Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin-up, the other one will always be observed to be spin-down and vice versa, this despite the fact that it is impossible to predict, according to quantum mechanics, which set of measurements will be observed. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. However, at this time classical information cannot be transmitted through entanglement faster than the speed of light. Fig. ... A quantum state is any possible state in which a quantum mechanical system can be. ... Wikiquote has a collection of quotations related to: Space Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. ... In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ... Information is a word which has many different meanings in everyday usage and in specialized contexts, but as a rule, the concept is closely related to others such as data, instruction, knowledge, meaning, communication, representation, and mental stimulus. ...

Quantum entanglement is the basis for emerging technologies such as quantum computing and quantum cryptography, and has been used for experiments in quantum teleportation. At the same time, it produces some of the more theoretically and philosophically disturbing aspects of the theory, as one can show that the correlations predicted by quantum mechanics are inconsistent with the seemingly obvious principle of local realism, which is that information about the state of a system should only be mediated by interactions in its immediate surroundings. Different views of what is actually occurring in the process of quantum entanglement give rise to different interpretations of quantum mechanics. See also: Innovation By the mid 20th century humans had achieved a level of technological mastery sufficient to leave the surface of the planet for the first time and explore space. ... Molecule of alanine used in NMR implementation of error correction. ... Quantum cryptography is an approach to securing communications based on certain phenomena of quantum physics. ... Quantum teleportation is a technique discussed in quantum information science to transfer a quantum state to an arbitrarily distant location using an entangled state and the transmission of some classical information. ... In physics, the principle of locality is that distant objects cannot have direct influence on one another: an object is influenced directly only by its immediate surroundings. ... An interpretation of quantum mechanics is an attempt to answer the question: what exactly is quantum mechanics talking about? Quantum mechanics has been described as the most precisely tested and most successful theory in the history of science (c. ...

Background

Entanglement is one of the properties of quantum mechanics which caused Einstein and others to dislike the theory. In 1935, Einstein, Podolsky, and Rosen formulated the EPR paradox, demonstrating that entanglement makes quantum mechanics a non-local theory. Einstein famously derided entanglement as "spooky action at a distance." Albert Einstein photographed by Oren J. Turner in 1947. ... 1935 (MCMXXXV) was a common year starting on Tuesday (link will take you to calendar). ... In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ... Nathan Rosen (March 22, 1909 – December 18, 1995) was a physicist. ... In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ... In physics, action at a distance is the instantaneous interaction of two objects which are separated in space; the term was coined as spooky action at a distance by Albert Einstein. ...

On the other hand, quantum mechanics has been highly successful in producing correct experimental predictions, and the strong correlations associated with the phenomenon of quantum entanglement have in fact been observed. One apparent way to explain quantum entanglement is an approach known as "hidden variable theory", in which unknown deterministic microscopic parameters would cause the correlations. However, in 1964 Bell showed that such a theory could not be "local", the quantum entanglement predicted by quantum mechanics being experimentally distinguishable from a broad class of local hidden-variable theories. Results of subsequent experiments have overwhelmingly supported quantum mechanics. It is known that there are a number of loopholes in these experiments, but these are generally considered to be of minor importance. For more information, see the article on the Bell test experiments. In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event. ... For the Nintendo 64 emulator, see 1964 (Emulator). ... John Bell (left) and Martinus Veltman (right) discussing Physics at CERN John S. Bell (June 28, 1928 – October 1, 1990) was a physicist who became well known as the originator of Bells Theorem, regarded by some in the quantum physics community as one of the most important theorems of... In quantum mechanics, Bells Theorem states that a Bell inequality must be obeyed under any local hidden variable theory but can in certain circumstance be violated under quantum mechanics(QM). ...

Entanglement produces some interesting interactions with the principle of relativity that states that information cannot be transferred faster than the speed of light. Although two entangled systems can interact across large spatial separations, no useful information can be transmitted in this way, so causality cannot be violated through entanglement. This occurs for two subtle reasons: (i) quantum mechanical measurements yield probabilistic results, and (ii) the no cloning theorem forbids the statistical inspection of entangled quantum states. It is possible to reconcile the speed of light with Bell violations by interpreting the two entangled particles as opposite ends of a wormhole; however, if entangling two particles creates a wormhole, then this change in topology implies causal violations. In physics, the term relativity is used in several, related contexts: Galileo first developed the principle of relativity, which is the postulate that the laws of physics are the same for all observers. ... Although causality, the relationship between causes and effects, is often examined in the fields of philosophy, computer science, and statistics, it has a place in the study of physics as well. ... The word probability derives from the Latin probare (to prove, or to test). ... The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unknown quantum state. ...

Although no information can be transmitted through entanglement alone, it is possible to transmit information using a set of entangled states used in conjunction with a classical information channel. This process is known as quantum teleportation. Despite its name, quantum teleportation cannot be used to transmit information faster than light, because a classical information channel is involved. Quantum teleportation is a technique discussed in quantum information science to transfer a quantum state to an arbitrarily distant location using an entangled state and the transmission of some classical information. ... In quantum information science, classical information channel (often called simply classical channel) is a communication channel that can be used to transmit classical information (as opposed to quantum channel which can transmit quantum information). ...

Formalism

The following discussion builds on the theoretical framework developed in the articles bra-ket notation and mathematical formulation of quantum mechanics. Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as Hilbert spaces and operators on these spaces. ...

Consider two noninteracting systems A and B, with respective Hilbert spaces H_A and H_B. The Hilbert space of the composite system is the tensor product In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

If the first system is in state and the second in state , the state of the composite system is

which is often also written as

States of the composite system which can be represented in this form are called separable states.

Pick observables (and corresponding Hermitian operators) Ω_A acting on H_A, and Ω_B acting on H_B. According to the spectral theorem, we can find a basis for H_A composed of eigenvectors of Ω_A, and a basis for H_B composed of eigenvectors of Ω_B. We can then write the above pure state as A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

,

for some choice of complex coefficients a_i and b_j. This is not the most general state of , which has the form

.

If such a state is not separable, it is known as an entangled state.

For example, given two basis vectors of H_A and two basis vectors of H_B, 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 pure state. Instead, their states are superposed with one another. In this sense, the systems are "entangled".

Now suppose Alice is an observer for system A, and Bob is an observer for system B. If Alice performs the measurement Ω_A, there are two possible outcomes, occurring with equal probability:

1. Alice measures 0, and the state of the system collapses to
2. Alice measures 1, and the state of the system collapses to .

If the former occurs, any subsequent measurement of Ω_B performed by Bob always returns 1. If the latter occurs, Bob's measurement always returns 0. Thus, system B has been altered by Alice performing her measurement on system A., even if the systems A and B are spatially separated. This is the foundation of the EPR paradox. In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ...

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. (There is a possible loophole: if Bob could make multiple duplicate copies of the state he receives, he could obtain information by collecting statistics. This loophole is closed by the no cloning theorem, which forbids the creation of duplicate states.) Causality is thus preserved, as we claimed above. The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unknown quantum state. ...

Entropy

Quantifying entanglement is an important step towards better understanding the phenomenon of entropy. The method by which density matrices are arrived at, provides us with a formal measure of entanglement. For other senses of the term entropy, see entropy (disambiguation). ... A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. ...

Consider as above systems A and B each with a Hilbert space H_A, H_B. 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 projection operator onto this state. The state of A is the partial trace of ρ_T over the basis of system B: In mathematics, a projection operator on a vector space is an idempotent linear transformation. ... In linear algebra and functional analysis, the partial trace is a generalization of the trace. ...

.

For example, the density matrix of A for the entangled state discussed above is

and the density matrix of A for the pure state discussed above is

.

This is simply the projection operator of |ψ〉_A. Note that the density matrix of the composite system, ρ_T, also takes this form. This is unsurprising, since we assumed that the state of the composite system is pure.

Given a general density matrix ρ, we can calculate the quantity

where k is Boltzmann's constant, and the trace is taken over the space H in which ρ acts. It turns out that S is precisely the entropy of the system corresponding to H. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... For other senses of the term entropy, see entropy (disambiguation). ...

The entropy of any pure state is zero, which is unsurprising since there is no uncertainty about the state of the system. The entropy of any of the two subsystems of the entangled state discussed above is kln2 (which can be shown to be the maximum entropy for a one-level system). If the overall system is pure, the entropy of its subsystems can be used to measure its degree of entanglement with the other subsystems.

It can also be shown that unitary operators acting on a state (such as the time evolution operator obtained from the Schrödinger equation) leave the entropy unchanged. This associates the reversibility of a process with its resulting entropy change, which is a deep result linking quantum mechanics to information theory and thermodynamics. In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... // Introduction Information theory is the mathematical theory of data communication and storage generally considered to have been founded in 1948 by Claude E. Shannon. ... Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of temperature, pressure, and volume changes on physical systems at the macroscopic scale. ...

Ensembles

The language of density matrices is also used to describe quantum ensembles, or a collection of identical quantum systems. In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ...

Consider a "black-box" apparatus that spits electrons towards an observer. The electrons' Hilbert spaces are identical. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then called a pure ensemble. Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ...

However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state (spins aligned in the positive direction), and the other with state (spins aligned in the negative direction.) Generally, there can be any number of populations, each corresponding to a different state. This is a mixed ensemble. In physics, spin is an intrinsic angular momentum associated with microscopic particles. ...

We can describe an ensemble as a collection of populations with weights w_i and corresponding states . The density matrix of the ensemble is defined as

.

All the above results for density matrices and the quantum entropy remain valid with this definition. As one can see from the above definition a density matrix is a probability distribution over projection operators onto pure states. This means that in all cases where there is less than total information about the state of a quantum system we need density matrices to represent the state. For a mixed state, where we do not know which vector in the Hilbert space to associate with the quantum system, we cannot use the reduced von Neumann entropy as a measure of entanglement. This is because the uncertainty in the mixed state gives us entropy in itself, irrespective of whether or not the state is entangled.

An entangled state is defined as a state that is not separable. A separable state can be written as a probability distribution over uncorrelated states, product states,

.

No product states have any correlations between them at all. A probability distribution over these is the same as correlations in a classical system due to uncertainty. A quantum state that cannot be written in this way is entangled. Finding out whether or not a general state is entangled is a hard problem.

Applications of entanglement

Entanglement has many applications in quantum information theory. These include quantum state teleportation and superdense coding. Quantum information science is a field of research at the interface of quantum mechanics and computer science. ... Quantum teleportation is a technique discussed in quantum information science to transfer a quantum state to an arbitrarily distant location using an entangled state and the transmission of some classical information. ... Superdense coding is a technique used in quantum information theory. ...

The Reeh-Schlieder theorem of quantum field theory is sometimes seen as the QFT analogue of Quantum entanglement. The Reeh-Schlieder theorem is a result of relativistic local quantum field theory, stating that the vacuum is a cyclic vector for the field algebra of any open set in Minkowski space. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...