,}$

where ${\displaystyle H}$, the momentum operator ${\displaystyle p}$ and the position operator ${\displaystyle x}$ are self-adjoint operators on the Hilbert space of square-integrable functions on the real line. The energy eigenstates solve the time-independent Schrödinger equation:

${\displaystyle H|n⟩=E_n|n⟩.}$

These eigenvalues can be shown to be given by

${\displaystyle E_n=\hslash \omega \left(n+\frac{1}{2}\right),}$

and these values give the possible numerical outcomes of an energy measurement upon the oscillator. The set of possible outcomes of a position measurement on a harmonic oscillator is continuous, and so predictions are stated in terms of a probability density function ${\displaystyle P(x)}$ that gives the probability of the measurement outcome lying in the infinitesimal interval from ${\displaystyle x}$ to ${\displaystyle x+dx}$.

History of the measurement concept

The "old quantum theory"

Main article: Old quantum theory

The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as a semi-classical approximation to modern quantum mechanics. Notable results from this period include Max Planck's calculation of the blackbody radiation spectrum, Albert Einstein's explanation of the photoelectric effect, Einstein and Peter Debye's work on the specific heat of solids, Niels Bohr and Hendrika van Leeuwen's proof that classical physics cannot account for magnetism, Bohr's model of the hydrogen atom and Arnold Sommerfeld's extension of the Bohr model to include relativistic effects.

Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result.

The Stern–Gerlach experiment, proposed in 1921 and implemented in 1922, became a prototypical example of a quantum measurement having a discrete set of possible outcomes. In the original experiment, silver atoms were sent through a spatially varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment are deflected, due to the magnetic field gradient, from a straight path. The screen reveals discrete points of accumulation, rather than a continuous distribution, owing to the particles' quantized spin.

Transition to the "new" quantum theory

A 1925 paper by Werner Heisenberg, known in English as "Quantum theoretical re-interpretation of kinematic and mechanical relations", marked a pivotal moment in the maturation of quantum physics. Heisenberg sought to develop a theory of atomic phenomena that relied only on "observable" quantities. At the time, and in contrast with the later standard presentation of quantum mechanics, Heisenberg did not regard the position of an electron bound within an atom as "observable". Instead, his principal quantities of interest were the frequencies of light emitted or absorbed by atoms.

The uncertainty principle dates to this period. It is frequently attributed to Heisenberg, who introduced the concept in analyzing a thought experiment where one attempts to measure an electron's position and momentum simultaneously. However, Heisenberg did not give precise mathematical definitions of what the "uncertainty" in these measurements meant. The precise mathematical statement of the position-momentum uncertainty principle is due to Earle Hesse Kennard, Wolfgang Pauli, and Hermann Weyl, and its generalization to arbitrary pairs of noncommuting observables is due to Howard P. Robertson and Erwin Schrödinger.

Writing ${\displaystyle x}$ and ${\displaystyle p}$ for the self-adjoint operators representing position and momentum respectively, a standard deviation of position can be defined as

${\displaystyle {\sigma }_x=\sqrt{⟨x^2⟩-⟨x{⟩}^2},}$

and likewise for the momentum:

${\displaystyle {\sigma }_p=\sqrt{⟨p^2⟩-⟨p{⟩}^2}.}$

The Kennard–Pauli–Weyl uncertainty relation is

${\displaystyle {\sigma }_x{\sigma }_p\ge \frac{\hslash }{2}.}$

This inequality means that no preparation of a quantum particle can imply simultaneously precise predictions for a measurement of position and for a measurement of momentum. The Robertson inequality generalizes this to the case of an arbitrary pair of self-adjoint operators ${\displaystyle A}$ and ${\displaystyle B}$. The commutator of these two operators is

${\displaystyle [A,B]=AB-BA,}$

and this provides the lower bound on the product of standard deviations:

${\displaystyle {\sigma }_A{\sigma }_B\ge \left|\frac{1}{2i}⟨[A,B]⟩\right|=\frac{1}{2}\left|⟨[A,B]⟩\right|.}$

Substituting in the canonical commutation relation ${\displaystyle [x,p]=i\hslash }$, an expression first postulated by Max Born in 1925, recovers the Kennard–Pauli–Weyl statement of the uncertainty principle.

From uncertainty to no-hidden-variables

Main articles: EPR paradox, Bell's theorem, and Bell test

The existence of the uncertainty principle naturally raises the question of whether quantum mechanics can be understood as an approximation to a more exact theory. Do there exist "hidden variables", more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory can provide? A collection of results, most significantly Bell's theorem, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics.

John Stewart Bell published the theorem now known by his name in 1964, investigating more deeply a thought experiment originally proposed in 1935 by Einstein, Boris Podolsky and Nathan Rosen. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then the results of a Bell test will be constrained in a particular, quantifiable way. If a Bell test is performed in a laboratory and the results are not thus constrained, then they are inconsistent with the hypothesis that local hidden variables exist. Such results would support the position that there is no way to explain the phenomena of quantum mechanics in terms of a more fundamental description of nature that is more in line with the rules of classical physics. Many types of Bell test have been performed in physics laboratories, often with the goal of ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. This is known as "closing loopholes in Bell tests". To date, Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave.

Quantum systems as measuring devices

The Robertson–Schrödinger uncertainty principle establishes that when two observables do not commute, there is a tradeoff in predictability between them. The Wigner–Araki–Yanase theorem demonstrates another consequence of non-commutativity: the presence of a conservation law limits the accuracy with which observables that fail to commute with the conserved quantity can be measured. Further investigation in this line led to the formulation of the Wigner–Yanase skew information.

Historically, experiments in quantum physics have often been described in semiclassical terms. For example, the spin of an atom in a Stern–Gerlach experiment might be treated as a quantum degree of freedom, while the atom is regarded as moving through a magnetic field described by the classical theory of Maxwell's equations. But the devices used to build the experimental apparatus are themselves physical systems, and so quantum mechanics should be applicable to them as well. Beginning in the 1950s, Léon Rosenfeld, Carl Friedrich von Weizsäcker and others tried to develop consistency conditions that expressed when a quantum-mechanical system could be treated as a measuring apparatus. One proposal for a criterion regarding when a system used as part of a measuring device can be modeled semiclassically relies on the Wigner function, a quasiprobability distribution that can be treated as a probability distribution on phase space in those cases where it is everywhere non-negative.

Decoherence

Main article: Quantum decoherence

A quantum state for an imperfectly isolated system will generally evolve to be entangled with the quantum state for the environment. Consequently, even if the system's initial state is pure, the state at a later time, found by taking the partial trace of the joint system-environment state, will be mixed. This phenomenon of entanglement produced by system-environment interactions tends to obscure the more exotic features of quantum mechanics that the system could in principle manifest. Quantum decoherence, as this effect is known, was first studied in detail during the 1970s. (Earlier investigations into how classical physics might be obtained as a limit of quantum mechanics had explored the subject of imperfectly isolated systems, but the role of entanglement was not fully appreciated.) A significant portion of the effort involved in quantum computing research is to avoid the deleterious effects of decoherence.

To illustrate, let ${\displaystyle {\rho }_S}$ denote the initial state of the system, ${\displaystyle {\rho }_E}$ the initial state of the environment and ${\displaystyle H}$ the Hamiltonian specifying the system-environment interaction. The density operator ${\displaystyle {\rho }_E}$ can be diagonalized and written as a linear combination of the projectors onto its eigenvectors:

${\displaystyle {\rho }_E=\sum _i{p}_i|{\psi }_i⟩⟨{\psi }_i|.}$

Expressing time evolution for a duration ${\displaystyle t}$ by the unitary operator ${\displaystyle U=e^{-iHt/\hslash }}$, the state for the system after this evolution is

${\displaystyle {\rho }_S^{\prime }={\mathrm{t}\mathrm{r}}_{E}U\left[{\rho }_S\otimes \left(\sum _i{p}_i|{\psi }_i⟩⟨{\psi }_i|\right)\right]U^{†},}$

which evaluates to

${\displaystyle {\rho }_S^{\prime }=\sum _{ij}\sqrt{p_i}⟨{\psi }_j|U|{\psi }_i⟩{\rho }_S\sqrt{p_i}⟨{\psi }_i|U^{†}|{\psi }_j⟩.}$

The quantities surrounding ${\displaystyle {\rho }_S}$ can be identified as Kraus operators, and so this defines a quantum channel.

Specifying a form of interaction between system and environment can establish a set of "pointer states," states for the system that are (approximately) stable, apart from overall phase factors, with respect to environmental fluctuations. A set of pointer states defines a preferred orthonormal basis for the system's Hilbert space.

Quantum information and computation

Quantum information science studies how information science and its application as technology depend on quantum-mechanical phenomena. Understanding measurement in quantum physics is important for this field in many ways, some of which are briefly surveyed here.

Measurement, entropy, and distinguishability

The von Neumann entropy is a measure of the statistical uncertainty represented by a quantum state. For a density matrix ${\displaystyle \rho }$, the von Neumann entropy is

${\displaystyle S(\rho )=-\mathrm{t}\mathrm{r}(\rho \log \rho );}$

writing ${\displaystyle \rho }$ in terms of its basis of eigenvectors,

${\displaystyle \rho =\sum _i{\lambda }_i|i⟩⟨i|,}$

the von Neumann entropy is

${\displaystyle S(\rho )=-\sum _i{\lambda }_i\log {\lambda }_i.}$

This is the Shannon entropy of the set of eigenvalues interpreted as a probability distribution, and so the von Neumann entropy is the Shannon entropy of the random variable defined by measuring in the eigenbasis of ${\displaystyle \rho }$. Consequently, the von Neumann entropy vanishes when ${\displaystyle \rho }$ is pure. The von Neumann entropy of ${\displaystyle \rho }$ can equivalently be characterized as the minimum Shannon entropy for a measurement given the quantum state ${\displaystyle \rho }$, with the minimization over all POVMs with rank-1 elements.

Many other quantities used in quantum information theory also find motivation and justification in terms of measurements. For example, the trace distance between quantum states is equal to the largest difference in probability that those two quantum states can imply for a measurement outcome:

${\displaystyle \frac{1}{2}||\rho -\sigma ||=\underset{0\le E\le I}{max}[\mathrm{t}\mathrm{r}(E\rho )-\mathrm{t}\mathrm{r}(E\sigma )].}$

Similarly, the fidelity of two quantum states, defined by

${\displaystyle F(\rho ,\sigma )={\left(\mathrm{Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^2,}$

expresses the probability that one state will pass a test for identifying a successful preparation of the other. The trace distance provides bounds on the fidelity via the Fuchs–van de Graaf inequalities:

${\displaystyle 1-\sqrt{F(\rho ,\sigma )}\le \frac{1}{2}||\rho -\sigma ||\le \sqrt{1-F(\rho ,\sigma )}.}$

Quantum circuits

Main article: Quantum circuit

Circuit representation of measurement. The single line on the left-hand side stands for a qubit, while the two lines on the right-hand side represent a classical bit.

Quantum circuits are a model for quantum computation in which a computation is a sequence of quantum gates followed by measurements. The gates are reversible transformations on a quantum mechanical analog of an n-bit register. This analogous structure is referred to as an n-qubit register. Measurements, drawn on a circuit diagram as stylized pointer dials, indicate where and how a result is obtained from the quantum computer after the steps of the computation are executed. Without loss of generality, one can work with the standard circuit model, in which the set of gates are single-qubit unitary transformations and controlled NOT gates on pairs of qubits, and all measurements are in the computational basis.

Measurement-based quantum computation

Main article: One-way quantum computer

Measurement-based quantum computation (MBQC) is a model of quantum computing in which the answer to a question is, informally speaking, created in the act of measuring the physical system that serves as the computer.

Quantum tomography

Main article: Quantum tomography

Quantum state tomography is a process by which, given a set of data representing the results of quantum measurements, a quantum state consistent with those measurement results is computed. It is named by analogy with tomography, the reconstruction of three-dimensional images from slices taken through them, as in a CT scan. Tomography of quantum states can be extended to tomography of quantum channels and even of measurements.

Quantum metrology

Main article: Quantum metrology

Quantum metrology is the use of quantum physics to aid the measurement of quantities that, generally, had meaning in classical physics, such as exploiting quantum effects to increase the precision with which a length can be measured. A celebrated example is the introduction of squeezed light into the LIGO experiment, which increased its sensitivity to gravitational waves.

Laboratory implementations

The range of physical procedures to which the mathematics of quantum measurement can be applied is very broad. In the early years of the subject, laboratory procedures involved the recording of spectral lines, the darkening of photographic film, the observation of scintillations, finding tracks in cloud chambers, and hearing clicks from Geiger counters. Language from this era persists, such as the description of measurement outcomes in the abstract as "detector clicks".

The double-slit experiment is a prototypical illustration of quantum interference, typically described using electrons or photons. The first interference experiment to be carried out in a regime where both wave-like and particle-like aspects of photon behavior are significant was G. I. Taylor's test in 1909. Taylor used screens of smoked glass to attenuate the light passing through his apparatus, to the extent that, in modern language, only one photon would be illuminating the interferometer slits at a time. He recorded the interference patterns on photographic plates; for the dimmest light, the exposure time required was roughly three months. In 1974, the Italian physicists Pier Giorgio Merli [it], Gian Franco Missiroli, and Giulio Pozzi implemented the double-slit experiment using single electrons and a television tube. A quarter-century later, a team at the University of Vienna performed an interference experiment with buckyballs, in which the buckyballs that passed through the interferometer were ionized by a laser, and the ions then induced the emission of electrons, emissions which were in turn amplified and detected by an electron multiplier.

Modern quantum optics experiments can employ single-photon detectors. For example, in the "BIG Bell test" of 2018, several of the laboratory setups used single-photon avalanche diodes. Another laboratory setup used superconducting qubits. The standard method for performing measurements upon superconducting qubits is to couple a qubit with a resonator in such a way that the characteristic frequency of the resonator shifts according to the state for the qubit, and detecting this shift by observing how the resonator reacts to a probe signal.

Interpretations of quantum mechanics

Main article: Interpretations of quantum mechanics

Niels Bohr and Albert Einstein, pictured here at Paul Ehrenfest's home in Leiden (December 1925), had a long-running collegial dispute about what quantum mechanics implied for the nature of reality.

Despite the consensus among scientists that quantum physics is in practice a successful theory, disagreements persist on a more philosophical level. Many debates in the area known as quantum foundations concern the role of measurement in quantum mechanics. Recurring questions include which interpretation of probability theory is best suited for the probabilities calculated from the Born rule; and whether the apparent randomness of quantum measurement outcomes is fundamental, or a consequence of a deeper deterministic process. Worldviews that present answers to questions like these are known as "interpretations" of quantum mechanics; as the physicist N. David Mermin once quipped, "New interpretations appear every year. None ever disappear."

A central concern within quantum foundations is the "quantum measurement problem," though how this problem is delimited, and whether it should be counted as one question or multiple separate issues, are contested topics. Of primary interest is the seeming disparity between apparently distinct types of time evolution. Von Neumann declared that quantum mechanics contains "two fundamentally different types" of quantum-state change. First, there are those changes involving a measurement process, and second, there is unitary time evolution in the absence of measurement. The former is stochastic and discontinuous, writes von Neumann, and the latter deterministic and continuous. This dichotomy has set the tone for much later debate. Some interpretations of quantum mechanics find the reliance upon two different types of time evolution distasteful and regard the ambiguity of when to invoke one or the other as a deficiency of the way quantum theory was historically presented. To bolster these interpretations, their proponents have worked to derive ways of regarding "measurement" as a secondary concept and deducing the seemingly stochastic effect of measurement processes as approximations to more fundamental deterministic dynamics. However, consensus has not been achieved among proponents of the correct way to implement this program, and in particular how to justify the use of the Born rule to calculate probabilities. Other interpretations regard quantum states as statistical information about quantum systems, thus asserting that abrupt and discontinuous changes of quantum states are not problematic, simply reflecting updates of the available information. Of this line of thought, Bell asked, "Whose information? Information about what?" Answers to these questions vary among proponents of the informationally-oriented interpretations.