**Quantum entanglement** occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description (

state), which is indefinite in terms of important factors such as position,

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

,

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

, polarization, etc. According to the

Copenhagen interpretationThe Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

of

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, their shared state is

**indefinite** until measured. Quantum entanglement is a form of

quantum superpositionQuantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

. When a measurement is made and it causes one member of such a pair to take on a definite value (e.g., clockwise spin), the other member of this entangled pair will at any subsequent time be found to have taken the complementary value (e.g., counterclockwise spin). Thus, there is a correlation between the results of measurements performed on entangled pairs, and this occurs even though the entangled pair may have been separated by arbitrarily large distances.

This behavior is theoretically coherent and has been demonstrated experimentally, and it is accepted by the physics community. However there is some debate about a possible

underlying mechanismHistorically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...

that enables this correlation to occur even when the separation distance is large. The difference in opinion derives from espousal of various interpretations of quantum mechanics.

Research into quantum entanglement was initiated by the

EPR paradoxThe EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

paper of

Albert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

,

Boris PodolskyBoris Yakovlevich Podolsky , was an American physicist of Russian Jewish descent.-Education:In 1896, Boris Podolsky was born into a poor Jewish family in Taganrog, in what was then the Russian Empire, and he moved to the United States in 1913...

and

Nathan RosenNathan Rosen was an American-Israeli physicist noted for his study on the structure of the hydrogen molecule and his work with Albert Einstein and Boris Podolsky on entangled wave functions and the EPR paradox.-Background:Nathan Rosen was born into a Jewish family in Brooklyn, New York...

in 1935, and several papers by

Erwin SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

shortly thereafter. Although these first studies focused on the counterintuitive properties of entanglement, with the aim of criticizing quantum mechanics, eventually entanglement was verified experimentally, and recognized as a valid, fundamental feature of quantum mechanics; the focus of the research has now changed to its utilization as a resource for communication and computation.

## History

The counterintuitive predictions of quantum mechanics about strongly correlated systems were first discussed by Albert Einstein in 1935, in a joint paper with Boris Podolsky and Nathan Rosen. In this study, they formulated the

EPR paradoxThe EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

, a thought experiment that attempted to show that

quantum mechanical theoryQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

was

incompleteIncompleteness of quantum physics is the assertion that the state of a physical system, as formulated by quantum mechanics, does not give a complete description for the system, assuming the usual philosophical requirements ....

. They wrote:

We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.

However, they did not coin the word

*entanglement* nor did they generalize the special properties of the state they considered. Following the EPR paper,

Erwin SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

wrote a letter (in German) to Einstein in which he used the word

*Verschränkung* (translated by himself as

*entanglement*) "to describe the correlations between two particles that interact and then separate, as in the EPR experiment". He shortly thereafter published a seminal paper defining and discussing the notion, and terming it "entanglement". In the paper he recognized the importance of the concept, and stated:

As with Einstein, Schrödinger was dissatisfied with the concept of entanglement, because it seemed to violate the speed limit on the transmission of information implicit in the

theory of relativityThe theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

. Einstein later famously derided entanglement as "

*spukhafte Fernwirkung*" or "spooky

action at a distanceIn physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant...

".

The EPR paper generated significant interest among physicists and inspired much discussion about the foundations of quantum mechanics (perhaps most famously Bohm's interpretation of quantum mechanics), but relatively little other published work. So despite the interest, the flaw in EPR's argument was not discovered until 1964, when

John Stewart BellJohn Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...

demonstrated precisely how one of their key assumptions, the

principle of localityIn physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...

, conflicted with quantum theory. Specifically, he demonstrated an upper limit, known as Bell's inequality, regarding the strength of correlations that can be produced in any theory obeying local realism, and he showed that quantum theory predicts violations of this limit for certain entangled systems. His inequality is experimentally testable, and there have been numerous

relevant experimentsThe Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality...

, starting with the pioneering work of Freedman and Clauser in 1972 and Aspect's experiments in 1982. They have all shown agreement with quantum mechanics rather than the principle of local realism. However, the issue is not finally settled, for each of these experimental tests has left open at least one loophole by which it is possible to question the validity of the results.

The work of Bell raised the possibility of using these super strong correlations as a resource for communication. It led to the discovery of quantum key distribution protocols, most famously

BB84BB84 is a quantum key distribution scheme developed by Charles Bennett and Gilles Brassard in 1984. It is the first quantum cryptography protocol. The protocol is provably secure, relying on the quantum property that information gain is only possible at the expense of disturbing the signal if the...

by Bennet and Brassard and E91 by Artur Ekert. Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

## Concept

Quantum systemsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

can become entangled through various types of interactions (see section on methods below). If entangled, one object cannot be fully described without considering the other(s). They remain in a

quantum superpositionQuantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

and share a single

quantum state until a measurement is made.

An example of entanglement occurs when

subatomic particleIn physics or chemistry, subatomic particles are the smaller particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...

s

decayParticle decay is the spontaneous process of one elementary particle transforming into other elementary particles. During this process, an elementary particle becomes a different particle with less mass and an intermediate particle such as W boson in muon decay. The intermediate particle then...

into other particles. These decay events obey the various conservation laws, and as a result, pairs of particles can be generated so that they are in some specific quantum states. For instance, a pair of these particles may be generated having a two-state

spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

: one must be spin up and the other must be spin down. This type of entangled pair, where the particles always have opposite spin, is known as the

*spin anti-correlated* case, and if the probabilities for measuring each spin are equal, the pair is said to be in the singlet state.

If each of two hypothetical experimenters,

Alice and BobThe names Alice and Bob are commonly used placeholder names for archetypal characters in fields such as cryptography and physics. The names are used for convenience; for example, "Alice sends a message to Bob encrypted with his public key" is easier to follow than "Party A sends a message to Party...

, has one of the particles that form an entangled pair, and Alice measures the spin of her particle, the measurement will be entirely unpredictable, with a 50% probability of the spin being up or down. But if Bob subsequently measures the spin of his particle, the measurement will be entirely predictable―always opposite to Alice's, hence perfectly anti-correlated.

So far, the correlation seen in this experiment can be simulated classically. To make an analogous experiment, a coin might be sliced along the circumference into two half-coins, in such a way that each half-coin is either "heads" or "tails", and each half-coin put in a separate envelope and distributed respectively to Alice and to Bob, randomly. If Alice then "measures" her half-coin, by opening her envelope, for her the measurement will be unpredictable, with a 50% probability of her half-coin being "heads" or "tails", and Bob's "measurement" of his half-coin will always be opposite, hence perfectly anti-correlated.

However, with quantum entanglement, if Alice and Bob measure the spin of their particles in directions other than just up or down, with the directions chosen to form a Bell's inequality, they can now observe a correlation that is fundamentally stronger than anything that is achievable in classical physics. Here, the classical simulation of the experiment breaks down because there are no "directions" other than heads or tails to be measured in the coins.

One might imagine that using a

dieA die is a small throwable object with multiple resting positions, used for generating random numbers...

instead of a coin could solve the problem, but the fundamental issue about measuring spin in different directions is that these measurements can't have definite values at the same time―they are incompatible. In classical physics this does not make sense, since any number of properties can be measured simultaneously with arbitrary accuracy. Bell's theorem implies, and it has been proven mathematically, that compatible measurements can't show Bell-like correlations, and thus entanglement is a fundamentally non-classical phenomenon.

Experimental results have demonstrated that effects due to entanglement travel at least thousands of times faster than the speed of light. In another experiment, the measurements of the entangled particles were made in moving, relativistic reference frames in which each respective measurement occurred before the other, and the measurement results remained correlated. Although the "effect" of quantum entanglement appears to exceed the speed of light, there is no violation of special relativity or causality which declares that information cannot be transferred faster than the speed of light. See EPR Paradox for an explanation about how causality is not violated.

## Entanglement, non-locality and hidden variables

There is much confusion about the meaning of entanglement,

non-localityQuantum nonlocality is the phenomenon by which measurements made at a microscopic level necessarily refute one or more notions that are regarded as intuitively true in classical mechanics...

and

hidden variablesHistorically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...

and how they relate to each other. As described above, entanglement is an experimentally verified and accepted property of nature. Non-locality and hidden variables are two proposed mechanisms that enable the effects of entanglement.

If the objects are indeterminate until one of them is measured, then the question becomes, "How can one account for something that was at one point indefinite with regard to its spin (or whatever is in this case the subject of investigation) suddenly becoming definite in that regard even though no physical interaction with the second object occurred, and, if the two objects are sufficiently far separated, could not even have had the time needed for such an interaction to proceed from the first to the second object?" The answer to the latter question involves the issue of

localityIn physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...

, i.e., whether for a change to occur in something the agent of change has to be in physical contact (at least via some intermediary such as a

field forceIn physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

) with the thing that changes. Study of entanglement brings into sharp focus the dilemma between locality and the completeness or lack of completeness of quantum mechanics.

In the media and popular science, quantum non-locality is often portrayed as being equivalent to entanglement. While it is true that a bipartite quantum state must be entangled in order for it to produce non-local correlations, there exist entangled states which do not produce such correlations. A well-known example of this is the

Werner stateA Werner state is a -dimensional bipartite quantum state that is invariant under all unitary operators of the form U \otimes U. That is, it is a quantum state ρ that satisfies\rho = \rho...

that is entangled for certain values of

, but can always be described using local hidden variables. In short, entanglement of a two-party state is necessary but not sufficient for that state to be non-local. It is important to recognise that entanglement is more commonly viewed as an algebraic concept, noted for being a precedent to non-locality as well as

quantum teleportationQuantum teleportation, or entanglement-assisted teleportation, is a process by which a qubit can be transmitted exactly from one location to another, without the qubit being transmitted through the intervening space...

and

superdense codingSuperdense coding is a technique used in quantum information theory to send two bits of classical information using only one qubit, with the aid of entanglement.- Overview :...

, whereas non-locality is defined according to experimental statistics and is much more involved with the foundations and interpretations of quantum mechanics

## 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 spontaneous parametric down-conversion to generate a pair of photons entangled in polarisation. Other methods include the use of a fiber coupler to confine and mix photons, use of

quantum dotA quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...

s to trap electrons until decay occurs, use of the

Hong-Ou-Mandel effectThe Hong–Ou–Mandel effect is a two-photon interference effect in quantum optics. The effect was demonstrated experimentally by Hong, Ou, and Mandel in 1987. The effect occurs when two identical single-photon wave packets enter a 50:50 beam splitter, one in each input port...

, etc. In the earliest tests of Bell's theorem, the entangled particles were generated using atomic cascades.

It's also possible to create entanglement between quantum systems that never directly interacted, through the use of entanglement swapping.

## Applications of entanglement

Entanglement has many applications in quantum information theory. With the aid of entanglement, otherwise impossible tasks may be achieved. Among the best-known applications of entanglement are

superdense codingSuperdense coding is a technique used in quantum information theory to send two bits of classical information using only one qubit, with the aid of entanglement.- Overview :...

and

quantum teleportationQuantum teleportation, or entanglement-assisted teleportation, is a process by which a qubit can be transmitted exactly from one location to another, without the qubit being transmitted through the intervening space...

. Not all researchers agree that entanglement is vital to the functioning of a

quantum computerA quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from traditional computers based on transistors...

. Entanglement is used in some protocols of

quantum cryptographyQuantum key distribution uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages...

.

The Reeh-Schlieder theorem of

quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

is sometimes seen as an analogue of quantum entanglement.

## Quantum mechanical framework

The following subsections are for those with a good working knowledge of

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, including familiarity with the theoretical framework developed in the articles:

bra-ket notationBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

and

mathematical formulation of quantum mechanicsThe mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

.

### Pure states

Consider two noninteracting systems

and

, with respective

Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s

and

. The Hilbert space of the composite system is the

tensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

If the first system is in state

and the second in state

, the state of the composite system is

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

*separable states*In quantum mechanics, separable quantum states are states without quantum entanglement.- Separable pure states :For simplicity, the following assumes all relevant state spaces are finite dimensional...

, or (in the simplest case)

*product state*In quantum mechanics, a product state, or simply separable state, is a special type of separable state in which there are neither classical nor quantum correlations. Precisely, it is a joint state of two systems which can be expressed as...

s.

Not all states are separable states (and thus product states). Fix a

basisIn linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

for

and a basis

for

. The most general state in

is of the form

.

This state is separable if

yielding

and

It is inseparable if

If a state is inseparable, it is called an

*entangled state*.

For example, given two basis vectors

of

and two basis vectors

of

, the following is an entangled state:

.

If the composite system is in this state, it is impossible to attribute to either system

or system

a definite pure state. Another way to say this is that while the

von Neumann entropyIn quantum statistical mechanics, von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantum mechanics....

of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". This has specific empirical ramifications for interferometry. It is worthwhile to note that the above example is one of four Bell states, which are (maximally) entangled pure states (pure states of the

space, but which cannot be separated into pure states of each

and

).

Now suppose Alice is an observer for system

, and Bob is an observer for system

. 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:

- 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 paradoxThe EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

.

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 theoremIn quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics such as the EPR paradox or...

.

### 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 matrixIn quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

, which is a positive matrix, or a

trace classIn mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis....

when the state space is infinite dimensional, and has trace 1. Again, by the

spectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

, such a matrix takes the general form:

where the

's sum up to 1, and in the infinite dimensional case, we would take the closure of such states in the trace norm. We can interpret

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.

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 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. The general bipartite case has been shown to be

NP-hardNP-hard , in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H...

. For the

and

cases, a necessary and sufficient criterion for separability is given by the famous

Positive Partial Transpose (PPT)The Peres-Horodecki criterion is a necessary condition, for the joint density matrix \rho of two quantum mechanical systems A and B, to be separable. It is also called the PPT criterion, for positive partial transpose. In the 2x2 and 2x3 dimensional cases the condition is also sufficient...

condition.

Experimentally, a mixed ensemble might be realized as follows. Consider a "black-box" apparatus that spits

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s towards an observer. The electrons' Hilbert spaces are

identicalIdentical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

. 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

spinsIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

aligned in the positive

direction, and the other with state

with spins aligned in the negative

direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

### Reduced density matrices

The idea of a reduced density matrix was introduced by

Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

in 1930. Consider as above systems

and

each with a Hilbert space

,

. 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

. However, it still is possible to associate a density matrix. Let

.

which is the projection operator onto this state. The state of

is the

partial traceIn linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function...

of

over the basis of system

:

.

is sometimes called the reduced density matrix of

on subsystem

*A*. Colloquially, we "trace out" system

*B* to obtain the reduced density matrix on

*A*.

For example, the reduced density matrix of

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

for the pure product state

discussed above is

In general, a bipartite pure state

*ρ* is entangled if and only if one, meaning both, of its reduced states are mixed states. Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is one dimensional

AKLT spin chainThe AKLT model is an extension of the one-dimensional quantum Heisenberg spin model. The proposal and exact solution of this model by Affleck, Lieb, Kennedy and Tasaki provided crucial insight into the physics of the spin 1 Heisenberg chain...

: the ground state can be divided into a block and environment. The reduced density matrix of the block is proportional to a projector to a degenerate ground state of another Hamiltonian.

The reduced density matrix also was evaluated for

XY spin chainsThe Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...

, where it has full rank. It was proved that in thermodynamic limit, the spectrum of the reduced density matrix of large block of spins is exact geometric sequence in this case.

### 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 information theory, the Shannon entropy,

is associated to a probability distribution,

, in the following way:

.

Since a mixed state ρ is a probability distribution over an ensemble, this leads naturally to the definition of the

von Neumann entropyIn quantum statistical mechanics, von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantum mechanics....

:

.

In general, one uses the

Borel functional calculusIn functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...

to calculate

. If ρ acts on a finite dimensional Hilbert space and has eigenvalues

, the Shannon entropy is recovered:

.

Since an event of probability 0 should not contribute to the entropy, and given that

, the convention is adopted that

. This extends to the infinite dimensional case as well: if ρ has

spectral resolutionIn mathematics, particularly functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a Hilbert space...

, assume the same convention when calculating

As in

statistical mechanicsEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

, 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

(which can be shown to be the maximum entropy for

mixed states).

#### 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 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

entropyIn classical statistical mechanics, the entropy function earlier introduced by Clausius is changed to statistical entropy using probability theory...

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 ). For example, by properties of the

Borel functional calculusIn functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...

, we see that for any

unitary operatorIn functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...

*U*,

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

HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

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

thermodynamicsThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

.

Rényi entropyIn information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system...

also can be used as a measure of entanglement.

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