Quantum entanglement is the phenomenon that occurs when a group of
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
s are generated, interact, or share spatial proximity in a way such that the
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.
Measurements of
physical properties such as
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
,
momentum,
spin, and
polarization performed on entangled particles can, in some cases, be found to be perfectly
correlated
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...
. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior gives rise to seemingly
paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
ical effects: any measurement of a particle's properties results in an irreversible
wave function collapse
In quantum mechanics, wave function collapse occurs when a wave function—initially in a quantum superposition, superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is ...
of that particle and changes the original quantum state. With entangled particles, such measurements affect the entangled system as a whole.
Such phenomena were the subject of a 1935 paper by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
,
Boris Podolsky
Boris Yakovlevich Podolsky (russian: link=no, Бори́с Я́ковлевич Подо́льский; June 29, 1896 – November 28, 1966) was a Russian-American physicist of Jewish descent, noted for his work with Albert Einstein and Nathan ...
, and
Nathan Rosen
Nathan Rosen (Hebrew: נתן רוזן; March 22, 1909 – December 18, 1995) was an American-Israeli physicist noted for his study on the structure of the hydrogen atom and his work with Albert Einstein and Boris Podolsky on entangled wave functio ...
,
[
] and several papers by
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
shortly thereafter,
[
][
] describing what came to be known as the
EPR paradox
EPR may refer to:
Science and technology
* EPR (nuclear reactor), European Pressurised-Water Reactor
* EPR paradox (Einstein–Podolsky–Rosen paradox), in physics
* Earth potential rise, in electrical engineering
* East Pacific Rise, a mid-oc ...
. Einstein and others considered such behavior impossible, as it violated the
local realism view of causality (Einstein referring to it as "spooky
action at a distance
In physics, action at a distance is the concept that an object can be affected without being physically touched (as in mechanical contact) by another object. That is, it is the non-local interaction of objects that are separated in space.
Non- ...
") and argued that the accepted formulation of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
must therefore be incomplete.
Later, however, the counterintuitive predictions of quantum mechanics were verified
in tests where polarization or spin of entangled particles was measured at separate locations, statistically violating
Bell's inequality. In earlier tests, it could not be ruled out that the result at one point could have been
subtly transmitted to the remote point, affecting the outcome at the second location.
[Francis, Matthew.]
Quantum entanglement shows that reality can't be local
''Ars Technica'', 30 October 2012 However, so-called "loophole-free" Bell tests have been performed where the locations were sufficiently separated that communications at the speed of light would have taken longer—in one case, 10,000 times longer—than the interval between the measurements.
According to ''some''
interpretations of quantum mechanics
An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraord ...
, the effect of one measurement occurs instantly. Other interpretations which do not recognize
wavefunction collapse
In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an ''observa ...
dispute that there is any "effect" at all. However, all interpretations agree that entanglement produces
correlation between the measurements and that the
mutual information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such ...
between the entangled particles can be exploited, but that any transmission of information at
faster-than-light speeds is impossible.
Quantum entanglement has been demonstrated experimentally with
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s,
[Carl A. Kocher, Ph.D. Thesis (University of California at Berkeley, 1967). ]
Polarization Correlation of Photons Emitted in an Atomic Cascade
' neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s,
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
s,
[ See als]
free online access version
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
s as large as
buckyballs, and even small diamonds. The utilization of entanglement in
communication
Communication (from la, communicare, meaning "to share" or "to be in relation with") is usually defined as the transmission of information. The term may also refer to the message communicated through such transmissions or the field of inqui ...
,
computation and
quantum radar is a very active area of research and development.
History
In 1935,
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
,
Boris Podolsky
Boris Yakovlevich Podolsky (russian: link=no, Бори́с Я́ковлевич Подо́льский; June 29, 1896 – November 28, 1966) was a Russian-American physicist of Jewish descent, noted for his work with Albert Einstein and Nathan ...
and
Nathan Rosen
Nathan Rosen (Hebrew: נתן רוזן; March 22, 1909 – December 18, 1995) was an American-Israeli physicist noted for his study on the structure of the hydrogen atom and his work with Albert Einstein and Boris Podolsky on entangled wave functio ...
published a paper on the counterintuitive predictions that quantum mechanics makes for pairs of objects prepared together in a particular way.
In this study, the three formulated the
Einstein–Podolsky–Rosen paradox
The Einstein–Podolsky–Rosen (EPR) paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen, with which they argued that the description of physical reality provided by quantum mechanics was inco ...
(EPR paradox), a
thought experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences.
History
The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...
that attempted to show that "the
quantum-mechanical description of physical reality given by wave functions is not complete."
However, the three scientists did not coin the word ''entanglement'', nor did they generalize the special properties of the
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
they considered. Following the EPR paper,
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
wrote a letter to Einstein in
German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
in which he used the word ''Verschränkung'' (translated by himself as ''entanglement'') "to describe the correlations between two particles that interact and then separate, as in the EPR experiment."
[Kumar, M., ''Quantum'', Icon Books, 2009, p. 313.]
Schrödinger shortly thereafter published a seminal paper defining and discussing the notion of "entanglement." In the paper, he recognized the importance of the concept, and stated:
"I would not call
ntanglement''one'' but rather ''the'' characteristic trait of quantum mechanics, the one that enforces its entire departure from
classical lines of thought."
Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, because it seemed to violate the speed limit on the transmission of information implicit in the
theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
. Einstein later famously derided entanglement as "''spukhafte Fernwirkung''"
[Letter from Einstein to Max Born, 3 March 1947; ''The Born-Einstein Letters; Correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955'', Walker, New York, 1971. (cited in )] or "
spooky action at a distance
In physics, action at a distance is the concept that an object can be affected without being physically touched (as in mechanical contact) by another object. That is, it is the non-local interaction of objects that are separated in space.
Non- ...
."
The EPR paper generated significant interest among physicists, which inspired much discussion about the foundations of quantum mechanics (perhaps most famously
Bohm's interpretation of quantum mechanics), but produced relatively little other published work. Despite the interest, the weak point in EPR's argument was not discovered until 1964, when
John Stewart Bell
John Stewart Bell FRS (28 July 1928 – 1 October 1990) was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum physics regarding hidden-variable theories.
In 2022, the Nobel Prize in Phy ...
proved that one of their key assumptions, the
principle of locality
In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of ins ...
, as applied to the kind of hidden variables interpretation hoped for by EPR, was mathematically inconsistent with the predictions of quantum theory.
Specifically, Bell demonstrated an upper limit, seen in
Bell's inequality, regarding the strength of correlations that can be produced in any theory obeying
local realism, and showed that quantum theory predicts violations of this limit for certain entangled systems. His inequality is experimentally testable, and there have been numerous
relevant experiments, starting with the pioneering work of
Stuart Freedman and
John Clauser
John Francis Clauser (; born December 1, 1942) is an American theoretical and experimental physicist known for contributions to the foundations of quantum mechanics, in particular the Clauser–Horne–Shimony–Holt inequality.
Clauser was a ...
in 1972
and
Alain Aspect
Alain Aspect (; born 15 June 1947) is a French physicist noted for his experimental work on quantum entanglement.
Aspect was awarded the 2022 Nobel Prize in Physics, jointly with John Clauser and Anton Zeilinger, "for experiments with entangl ...
's experiments in 1982. An early experimental breakthrough was due to Carl Kocher,
who already in 1967 presented an apparatus in which two photons successively emitted from a calcium atom were shown to be entangled – the first case of entangled visible light. The two photons passed diametrically positioned parallel polarizers with higher probability than classically predicted but with correlations in quantitative agreement with quantum mechanical calculations. He also showed that the correlation varied as the squared
cosine of the angle between the polarizer settings
and decreased exponentially with time lag between emitted photons.
Kocher’s apparatus, equipped with better polarizers, was used by Freedman and Clauser who could confirm the cosine-squared dependence and use it to demonstrate a violation of Bell’s inequality for a set of fixed angles.
All these experiments have shown agreement with quantum mechanics rather than the principle of local realism.
For decades, each had left open at least one
loophole
A loophole is an ambiguity or inadequacy in a system, such as a law or security, which can be used to circumvent or otherwise avoid the purpose, implied or explicitly stated, of the system.
Originally, the word meant an arrowslit, a narrow ver ...
by which it was possible to question the validity of the results. However, in 2015 an experiment was performed that simultaneously closed both the detection and locality loopholes, and was heralded as "loophole-free"; this experiment ruled out a large class of local realism theories with certainty.
Aspect writes that "... no experiment ... can be said to be totally loophole-free," but he says the experiments "remove the last doubts that we should renounce" local hidden variables, and refers to examples of remaining loopholes as being "far fetched" and "foreign to the usual way of reasoning in physics."
Bell's work raised the possibility of using these super-strong correlations as a resource for communication. It led to the 1984 discovery of
quantum key distribution
Quantum key distribution (QKD) is a secure communication method which implements a cryptographic protocol involving components of quantum mechanics. It enables two parties to produce a shared random secret key known only to them, which can then b ...
protocols, most famously
BB84
BB84 is a quantum key distribution scheme developed by Charles Bennett and Gilles Brassard in 1984. It is the first quantum cryptography protocol. The protocol is provably secure, relying on two conditions: (1) the quantum property that informat ...
by
Charles H. Bennett and
Gilles Brassard
Gilles Brassard, is a faculty member of the Université de Montréal, where he has been a Full Professor since 1988 and Canada Research Chair since 2001.
Education and early life
Brassard received a Ph.D. in Computer Science from Cornell Unive ...
and
E91 by
Artur Ekert
Artur Konrad Ekert FRS (born 19 September 1961) is a Polish professor of quantum physics at the Mathematical Institute, University of Oxford, professorial fellow in quantum physics and cryptography at Merton College, Oxford, Lee Kong Chian C ...
. Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.
In 2022, the Nobel Prize in Physics was awarded to Aspect, Clauser, and
Anton Zeilinger
Anton Zeilinger (; born 20 May 1945) is an Austrian quantum physicist and Nobel laureate in physics of 2022. Zeilinger is professor of physics emeritus at the University of Vienna and senior scientist at the Institute for Quantum Optics and ...
"for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".
Concept
Meaning of entanglement
An entangled system is defined to be one whose
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole. In entanglement, one constituent cannot be fully described without considering the other(s). The state of a composite system is always expressible as a sum, or
superposition, of products of states of local constituents; it is entangled if this sum cannot be written as a single product term.
Quantum
systems can become entangled through various types of interactions. For some ways in which entanglement may be achieved for experimental purposes, see the section below on
methods
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to:
*Scien ...
. Entanglement is broken when the entangled particles
decohere
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave ...
through interaction with the environment; for example, when a measurement is made.
[Asher Peres, '' Quantum Theory: Concepts and Methods'', Kluwer, 1993; p. 115.]
As an example of entanglement: a
subatomic particle decays into an entangled pair of other particles. The decay events obey the various
conservation laws
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
, and as a result, the measurement outcomes of one daughter particle must be highly correlated with the measurement outcomes of the other daughter particle (so that the total momenta, angular momenta, energy, and so forth remains roughly the same before and after this process). For instance, a
spin-zero particle could decay into a pair of spin-1/2 particles. Since the total spin before and after this decay must be zero (conservation of angular momentum), whenever the first particle is measured to be
spin up on some axis, the other, when measured on the same axis, is always found to be
spin down. (This is called the spin anti-correlated case; and if the prior probabilities for measuring each spin are equal, the pair is said to be in the
singlet state
In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired. The term 'singlet' originally meant a linked set of particles whose net angular momentum is zero, that is, whose overall spin quantum number s=0. A ...
.)
The above result may or may not be perceived as surprising. A classical system would display the same property, and a
hidden variable theory
In physics, hidden-variable theories are proposals to provide explanations of quantum mechanical phenomena through the introduction of (possibly unobservable) hypothetical entities. The existence of fundamental indeterminacy for some measurem ...
would certainly be required to do so, based on conservation of angular momentum in classical and quantum mechanics alike. The difference is that a classical system has definite values for all the observables all along, while the quantum system does not. In a sense to be discussed below, the quantum system considered here seems to acquire a probability distribution for the outcome of a measurement of the spin along any axis of the other particle upon measurement of the first particle. This probability distribution is in general different from what it would be without measurement of the first particle. This may certainly be perceived as surprising in the case of spatially separated entangled particles.
Paradox
The paradox is that a measurement made on either of the particles apparently collapses the state of the entire entangled system—and does so instantaneously, before any information about the measurement result could have been communicated to the other particle (assuming that information cannot travel
faster than light
Faster-than-light (also FTL, superluminal or supercausal) travel and communication are the conjectural propagation of matter or information faster than the speed of light (). The special theory of relativity implies that only particles with zero ...
) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. In the
Copenhagen interpretation
The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as feat ...
, the result of a spin measurement on one of the particles is a collapse into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is taken to be random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it too is measured.
The distance and timing of the measurements can be chosen so as to make the interval between the two measurements
spacelike
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, hence, any causal effect connecting the events would have to travel faster than light. According to the principles of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
, it is not possible for any information to travel between two such measuring events. It is not even possible to say which of the measurements came first. For two spacelike separated events and there are
inertial frames in which is first and others in which is first. Therefore, the correlation between the two measurements cannot be explained as one measurement determining the other: different observers would disagree about the role of cause and effect.
(In fact similar paradoxes can arise even without entanglement: the position of a single particle is spread out over space, and two widely separated detectors attempting to detect the particle in two different places must instantaneously attain appropriate correlation, so that they do not both detect the particle.)
Hidden variables theory
A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "hidden variables". The state of the particles being measured contains some
hidden variables
Hidden variables may refer to:
* Confounding, in statistics, an extraneous variable in a statistical model that correlates (directly or inversely) with both the dependent variable and the independent variable
* Hidden transformation, in computer ...
, whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. Einstein and others (see the previous section) originally believed this was the only way out of the paradox, and the accepted quantum mechanical description (with a random measurement outcome) must be incomplete.
Violations of Bell's inequality
Local hidden variable theories fail, however, when measurements of the spin of entangled particles along different axes are considered. If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the
local realist or hidden variables view were correct, the results would always satisfy
Bell's inequality. A
number of experiments have shown in practice that Bell's inequality is not satisfied. However, prior to 2015, all of these had loophole problems that were considered the most important by the community of physicists. When measurements of the entangled particles are made in moving
relativistic reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.
[Some of the history of both referenced Zbinden, et al. experiments is provided in Gilder, L., ''The Age of Entanglement'', Vintage Books, 2008, pp. 321–324.]
The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are
incompatible in the sense that these measurements' maximum simultaneous precision is constrained by the
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating correlations, and thus entanglement is a fundamentally non-classical phenomenon.
Notable experimental results proving quantum entanglement
The first experiment that verified Einstein's ''spooky action at a distance'' or entanglement was successfully corroborated in a lab by
Chien-Shiung Wu and a colleague named I. Shaknov in 1949, and was published on new year's day in 1950. The result specifically proved the quantum correlations of a pair of photons. In experiments in 2012 and 2013, polarization correlation was created between photons that never coexisted in time.
The authors claimed that this result was achieved by
entanglement swapping between two pairs of entangled photons after measuring the polarization of one photon of the early pair, and that it proves that quantum non-locality applies not only to space but also to time.
In three independent experiments in 2013, it was shown that
classically communicated separable quantum states can be used to carry entangled states. The first loophole-free Bell test was held by
Ronald Hanson of the
Delft University of Technology
Delft University of Technology ( nl, Technische Universiteit Delft), also known as TU Delft, is the oldest and largest Dutch public technical university, located in Delft, Netherlands. As of 2022 it is ranked by QS World University Rankings among ...
in 2015, confirming the violation of Bell inequality.
In August 2014, Brazilian researcher Gabriela Barreto Lemos and team were able to "take pictures" of objects using photons that had not interacted with the subjects, but were entangled with photons that did interact with such objects. Lemos, from the University of Vienna, is confident that this new quantum imaging technique could find application where low light imaging is imperative, in fields like biological or medical imaging.
Since 2016, various companies, for example IBM and Microsoft, have successfully created quantum computers that allowed developers and tech enthusiasts to freely experiment with concepts of quantum mechanics including quantum entanglement.
Mystery of time
There have been suggestions to look at the concept of time as an
emergent phenomenon
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergence ...
that is a side effect of quantum entanglement.
In other words, time is an entanglement phenomenon, which places all equal clock readings (of correctly prepared clocks, or of any objects usable as clocks) into the same history. This was first fully theorized by
Don Page and
William Wootters
William "Bill" Kent Wootters () is an American theoretical physicist, and one of the founders of the field of quantum information theory. In a 1982 joint paper with Wojciech H. Zurek, Wootters proved the no cloning theorem, at the same time as D ...
in 1983.
The
Wheeler–DeWitt equation
The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general ...
that combines general relativity and quantum mechanics – by leaving out time altogether – was introduced in the 1960s and it was taken up again in 1983, when Page and Wootters made a solution based on quantum entanglement. Page and Wootters argued that entanglement can be used to measure time.
Emergent gravity
Based on
AdS/CFT correspondence
In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter s ...
,
Mark Van Raamsdonk suggested that
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the space-time.
Induced gravity can emerge from the entanglement first law.
Non-locality and entanglement
In the media and popular science,
quantum non-locality is often portrayed as being equivalent to entanglement. While this is true for pure bipartite quantum states, in general entanglement is only necessary for non-local correlations, but there exist mixed entangled states that do not produce such correlations.
A well-known example is the
Werner state A Werner state is a -dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form U \otimes U. That is, it is a bipartite quantum state \rho_ that satisfies
:\rho_ = (U \otimes U) \rho_ (U^\dagger \ot ...
s that are entangled for certain values of
, but can always be described using local hidden variables.
Moreover, it was shown that, for arbitrary numbers of particles, there exist states that are genuinely entangled but admit a local model.
The mentioned proofs about the existence of local models assume that there is only one copy of the quantum state available at a time. If the particles are allowed to perform local measurements on many copies of such states, then many apparently local states (e.g., the qubit Werner states) can no longer be described by a local model. This is, in particular, true for all
distillable states. However, it remains an open question whether all entangled states become non-local given sufficiently many copies.
In short, entanglement of a state shared by two particles is necessary but not sufficient for that state to be non-local. It is important to recognize that entanglement is more commonly viewed as an algebraic concept, noted for being a prerequisite to non-locality as well as to
quantum teleportation and to
superdense coding
In quantum information theory, superdense coding (also referred to as ''dense coding'') is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assum ...
, whereas non-locality is defined according to experimental statistics and is much more involved with the
foundations and
interpretations of quantum mechanics
An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraord ...
.
Quantum mechanical framework
The following subsections are for those with a good working knowledge of the formal, mathematical description of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, including familiarity with the formalism and theoretical framework developed in the articles:
bra–ket notation and
mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which ...
.
Pure states
Consider two arbitrary quantum systems and , with respective
Hilbert spaces and . The Hilbert space of the composite system is the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
:
If the first system is in state
and the second in state
, the state of the composite system is
:
States of the composite system that can be represented in this form are called
separable state
In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determinin ...
s, or
product state
In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determinin ...
s.
Not all states are separable states (and thus product states). Fix a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
for and a basis
for . The most general state in is of the form
:
.
This state is separable if there exist vectors