Quantum entanglement is the phenomenon wherein thequantum state of eachparticle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity betweenclassical physics andquantum physics: entanglement is a primary feature of quantum mechanics not present inclassical mechanics.[1]: 867
Measurements ofphysical properties such asposition,momentum,spin, andpolarization performed on entangled particles can, in some cases, be found to be perfectlycorrelated. 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. This behavior gives rise to seeminglyparadoxical effects: any measurement of a particle's properties results in an apparent and irreversiblewave function collapse of that particle and changes the original quantum state. With entangled particles, such measurements affect the entangled system as a whole.
Later, the counterintuitive predictions of quantum mechanics were verified in tests where polarization or spin of entangled particles were measured at separate locations, statistically violatingBell's inequality.[6][7][8][9] This established that the correlations produced from quantum entanglement cannot be explained in terms oflocal hidden variables, i.e., properties contained within the individual particles themselves.
Albert Einstein and Niels Bohr engaged in a long-running collegial dispute over the interpretation of quantum mechanics, now known as theBohr–Einstein debates. During these debates, Einstein introduced athought experiment involving a box that emits a photon. He noted that the experimenter's choice of which measurement to make on the box would change what can be predicted about the photon, even when the photon is very far away. This argument, which Einstein had formulated by 1931, was an early recognition of what would later be called entanglement.[20] That same year,Hermann Weyl observed in his textbook ongroup theory and quantum mechanics that quantum systems composed of multiple interacting parts exhibit a kind ofGestalt, in which "the whole is greater than the sum of its parts".[21][22] In 1932,Erwin Schrödinger derived the defining equations of quantum entanglement but left them unpublished.[23] In 1935,Grete Hermann studied the mathematics of an electron interacting with a photon and noted the phenomenon that would later be called entanglement.[24] Later that same year, Einstein,Boris Podolsky andNathan Rosen published a paper on what is now known as theEinstein–Podolsky–Rosen (EPR) paradox, a thought experiment that attempted to show that "thequantum-mechanical description of physical reality given by wave functions is not complete".[2] Their thought experiment considered two systems that interact and then separate, and they argued that, afterward, quantum mechanics could not describe the two systems individually.
Shortly after this paper appeared,Erwin Schrödinger wrote a letter to Einstein inGerman in which he used the wordVerschränkung (translated by himself asentanglement) to describe situations like that of the EPR scenario.[25] Schrödinger followed up with a full paper defining and discussing the notion ofentanglement,[26] saying "I would not call [entanglement]one but ratherthe characteristic trait of quantum mechanics, the one that enforces its entire departure fromclassical lines of thought."[3] 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 thetheory of relativity.[27] Einstein later disparaged quantum mechanics for seemingly exhibiting "spukhafte Fernwirkung"[28] or "spooky action at a distance", meaning the acquisition of a value of a property at one location resulting from a measurement at a distant location.[29]
In 1946,John Archibald Wheeler suggested studying thepolarization of pairs ofgamma-ray photons produced by electron–positron annihilation.[30]Chien-Shiung Wu and I. Shaknov carried out this experiment in 1949,[31] thereby demonstrating that the entangled particle pairs considered by EPR could be created in the laboratory.[32]
Despite Schrödinger's claim of its importance, little work on entanglement was published for decades after his paper was published.[26] In 1964John S. Bell demonstrated an upper limit, seen inBell's inequality, regarding the strength of correlations that can be produced in any theory obeyinglocal realism, and showed that quantum theory predicts violations of this limit for certain entangled systems.[33][34]: 405 His inequality is experimentally testable, and there have been numerousrelevant experiments, starting with the pioneering work ofStuart Freedman andJohn Clauser in 1972[6] andAlain Aspect's experiments in 1982.[35][36][37]
While Bell actively discouraged students from pursuing work like his as too esoteric, after a talk at Oxford a student namedArtur Ekert suggested that the violation of a Bell inequality could be used as a resource for communication.[38]: 315 Ekert followed up by publishing aquantum key distribution protocol calledE91 based on it.[39][1]: 874
In 1992, the entanglement concept was leveraged to proposequantum teleportation,[40] an effect that was realized experimentally in 1997.[41][42][43]
In 2022, theNobel Prize in Physics was awarded to Aspect, Clauser, and Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".[46]
Just asenergy is a resource that facilitates mechanical operations, entanglement is a resource that facilitates performing tasks that involve communication and computation.[47]: 106 [48]: 218 [49]: 435 [50] The mathematical definition of entanglement can be paraphrased as saying that maximal knowledge about the whole of a system does not imply maximal knowledge about the individual parts of that system.[51] If the quantum state that describes a pair of particles is entangled, then the results of measurements upon one half of the pair can be strongly correlated with the results of measurements upon the other. However, entanglement is not the same as "correlation" as understood in classical probability theory and in daily life. Instead, entanglement can be thought of aspotential correlation that can be used to generate actual correlation in an appropriate experiment.[52]: 130 The correlations generated from an entangled quantum state cannot in general be replicated by classical probability.[53]: 33
An example of entanglement is asubatomic particle thatdecays into an entangled pair of other particles. The decay events obey the variousconservation laws, 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, aspin-zero particle could decay into a pair of spin-1/2 particles. If there is no orbital angular momentum, the total spin angular momentum after this decay must be zero (by theconservation of angular momentum). Whenever the first particle is measured to bespin up on some axis, the other, when measured on the same axis, is always found to bespin down. This is called the spin anti-correlated case and the pair is said to be in thesinglet state. Perfect anti-correlations like this could be explained by "hidden variables" within the particles. For example, we could hypothesize that the particles are made in pairs such that one carries a value of "up" while the other carries a value of "down". Then, knowing the result of the spin measurement upon one particle, we could predict that the other will have the opposite value. Bell illustrated this with a story about a colleague, Bertlmann, who always wore socks with mismatching colors. "Which colour he will have on a given foot on a given day is quite unpredictable," Bell wrote, but upon observing "that the first sock is pink you can be already sure that the second sock will not be pink."[54] Revealing the remarkable features of quantum entanglement requires considering multiple distinct experiments, such as spin measurements along different axes, and comparing the correlations obtained in these different configurations.[55]: §18.8
Quantumsystems can become entangled through various types of interactions. For some ways in which entanglement may be achieved for experimental purposes, see the section below onmethods. Entanglement is broken when the entangled particlesdecohere through interaction with the environment; for example, when a measurement is made. In more detail, this process involves the particles becoming entangled with the environment, as a consequence of which, the quantum state describing the particles themselves is no longer entangled.[56]: 369 [57]
Mathematically, an entangled system can be defined to be one whose quantum state 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. When entanglement is present, one constituent cannot be fully described without considering the other(s).[58]: 18–19 [55]: §1.5 The state of a composite system is always expressible as a sum, orsuperposition, of products of states of local constituents; it is entangled if this sum cannot be written as a single product term.[48]: 39
The singlet state described above is the basis for one version of the EPR paradox. In this variant, introduced byDavid Bohm, a source emits particles and sends them in opposite directions. The state describing each pair is entangled.[59] In the standard textbook presentation of quantum mechanics, performing a spin measurement on one of the particles causes the wave function for the whole pair tocollapse into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is 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.[55]: §18.8 [12]: 447–448
The distance and timing of the measurements can be chosen so as to make the interval between the two measurementsspacelike, hence, any causal effect connecting the events would have to travel faster than light. According to the principles ofspecial relativity, 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 eventsx1 andx2 there areinertial frames in whichx1 is first and others in whichx2 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.[60]
A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "hidden variables".[61] The state of the particles being measured contains some hidden variables, 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 local realist or hidden variables view were correct, the results would always satisfyBell's inequality. Anumber of experiments have shown in practice that Bell's inequality is not satisfied.[6][62][63][64] Moreover, when measurements of the entangled particles are made in movingrelativistic reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.[65][38]: 321–324
The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they areincompatible in the sense that these measurements' maximum simultaneous precision is constrained by theuncertainty principle. 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,[66] and thus entanglement is a fundamentally non-classical phenomenon.
As discussed above, entanglement is necessary to produce a violation of aBell inequality. However, the mere presence of entanglement alone is insufficient,[67] as Bell himself noted in his 1964 paper.[33] This is demonstrated, for example, byWerner states, which are a family of states describing pairs of particles. For appropriate choices of the key parameter that identifies a given Werner state within the full set thereof, the Werner states exhibit entanglement. Yet pairs of particles described by Werner states always admit a local hidden variable model. In other words, these states cannot power the violation of a Bell inequality, despite possessing entanglement.[68] This can be generalized from pairs of particles to larger collections as well.[69]
The violation of Bell inequalities is often calledquantum nonlocality. This term is not without controversy.[70] It is sometimes argued that using the termnonlocality carries the unwarranted implication that the violation of Bell inequalities must be explained by physical, faster-than-light signals.[71] In other words, the failure of local hidden-variable models to reproduce quantum mechanics is not necessarily a sign of true nonlocality in quantum mechanics itself.[72][73][74] Despite these reservations, the termnonlocality has become a widespread convention.[71]
Consider two arbitrary quantum systemsA andB, with respectiveHilbert spacesHA andHB. The Hilbert space of the composite system is thetensor product
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 states, orproduct states. However, not all states of the composite system are separable. Fix abasis forHA and a basis forHB. The most general state inHA ⊗HB is of the form
.
This state is separable if there exist vectors so that yielding and It is inseparable if for any vectors at least for one pair of coordinates we have If a state is inseparable, it is called an 'entangled state'.[48]: 218 [55]: §1.5
For example, given two basis vectors ofHA and two basis vectors ofHB, the following is an entangled state:
If the composite system is in this state, it is impossible to attribute to either systemA or systemB a definitepure state. Another way to say this is that while thevon Neumann entropy 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". The above example is one of fourBell states, which are (maximally) entangled pure states (pure states of theHA ⊗HB space, but which cannot be separated into pure states of eachHA andHB).[55]: §18.6
Now suppose Alice is an observer for systemA, and Bob is an observer for systemB. If in the entangled state given above Alice makes a measurement in the eigenbasis ofA, there are two possible outcomes, occurring with equal probability: Alice can obtain the outcome 0, or she can obtain the outcome 1. If she obtains the outcome 0, then she can predict with certainty that Bob's result will be 1. Likewise, if she obtains the outcome 1, then she can predict with certainty that Bob's result will be 0. In other words, the results of measurements on the two qubits will be perfectly anti-correlated. This remains true even if the systemsA andB are spatially separated. This is the foundation of the EPR paradox.[47]: 113–114
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, seeno-communication theorem.
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 adensity matrix, which is apositive-semidefinite matrix, or atrace class when the state space is infinite-dimensional, and which has trace 1. By thespectral theorem, such a matrix takes the general form:
where thewi are positive-valued probabilities (they sum up to 1), the vectorsαi are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpretρ as representing an ensemble where 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 needdensity matrices to represent the state.[56]: 73–74 [53]: 13–15 [55]: §22.2
Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spitselectrons towards an observer. The electrons' Hilbert spaces areidentical. 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 spins aligned in the positivez direction, and the other with state with spins aligned in the negativey 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 onHA ⊗HB. That is, it has the general form
where thewi 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[77]: 131–132
where thewi are positively valued probabilities and thes ands are themselves mixed states (density operators) on the subsystemsA andB 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 and 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 beNP-hard.[78] For the2 × 2 and2 × 3 cases, a necessary and sufficient criterion for separability is given by the famousPositive Partial Transpose (PPT) condition.[79]
The idea of a reduced density matrix was introduced byPaul Dirac in 1930.[80] Consider as above systemsA andB each with a Hilbert spaceHA, 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 systemA. However, it still is possible to associate a density matrix. Let
The sum occurs over and the identity operator in.ρA is sometimes called the reduced density matrix ofρ on subsystemA. Colloquially, we "trace out" or "trace over" systemB to obtain the reduced density matrix onA.[48]: 207–212 [51]: 133 [55]: §22.4
For example, the reduced density matrix ofA for the entangled state
This demonstrates that the reduced density matrix for an entangled pure ensemble is a mixed ensemble. In contrast, the density matrix ofA for the pure product state discussed above is[47]: 106
the projection operator onto.
In general, a bipartite pure stateρ is entangled if and only if its reduced states are mixed rather than pure.[51]: 131
In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows implementing valuable transformations.[81][82] The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labelled "A" and "B" on each of which arbitraryquantum operations can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations calledLOCC (local operations and classical communication). These operations do not allow the production of entangled states between systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations.
If Alice and Bob share an entangled state, Alice can tell Bob over a telephone call how to reproduce a quantum state she has in her lab. Alice performs a joint measurement on together with her half of the entangled state and tells Bob the results. Using Alice's results Bob operates on his half of the entangled state to make it equal to. Since Alice's measurement necessarily erases the quantum state of the system in her lab, the state is not copied, but transferred: it is said to be "teleported" to Bob's laboratory through this protocol.[47]: 27 [1]: 875 [83]
Entanglement of states from independent sources can be swapped through Bell state measurement.[84]: 341
Entanglement swapping is variant of teleportation that allows two parties that have never interacted to share an entangled state. The swapping protocol begins with two EPR sources. One source emits an entangled pair of particles A and B, while the other emits a second entangled pair of particles C and D. Particles B and C are subjected to a measurement in the basis of Bell states. The state of the remaining particles, A and D, collapses to a Bell state, leaving them entangled despite never having interacted with each other.[1][85]
An interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.[1]
Quantum states describing systems made of more than two pieces can also be entangled. An example for a three-qubit system is theGreenberger–Horne–Zeilinger (GHZ) state,Another three-qubit example is theW state:Tracing out any one of the three qubits turns the GHZ state into a separable state, whereas the result of tracing over any of the three qubits in the W state is still entangled. This illustrates how multipartite entanglement is a more complicated topic than bipartite entanglement: systems composed of three or more parts can exhibit multiple qualitatively different types of entanglement.[49]: 493–497 A single particle cannot be maximally entangled with more than a particle at a time, a property calledmonogamy.[86]
Not all quantum states are equally valuable as a resource. One method to quantify this value is to use anentanglement measure that assigns a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are:
If two states can be transformed into each other by a local unitary operation, they are said to be in the sameLU class. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state).[87][88]
If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states and in the same SLOCC class are equally powerful, since one can transform each into the other, but since the transformations and may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like) and the separable ones (i.e., product states like).[89][90]
Instead of considering transformations of single copies of a state (like) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when is impossible by LOCC, but is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state into at least one pure entangled state. States that have this property are calleddistillable. These states are the most useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable; those that are not are called "bound entangled".[91][1]
A different entanglement classification is based on what the quantum correlations present in a state allow A and B to do: one distinguishes three subsets of entangled states: (1) thenon-local states, which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) thesteerable states that contain sufficient correlations for A to modify ("steer") by local measurements the conditional reduced state of B in such a way, that A can prove to B that the state they possess is indeed entangled, and finally (3) those entangled states that are neither non-local nor steerable. All three sets are non-empty.[92]
Since a mixed stateρ is a probability distribution over an ensemble, this leads naturally to the definition of thevon Neumann entropy:[56]: 264
which can be expressed in terms of theeigenvalues ofρ:
.
Since an event of probability 0 should not contribute to the entropy, and given that
the convention0 log(0) = 0 is adopted. When a pair of particles is described by the spin singlet state discussed above, the von Neumann entropy of either particle islog(2), which can be shown to be the maximum entropy for2 × 2 mixed states.[53]: 15
Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.[94][95] If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.[96]
It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n, ..., 1/n}.[47]: 505 Therefore, a bipartite pure stateρ ∈HA ⊗HB is said to be amaximally entangled state if the reduced state of each subsystem ofρ is the diagonal matrix[97]
For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.[49]: 471
Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned,entanglement entropy is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature[94] and no single one is standard.
Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult (NP-hard) to compute for mixed states as the dimension of the entangled system grows.[99]
In August 2014, Brazilian researcher Gabriela Barreto Lemos, from the University of Vienna, 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.[105] The idea has been adapted to make infrared images using only standard cameras that are insensitive to infrared.[106]
These four pure states are all maximally entangled and form anorthonormalbasis of the Hilbert space of the two qubits.[48]: 38–39 [47]: 98 They provide examples of how quantum mechanics can violateBell-type inequalities.[48]: 62 [47]: 116
which reduces to the Bell state forM = 2. The traditional GHZ state was defined forM = 3. GHZ states are occasionally extended toqudits, i.e., systems ofd rather than 2 dimensions.[107][108]
Also forM > 2 qubits, there arespin squeezed states, a class ofsqueezed coherent states satisfying certain restrictions on the uncertainty of spin measurements, which are necessarily entangled.[109] Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement.[110]
This is like the Bell state except the basis states and have been replaced with "theN photons are in one mode" and "theN photons are in the other mode".[111]
Finally, there also existtwin Fock states for bosonic modes, which can be created by feeding aFock state into two arms leading to a beam splitter. They are the sum of multiple NOON states, and can be used to achieve theHeisenberg limit.[112]
For the appropriately chosen measures of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled.[113][111][114]
It is also possible to create entanglement between quantum systems that never directly interacted, through the use ofentanglement swapping. Two independently prepared, identical particles may also be entangled if their wave functions merely spatially overlap, at least partially.[120]
A density matrixρ is called separable if it can be written as a convex sum of product states, namelywith probabilities. By definition, a state is entangled if it is not separable.
For 2-qubit and qubit-qutrit systems (2 × 2 and 2 × 3 respectively) the simplePeres–Horodecki criterion provides both a necessary and a sufficient criterion for separability, and thus—inadvertently—for detecting entanglement. However, for the general case, the criterion is merely a necessary one for separability, as the problem becomes NP-hard when generalized.[121][122] Other separability criteria include (but not limited to) therange criterion,reduction criterion, and those based on uncertainty relations.[123][124][125][126] See Ref.[127] for a review of separability criteria in discrete-variable systems and Ref.[128] for a review on techniques and challenges in experimental entanglement certification in discrete-variable systems.
A numerical approach to the problem is suggested byJon Magne Leinaas,Jan Myrheim andEirik Ovrum in their paper "Geometrical aspects of entanglement".[129] Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached.
In continuous variable systems, the Peres–Horodecki criterion also applies. Specifically, Simon[130] formulated a particular version of the Peres–Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for-mode Gaussian states (see Ref.[131] for a seemingly different but essentially equivalent approach). It was later found[132] that Simon's condition is also necessary and sufficient for-mode Gaussian states, but no longer sufficient for-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators[133][134] or by using entropic measures.[135][136]
There is a fundamental conflict, referred to as theproblem of time, between the way the concept oftime is used in quantum mechanics, and the role it plays ingeneral relativity. In standard quantum theories time acts as an independent background through which states evolve, while general relativity treats time as a dynamical variable which relates directly with matter. Part of the effort to reconcile these approaches to time results in theWheeler–DeWitt equation, which predicts the state of the universe is timeless or static, contrary to ordinary experience.[137] Work started byDon Page andWilliam Wootters[138][139][140] suggests that the universe appears to evolve for observers on the inside because of energy entanglement between an evolving system and a clock system, both within the universe.[137] In this way the overall system can remain timeless while parts experience time via entanglement. The issue remains an open question closely related to attempts at theories ofquantum gravity.[141][142]
In general relativity, gravity arises from the curvature of spacetime and that curvature derives from the distribution of matter. However, matter is governed by quantum mechanics. Integration of these two theories faces many problems. In an (unrealistic) model space called theanti-de Sitter space, theAdS/CFT correspondence allows a quantum gravitational system to be related to a quantum field theory without gravity.[143] Using this correspondence,Mark Van Raamsdonk suggested thatspacetime arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the spacetime.[144]
ABell test, also known asBell inequality test orBell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics against the hypothesis of local hidden variables. These tests empirically evaluate the implications ofBell's theorem. To date, all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave. Many types of Bell tests have been performed in physics laboratories, often with the goal of ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. This is known as "closing loopholes in Bell tests". 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.[9] However, so-called "loophole-free" Bell tests have since 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.[8][7][15][36]
In 2017, Yin et al. reported setting a new quantum entanglement distance record of 1,203 km, demonstrating the survival of a two-photon pair and a violation of a Bell inequality, reaching aCHSH valuation of2.37±0.09, under strict Einstein locality conditions, from theMicius satellite to bases in Lijian, Yunnan and Delingha, Qinghai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude.[145][146]
In 2023 theLHC using techniques fromquantum tomography measured entanglement at the highest energy so far,[147][148][149] a rare intersection between quantum information and high energy physics based on theoretical work first proposed in 2021.[150] The experiment was carried by theATLAS detector measuring the spin of top-quark pair production and the effect was observed with a more than 5σ level of significance, the top quark is the heaviest known particle and therefore has a very short lifetime ( ≈10−25 s) being the only quark that decays before undergoinghadronization (~ 10−23 s) and spin decorrelation (~ 10−21 s), so the spin information is transferred without much loss to the leptonic decays products that will be caught by the detector.[151] Thespin polarization and correlation of the particles was measured and tested for entanglement withconcurrence as well as thePeres–Horodecki criterion and subsequently the effect has been confirmed too in theCMS detector.[152][153]
Physicists atBrookhaven National Laboratory demonstrated quantum entanglement withinprotons, showingquarks andgluons are interdependent rather than isolated particles.[158] Using high-energy electron-proton collisions, they revealed maximal entanglement, reshaping our understanding of proton structure.[159]
^Physicist John Bell depicts the Einstein camp in this debate in his article entitled "Bertlmann's socks and the nature of reality", p. 143 ofSpeakable and unspeakable in quantum mechanics: "For EPR that would be an unthinkable 'spooky action at a distance'. To avoid such action at a distance they have to attribute, to the space-time regions in question, real properties in advance of observation, correlated properties, which predetermine the outcomes of these particular observations. Since these real properties, fixed in advance of observation, are not contained in quantum formalism, that formalism for EPR is incomplete. It may be correct, as far as it goes, but the usual quantum formalism cannot be the whole story." And again on p. 144 Bell says: "Einstein had no difficulty accepting that affairs in different places could be correlated. What he could not accept was that an intervention at one place could influence, immediately, affairs at the other." Downloaded 5 July 2011 fromBell, J. S. (1987).Speakable and Unspeakable in Quantum Mechanics(PDF).CERN.ISBN0-521-33495-0. Archived fromthe original(PDF) on 12 April 2015.
^Heathcote, Adrian (2021). "Multiplicity and indiscernability".Synthese.198 (9):8779–8808.doi:10.1007/s11229-020-02600-8.For Weyl clearly anticipated entanglement by noting that the pure state of a coupled system need not be determined by the states of the composites [...] Weyl deserves far more credit than he has received for laying out the basis for entanglement — more than six years before Schrödinger coined the term.
^Christandl, Matthias (2006).The Structure of Bipartite Quantum States – Insights from Group Theory and Cryptography (PhD thesis). University of Cambridge. pp. vi, iv.arXiv:quant-ph/0604183.Bibcode:2006PhDT.......289C.
^Filk, Thomas (2016). "Carl Friedrich von Weizsäcker's 'Ortsbestimmung eines Elektrons' and its Influence on Grete Hermann". In Crull, Elise; Bacciagaluppi, Guido (eds.).Grete Hermann – Between Physics and Philosophy. Studies in History and Philosophy of Science. Vol. 42. Springer. p. 76.doi:10.1007/978-94-024-0970-3_5.ISBN978-94-024-0968-0.
^Kumar, Manjit (2010).Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality. W. W. Norton & Company. p. 313.ISBN978-0-393-07829-9.
^Bokulich, Alisa; Jaeger, Gregg, eds. (2010). "Introduction".Philosophy of Quantum Information and Entanglement. Cambridge University Press. p. xv.ISBN978-0-511-67655-0.
^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 inHobson, M. P.; et al. (1998). "Quantum Entanglement and Communication Complexity".SIAM J. Comput.30 (6):1829–1841.CiteSeerX10.1.1.20.8324.)
^abcGilder, Louisa (2009).The age of entanglement: when quantum physics was reborn (1. Vintage Book ed.). New York, NY: Vintage Books.ISBN978-1-4000-9526-1.
^Żukowski, Marek (2017). "Bell's Theorem Tells Us Not What Quantum Mechanics is, but What Quantum Mechanics is Not". In Bertlmann, Reinhold;Zeilinger, Anton (eds.).Quantum [Un]Speakables II. The Frontiers Collection. Cham: Springer International Publishing. pp. 175–185.arXiv:1501.05640.doi:10.1007/978-3-319-38987-5_10.ISBN978-3-319-38985-1.
^Summers, Stephen J. (2011). "Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State". In Halvorson, Hans (ed.).Deep Beauty: Understanding the Quantum World through Mathematical Innovation. Cambridge University Press. pp. 317–341.arXiv:0802.1854.ISBN978-1-139-49922-4.
^Pirandola, S.; U. L. Andersen; L. Banchi; M. Berta; D. Bunandar; R. Colbeck; D. Englund; T. Gehring; C. Lupo; C. Ottaviani; J. L. Pereira; M. Razavi; J. Shamsul Shaari; M. Tomamichel; V. C. Usenko; G. Vallone; P. Villoresi; P. Wallden (2020). "Advances in quantum cryptography".Adv. Opt. Photon.12 (4):1012–1236.arXiv:1906.01645.Bibcode:2020AdOP...12.1012P.doi:10.1364/AOP.361502.S2CID174799187.
^Caves, Carlton M.; Fuchs, Christopher A.; Schack, Rüdiger (20 August 2002). "Unknown quantum states: The quantum de Finetti representation".Journal of Mathematical Physics.43 (9):4537–4559.arXiv:quant-ph/0104088.Bibcode:2002JMP....43.4537C.doi:10.1063/1.1494475.Mermin was the first to point out the interesting properties of this three-system state, following the lead of D. M. Greenberger, M. Horne, and A. Zeilinger, "Going beyond Bell's Theorem," in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 69, where a similar four-system state was proposed.
^Shadbolt, P. J.; Verde, M. R.; Peruzzo, A.; Politi, A.; Laing, A.; Lobino, M.; Matthews, J. C. F.; Thompson, M. G.; O'Brien, J. L. (2012). "Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit".Nature Photonics.6 (1):45–59.arXiv:1108.3309.Bibcode:2012NaPho...6...45S.doi:10.1038/nphoton.2011.283.S2CID56206588.
^Gurvits, L. (2003). "Classical deterministic complexity of Edmonds' problem and quantum entanglement".Proceedings of the 35th ACM Symposium on Theory of Computing. New York: ACM Press. pp. 10–19.doi:10.1145/780542.780545.ISBN1-58113-674-9.
^CMS Collaboration (6 June 2024). "Observation of quantum entanglement in top quark pair production in proton–proton collisions at√s = 13 TeV".Reports on Progress in Physics.87 (11).arXiv:2406.03976.doi:10.1088/1361-6633/ad7e4d.PMID39315475.
^CMS Collaboration (17 September 2024). "Measurements of polarization and spin correlation and observation of entanglement in top quark pairs using <math><mrow>lepton+\text{jets}</mrow></math> events from proton-proton collisions at <math>\sqrt{s}=13\text{ }\text{ }TeV</math>".Physical Review D.110 (11) 112016.arXiv:2409.11067.doi:10.1103/PhysRevD.110.112016.
