Quantum entanglement
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Quantum mechanics |
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Quantum entanglement is the phenomenon of a group of
Such phenomena were the subject of a 1935 paper by
Later, however, the counterintuitive predictions of quantum mechanics were verified
According to some
Quantum entanglement has been demonstrated experimentally with
History
In 1935,
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:
The EPR paper generated significant interest among physicists, which inspired much discussion about the foundations of quantum mechanics and Bohm's interpretation in particular, 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 proved that one of their key assumptions, the principle of locality, 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
An early experimental breakthrough was due to Carl Kocher,[12][13] 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[13] and decreased exponentially with time lag between emitted photons.[24] 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.[22] 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
Bell's work raised the possibility of using these super-strong correlations as a resource for communication. It led to the 1984 discovery of
In 2022, the Nobel Prize in Physics was awarded to Alain Aspect, John Clauser, and Anton Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".[29]
Concept
Meaning of entanglement
An entangled system is 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. 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. Entanglement is broken when the entangled particles decohere through interaction with the environment; for example, when a measurement is made.[30]
As an example of entanglement: a
The above result may or may not be perceived as surprising. A classical system would display the same property, and a
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
The distance and timing of the measurements can be chosen so as to make the interval between the two measurements
(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".[32] 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.
Violations of Bell's inequality
The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are
Notable experimental results proving quantum entanglement
The first experiment that verified Einstein's spooky action at a distance (entanglement) was successfully corroborated in a lab by Chien-Shiung Wu and colleague 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.[38] In experiments in 2012 and 2013, polarization correlation was created between photons that never coexisted in time.[39][40] 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.[41] The first loophole-free Bell test was held by Ronald Hanson of the Delft University of Technology in 2015, confirming the violation of Bell inequality.[42]
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 such as biological or medical imaging.[43]
Since 2016, various companies, for example IBM and Microsoft, have created quantum computers that allowed developers and tech enthusiasts to freely experiment with concepts of quantum mechanics including quantum entanglement.[44]
Emergence of time from quantum entanglement
There is a fundamental conflict, referred to as the problem of time, between the way the concept of time is used in quantum mechanics, and the role it plays in general relativity. In standard quantum theories time acts as an independent background through which states evolve, with the Hamiltonian operator acting as the generator of infinitesimal translations of quantum states through time.[45]
In contrast, general relativity treats time as a dynamical variable which relates directly with matter and moreover requires the Hamiltonian constraint to vanish. In quantized general relativity, the quantum version of the Hamiltonian constraint using metric variables, leads to the Wheeler–DeWitt equation:
where is the Hamiltonian constraint and stands for the
The emergence of time was also proposed as arising from quantum correlations between an evolving system and a reference quantum clock system, the concept of system-time entanglement is introduced as a quantifier of the actual distinguishable evolution undergone by the system.[48][49] [50][51]
Emergent gravity
Based on AdS/CFT correspondence, Mark Van Raamsdonk suggested that spacetime arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the space-time.[52] Induced gravity can emerge from the entanglement first law.[53][54]
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.[55] A well-known example is the Werner states that are entangled for certain values of , but can always be described using local hidden variables.[56] Moreover, it was shown that, for arbitrary numbers of particles, there exist states that are genuinely entangled but admit a local model.[57]
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.[58]
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, whereas non-locality is defined according to experimental statistics and is much more involved with the foundations and interpretations of quantum mechanics.[59]
Quantum-mechanical framework
The following subsections are for those with a good working knowledge of the formal, mathematical description of quantum mechanics, including familiarity with the formalism and theoretical framework developed in the articles: bra–ket notation and mathematical formulation of quantum mechanics.
Pure states
Consider two arbitrary quantum systems A and B, with respective Hilbert spaces HA and HB. The Hilbert space of the composite system is the tensor 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, or
Not all states are separable states (and thus product states). Fix a basis 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 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'.
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
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:[61]
- Alice measures 0, and the state of the system collapses to .
- Alice measures 1, and the state of the system collapses to .
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 EPR paradox.
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 no-communication theorem.
Ensembles
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
where the wi 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 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 density matrices to represent the state.
Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits
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 wi 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[62]: 131–132
where the wi are positively valued probabilities and the 's and '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 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 be
Reduced density matrices
The idea of a reduced density matrix was introduced by Paul Dirac in 1930.[65] 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
The sum occurs over and the identity operator in . 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
discussed above is
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 discussed above is
- .
In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure.
Two applications that use them
Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional
The reduced density matrix also was evaluated for
Entanglement as a resource
In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows implementing valuable transformations.[68][69] The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labeled "A" and "B" on each of which arbitrary quantum 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 called LOCC (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. For example, 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.[70]
Classification of entanglement
Not all quantum states are equally valuable as a resource. To quantify this value, different entanglement measures (see below) can be used, that assign 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 same LU 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).[71][72]
- 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 I can transform one into the other and then do whatever it allows me to do), 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 ).[73][74]
- 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 called distillable. 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'.[75][70]
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) the non-local states, which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) the steerable 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.[76]
Entropy
In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.
Definition
In classical
Since a mixed state ρ is a probability distribution over an ensemble, this leads naturally to the definition of the von Neumann entropy:
In general, one uses the Borel functional calculus to calculate a non-polynomial function such as log2(ρ). If the nonnegative operator ρ acts on a finite-dimensional Hilbert space and has eigenvalues , log2(ρ) turns out to be nothing more than the operator with the same eigenvectors, but the eigenvalues . The Shannon entropy is then:
- .
Since an event of probability 0 should not contribute to the entropy, and given that
the convention 0 log(0) = 0 is adopted. This extends to the infinite-dimensional case as well: if ρ has spectral resolution
assume the same convention when calculating
As in statistical mechanics, the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is log(2) (which can be shown to be the maximum entropy for 2 × 2 mixed states).
As a measure of entanglement
Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.[78][79] 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.[80]
It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n,...,1/n}. Therefore, a bipartite pure state ρ ∈ HA ⊗ HB is said to be a maximally entangled state if the reduced state of each subsystem of ρ is the diagonal matrix
For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.
As an aside, the information-theoretic definition is closely related to
Indeed, without this property, the von Neumann entropy would not be well-defined.
In particular, U could be the time evolution operator of the system, i.e.,
where H is the Hamiltonian of the system. Here the entropy is unchanged.
The reversibility of a process is associated with the resulting entropy change, i.e., a process is reversible if, and only if, it leaves the entropy of the system invariant. Therefore, the march of the
Rényi entropy also can be used as a measure of entanglement.
Nevertheless, on 23 January 2023, physicists reported, that, after all, there is no second law of entanglement manipulation. In the words of the researchers, "no direct counterpart to the second law of thermodynamics can be established".[83]
Entanglement measures
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[78] and no single one is standard.
- Entanglement cost
- Distillable entanglement
- Entanglement of formation
- Concurrence
- Relative entropy of entanglement
- Squashed entanglement
- Logarithmic negativity
Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult (
Quantum field theory
The
Applications
Entanglement has many applications in
Among the best-known applications of entanglement are superdense coding and quantum teleportation.[85]
Most researchers believe that entanglement is necessary to realize
Entanglement is used in some protocols of quantum cryptography,[87][88] but to prove the security of quantum key distribution (QKD) under standard assumptions does not require entanglement.[89] However, the device independent security of QKD is shown exploiting entanglement between the communication partners.[90]
Entangled states
There are several canonical entangled states that appear often in theory and experiments.
For two
These four pure states are all maximally entangled (according to the
For M>2 qubits, the GHZ state is
which reduces to the Bell state for . The traditional GHZ state was defined for . GHz states are occasionally extended to
Also for M>2 qubits, there are
For two bosonic modes, a NOON state is
This is like the Bell state except the basis kets 0 and 1 have been replaced with "the N photons are in one mode" and "the N photons are in the other mode".
Finally, there also exist twin Fock states for bosonic modes, which can be created by feeding a Fock state into two arms leading to a beam splitter. They are the sum of multiple of NOON states, and can be used to achieve the Heisenberg limit.[93]
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. The partially entangled states are generally easier to prepare experimentally.
Methods of creating entanglement
Entanglement is usually created by direct interactions between subatomic particles. These interactions can take numerous forms. One of the most commonly used methods is
It is also possible to create entanglement between quantum systems that never directly interacted, through the use of entanglement swapping. Two independently prepared, identical particles may also be entangled if their wave functions merely spatially overlap, at least partially.[98]
Testing a system for entanglement
A density matrix ρ is called separable if it can be written as a convex sum of product states, namely
For 2-Qubit and Qubit-Qutrit systems (2 × 2 and 2 × 3 respectively) the simple Peres–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.[99][100] Other separability criteria include (but not limited to) the range criterion, reduction criterion, and those based on uncertainty relations.[101][102][103][104] See Ref.[105] for a review of separability criteria in discrete-variable systems and Ref.[106] for a review on techniques and challenges in experimental entanglement certification in discrete-variable systems.
A numerical approach to the problem is suggested by
In continuous variable systems, the Peres-Horodecki criterion also applies. Specifically, Simon[108] 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.[109] for a seemingly different but essentially equivalent approach). It was later found[110] 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[111][112] or by using entropic measures.[113][114]
In 2016, China launched the world's first quantum communications satellite.[115] The $100m Quantum Experiments at Space Scale (QUESS) mission was launched on 16 Aug 2016, from the Jiuquan Satellite Launch Center in northern China at 01:40 local time.[citation needed]
For the next two years, the satellite – nicknamed "Micius" after the ancient Chinese philosopher – will demonstrate the feasibility of quantum communication between Earth and space, and test quantum entanglement over unprecedented distances.[citation needed]
In the 16 June 2017, issue of Science, Yin et al. report 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 a CHSH valuation of 2.37 ± 0.09, under strict Einstein locality conditions, from the Micius satellite to bases in Lijian, Yunnan and Delingha, Quinhai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude.[116][117]
Naturally entangled systems
The electron shells of multi-electron atoms always consist of entangled electrons. The correct ionization energy can be calculated only by consideration of electron entanglement.[118]
Photosynthesis
It has been suggested that in the process of
However, critical follow-up studies question the interpretation of these results and assign the reported signatures of electronic quantum coherence to nuclear dynamics in the chromophores or to the experiments being performed at cryogenic rather than physiological temperatures.[121][122][123][124][125][126][127]
Entanglement of macroscopic objects
In 2020, researchers reported the quantum entanglement between the motion of a millimeter-sized mechanical oscillator and a disparate distant spin system of a cloud of atoms.[128][129] Later work complemented this work by quantum-entangling two mechanical oscillators.[130][131][132]
Entanglement of elements of living systems
In October 2018, physicists reported producing quantum entanglement using
Living organisms (green sulphur bacteria) have been studied as mediators to create quantum entanglement between otherwise non-interacting light modes, showing high entanglement between light and bacterial modes, and to some extent, even entanglement within the bacteria.[135]
In December 2023, physicists, for the first time, report the entanglement of individual molecules, which may have significant applications in quantum computing.[136]
See also
- Bound entanglement
- Concurrence
- CNOT gate
- Einstein's thought experiments
- Entanglement distillation
- Entanglement witness
- ER = EPR
- Faster-than-light communication
- Multipartite entanglement
- Normally distributed and uncorrelated does not imply independent
- Pauli exclusion principle
- Quantum coherence
- Quantum computing
- Quantum discord
- Quantum network
- Quantum phase transition
- Quantum pseudo-telepathy
- Quantum teleportation
- Retrocausality
- Separable state
- Spontaneous parametric down-conversion
- Squashed entanglement
- Stern–Gerlach experiment
- Ward's probability amplitude
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Further reading
- Albert, David Z.; Galchen, Rivka (2009). "Was Einstein Wrong?: A Quantum Threat to Special Relativity". PMID 19253771.
- Bengtsson I.; Życzkowski K. (2006). "Geometry of Quantum States". An Introduction to Quantum Entanglement. Cambridge: Cambridge University Press. second, revised edition (2017)
- Bub, Jeffrey (2019). "Quantum Entanglement and Information". Stanford Encyclopedia of Philosophy. Stanford, California: Stanford University.
- Cramer JG (2015). The Quantum Handshake: Entanglement, Nonlocality and Transactions. Springer Verlag. ISBN 978-3-319-24642-0.
- ISBN 978-0-7503-2226-3.
- Gühne O, Tóth G (2009). "Entanglement detection". S2CID 119288569.
- Bhaskara VS, Panigrahi PK (2017). "Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange's identity and wedge product". Quantum Information Processing. 16 (5): 118. S2CID 43754114.
- Swain SN, Bhaskara VS, Panigrahi PK (2022). "Generalized entanglement measure for continuous-variable systems". Phys. Rev. A. 105 (5): 052441. S2CID 239885759.
- Jaeger G (2009). Entanglement, Information, and the Interpretation of Quantum Mechanics. Heildelberg: Springer. ISBN 978-3-540-92127-1.
- Steward EG (2008). Quantum Mechanics: Its Early Development and the Road to Entanglement. Imperial College Press. ISBN 978-1-86094-978-4.
External links
- Explanatory video by Scientific American magazine
- Entanglement experiment with photon pairs – interactive
- Audio – Cain/Gay (2009) Astronomy Cast Entanglement
- "Spooky Actions at a Distance?": Oppenheimer Lecture, Prof. David Mermin (Cornell University) Univ. California, Berkeley, 2008. Non-mathematical popular lecture on YouTube, posted Mar 2008
- "Quantum Entanglement versus Classical Correlation" (Interactive demonstration)