Quantum foundations is a discipline of science that seeks to understand the most counter-intuitive aspects of quantum theory, reformulate it and even propose new generalizations thereof. Contrary to other physical theories, such as general relativity, the defining axioms of quantum theory are quite ad hoc,
with no obvious physical intuition. While they lead to the right
experimental predictions, they do not come with a mental picture of the
world where they fit.
There exist different approaches to resolve this conceptual gap:
First, one can put quantum physics in contraposition with classical physics: by identifying scenarios, such as Bell experiments,
where quantum theory radically deviates from classical predictions, one
hopes to gain physical insights on the structure of quantum physics.
Second, one can attempt to find a re-derivation of the quantum formalism in terms of operational axioms.
Third, one can search for a full correspondence between the
mathematical elements of the quantum framework and physical phenomena:
any such correspondence is called an interpretation.
Fourth, one can renounce quantum theory altogether and propose a different model of the world.
Research in quantum foundations is structured along these roads.
Two or more separate parties conducting measurements over a quantum
state can observe correlations which cannot be explained with any local hidden variable theory. Whether this should be regarded as proving that the physical world itself is "nonlocal" is a topic of debate, but the terminology of "quantum nonlocality" is commonplace.
Nonlocality research efforts in quantum foundations focus on determining
the exact limits that classical or quantum physics enforces on the
correlations observed in a Bell experiment or more complex causal
scenarios. This research program has so far provided a generalization of Bell's
theorem that allows falsifying all classical theories with a
superluminal, yet finite, hidden influence.
Nonlocality can be understood as an instance of quantum contextuality.
A situation is contextual when the value of an observable depends on
the context in which it is measured (namely, on which other observables
are being measured as well). The original definition of measurement
contextuality can be extended to state preparations and even general
physical transformations.
Epistemic models for the quantum wave-function
A
physical property is epistemic when it represents our knowledge or
beliefs on the value of a second, more fundamental feature. The
probability of an event to occur is an example of an epistemic property.
In contrast, a non-epistemic or ontic variable captures the notion of a
“real” property of the system under consideration.
There is an on-going debate on whether the wave-function represents the epistemic state of a yet to be discovered ontic variable or, on the contrary, it is a fundamental entity. Under some physical assumptions, the Pusey–Barrett–Rudolph (PBR) theorem demonstrates the inconsistency of quantum states as epistemic states, in the sense above. Note that, in QBism and Copenhagen-type views, quantum states are still regarded as epistemic, not with respect
to some ontic variable, but to one's expectations about future
experimental outcomes. The PBR theorem does not exclude such epistemic
views on quantum states.
Axiomatic reconstructions
Some
of the counter-intuitive aspects of quantum theory, as well as the
difficulty to extend it, follow from the fact that its defining axioms
lack a physical motivation. An active area of research in quantum
foundations is therefore to find alternative formulations of quantum
theory which rely on physically compelling principles. Those efforts
come in two flavors, depending on the desired level of description of
the theory: the so-called Generalized Probabilistic Theories approach
and the Black boxes approach.
The framework of generalized probabilistic theories
Generalized Probabilistic Theories (GPTs) are a general framework to
describe the operational features of arbitrary physical theories.
Essentially, they provide a statistical description of any experiment
combining state preparations, transformations and measurements. The
framework of GPTs can accommodate classical and quantum physics, as well
as hypothetical non-quantum physical theories which nonetheless possess
quantum theory's most remarkable features, such as entanglement or
teleportation. Notably, a small set of physically motivated axioms is enough to single out the GPT representation of quantum theory.
L. Hardy introduced the concept of GPT in 2001, in an attempt to re-derive quantum theory from basic physical principles. Although Hardy's work was very influential (see the follow-ups below),
one of his axioms was regarded as unsatisfactory: it stipulated that, of
all the physical theories compatible with the rest of the axioms, one
should choose the simplest one. The work of Dakic and Brukner eliminated this “axiom of simplicity” and provided a reconstruction of quantum theory based on three physical principles. This was followed by the more rigorous reconstruction of Masanes and Müller.
Axioms common to these three reconstructions are:
The subspace axiom: systems which can store the same amount of information are physically equivalent.
Local tomography: to characterize the state of a composite system it is enough to conduct measurements at each part.
Reversibility: for any two extremal states [i.e., states which are
not statistical mixtures of other states], there exists a reversible
physical transformation that maps one into the other.
An alternative GPT reconstruction proposed by Chiribella, D'Ariano and Perinotti around the same time is also based on the
Purification axiom: for any state of a physical system A there exists a bipartite physical system and an extremal state (or purification) such that is the restriction of to system . In addition, any two such purifications of can be mapped into one another via a reversible physical transformation on system .
The use of purification to characterize quantum theory has been criticized on the grounds that it also applies in the Spekkens toy model.
To the success of the GPT approach, it can be countered that all
such works just recover finite dimensional quantum theory. In addition,
none of the previous axioms can be experimentally falsified unless the
measurement apparatuses are assumed to be tomographically complete.
Categorical Quantum Mechanics (CQM) or Process Theories are a general
framework to describe physical theories, with an emphasis on processes
and their compositions. It was pioneered by Samson Abramsky and Bob Coecke.
Besides its influence in quantum foundations, most notably the use of a
diagrammatic formalism, CQM also plays an important role in quantum
technologies, most notably in the form of ZX-calculus.
It also has been used to model theories outside of physics, for
example the DisCoCat compositional natural language meaning model.
In the black box or device-independent framework, an experiment is
regarded as a black box where the experimentalist introduces an input
(the type of experiment) and obtains an output (the outcome of the
experiment). Experiments conducted by two or more parties in separate
labs are hence described by their statistical correlations alone.
From Bell's theorem,
we know that classical and quantum physics predict different sets of
allowed correlations. It is expected, therefore, that far-from-quantum
physical theories should predict correlations beyond the quantum set. In
fact, there exist instances of theoretical non-quantum correlations
which, a priori, do not seem physically implausible. The aim of device-independent reconstructions is to show that all such
supra-quantum examples are precluded by a reasonable physical principle.
The physical principles proposed so far include no-signalling, Non-Trivial Communication Complexity, No-Advantage for Nonlocal computation, Information Causality, Macroscopic Locality, and Local Orthogonality. All these principles limit the set of possible correlations in
non-trivial ways. Moreover, they are all device-independent: this means
that they can be falsified under the assumption that we can decide if
two or more events are space-like separated. The drawback of the
device-independent approach is that, even when taken together, all the
afore-mentioned physical principles do not suffice to single out the set
of quantum correlations. In other words: all such reconstructions are partial.
An interpretation of quantum theory is a correspondence between the
elements of its mathematical formalism and physical phenomena. For
instance, in the pilot wave theory, the quantum wave function
is interpreted as a field that guides the particle trajectory and
evolves with it via a system of coupled differential equations. Most
interpretations of quantum theory stem from the desire to solve the quantum measurement problem.
Extensions of quantum theory
In
an attempt to reconcile quantum and classical physics, or to identify
non-classical models with a dynamical causal structure, some
modifications of quantum theory have been proposed.
Collapse models
Collapse models posit the existence of natural processes which periodically localize the wave-function. Such theories provide an explanation to the nonexistence of superpositions of macroscopic objects, at the cost of abandoning unitarity and exact energy conservation.
Quantum measure theory
In Sorkin's
quantum measure theory (QMT), physical systems are not modeled via
unitary rays and Hermitian operators, but through a single matrix-like
object, the decoherence functional. The entries of the decoherence functional determine the feasibility to
experimentally discriminate between two or more different sets of
classical histories, as well as the probabilities of each experimental
outcome. In some models of QMT the decoherence functional is further
constrained to be positive semidefinite (strong positivity). Even under
the assumption of strong positivity, there exist models of QMT which
generate stronger-than-quantum Bell correlations.
Acausal quantum processes
The
formalism of process matrices starts from the observation that, given
the structure of quantum states, the set of feasible quantum operations
follows from positivity considerations. Namely, for any linear map from
states to probabilities one can find a physical system where this map
corresponds to a physical measurement. Likewise, any linear
transformation that maps composite states to states corresponds to a
valid operation in some physical system. In view of this trend, it is
reasonable to postulate that any high-order map from quantum instruments
(namely, measurement processes) to probabilities should also be
physically realizable. Any such map is termed a process matrix. As shown by Oreshkov et al., some process matrices describe situations where the notion of global causality breaks.
The starting point of this claim is the following mental experiment: two parties, Alice and Bob,
enter a building and end up in separate rooms. The rooms have ingoing
and outgoing channels from which a quantum system periodically enters
and leaves the room. While those systems are in the lab, Alice and Bob
are able to interact with them in any way; in particular, they can
measure some of their properties.
Since Alice and Bob's interactions can be modeled by quantum
instruments, the statistics they observe when they apply one instrument
or another are given by a process matrix. As it turns out, there exist
process matrices which would guarantee that the measurement statistics
collected by Alice and Bob is incompatible with Alice interacting with
her system at the same time, before or after Bob, or any convex
combination of these three situations. Such processes are called acausal.
In quantum mechanics each measurable physical quantity of a quantum system is called an observable which, for example, could be the position and the momentum but also energy , components of spin (), and so on. The observable acts as a linear function on the states of the system; its eigenvectors correspond to the quantum state (i.e. eigenstate) and the eigenvalues
to the possible values of the observable. The collection of
eigenstates/eigenvalue pairs represent all possible values of the
observable. Writing for an eigenstate and for the corresponding observed value, any arbitrary state of the quantum system can be expressed as a vector using bra–ket notation:
The kets specify the different available quantum "alternatives", i.e., particular quantum states.
The wave function
is a specific representation of a quantum state. Wave functions can
therefore always be expressed as eigenstates of an observable though the
converse is not necessarily true.
Collapse
To
account for the experimental result that repeated measurements of a
quantum system give the same results, the theory postulates a "collapse"
or "reduction of the state vector" upon observation, abruptly converting an arbitrary state into a single component eigenstate of the observable:
where the arrow represents a measurement of the observable corresponding to the basis. For any single event, only one eigenvalue is measured, chosen randomly from among the possible values.
Meaning of the expansion coefficients
The complex coefficients in the expansion of a quantum state in terms of eigenstates ,
can be written as an (complex) overlap of the corresponding eigenstate and the quantum state:
They are called the probability amplitudes. The square modulus is the probability that a measurement of the observable yields the eigenstate . The sum of the probability over all possible outcomes must be one:
As examples, individual counts in a double slit experiment
with electrons appear at random locations on the detector; after many
counts are summed the distribution shows a wave interference pattern. In a Stern-Gerlach experiment
with silver atoms, each particle appears in one of two areas
unpredictably, but the final conclusion has equal numbers of events in
each area.
This statistical aspect of quantum measurements differs fundamentally from classical mechanics.
In quantum mechanics the only information we have about a system is its
wave function and measurements of its wave function can only give
statistical information.
Terminology
The
two terms "reduction of the state vector" (or "state reduction" for
short) and "wave function collapse" are used to describe the same
concept. A quantum state is a mathematical description of a quantum system; a quantum state vector uses Hilbert space vectors for the description. Reduction of the state vector replaces the full state vector with a single eigenstate of the observable.
The term "wave function" is typically used for a different
mathematical representation of the quantum state, one that uses spatial
coordinates also called the "position representation". When the wave function representation is used, the "reduction" is called "wave function collapse".
The measurement problem
The
Schrödinger equation describes quantum systems but does not describe
their measurement. Solution to the equations include all possible
observable values for measurements, but measurements only result in one
definite outcome. This difference is called the measurement problem
of quantum mechanics. To predict measurement outcomes from quantum
solutions, the orthodox interpretation of quantum theory postulates wave
function collapse and uses the Born rule to compute the probable outcomes. Despite the widespread quantitative success of these postulates
scientists remain dissatisfied and have sought more detailed physical
models. Rather than suspending the Schrödinger equation during the
process of measurement, the measurement apparatus should be included and
governed by the laws of quantum mechanics.
Physical approaches to collapse
Quantum
theory offers no dynamical description of the "collapse" of the wave
function. Viewed as a statistical theory, no description is expected. As
Fuchs and Peres put it, "collapse is something that happens in our
description of the system, not to the system itself".
Various interpretations of quantum mechanics attempt to provide a physical model for collapse.Three treatments of collapse can be found among the common
interpretations. The first group includes hidden-variable theories like de Broglie–Bohm theory; here random outcomes only result from unknown values of hidden variables. Results from tests of Bell's theorem
shows that these variables would need to be non-local. The second group
models measurement as quantum entanglement between the quantum state
and the measurement apparatus. This results in a simulation of classical
statistics called quantum decoherence. This group includes the many-worlds interpretation and consistent histories
models. The third group postulates additional, but as yet undetected,
physical basis for the randomness; this group includes for example the objective-collapse interpretations.
While models in all groups have contributed to better understanding of
quantum theory, no alternative explanation for individual events has
emerged as more useful than collapse followed by statistical prediction
with the Born rule.
The significance ascribed to the wave function varies from
interpretation to interpretation and even within an interpretation (such
as the Copenhagen interpretation).
If the wave function merely encodes an observer's knowledge of the
universe, then the wave function collapse corresponds to the receipt of
new information. This is somewhat analogous to the situation in
classical physics, except that the classical "wave function" does not
necessarily obey a wave equation. If the wave function is physically
real, in some sense and to some extent, then the collapse of the wave
function is also seen as a real process, to the same extent.
Quantum decoherence explains why a system interacting with an environment transitions from being a pure state, exhibiting superpositions, to a mixed state, an incoherent combination of classical alternatives. This transition is fundamentally reversible, as the combined state of
system and environment is still pure, but for all practical purposes
irreversible in the same sense as in the second law of thermodynamics:
the environment is a very large and complex quantum system, and it is
not feasible to reverse their interaction. Decoherence is thus very
important for explaining the classical limit
of quantum mechanics, but cannot explain wave function collapse, as all
classical alternatives are still present in the mixed state, and wave
function collapse selects only one of them.
The form of decoherence known as environment-induced superselection proposes that when a quantum system interacts with the environment, the superpositions apparently
reduce to mixtures of classical alternatives. The combined wave
function of the system and environment continue to obey the Schrödinger
equation throughout this apparent collapse. More importantly, this is not enough to explain actual wave function collapse, as decoherence does not reduce it to a single eigenstate.
History
The concept of wavefunction collapse was introduced by Werner Heisenberg in his 1927 paper on the uncertainty principle, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", and incorporated into the mathematical formulation of quantum mechanics by John von Neumann, in his 1932 treatise Mathematische Grundlagen der Quantenmechanik. Heisenberg did not try to specify exactly what the collapse of the
wavefunction meant. However, he emphasized that it should not be
understood as a physical process. Niels Bohr never mentions wave function collapse in his published work,
but he repeatedly cautioned that we must give up a "pictorial
representation". Despite the differences between Bohr and Heisenberg,
their views are often grouped together as the "Copenhagen
interpretation", of which wave function collapse is regarded as a key
feature.
In 1957 Hugh Everett III
proposed a model of quantum mechanics that dropped von Neumann's first
postulate. Everett observed that the measurement apparatus was also a
quantum system and its quantum interaction with the system under
observation should determine the results. He proposed that the
discontinuous change is instead a splitting of a wave function
representing the universe.
While Everett's approach rekindled interest in foundational quantum
mechanics, it left core issues unresolved. Two key issues relate to
origin of the observed classical results: what causes quantum systems to
appear classical and to resolve with the observed probabilities of the Born rule.
Beginning in 1970 H. Dieter Zeh sought a detailed quantum decoherence model for the discontinuous change without postulating collapse. Further work by Wojciech H. Zurek in 1980 lead eventually to a large number of papers on many aspects of the concept. Decoherence assumes that every quantum system interacts quantum
mechanically with its environment and such interaction is not separable
from the system, a concept called an "open system".
Decoherence has been shown to work very quickly and within a minimal
environment, but as yet it has not succeeded in a providing a detailed
model replacing the collapse postulate of orthodox quantum mechanics.
By explicitly dealing with the interaction of object and measuring instrument, von Neumann described a quantum mechanical measurement scheme consistent with wave function collapse. However, he did not prove the necessity
of such a collapse. Von Neumann's projection postulate was conceived
based on experimental evidence available during the 1930s, in particular
Compton scattering. Later work refined the notion of measurements into the more easily discussed first kind, that will give the same value when immediately repeated, and the second kind that give different values when repeated.
In social psychology, naïve realism is the human tendency to believe that we see the world around us objectively, and that people who disagree with us must be uninformed, irrational, or biased.
The term, as it is used in psychology today, was coined by social psychologistLee Ross and his colleagues in the 1990s. It is related to the philosophical concept of naïve realism, which is the idea that our senses allow us to perceive objects directly and without any intervening processes. Social psychologists in the mid-20th century argued against this stance and proposed instead that perception is inherently subjective.
Several prominent social psychologists have studied naïve realism experimentally, including Lee Ross, Andrew Ward, Dale Griffin, Emily Pronin, Thomas Gilovich, Robert Robinson, and Dacher Keltner. In 2010, the Handbook of Social Psychology recognized naïve realism as one of "four hard-won insights about human perception, thinking, motivation and behavior that ... represent important, indeed foundational, contributions of social psychology."
Main assumptions
Lee Ross
and fellow psychologist Andrew Ward have outlined three interrelated
assumptions, or "tenets", that make up naïve realism. They argue that
these assumptions are supported by a long line of thinking in social
psychology, along with several empirical studies. According to their
model, people:
Believe that they see the world objectively and without bias.
Expect that others will come to the same conclusions, so long as
they are exposed to the same information and interpret it in a rational
manner.
Assume that others who do not share the same views must be ignorant, irrational, or biased.
History of the concept
Naïve realism follows from a subjectivist tradition in modern social psychology, which traces its roots back to one of the field's founders, German-American psychologist Kurt Lewin. Lewin's ideas were strongly informed by Gestalt psychology, a 20th-century school of thought which focused on examining psychological phenomena in context, as parts of a whole.
From the 1920s through the 1940s, Lewin developed an approach for studying human behavior which he called field theory. Field theory proposes that a person's behavior is a function of the person and the environment. Lewin considered a person's psychological environment, or "life space", to be subjective and thus distinct from physical reality.
During this time period, subjectivist ideas also propagated throughout other areas of psychology. For example, the developmental psychologistJean Piaget argued that children view the world through an egocentric lens, and they have trouble separating their own beliefs from the beliefs of others.
In the 1940s and 1950s, early pioneers in social psychology applied the subjectivist view to the field of social perception.
In 1948, psychologists David Kretch and Richard Krutchfield argued that
people perceive and interpret the world according to their "own needs,
own connotations, own personality, own previously formed cognitive
patterns".
Social psychologist Gustav Ichheiser expanded on this idea, noting how biases in person perception lead to misunderstandings in social relations.
According to Ichheiser, "We tend to resolve our perplexity arising out
of the experience that other people see the world differently than we
see it ourselves by declaring that these others, in consequence of some
basic intellectual and moral defect, are unable to see things 'as they
really are' and to react to them 'in a normal way'. We thus imply, of
course, that things are in fact as we see them, and that our ways are
the normal ways."
Solomon Asch, a prominent social psychologist who was also brought up in the Gestalt tradition, argued that people disagree because they base their judgments on different construals, or ways of looking at various issues. However, they are under the illusion that their judgments about the
social world are objective. "This attitude, which has been aptly
described as naive realism, sees no problem in the fact of perception or
knowledge of the surroundings. Things are what they appear to be; they
have just the qualities that they reveal to sight and touch," he wrote
in his textbook Social Psychology in 1952. "This attitude, does not, however, describe the actual conditions of our knowledge of the surroundings."
Experimental evidence
"They saw a game"
In a seminal study in social psychology, which was published in a paper in 1954, students from Dartmouth and Princeton watched a video of a heated football game between the two schools. Though they looked at the same footage, fans from both schools
perceived the game very differently. The Princeton students "saw" the
Dartmouth team make twice as many infractions as their own team, and
they also saw the team make twice as many infractions compared to what
the Dartmouth students saw. Dartmouth students viewed the game as being
evenly-matched in violence, in which both sides were to blame. This
study revealed that two groups perceived an event subjectively. Each team believed they saw the event objectively and that the other side's perception of the event was blinded by bias.
False consensus effect
A 1977 study conducted by Ross and colleagues provided early evidence for a cognitive bias called the false consensus effect, which is the tendency for people to overestimate the extent to which others share the same views. This bias has been cited as supporting the first two tenets of naïve realism. In the study, students were asked whether they would wear a
sandwich-board sign, which said "Eat At Joe's" on it, around campus.
Then they were asked to indicate whether they thought other students
were likely to wear the sign, and what they thought about students who
were either willing to wear it or not. The researchers found that
students who agreed to wear the sign thought that the majority of
students would wear the sign, and they thought that refusing to wear the
sign was more revealing of their peers' personal attributes.
Conversely, students who declined to wear the sign thought that most
other students would also refuse, and that accepting the invitation was
more revealing of certain personality traits.
Hostile media effect
A phenomenon referred to as the hostile media effect demonstrates that partisans can view neutral events subjectively
according to their own needs and values, and make the assumption that
those who interpret the event differently are biased. For a study in
1985, pro-Israeli and pro-Arab students were asked to watch real news
coverage on the 1982 Sabra and Shatila massacre, a massive killing of Palestinian refugees (Vallone, Lee Ross and Lepper, 1985). Researchers found that partisans from both sides perceived the coverage
as being biased in favor of the opposite viewpoint, and believed that
the people in charge of the news program held the ideological views of
the opposite side.
"Musical tapping" study
More
empirical evidence for naïve realism came from psychologist Elizabeth
Newton's "musical tapping study" in 1990. For the study, participants
were designated either as "tappers" or as "listeners". The tappers were
told to tap out the rhythm of a well-known song, while the "listeners"
were asked to try to identify the song. While tappers expected that
listeners would guess the tune around 50 percent of the time, the
listeners were able to identify it only around 2.5 percent of the time.
This provided support for a failure in perspective-taking
on the side of the tappers, and an overestimation of the extent to
which others would share in "hearing" the song as it was tapped.
Wall Street Game
In 2004, Ross,
Liberman, and Samuels asked dorm resident advisors to nominate students
to participate in a study, and to indicate whether those students were
likely to cooperate or defect in the first round of the classic
decision-making game called the Prisoner's Dilemma.
The game was introduced to subjects in one of two ways: it was either
referred to as the "Wall Street Game" or as the "Community Game". The
researchers found that students in the "Community Game" condition were
twice as likely to cooperate, and that it did not seem to make a
difference whether students were previously categorized as "cooperators"
versus "defectors". This experiment demonstrated that the game's label
exerted more power on how the students played the game than the
subjects' personality traits. Furthermore, the study showed that the
dorm advisors did not make sufficient allowances for subjective interpretations of the game.
Consequences
Naïve
realism causes people to exaggerate differences between themselves and
others. Psychologists believe that it can spark and exacerbate conflict,
as well as create barriers to negotiation through several different mechanisms.
Bias blind spot
One consequence of naïve realism is referred to as the bias blind spot, which is the ability to recognize cognitive and motivational
biases in others while failing to recognize the impact of bias on the
self. In a study conducted by Pronin, Lin, and Ross (2002), Stanford students completed a questionnaire about various biases in social judgment. The participants indicated how susceptible they thought they were to
these biases compared to the average student. The researchers found that
the participants consistently believed that they were less likely to be
biased than their peers. In a follow-up study, students answered
questions about their personal attributes (e.g. how considerate they
were) compared to those of other students. The majority of students saw
themselves as falling above average on most traits, which provided
support for a cognitive bias known as the better-than-average effect.
The students then were told that 70 to 80 percent of people fall prey
to this bias. When asked about the accuracy of their self-assessments,
63 percent of the students argued that their ratings had been objective,
while 13 percent of students indicated they thought their ratings had
been too modest.
Fig.
1. Actual views (top), "circle's" perception of views (middle),
"triangle's" perception of views (bottom). (Modeled after similar
illustrations found in Robinson et al., 1995, and Ross & Ward,
1996.)
False polarization
When
an individual does not share our views, the third tenet of naïve
realism attributes this discrepancy to three possibilities. The
individual either has been exposed to a different set of information, is
lazy or unable to come to a rational conclusion, or is under a
distorting influence such as bias or self-interest. This gives rise to a phenomenon called false polarization, which
involves interpreting others' views as more extreme than they really
are, and leads to a perception of greater intergroup differences (see Fig. 1). People assume that they perceive the issue objectively, carefully
considering it from multiple views, while the other side processes
information in top-down fashion. For instance, in a study conducted by Robinson et al. in 1996, pro-life
and pro-choice partisans greatly overestimated the extremity of the
views of the opposite side, and also overestimated the influence of
ideology on others in their own group.
Reactive devaluation
The
assumption that others' views are more extreme than they are, can
create a barrier for conflict resolution. In a sidewalk survey conducted
in the 1980s, pedestrians evaluated a nuclear arms' disarmament
proposal (Stillinger et al., 1991). One group of participants was told that the proposal was made by American President Ronald Reagan, while others thought the proposal came from Soviet leader Mikhail Gorbachev.
The researchers found that 90 percent of the participants who thought
the proposal was from Reagan supported it, while only 44 percent in the
Gorbachev group indicated their support. This provided support for a
phenomenon called reactive devaluation,
which involves dismissing a concession from an adversary on the
assumption that the concession is either motivated by self-interest or
less valuable.
Some interpretations of quantum mechanics posit a central role for an observer of a quantum phenomenon. The quantum mechanical observer is tied to the issue of observer effect,
where a measurement necessarily requires interacting with the physical
object being measured, affecting its properties through the interaction.
The term "observable" has gained a technical meaning, denoting a Hermitian operator that represents a measurement.
Foundation
The theoretical foundation of the concept of measurement in quantum mechanics is a contentious issue deeply connected to the many interpretations of quantum mechanics. A key focus point is that of wave function collapse, for which several popular interpretations assert that measurement causes a discontinuous change into an eigenstate of the operator associated with the quantity that was measured, a change which is not time-reversible.
More explicitly, the superposition principle (ψ = Σnanψn) of quantum physics dictates that for a wave function ψ, a measurement will result in a state of the quantum system of one of the m possible eigenvalues fn , n = 1, 2, ..., m, of the operator ∧F which is in the space of the eigenfunctions ψn , n = 1, 2, ..., m.
Once one has measured the system, one knows its current state;
and this prevents it from being in one of its other states — it has
apparently decohered from them without prospects of future strong quantum interference. This means that the type of measurement one performs on the system affects the end-state of the system.
An experimentally studied situation related to this is the quantum Zeno effect,
in which a quantum state would decay if left alone, but does not decay
because of its continuous observation. The dynamics of a quantum system
under continuous observation are described by a quantum stochastic master equation known as the Belavkin equation. Further studies have shown that even observing the results after the
photon is produced leads to collapsing the wave function and loading a
back-history as shown by delayed choice quantum eraser.
When discussing the wave function ψ
which describes the state of a system in quantum mechanics, one should
be cautious of a common misconception that assumes that the wave
function ψ
amounts to the same thing as the physical object it describes. This
flawed concept must then require existence of an external mechanism,
such as a measuring instrument, that lies outside the principles
governing the time evolution of the wave function ψ, in order to account for the so-called "collapse of the wave function" after a measurement has been performed. But the wave function ψ is not a physical object like, for example, an atom, which has an observable mass, charge and spin, as well as internal degrees of freedom. Instead, ψ is an abstract mathematical function that contains all the statistical
information that an observer can obtain from measurements of a given
system. In this case, there is no real mystery in that this mathematical
form of the wave function ψ must change abruptly after a measurement has been performed.
A consequence of Bell's theorem is that measurement on one of two entangled particles can appear to have a nonlocal effect on the other particle. Additional problems related to decoherence arise when the observer is modeled as a quantum system.
Of course the introduction of the observer must not be
misunderstood to imply that some kind of subjective features are to be
brought into the description of nature. The observer has, rather, only
the function of registering decisions, i.e., processes in space and
time, and it does not matter whether the observer is an apparatus or a
human being; but the registration, i.e., the transition from the
"possible" to the "actual," is absolutely necessary here and cannot be
omitted from the interpretation of quantum theory.
Niels Bohr, also a founder of the Copenhagen interpretation, wrote:
all unambiguous information concerning atomic objects is
derived from the permanent marks such as a spot on a photographic plate,
caused by the impact of an electron left on the bodies which define the
experimental conditions. Far from involving any special intricacy, the
irreversible amplification effects on which the recording of the
presence of atomic objects rests rather remind us of the essential
irreversibility inherent in the very concept of observation. The
description of atomic phenomena has in these respects a perfectly
objective character, in the sense that no explicit reference is made to
any individual observer and that therefore, with proper regard to
relativistic exigencies, no ambiguity is involved in the communication
of information.
Likewise, Asher Peres stated that "observers" in quantum physics are
similar to the ubiquitous "observers" who send and receive light signals in special relativity.
Obviously, this terminology does not imply the actual presence of human
beings. These fictitious physicists may as well be inanimate automata
that can perform all the required tasks, if suitably programmed.
Critics of the special role of the observer also point out that
observers can themselves be observed, leading to paradoxes such as that
of Wigner's friend; and that it is not clear how much consciousness is required. As John Bell
inquired, "Was the wave function waiting to jump for thousands of
millions of years until a single-celled living creature appeared? Or did
it have to wait a little longer for some highly qualified measurer—with
a PhD?"
Anthropocentric interpretation
The prominence of seemingly subjective or anthropocentric ideas like "observer" in the early development of the theory has been a continuing source of disquiet and philosophical dispute. A number of new-age religious or philosophical views give the observer a
more special role, or place constraints on who or what can be an
observer. As an example of such claims, Fritjof Capra
declared, "The crucial feature of atomic physics is that the human
observer is not only necessary to observe the properties of an object,
but is necessary even to define these properties." There is no credible peer-reviewed research that backs such claims.
The uncertainty principle has been frequently confused with the observer effect, evidently even by its originator, Werner Heisenberg. The uncertainty principle in its standard form describes how precisely
it is possible to measure the position and momentum of a particle at
the same time. If the precision in measuring one quantity is increased,
the precision in measuring the other decreases.
An alternative version of the uncertainty principle, more in the spirit of an observer effect, fully accounts for the disturbance the observer has on a system and the
error incurred, although this is not how the term "uncertainty
principle" is most commonly used in practice.
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement
of a quantum state. The result is a prediction for the system
represented by the state. Knowledge of the quantum state, and the rules
for the system's evolution in time, exhausts all that can be known about
a quantum system.
Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are
wave functions describing quantum systems using position or momentum variables and
Historical, educational, and application-focused problems typically
feature wave functions; modern professional physics uses the abstract
vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories with special properties include stationary states for time independence and quantum vacuum states in quantum field theory.
From the states of classical mechanics
As a tool for physics, quantum states grew out of states in classical mechanics. A classical dynamical state consists of a set of dynamical variables with well-defined real values at each instant of time. For example, the state of a cannon ball would consist of its position
and velocity. The state values evolve under equations of motion and thus
remain strictly determined. If we know the position of a cannon and the
exit velocity of its projectiles, then we can use equations containing
the force of gravity to predict the trajectory of a cannon ball
precisely.
Similarly, quantum states consist of sets of dynamical variables
that evolve under equations of motion. However, the values derived from
quantum states are complex numbers, quantized, limited by uncertainty relations, and only provide a probability distribution
for the outcomes for a system. These constraints alter the nature of
quantum dynamic variables. For example, the quantum state of an electron
in a double-slit experiment
would consist of complex values over the detection region and, when
squared, only predict the probability distribution of electron counts
across the detector.
Role in quantum mechanics
The
process of describing a quantum system with quantum mechanics begins
with identifying a set of variables defining the quantum state of the
system. The set will contain compatible and incompatible variables. Simultaneous measurement of a complete set of compatible variables prepares the system in a unique state. The state then evolves deterministically according to the equations of motion. Subsequent measurement of the state produces a sample from a probability distribution predicted by the quantum mechanical operator corresponding to the measurement.
The fundamentally statistical or probabilisitic nature of quantum
measurements changes the role of quantum states in quantum mechanics
compared to classical states in classical mechanics. In classical
mechanics, the initial state of one or more bodies is measured; the
state evolves according to the equations of motion; measurements of the
final state are compared to predictions. In quantum mechanics, ensembles
of identically prepared quantum states evolve according to the
equations of motion and many repeated measurements are compared to
predicted probability distributions.
Measurements, macroscopic operations on quantum states, filter the state.
Whatever the input quantum state might be, repeated identical
measurements give consistent values. For this reason, measurements
'prepare' quantum states for experiments, placing the system in a
partially defined state. Subsequent measurements may either further
prepare the system – these are compatible measurements – or it may alter
the state, redefining it – these are called incompatible or
complementary measurements. For example, we may measure the momentum of a
state along the
axis any number of times and get the same result, but if we measure the
position after once measuring the momentum, subsequent measurements of
momentum are changed. The quantum state appears unavoidably altered by
incompatible measurements. This is known as the uncertainty principle.
The quantum state after a measurement is in an eigenstate corresponding to that measurement and the value measured.
Other aspects of the state may be unknown. Repeating the measurement
will not alter the state. In some cases, compatible measurements can
further refine the state, causing it to be an eigenstate corresponding
to all these measurements. A full set of compatible measurements produces a pure state. Any state that is not pure is called a mixed state as discussed in more depth below.
The eigenstate solutions to the Schrödinger equation
can be formed into pure states. Experiments rarely produce pure
states. Therefore statistical mixtures of solutions must be compared to
experiments.
Representations
The same physical quantum state can be expressed mathematically in different ways called representations. The position wave function is one representation often seen first in
introductions to quantum mechanics. The equivalent momentum wave
function is another wave function based representation. Representations
are analogous to coordinate systems or similar mathematical devices like parametric equations. Selecting a representation will make some aspects of a problem easier at the cost of making other things difficult.
In formal quantum mechanics (see § Formalism in quantum physics below) the theory develops in terms of abstract 'vector space',
avoiding any particular representation. This allows many elegant
concepts of quantum mechanics to be expressed and to be applied even in
cases where no classical analog exists.
Wave functions represent quantum states, particularly when they are functions of position or of momentum. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed. The wave function is a complex-valued function of any complete set of commuting or compatible degrees of freedom. For example, one set could be the spatial coordinates of an electron.
Preparing a system by measuring the complete set of compatible observables produces a pure quantum state. More common, incomplete preparation produces a mixed quantum state. Wave function solutions of Schrödinger's equations of motion
for operators corresponding to measurements can readily be expressed as
pure states; they must be combined with statistical weights matching
experimental preparation to compute the expected probability
distribution.
Pure states of wave functions
Probability densities for the electron of a hydrogen atom in different quantum states.
The postulates of quantum mechanics state that pure states, at a given time t, correspond to vectors in a separablecomplexHilbert space, while each measurable physical quantity (such as the energy or momentum of a particle) is associated with a mathematical operator called the observable. The operator serves as a linear function that acts on the states of the system. The eigenvalues
of the operator correspond to the possible values of the observable.
For example, it is possible to observe a particle with a momentum of
1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator
is 1 kg⋅m/s. The corresponding eigenvector (which physicists call an eigenstate) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s.
On the other hand, a pure state described as a superposition of multiple different eigenstates does in general have quantum uncertainty for the given observable. Using bra–ket notation, this linear combination of eigenstates can be represented as:
The coefficient that corresponds to a particular state in the linear
combination is a complex number, thus allowing interference effects
between states. The coefficients are time dependent. How a quantum state
changes in time is governed by the time evolution operator.
Mixed states of wave functions
A
mixed quantum state corresponds to a probabilistic mixture of pure
states; however, different distributions of pure states can generate
equivalent (i.e., physically indistinguishable) mixed states. A mixture of quantum states is again a quantum state.
A mixed state for electron spins, in the density-matrix formulation, has the structure of a matrix that is Hermitian and positive semi-definite, and has trace 1. A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement:
which involves superposition
of joint spin states for two particles with spin 1/2. The singlet state
satisfies the property that if the particles' spins are measured along
the same direction then either the spin of the first particle is
observed up and the spin of the second particle is observed down, or the
first one is observed down and the second one is observed up, both
possibilities occurring with equal probability.
A pure quantum state can be represented by a ray in a projective Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces. The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states. Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle:
a state that implies a narrow spread of possible outcomes for one
experiment necessarily implies a wide spread of possible outcomes for
another.
Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a statistical ensemble
of independent systems. Statistical mixtures represent the degree of
knowledge whilst the uncertainty within quantum mechanics is
fundamental. Mathematically, a statistical mixture is not a combination
using complex coefficients, but rather a combination using real-valued,
positive probabilities of different states . A number represents the probability of a randomly selected system being in the state . Unlike the linear combination case each system is in a definite eigenstate.
The expectation value of an observable A
is a statistical mean of measured values of the observable. It is this
mean, and the distribution of probabilities, that is predicted by
physical theories.
There is no state that is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) are known exactly; at least one of them will have a range of possible values.[a] This is the content of the Heisenberg uncertainty relation.
Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state. More precisely: After measuring an observable A, the system will be in an eigenstate of A;
thus the state has changed, unless the system was already in that
eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences, however, as follows.
Consider two incompatible observables, A and B, where A corresponds to a measurement earlier in time than B. Suppose that the system is in an eigenstate of B at the experiment's beginning. If we measure only B, all runs of the experiment will yield the same result.
If we measure first A and then B in the same run of the experiment, the system will transfer to an eigenstate of A after the first measurement, and we will generally notice that the results of B are statistical. Thus: Quantum mechanical measurements influence one another, and the order in which they are performed is important.
Another feature of quantum states becomes relevant if we consider
a physical system that consists of multiple subsystems; for example, an
experiment with two particles rather than one. Quantum physics allows
for certain states, called entangled states, that show certain
statistical correlations between measurements on the two particles which
cannot be explained by classical theory. For details, see Quantum entanglement. These entangled states lead to experimentally testable properties (Bell's theorem)
that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.
Schrödinger picture vs. Heisenberg picture
One can take the observables to be dependent on time, while the state σ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observables P(t), Q(t).) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. (This approach was taken in the earlier part of the discussion above, with a time-varying state .) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention.
Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics
is usually formulated in terms of the Schrödinger picture, the
Heisenberg picture is often preferred in a relativistic context, that
is, for quantum field theory. Compare with Dirac picture.
Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1.
Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If a vector in a complex Hilbert space can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in are said to correspond to the same ray in the projective Hilbert space of . Note that although the word ray is used, properly speaking, a point in the projective Hilbert space corresponds to a line passing through the origin of the Hilbert space, rather than a half-line, or ray in the geometrical sense.
The angular momentum has the same dimension (M·L2·T−1) as the Planck constant and, at quantum scale, behaves as a discrete
degree of freedom of a quantum system. Most particles possess a kind of
intrinsic angular momentum that does not appear at all in classical
mechanics and arises from Dirac's relativistic generalization of the
theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the group representations of the Lie group
SU(2) are used to describe this additional freedom. For a given
particle, the choice of representation (and hence the range of possible
values of the spin observable) is specified by a non-negative number S that, in units of the reduced Planck constantħ, is either an integer (0, 1, 2, ...) or a half-integer (1/2, 3/2, 5/2, ...). For a massive particle with spin S, its spin quantum numberm always assumes one of the 2S + 1 possible values in the set
As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).
The quantum state of a system of N particles, each potentially
with spin, is described by a complex-valued function with four
variables per particle, corresponding to 3 spatial coordinates and spin, e.g.
Here, the spin variables mν assume values from the set
where is the spin of νth particle. for a particle that does not exhibit spin.
The treatment of identical particles is very different for bosons (particles with integer spin) versus fermions (particles with half-integer spin). The above N-particle
function must either be symmetrized (in the bosonic case) or
anti-symmetrized (in the fermionic case) with respect to the particle
numbers. If not all N particles are identical, but some of them
are, then the function must be (anti)symmetrized separately over the
variables corresponding to each group of identical variables, according
to its statistics (bosonic or fermionic).
Electrons are fermions with S = 1/2, photons (quanta of light) are bosons with S = 1 (although in the vacuum they are massless and can't be described with Schrödinger mechanics).
When symmetrization or anti-symmetrization is unnecessary, N-particle spaces of states can be obtained simply by tensor products of one-particle spaces, to which we will return later.
In general, the expression for probability always consist of a relation between the quantum state and a portion of the spectrum of the dynamical variable (i.e. random variable) being observed. For example, the situation above describes the discrete case as eigenvalues belong to the point spectrum. Likewise, the wave function is just the eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s) ; the energy of the system.
An example of the continuous case is given by the position operator. The probability measure for a system in state is given by:
where
is the probability density function for finding a particle at a given
position. These examples emphasize the distinction in charactertistics
between the state and the observable. That is, whereas is a pure state belonging to , the (generalized) eigenvectors of the position operator do not.
Though closely related, pure states are not the same as bound states belonging to the pure point spectrum of an observable with no quantum uncertainty. A particle is said to be in a bound state if it remains localized in a bounded region of space for all times. A pure state is called a bound state if and only if for every there is a compact set such that
for all . The integral represents the probability that a particle is found in a bounded region at any time . If the probability remains arbitrarily close to then the particle is said to remain in .
For example, non-normalizable solutions of the free Schrödinger equation can be expressed as functions that are normalizable, using wave packets. These wave packets belong to the pure point spectrum of a corresponding projection operator which, mathematically speaking, constitutes an observable. However, they are not bound states.
As mentioned above, quantum states may be superposed. If and are two kets corresponding to quantum states, the ket
is also a quantum state of the same system. Both and can be complex numbers; their relative amplitude and relative phase will influence the resulting quantum state.
Writing the superposed state using
and defining the norm of the state as:
and extracting the common factors gives:
The overall phase factor in front has no physical effect. Only the relative phase affects the physical nature of the superposition.
One example of superposition is the double-slit experiment, in which superposition leads to quantum interference. Another example of the importance of relative phase is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.
A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see Quantum statistical mechanics).
Mixed states arise in quantum mechanics in two different
situations: first, when the preparation of the system is not fully
known, and thus one must deal with a statistical ensemble of possible preparations; and second, when one wants to describe a physical system which is entangled
with another, as its state cannot be described by a pure state. In the
first case, there could theoretically be another person who knows the
full history of the system, and therefore describe the same system as a
pure state; in this case, the density matrix is simply used to represent
the limited knowledge of a quantum state. In the second case, however,
the existence of quantum entanglement theoretically prevents the
existence of complete knowledge about the subsystem, and it's impossible
for any person to describe the subsystem of an entangled pair as a pure
state.
Mixed states inevitably arise from pure states when, for a composite quantum system with an entangled state on it, the part is inaccessible to the observer. The state of the part is expressed then as the partial trace over .
A mixed state cannot be described with a single ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Density matrices can describe both mixed and
pure states, treating them on the same footing. Moreover, a mixed
quantum state on a given quantum system described by a Hilbert space can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system for a sufficiently large Hilbert space .
The density matrix describing a mixed state is defined to be an operator of the form
where ps is the fraction of the ensemble in each pure state The density matrix can be thought of as a way of using the one-particle formalism
to describe the behavior of many similar particles by giving a
probability distribution (or ensemble) of states that these particles
can be found in.
A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ2 is equal to 1 if the state is pure, and less than 1 if the state is mixed. Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.
The rules for
measurement in quantum mechanics are particularly simple to state in
terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable A is given by
where and are eigenkets and eigenvalues, respectively, for the operator A, and "tr" denotes trace. It is important to note that two types of averaging are occurring, one (over ) being the usual expected value of the observable when the quantum is in state , and the other (over ) being a statistical (said incoherent) average with the probabilities ps that the quantum is in those states.