It has been suggested that Quantum coherence be merged into this article or section. (Discuss)

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. But quantum entanglement does not enable the transmission of classical information faster than the speed of light (see discussion in next section below). Fig. ... A quantum state is any possible state in which a quantum mechanical system can be. ... A physical body is an object which can be described by the theories of classical mechanics, or quantum mechanics, and experimented upon by physical instruments. ... Space has been an interest for philosophers and scientists for much of human history. ... Positive linear correlations between 1000 pairs of numbers. ... A physical property is any aspect of an object or substance that can be measured or perceived without changing its identity. ... System (from Latin systēma, in turn from Greek sustēma) is a set of entities, real or abstract, comprising a whole where each component interacts with or is related to at least one other component. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... A prediction or forecast is a statement or claim that a particular event will occur in the future. ... Various meters Measurement is the estimation or determination of extent, dimension or capacity, usually in relation to some standard or unit of measurement. ... The ASCII codes for the word Wikipedia represented in binary, the numeral system most commonly used for encoding computer information. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation in a vacuum, not just visible light. ...

Quantum entanglement has applications in the emerging technologies of quantum computing and quantum cryptography, and has been used to experimentally realize quantum teleportation. At the same time, it prompts some of the more philosophically oriented discussions concerning quantum theory. The correlations predicted by quantum mechanics, and observed in experiment, reject the 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 can be related to different interpretations of quantum mechanics. By the mid 20th century humans had achieved a mastery of technology sufficient to leave the surface of the Earth for the first time and explore space. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ... Quantum cryptography is an approach based on quantum physics for secure communications. ... In quantum information, quantum teleportation, or entanglement-assisted teleportation, is a technique that transfers a quantum state to an arbitrarily distant location using a distributed entangled state and the transmission of some classical information. ... For other uses, see Philosophy (disambiguation). ... 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. ... It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ...

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, a quantum-mechanical thought experiment with a highly counterintuitive and apparently nonlocal outcome. Einstein famously derided entanglement as "Spukhafte Fernwirkung" or "spooky action at a distance." Albert Einstein ( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... 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 challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. ... Physical theories are said to exhibit nonlocality if it is not possible to treat widely separated systems as independent. ... In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. ...

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, shared, local parameters would cause the correlations. However, in 1964 Bell derived an upper limit, known as Bell's inequality, on the strength of correlations for any theory obeying "local realism" (see principle of locality). Quantum entanglement can lead to stronger correlations that violate this limit, so that quantum entanglement is experimentally distinguishable from a broad class of local hidden-variable theories. Results of subsequent experiments have overwhelmingly supported quantum mechanics. Although there are a number of known loopholes in these experiments, high-efficiency and high-visibility experiments are now in progress which should confirm or disaffirm the influence of those loopholes. For more information, see the article on Bell test experiments. In physics, the hidden variable theory is espoused by a minority of physicists who argue that the statistical nature of quantum mechanics indicates that QM is incomplete. ... This article or section is not written in the formal tone expected of an encyclopedia article. ... Bells theorem is the most famous legacy of the late John Bell. ... 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. ... In quantum mechanics, Bells Theorem states that a Bell inequality must be obeyed under any local hidden variable theory but can in certain circumstances be violated under quantum mechanics (QM). ...

Observations on entangled states naively appear to conflict with the property of relativity that information cannot be transferred faster than the speed of light. Although two entangled systems appear to interact across large spatial separations, no useful information can be transmitted in this way, so causality cannot be violated through entanglement. This is the statement of no communication theorem. Two-dimensional analogy of space-time distortion described in General Relativity. ... 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. ... In quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. ...

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 required. In quantum information, quantum teleportation, or entanglement-assisted teleportation, is a technique that transfers a quantum state to an arbitrarily distant location using a distributed 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). ...

Pure States

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. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ...

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 a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, 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, or product states. Separable quantum states are those without Quantum entanglement. ...

Not all states are product states. Fix a basis for H_A and a basis for H_B. The most general state in is of the form In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

.

If a state is not separable, it is called 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 makes a measurement in the eigenbasis of 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 performed by Bob, in the same basis, will always return 1. If the latter occurs, 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. In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. ...

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. In quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. ...

In some formal mathematical settings, it is noted that the correct setting for pure states in quantum mechanics is projective Hilbert space endowed with the Fubini-Study metric. The product of two pure states is then given by the Segre embedding. In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by v ~ w when v = λw with λ a scalar, that... In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ... In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety. ...

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 a large number of copies of the same system, then the state of this ensemble is described by a density matrix, which is a positive matrix (or trace class, when the state space is infinite dimensional) and has trace 1. Again, by the spectral theorem, such a matrix takes the general form: A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ... A nonnegative matrix is a matrix where all the elements are equal to or above zero A positive matrix is defined similarly. ... In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ...

,

where the w_i's sum up to 1 (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 w_i 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 (see experiment discussed below).

Following the definition in previous section, for a bipartite composite system, mixed states are just density matrices on . Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as

,where 's and 's are they themselves states on the subsystems A and B respectively. In other words, a state is separable if it is probability distribution over uncorrelated states, or product states. We can assume without loss of generality that and are 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. Formally, it has been shown to be NP-hard. For the and cases, a necessary and sufficient criterion for separability is given by the famous PPT (Positive Partial Transpose) condition. In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing... The Peres-Horodecki criterion is a necessary condition, for the joint density matrix of two systems and , to be separable. ...

Experimentally, a mixed ensemble might be realized as follows. 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 a pure ensemble. 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. Therefore we now have a mixed ensemble. e- redirects here. ... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...

Reduced Density Matrices

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: Template:Unite See also projection (linear algebra). ... In linear algebra and functional analysis, the partial trace is a generalization of the trace. ...

.

ρ_A 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 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 one, therefore both, of its reduced states are mixed states.

Entropy

In this section we briefly discuss entropy of a mixed state and how it can be viewed as a measure of entanglement.

Definition

In classical information theory, to a probability distribution , one can associate the Shannon entropy: Entropy of a Bernoulli trial as a function of success probability. ...

where the logarithm is taken in base 2.

Since one can think of a mixed state ρ as a probability distribution over an ensemble, this leads naturally to the definition of the von Neumann entropy: Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ...

where the logarithm is again taken in base 2. In general, to calculate , one would use the Borel functional calculus. If ρ acts on a finite dimensional Hilbert space and has eigenvalues , then we recover the Shannon entropy: In functional analysis, the Borel functional calculus is a functional calculus (i. ...

.

Since an event of probability 0 should not contribute to the entropy, we adopt the convention that . This extends to the infinite dimensional case as well: if ρ has spectral resolution , then we assume the same convention when calculating In mathematics, projection-valued measures are used to express results in spectral theory. ...

As in statistical mechanics, one can say that 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 log2 (which can be shown to be the maximum entropy for mixed states). Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ...

As a measure of entanglement

Entropy provides one tool which can be used to quantify entanglement (although other entanglement measures exist). 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.

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

is said to be a maximally entangled state if there exists some local bases on H such that the reduced state of ρ is the diagonal matrix

For mixed states, the reduced von Neumann entropy is not the unique entanglement measure.

As an aside, the information-theoretic definition is closely related to entropy in the sense of statistical mechanics (comparing the two definitions, we note that, in the present context, it is customary to set the Boltzmann constant k = 1). For example, by properties of the Borel functional calculus, we see that for any unitary operator U, In this section we discuss the statistical nature of entropy. ... Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... In functional analysis, the Borel functional calculus is a functional calculus (i. ... 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. ...

.

Indeed, without the above 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. This associates the reversibility of a process with its resulting entropy change, i.e. a process is reversible if and only if it leaves the entropy of the system invariant. This provides a connection between quantum information theory and thermodynamics. The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Quantum information science is a field of research at the interface of quantum mechanics and computer science. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...

Applications of entanglement

Entanglement has many applications in quantum information theory. Mixed state entanglement can be viewed as a resource for quantum communication. With the aid of entanglement, otherwise impossible tasks may be achieved. Among the most well known such applications of entanglement are superdense coding and quantum state teleportation. Efforts to quantify this resource are often termed entanglement theory. See for example Entanglement Theory Tutorials. Quantum information science is a field of research at the interface of quantum mechanics and computer science. ... In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. ... Superdense coding is a technique used in quantum information theory. ... In quantum information, quantum teleportation, or entanglement-assisted teleportation, is a technique that transfers a quantum state to an arbitrarily distant location using a distributed entangled state and the transmission of some classical information. ...

Quantum computers use entanglement and superposition. The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ...

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. ...

See also

• Entanglement witness
• Fubini-Study metric
• Separable states
• Squashed entanglement
• Quantum coherence
• Action at a distance (physics)

In quantum information theory, an entanglement witness is an object of geometric nature which distinguishes an entangled state from separable ones. ... In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ... Separable quantum states are those without Quantum entanglement. ... Squashed entanglement, also called CMI entanglement (CMI can be pronounced see me), is an information theoretic measure of quantum entanglement for a bipartite quantum system. ... Coherence is the property of wave-like states that enables them to exhibit interference. ... In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. ...

References

• M. Horodecki, P. Horodecki, R. Horodecki, "Separability of Mixed States: Necessary and Sufficient Conditions", Physics Letters A 210, 1996.

• L. Gurvits, "Classical deterministic complexity of Edmonds' Problem and quantum entanglement", Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, 2003.

External links

• An Interview With Brian Clegg, Author of "The God Effect : Quantum Entanglement, Science's Strangest Phenomenon" California Literary Review
• Multiple entanglement and quantum repeating
• How to entangle photons experimentally
• Quantum Entanglement
• Recorded research seminars at Imperial relating to quantum entanglement