Quantum Entanglement is a phenomena that describes how the measurements of spin, momentum, and position can all be same between two unique particles.The main issue is that entanglement can only exist for brief moments because any sort of observation or change in environment can break it.Entanglement is interesting because it can fit well into other theories as well. One example is Hawking Radiation. The very premise of the theory is that an Entangled Photon Pair is separated at the surface of a Black Hole. How interesting would it be if humans could be entangled with each other?

What is Quantum Entanglement?

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Science & Nature

Quantum entanglement is a physical phenomenon that occurs when a group of <a href="https://en.m.wikipedia.org/wiki/Particle">particles </a>are generated, interact, or share spatial proximity in a way such that the <a href="https://en.m.wikipedia.org/wiki/Quantum_state">quantum state </a>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 <a href="https://en.m.wikipedia.org/wiki/Principle_of_locality">disparity between classical and quantum physics </a>: entanglement is a primary feature of quantum mechanics lacking in classical mechanics.

<a href="https://en.m.wikipedia.org/wiki/Measurement#Quantum_mechanics">Measurements </a>of <a href="https://en.m.wikipedia.org/wiki/Physical_properties">physical properties </a>such as <a href="https://en.m.wikipedia.org/wiki/Position_(vector)">position </a>, <a href="https://en.m.wikipedia.org/wiki/Momentum">momentum </a>, <a href="https://en.m.wikipedia.org/wiki/Spin_(physics)">spin </a>, and <a href="https://en.m.wikipedia.org/wiki/Polarization_(waves)">polarization </a>performed on entangled particles can, in some cases, be found to be perfectly <a href="https://en.m.wikipedia.org/wiki/Correlated">correlated </a>. 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 counterclockwise. However, this behavior gives rise to seemingly <a href="https://en.m.wikipedia.org/wiki/Paradox">paradoxical </a>effects: any measurement of a particle's properties results in an irreversible <a href="https://en.m.wikipedia.org/wiki/Wave_function_collapse">wave function collapse </a>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 <a href="https://en.m.wikipedia.org/wiki/Albert_Einstein">Albert Einstein </a>, <a href="https://en.m.wikipedia.org/wiki/Boris_Podolsky">Boris Podolsky </a>, and <a href="https://en.m.wikipedia.org/wiki/Nathan_Rosen">Nathan Rosen </a>,<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Einstein1935-1">[1] </a> and several papers by <a href="https://en.m.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger">Erwin Schrödinger </a>shortly thereafter,<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Schr%C3%B6dinger1935-2">[2] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Schr%C3%B6dinger1936-3">[3] </a> describing what came to be known as the <a href="https://en.m.wikipedia.org/wiki/EPR_paradox">EPR paradox </a>. Einstein and others considered such behavior impossible, as it violated the <a href="https://en.m.wikipedia.org/wiki/Local_realism">local realism </a>view of causality (Einstein referring to it as "spooky <a href="https://en.m.wikipedia.org/wiki/Action_at_a_distance">action at a distance </a>")<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-4">[4] </a> and argued that the accepted formulation of <a href="https://en.m.wikipedia.org/wiki/Quantum_mechanics">quantum mechanics </a>must therefore be incomplete.

Later, however, the counterintuitive predictions of quantum mechanics were verified<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-:0-5">[5] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-:1-6">[6] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-:2-7">[7] </a> in tests where polarization or spin of entangled particles was measured at separate locations, statistically violating <a href="https://en.m.wikipedia.org/wiki/Bell%27s_inequality">Bell's inequality </a>. In earlier tests, it couldn't be ruled out that the result at one point could have been <a href="https://en.m.wikipedia.org/wiki/Loopholes_in_Bell_test_experiments">subtly transmitted </a>to the remote point, affecting the outcome at the second location.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-:2-7">[7] </a> 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.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-:1-6">[6] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-:0-5">[5] </a>

According to some <a href="https://en.m.wikipedia.org/wiki/Interpretations_of_quantum_mechanics">interpretations of quantum mechanics </a>, the effect of one measurement occurs instantly. Other interpretations which don't recognize <a href="https://en.m.wikipedia.org/wiki/Wavefunction_collapse">wavefunction collapse </a>dispute that there is any "effect" at all. However, all interpretations agree that entanglement produces <a href="https://en.m.wikipedia.org/wiki/Correlation">correlation </a> between the measurements and that the <a href="https://en.m.wikipedia.org/wiki/Mutual_information">mutual information </a>between the entangled particles can be exploited, but that any transmission of information at faster-than-light speeds is impossible.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-8">[8] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Griffiths2004-9">[9] </a>

Quantum entanglement has been demonstrated experimentally with <a href="https://en.m.wikipedia.org/wiki/Photon">photons </a>,<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Kocher1-10">[10] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Kocherphd-11">[11] </a> <a href="https://en.m.wikipedia.org/wiki/Neutrino">neutrinos </a>,<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-12">[12] </a> <a href="https://en.m.wikipedia.org/wiki/Electron">electrons </a>,<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-NTR-20151021-13">[13] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-NYT-20151021-14">[14] </a> <a href="https://en.m.wikipedia.org/wiki/Molecule">molecules </a>as large as <a href="https://en.m.wikipedia.org/wiki/Buckyball">buckyballs </a>,<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-15">[15] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-16">[16] </a> and even small diamonds.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-17">[17] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-18">[18] </a> The utilization of entanglement in <a href="https://en.m.wikipedia.org/wiki/Quantum_communication">communication </a>, <a href="https://en.m.wikipedia.org/wiki/Quantum_computing">computation </a>and <a href="https://en.m.wikipedia.org/wiki/Quantum_radar">quantum radar </a>is a very active area of research and development.

The counterintuitive predictions of quantum mechanics about strongly correlated systems were first discussed by <a href="https://en.m.wikipedia.org/wiki/Albert_Einstein">Albert Einstein </a>in 1935, in a joint paper with <a href="https://en.m.wikipedia.org/wiki/Boris_Podolsky">Boris Podolsky </a>and <a href="https://en.m.wikipedia.org/wiki/Nathan_Rosen">Nathan Rosen </a>.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Einstein1935-1">[1] </a> In this study, the three formulated the <a href="https://en.m.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox">Einstein–Podolsky–Rosen paradox </a>(EPR paradox), a <a href="https://en.m.wikipedia.org/wiki/Thought_experiment">thought experiment </a>that attempted to show that "the <a href="https://en.m.wikipedia.org/wiki/Quantum-mechanical">quantum-mechanical </a>description of physical reality given by wave functions is not complete."<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Einstein1935-1">[1] </a> However, the three scientists did not coin the word entanglement , nor did they generalize the special properties of the state they considered. Following the EPR paper, <a href="https://en.m.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger">Erwin Schrödinger </a>wrote a letter to Einstein in <a href="https://en.m.wikipedia.org/wiki/German_language">German </a>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."<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-MK-19">[19] </a>

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:<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Schr%C3%B6dinger1935-2">[2] </a> "I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from <a href="https://en.m.wikipedia.org/wiki/Classical_mechanics">classical </a>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 <a href="https://en.m.wikipedia.org/wiki/Theory_of_relativity">theory of relativity </a>.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-20">[20] </a> Einstein later famously derided entanglement as "spukhafte Fernwirkung "<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-spukhafte-21">[21] </a> or "spooky <a href="https://en.m.wikipedia.org/wiki/Action_at_a_distance_(physics)">action at a distance </a>."

The EPR paper generated significant interest among physicists, which inspired much discussion about the foundations of quantum mechanics (perhaps most famously <a href="https://en.m.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory">Bohm's interpretation </a>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 <a href="https://en.m.wikipedia.org/wiki/John_Stewart_Bell">John Stewart Bell </a>proved that one of their key assumptions, the <a href="https://en.m.wikipedia.org/wiki/Principle_of_locality">principle of locality </a>, 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 <a href="https://en.m.wikipedia.org/wiki/Bell%27s_inequality">Bell's inequality </a>, regarding the strength of correlations that can be produced in any theory obeying <a href="https://en.m.wikipedia.org/wiki/Local_realism">local realism </a>, and showed that quantum theory predicts violations of this limit for certain entangled systems.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-22">[22] </a> His inequality is experimentally testable, and there have been numerous <a href="https://en.m.wikipedia.org/wiki/Bell_test_experiments">relevant experiments </a>, starting with the pioneering work of <a href="https://en.m.wikipedia.org/wiki/Stuart_Freedman">Stuart Freedman </a>and <a href="https://en.m.wikipedia.org/wiki/John_Clauser">John Clauser </a>in 1972<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Clauser-23">[23] </a> and <a href="https://en.m.wikipedia.org/wiki/Alain_Aspect">Alain Aspect </a>'s experiments in 1982.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-24">[24] </a> An early experimental breakthrough was due to Carl Kocher,<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Kocher1-10">[10] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Kocherphd-11">[11] </a> 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 only upon (as cosine square of) the angle between the polarizer settings<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Kocherphd-11">[11] </a> and decreased exponentially with time lag between emitted photons.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Kocher2-25">[25] </a> Kocher’s apparatus, equipped with better polarizers, was used by Freedman and Clauser who could confirm the cosine square dependence and use it to demonstrate a violation of Bell’s inequality for a set of fixed angles.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Clauser-23">[23] </a> 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 <a href="https://en.m.wikipedia.org/wiki/Loopholes_in_Bell_test_experiments">loophole </a>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.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-hanson-26">[26] </a> <a href="https://en.m.wikipedia.org/wiki/Alain_Aspect">Alain Aspect </a>notes that the "setting-independence loophole" – which he refers to as "far-fetched", yet, a "residual loophole" that "cannot be ignored" – has yet to be closed, and the free-will / <a href="https://en.m.wikipedia.org/wiki/Superdeterminism">superdeterminism </a> loophole is unclosable; saying "no experiment, as ideal as it is, can be said to be totally loophole-free."<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-27">[27] </a>

Bell's work raised the possibility of using these super-strong correlations as a resource for communication. It led to the 1984 discovery of <a href="https://en.m.wikipedia.org/wiki/Quantum_key_distribution">quantum key distribution </a>protocols, most famously <a href="https://en.m.wikipedia.org/wiki/BB84">BB84 </a>by <a href="https://en.m.wikipedia.org/wiki/Charles_H._Bennett_(computer_scientist)">Charles H. Bennett </a>and <a href="https://en.m.wikipedia.org/wiki/Gilles_Brassard">Gilles Brassard </a><a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-28">[28] </a> and <a href="https://en.m.wikipedia.org/wiki/E91_protocol">E91 </a>by <a href="https://en.m.wikipedia.org/wiki/Artur_Ekert">Artur Ekert </a>.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-29">[29] </a> Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

An entangled system is defined to be one whose <a href="https://en.m.wikipedia.org/wiki/Quantum_state">quantum state </a>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 <a href="https://en.m.wikipedia.org/wiki/Quantum_superposition">superposition </a>, of products of states of local constituents; it is entangled if this sum cannot be written as a single product term.

Quantum <a href="https://en.m.wikipedia.org/wiki/Physical_system">systems </a>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 <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#Methods_of_creating_entanglement">methods </a>. Entanglement is broken when the entangled particles <a href="https://en.m.wikipedia.org/wiki/Quantum_decoherence">decohere </a>through interaction with the environment; for example, when a measurement is made.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Peres1993-30">[30] </a>

As an example of entanglement: a <a href="https://en.m.wikipedia.org/wiki/Subatomic_particle">subatomic particle </a><a href="https://en.m.wikipedia.org/wiki/Particle_decay">decays </a>into an entangled pair of other particles. The decay events obey the various <a href="https://en.m.wikipedia.org/wiki/Conservation_laws">conservation laws </a>, 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 <a href="https://en.m.wikipedia.org/wiki/Spin_(physics)">spin </a>-zero particle could decay into a pair of spin-½ 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 <a href="https://en.m.wikipedia.org/wiki/Spin_(physics)#Direction">spin up </a>on some axis, the other, when measured on the same axis, is always found to be <a href="https://en.m.wikipedia.org/wiki/Spin_(physics)#Direction">spin down </a>. (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 <a href="https://en.m.wikipedia.org/wiki/Singlet_state">singlet state </a>.)

The above result may or may not be perceived as surprising. A classical system would display the same property, and a <a href="https://en.m.wikipedia.org/wiki/Hidden_variable_theory">hidden variable theory </a>(see below) 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.

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 <a href="https://en.m.wikipedia.org/wiki/Faster_than_light">faster than light </a>) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. In the <a href="https://en.m.wikipedia.org/wiki/Copenhagen_interpretation">Copenhagen interpretation </a>, 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.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-31">[31] </a>

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements <a href="https://en.m.wikipedia.org/wiki/Spacelike">spacelike </a>, hence, any causal effect connecting the events would have to travel faster than light. According to the principles of <a href="https://en.m.wikipedia.org/wiki/Special_relativity">special relativity </a>, 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 x 1 and x 2 there are <a href="https://en.m.wikipedia.org/wiki/Inertial_frame">inertial frames </a>in which x 1 is first and others in which x 2 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.)

A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "hidden variables".<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-32">[32] </a> The state of the particles being measured contains some <a href="https://en.m.wikipedia.org/wiki/Hidden-variable_theory">hidden variables </a>, 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.

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 <a href="https://en.m.wikipedia.org/wiki/Local_realism">local realist </a>or hidden variables view were correct, the results would always satisfy <a href="https://en.m.wikipedia.org/wiki/Bell%27s_inequality">Bell's inequality </a>. A <a href="https://en.m.wikipedia.org/wiki/Bell_test_experiments">number of experiments </a>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.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-33">[33] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-34">[34] </a> When measurements of the entangled particles are made in moving <a href="https://en.m.wikipedia.org/wiki/Special_relativity">relativistic </a>reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-35">[35] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-LG-36">[36] </a>

The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are <a href="https://en.m.wikipedia.org/wiki/Incompatible_observables">incompatible </a>in the sense that these measurements' maximum simultaneous precision is constrained by the <a href="https://en.m.wikipedia.org/wiki/Uncertainty_principle">uncertainty principle </a>. 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,<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-37">[37] </a> 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 <a href="https://en.m.wikipedia.org/wiki/Chien-Shiung_Wu">Chien-Shiung Wu </a>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.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-38">[38] </a> In experiments in 2012 and 2013, polarization correlation was created between photons that never coexisted in time.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Xiao-song2012-39">[39] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-40">[40] </a> The authors claimed that this result was achieved by <a href="https://en.m.wikipedia.org/wiki/Quantum_teleportation#Entanglement_swapping">entanglement swapping </a>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 <a href="https://en.m.wikipedia.org/wiki/Classical_physics">classically communicated </a><a href="https://en.m.wikipedia.org/wiki/Separable_state">separable quantum states </a>can be used to carry entangled states.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-41">[41] </a> The first loophole-free Bell test was held in TU Delft in 2015 confirming the violation of Bell inequality.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-42">[42] </a>

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.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-43">[43] </a>

From 2016 various companies like IBM, Microsoft etc. have successfully created quantum computers and allowed developers and tech enthusiasts to openly experiment with concepts of quantum mechanics including quantum entanglement.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-44">[44] </a>

There have been suggestions to look at the concept of time as an <a href="https://en.m.wikipedia.org/wiki/Emergent_phenomenon">emergent phenomenon </a>that is a side effect of quantum entanglement.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-45">[45] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-46">[46] </a> 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 <a href="https://en.m.wikipedia.org/wiki/Don_Page_(physicist)">Don Page </a>and <a href="https://en.m.wikipedia.org/wiki/William_Wootters">William Wootters </a>in 1983.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-47">[47] </a> The <a href="https://en.m.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation">Wheeler–DeWitt equation </a>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.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-medium.com-48">[48] </a>

Based on <a href="https://en.m.wikipedia.org/wiki/AdS/CFT_correspondence">AdS/CFT correspondence </a>, <a href="https://en.m.wikipedia.org/wiki/Mark_Van_Raamsdonk">Mark Van Raamsdonk </a>suggested that <a href="https://en.m.wikipedia.org/wiki/Spacetime">spacetime </a>arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the space-time.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-49">[49] </a> <a href="https://en.m.wikipedia.org/wiki/Induced_gravity">Induced gravity </a>can emerge from the entanglement first law.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-50">[50] </a> <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-51">[51] </a>

In the media and popular science, <a href="https://en.m.wikipedia.org/wiki/Quantum_nonlocality">quantum non-locality </a>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 href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Brunner-RMP2014-52">[52] </a> A well-known example is the <a href="https://en.m.wikipedia.org/wiki/Werner_state">Werner states </a>that are entangled for certain values of

, but can always be described using local hidden variables.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-werner1989-53">[53] </a> Moreover, it was shown that, for arbitrary numbers of parties, there exist states that are genuinely entangled but admit a local model.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-54">[54] </a> 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 parties 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 <a href="https://en.m.wikipedia.org/wiki/Entanglement_distillation">distillable </a>states. However, it remains an open question whether all entangled states become non-local given sufficiently many copies.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-55">[55] </a>

In short, entanglement of a state shared by two parties 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 <a href="https://en.m.wikipedia.org/wiki/Quantum_teleportation">quantum teleportation </a>and to <a href="https://en.m.wikipedia.org/wiki/Superdense_coding">superdense coding </a>, whereas non-locality is defined according to experimental statistics and is much more involved with the <a href="https://en.m.wikipedia.org/wiki/Quantum_foundations">foundations </a>and <a href="https://en.m.wikipedia.org/wiki/Interpretations_of_quantum_mechanics">interpretations of quantum mechanics </a>.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-56">[56] </a>

The following subsections are for those with a good working knowledge of the formal, mathematical description of <a href="https://en.m.wikipedia.org/wiki/Quantum_mechanics">quantum mechanics </a>, including familiarity with the formalism and theoretical framework developed in the articles: <a href="https://en.m.wikipedia.org/wiki/Bra%E2%80%93ket_notation">bra–ket notation </a>and <a href="https://en.m.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics">mathematical formulation of quantum mechanics </a>.

Consider two arbitrary quantum systems A and B , with respective <a href="https://en.m.wikipedia.org/wiki/Hilbert_space">Hilbert spaces </a>HA and HB . The Hilbert space of the composite system is the <a href="https://en.m.wikipedia.org/wiki/Tensor_product">tensor product </a>

, the state of the composite system is

States of the composite system that can be represented in this form are called <a href="https://en.m.wikipedia.org/wiki/Separable_state">separable states </a>, or <a href="https://en.m.wikipedia.org/wiki/Product_state">product states </a>.

Not all states are separable states (and thus product states). Fix a <a href="https://en.m.wikipedia.org/wiki/Basis_(linear_algebra)">basis </a>

for HA and a basis

for HB . The most general state in HA ⊗ HB is of the form

This state is separable if there exist vectors

It is inseparable if for any vectors

at least for one pair of coordinates

If a state is inseparable, it is called an 'entangled state'.

For example, given two basis vectors

of HA and two basis vectors

of HB , the following is an entangled state:

If the composite system is in this state, it is impossible to attribute to either system A or system B a definite <a href="https://en.m.wikipedia.org/wiki/Pure_state">pure state </a>. Another way to say this is that while the <a href="https://en.m.wikipedia.org/wiki/Von_Neumann_entropy">von Neumann entropy </a>of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". This has specific empirical ramifications for interferometry.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-JaegerEtAl95-57">[57] </a> The above example is one of four <a href="https://en.m.wikipedia.org/wiki/Bell_states">Bell states </a>, which are (maximally) entangled pure states (pure states of the HA ⊗ HB space, but which cannot be separated into pure states of each HA and HB ).

Now suppose Alice is an observer for system A , and Bob is an observer for system B . If in the entangled state given above Alice makes a measurement in the

eigenbasis of A , there are two possible outcomes, occurring with equal probability:<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-nielchuang-58">[58] </a>

<ol><li>Alice measures 0, and the state of the system collapses to <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc5ace129e7af51062d3f2e8a4d10bbd424fd41">. </li><li>Alice measures 1, and the state of the system collapses to <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76b669b971b79d549e2d788d168232699ec5599b">. </li> </ol>

If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system B has been altered by Alice performing a local measurement on system A . This remains true even if the systems A and B are spatially separated. This is the foundation of the <a href="https://en.m.wikipedia.org/wiki/EPR_paradox">EPR paradox </a>.

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see <a href="https://en.m.wikipedia.org/wiki/No-communication_theorem">no-communication theorem </a>.

As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a <a href="https://en.m.wikipedia.org/wiki/Density_matrix">density matrix </a>, which is a <a href="https://en.m.wikipedia.org/wiki/Positive-semidefinite_matrix">positive-semidefinite matrix </a>, or a <a href="https://en.m.wikipedia.org/wiki/Trace_class">trace class </a>when the state space is infinite-dimensional, and has trace 1. Again, by the <a href="https://en.m.wikipedia.org/wiki/Spectral_theorem">spectral theorem </a>, such a matrix takes the general form:

where the w i are positive-valued probabilities (they sum up to 1), the vectors αi are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret ρ as representing an ensemble where wi is the proportion of the ensemble whose states are

. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need <a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#Reduced_density_matrices">density matrices </a>to represent the state.

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits <a href="https://en.m.wikipedia.org/wiki/Electron">electrons </a>towards an observer. The electrons' Hilbert spaces are <a href="https://en.m.wikipedia.org/wiki/Identical_particles">identical </a>. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state

with <a href="https://en.m.wikipedia.org/wiki/Spin_(physics)">spins </a>aligned in the positive z direction, and the other with state

with spins aligned in the negative y direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

Following the definition above, for a bipartite composite system, mixed states are just density matrices on HA ⊗ HB . That is, it has the general form

where the w i are positively valued probabilities,

, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-Laloe-59">[59] </a> :131–132 ​

where the wi are positively valued probabilities and the

's are themselves mixed states (density operators) on the subsystems A and B respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that

are themselves pure ensembles. A state is then said to be entangled if it is not separable.

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be <a href="https://en.m.wikipedia.org/wiki/NP-hard">NP-hard </a>.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-60">[60] </a> For the 2 × 2 and 2 × 3 cases, a necessary and sufficient criterion for separability is given by the famous <a href="https://en.m.wikipedia.org/wiki/Peres-Horodecki_criterion">Positive Partial Transpose (PPT) </a>condition.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-61">[61] </a>

The idea of a reduced density matrix was introduced by <a href="https://en.m.wikipedia.org/wiki/Paul_Dirac">Paul Dirac </a>in 1930.<a href="https://en.m.wikipedia.org/wiki/Quantum_entanglement#cite_note-62">[62] </a> Consider as above systems A and B each with a Hilbert space HA , HB . Let the state of the composite system be

As indicated above, in general there is no way to associate a pure state to the component system A . However, it still is possible to associate a density matrix. Let

which is the <a href="https://en.m.wikipedia.org/wiki/Projection_operator">projection operator </a>onto this state. The state of A is the <a href="https://en.m.wikipedia.org/wiki/Partial_trace">partial trace </a>of ρT over the basis of system B :

. ρA is sometimes called the reduced density matrix of ρ on subsystem A . Colloquially, we "trace out" system B to obtain the reduced density matrix on A .

For example, the reduced density matrix of A for the entangled state

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of A for the pure product state

Quantum entanglement - Wikipedia

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