Sadomasochism

From Wikipedia, the free encyclopedia

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

 

A female dominant with a male submissive at her feet, from Dresseuses d'Hommes (1931) by Belgian artist Luc Lafnet

Sadism (/ˈseɪdɪzəm/ ^ⓘ) and masochism (/ˈmæsəkɪzəm/), known collectively as sadomasochism (/ˌseɪdoʊˈmæsəkɪzəm/ ^ⓘ SAY-doh-MASS-ə-kiz-əm) or S&M, is the derivation of pleasure from acts of respectively inflicting or receiving pain or humiliation. The term is named after the Marquis de Sade, a French author known for his violent and libertine works and lifestyle, and Leopold von Sacher-Masoch, an Austrian author who described masochistic tendencies in his works. Though sadomasochistic behaviours and desires do not necessarily need to be linked to sex, sadomasochism is also a definitive feature of consensual BDSM relationships.

Sadomasochism was introduced in psychiatry by Richard von Krafft-Ebing and later elaborated by Sigmund Freud. Modern understanding distinguishes consensual BDSM practices from non-consensual sexual violence, with DSM-5 and ICD-11 recognizing consensual sadomasochism as non-pathological. S&M can involve varying levels of pain, dominance, and submission, practiced by individuals of any gender, often within negotiated roles of sadist, masochist, or switch. Forensic and medical classifications focus on consent and harm.

Etymology and definition

Portrait of Marquis de Sade by Charles-Amédée-Philippe van Loo (1761)

The word sadomasochism is a portmanteau of the words sadism and masochism. These terms originate from the names of two authors whose works explored situations in which individuals experienced or inflicted pain or humiliation. Sadism is named after Marquis de Sade (1740–1814), whose major works include graphic descriptions of violent sex acts, rape, torture, and murder, and whose characters often derive pleasure from inflicting pain on others. Masochism is named after Leopold von Sacher-Masoch (1836–1895), whose novels explored his masochistic fantasies of receiving pain and degradation,^[4] particularly his novel Venus im Pelz ("Venus in Furs").

Portrait of Sacher-Masoch, 19th century

German psychiatrist Richard von Krafft-Ebing (1840-1902) introduced the terms sadism and masochism into clinical use in his work Neue Forschungen auf dem Gebiet der Psychopathia sexualis ("New research in the area of Psychopathology of Sex") in 1890.

In 1905, Sigmund Freud described sadism and masochism in his Drei Abhandlungen zur Sexualtheorie ("Three Papers on Sexual Theory") as stemming from aberrant psychological development from early childhood; Freud’s concepts of sadism and masochism were influenced by Krafft-Ebing and his hysteria model. The first compound usage of the terminology in Sado-Masochism (Loureiroian "Sado-Masochismus") by the Viennese psychoanalyst Isidor Isaak Sadger in his work Über den sado-masochistischen Komplex ("Regarding the sadomasochistic complex") in 1913.

Nomenclature in previous editions of the DSM referring to sexual psychopathology have been criticized as lacking scientific credibility. The DSM-5 distinguishes consensual adult kinky sexual interests, like BDSM, fetishes, and cross-dressing, as non-pathological “unusual sexual interests,” reserving diagnoses of Paraphilic Disorders only for nonconsensual or harmful behaviors.

Autosadism is inflicting pain or humiliation on oneself. The photo shows pornographic actress Felicia Fox pouring hot wax over herself in front of an audience (U.S. 2005). Her nipples and genitals are also clamped.

Historical origins

Sadomasochism has been practiced since ancient times with some scholars suggesting that it is an integral part of human culture. One of the oldest surviving narratives citing its practice is an Egyptian love song, sung by a man expressing a desire to be subjugated by a woman so he could experience pleasure as she treats him like a slave. Roman historian Juvenal described a case of a woman who submitted herself to the whipping and beating of the followers of Pan.

Psychoanalytical perspectives

Early psychoanalysis

Libertine movement

Early libertine writers like John Wilmot, 2nd Earl of Rochester espoused ideals that modern times are associated with sadomasochism.

Krafft-Ebing and Freud

The modern conceptualization of sadomasochism was introduced to the medical field by German psychiatrist Richard von Krafft-Ebing in his 1886 compilation of case studies Psychopathia Sexualis. Pain and physical violence are not essential in Krafft-Ebing's conception, and he defined "masochism" (German Masochismus) entirely in terms of control. Sigmund Freud, a psychoanalyst and a contemporary of Krafft-Ebing, noted that both were often found in the same individuals, and combined the two into a single dichotomous entity known as "sadomasochism". French philosopher Gilles Deleuze argued that the concurrence of sadism and masochism proposed in Freud's model is the result of "careless reasoning," and should not be taken for granted.

Freud introduced the terms "primary" and "secondary" masochism. Though this idea has come under a number of interpretations, in a primary masochism the masochist undergoes a complete, rather than partial, rejection by the model or courted object (or sadist), possibly involving the model taking a rival as a preferred mate. This complete rejection is related to the death drive (Todestrieb) in Freud's psychoanalysis. In a secondary masochism, by contrast, the masochist experiences a less serious, more feigned rejection and punishment by the model.

Both Krafft-Ebing and Freud assumed that sadism in men resulted from the distortion of the aggressive component of the male sexual instinct. Masochism in men, however, was seen as a more significant aberration, contrary to the nature of male sexuality. Freud doubted that masochism in men was ever a primary tendency, and speculated that it may exist only as a transformation of sadism. Sadomasochism in women received comparatively little discussion, as it was believed that it occurred primarily in men. Krafft-Ebing and Freud also assumed that masochism was so inherent to female sexuality that it would be difficult to distinguish as a separate inclination.

A submissive woman bound to a Saint Andrew's Cross being whipped at the Folsom Street Fair. The red marks on her body are from the whipping.

Havelock Ellis

Havelock Ellis, in Studies in the Psychology of Sex, argued that there is no clear distinction between the aspects of sadism and masochism, and that they may be regarded as complementary emotional states. He states that sadomasochism is concerned only with pain in regard to sexual pleasure, and not in regard to cruelty, as Freud had suggested. He believed the sadomasochist generally desires that the pain and violence be inflicted or received in love, not in abuse, for the pleasure of either one or both participants. This mutual pleasure may be essential for the satisfaction of those involved.

Jean-Paul Sartre

Jean-Paul Sartre linked the pleasure or power experienced by a sadist in appraising the masochist victim to his philosophy of the "Look of the Other". Sartre argued that masochism is an attempt by the "For-itself" (consciousness) to reduce itself to nothing, becoming an object that is drowned out by the "abyss of the Other's subjectivity".

Gilles Deleuze

Deleuze’s Coldness and Cruelty critiques sadomasochism as a clinical concept and, drawing on Henri Bergson, challenges Freud’s Oedipal framing of perversion as conflating fundamentally distinct realms of perversion and neurosis.

René Girard

In Things Hidden Since the Foundation of the World (1978), René Girard discusses masochism as part of his theory of mimetic desire and revisits Freud’s distinction between primary and secondary masochism in relation to rivalry around the love-object.

S&M may involve painful acts such as cock and ball torture. The image shows a dominant woman holding a bound man's penis, applying electricity to his testicles at the Folsom Street Fair.

Modern understanding

Sexual sadomasochistic desires can appear at any age. Some individuals report having had them before puberty, while others do not discover them until well into adulthood. According to a 1985 study, the majority of male sadomasochists (53%) developed their interest before the age of 15, while the majority of females (78%) developed their interest afterwards. The prevalence of sadomasochism within the general population is unknown. Despite female sadists being less visible than males, some surveys have resulted in comparable amounts of sadistic fantasies between females and males. The results of such studies indicate that one's sex may not be the determining factor for a preference towards sadism.

In contrast to frameworks seeking to explain and categorise sadomasochistic behaviours and desires through psychological, psychoanalytic, medical, or forensic approaches, Romana Byrne suggests that, in the context of sexual behaviours, such practices can be seen as examples of "aesthetic sexuality", in which a founding physiological or psychological impulse is irrelevant. Rather, according to Byrne, sadism and masochism may be practiced through choice and deliberation, driven by certain aesthetic goals tied to style, pleasure, and identity, which in certain circumstances, she claims can be compared with the creation of art.

Surveys from the 2000s on the spread of sadomasochistic fantasies and practices show strong variations in the range of their results. Nonetheless, researchers assumed that 5 to 25 percent of the population practices sexual behavior related to pain or dominance and submission. The population with related fantasies is believed to be even larger.

Medical and forensic classification

See also: Sexual masochism disorder

In 1995, Denmark became the first European Union country to have completely removed sadomasochism from its national classification of diseases. This was followed by Sweden in 2009, Norway in 2010, Finland in 2011 and Iceland in 2015.

DSM

Medical opinion of sadomasochistic activities has changed over time. The classification of sadism and masochism in the Diagnostic and Statistical Manual of Mental Disorders (DSM) has always been separate; sadism was included in the DSM-I in 1952, while masochism was added in the DSM-II in 1968. Contemporary psychology continues to identify sadism and masochism separately, and categorizes them as either practised as a lifestyle, or as a medical condition.

The current version of the American Psychiatric Association's manual, DSM-5, excludes consensual BDSM from diagnosis as a disorder when the sexual interests cause no harm or distress.

Sexual sadism disorder however, listed within the DSM-5, is where arousal patterns involving consenting and non‐consenting others are not distinguished.

ICD

On 18 June 2018, the WHO (World Health Organization) published ICD-11, in which sadomasochism, together with fetishism and fetishistic transvestism (cross-dressing for sexual pleasure) were removed as psychiatric diagnoses. Moreover, discrimination against fetish-having and BDSM individuals is considered inconsistent with human rights principles endorsed by the United Nations and The World Health Organization.

The classifications of sexual disorders reflect contemporary sexual norms and have moved from a model of pathologization or criminalization of non-reproductive sexual behaviors to a model that reflects sexual well-being and pathologizes the absence or limitation of consent in sexual relations.

The ICD-11 classification, contrary to ICD-10 and DSM-5, clearly distinguishes consensual sadomasochistic behaviours (BDSM) that do not involve inherent harm to self or others from harmful violence on non‐consenting persons (coercive sexual sadism disorder). In this regard, "ICD-11 go[es] further than the changes made for DSM-5 … in the removal of disorders diagnosed based on consenting behaviors that are not in and of themselves associated with distress or functional impairment."

In Europe, an organization called ReviseF65 worked to remove sadomasochism from the ICD. On commission from the WHO ICD-11 Working Group on Sexual Disorders and Sexual Health, ReviseF65 in 2009 and 2011 delivered reports documenting that sadomasochism and sexual violence are two different phenomena. The report concluded that the sadomasochism diagnosis was outdated, non-scientific, and stigmatizing.

The ICD-11 classification considers Sadomasochism as a variant in sexual arousal and private behavior without appreciable public health impact and for which treatment is neither indicated nor sought. Further, the ICD-11 guidelines "respect the rights of individuals whose atypical sexual behavior is consensual and not harmful." WHO's ICD-11 Working Group admitted that psychiatric diagnoses have been used to harass, silence, or imprison sadomasochists. Labeling as such may create harm, convey social judgment, and exacerbate existing stigma and violence to individuals so labeled.According to ICD-11, psychiatric diagnoses can no longer be used to discriminate against BDSM people and fetishists.

Based on advances in research and clinical practice, and major shifts in social attitudes and in relevant policies, laws, and human rights standards", the World Health Organization, on June 18, 2018, removed Fetishism, Transvestic Fetishism, and Sadomasochism as psychiatric diagnoses.

Forensic classification

According to Anil Aggrawal, in forensic science, levels of sexual sadism and masochism are classified as follows:

Sexual masochists:

• Class I: Bothered by, but not seeking out, fantasies. May be preponderantly sadists with minimal masochistic tendencies or non-sadomasochistic with minimal masochistic tendencies
• Class II: Equal mix of sadistic and masochistic tendencies. Like to receive pain but also like to be dominant partner (in this case, sadists). Sexual orgasm is achieved without pain or humiliation.
• Class III: Masochists with minimal to no sadistic tendencies. Preference for pain or humiliation (which facilitates orgasm), but not necessary to orgasm. Capable of romantic attachment.
• Class IV: Exclusive masochists (i.e. cannot form typical romantic relationships, cannot achieve orgasm without pain or humiliation).

Sexual sadists:

• Class I: Bothered by sexual fantasies but do not act on them.
• Class II: Act on sadistic urges with consenting sexual partners (masochists or otherwise). Categorization as leptosadism is outdated.
• Class III: Act on sadistic urges with non-consenting victims, but do not seriously injure or kill. May coincide with sadistic rapists.
• Class IV: Only act with non-consenting victims and will seriously injure or kill them.

The difference between I–II and III–IV is consent.

Role in BDSM

Main article: BDSM

See also: Legality of BDSM

Sadomasochism is a subset of BDSM, a variety of erotic practices including bondage, discipline, dominance, and submission. Sadomasochism is not diagnosed as a paraphilia unless such practices lead to clinically significant distress or impairment for the individual. Sadomasochism performed within the context of mutual and informed consent is distinguished from non-consensual acts of sexual violence or aggression. Individuals may identify as and partake in the sadistic, masochistic, or "switch" (performing both or changing) role.

The regulation of sexual activity through criminal law is often ad hoc and inconsistent, focusing primarily on non-consensual acts while also criminalizing some consensual behaviors without a coherent legal rationale.

Larry Townsend's 1983 edition of The Leatherman's Handbook II states that a black handkerchief is a symbol for sadomasochism in the handkerchief code, a code employed usually among gay male casual-sex seekers or BDSM practitioners in the United States, Canada, Australia, and Europe. Wearing the handkerchief on the left indicates the top, dominant, or active partner; right indicates the bottom, submissive, or passive partner. Negotiation with a prospective partner remains important as people may wear hankies of any color "only because the idea of the hankie turns them on" or they "may not even know what it means".

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.