# Dictionary Definition

entangled adj

1 deeply involved especially in something
complicated; "embroiled in the conflict"; "felt unwilling entangled
in their affairs" [syn: embroiled]

2 twisted together in a tangled mass; "toiled
through entangled growths of mesquite"

3 involved in difficulties

# User Contributed Dictionary

## English

### Pronunciation

### Verb form

entangled- past of entangle

### Adjective

entangled- tangled or twisted together
- confused or complicated
- (of two quantum states) correlated, even though physically separated

### See also

# Extensive Definition

Quantum entanglement is a quantum
mechanical phenomenon in which the
quantum
states of two or more objects are
linked together so that one object cannot be adequately described
without full mention of its counterpart — even though the
individual objects may be spatially separated. This
interconnection leads to correlations between
observable physical
properties of remote systems. For example, quantum
mechanics holds that states such as spin are
indeterminate until such time as some physical intervention is made
to measure the spin of the object in question. It is equally as
likely that any given particle will be observed to be spin-up as
that it will be spin-down. Measuring any number of particles will
result in an unpredictable series of measures that will tend more
and more closely to half up and half down. However, if this
experiment is done with entangled particles the results are quite
different. When two members of an entangled pair are measured, one
will always be spin-up and the other will be spin-down. The
distance between the two particles is irrelevant. In order to
explain this result, some have theorized that there are hidden
variables that account for the spin of each particle, and that
these hidden variables are determined when the entangled pair is
created. But, if this is so, then the hidden variables must stay in
communication no matter how far apart the particles are, the hidden
variable describing one particle must be able to change instantly
when the other is measured. If the hidden variables stop
interacting when they are far apart, the statistics of the
measurements obey an
inequality, which is violated both in quantum mechanics and in
experiments.

The phenomenon of wavefunction
collapse leads to the impression that measurements performed on
one system instantaneously influence other systems entangled with
the measured system, even when far apart. But quantum entanglement
does not
enable the transmission of classical information
faster than the speed of
light in quantum mechanics.

Quantum entanglement has applications in the
emerging technologies
of quantum
computing and quantum
cryptography, and has been used to realize quantum
teleportation experimentally. 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 and that the state of a system exists and is
well-defined before any measurement. Different views of what is
actually occurring in the process of quantum entanglement can be
related to different
interpretations of quantum mechanics. In the standard one, the
Copenhagen
interpretation, quantum mechanics is neither "real" (since
measurements do not state, but instead prepare properties of the
system) nor "local" (since the state vector |\psi\rangle comprises
the simultaneous probability amplitudes for all positions, e.g.
|\psi\rangle \to \psi(x,y,z)).

## Background

Entanglement is one of the properties of quantum
mechanics that 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, in
response to Niels Bohr's
advocacy of the belief that quantum mechanics as a theory was
complete. Einstein famously derided entanglement as "spukhafte
Fernwirkung" or "spooky
action at a distance." In fact, it was his belief that future
mathematicians would, in fact, discover that quantum entanglement
actually entailed nothing more or less than an error in their
calculations. As he once wrote: "I find the idea quite intolerable
that an electron exposed to radiation should choose of its own free
will, not only its moment to jump off, but also its direction. In
that case, I would rather be a cobbler, or even an employee in a
gaming house, than a physicist.”

On the other hand, quantum mechanics has been
highly successful in producing correct experimental predictions,
and the strong correlations predicted by the theory of quantum
entanglement have in fact been observed. One apparent way to
explain found correlations in line with the predictions of quantum
entanglement is an approach known as "local
hidden variable theory", in which unknown, shared, local
parameters would cause the correlations. However, in 1964 John
Stewart Bell derived an upper limit, known as Bell's
inequality, on the strength of correlations for any theory
obeying "local
realism." 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. There may be
experimental problems that affect the validity of these
experimental findings, known as "loopholes".
High-efficiency and high-visibility experiments are now in progress
that should confirm or invalidate the existence of those loopholes.
For more information, see the article on experimental
tests of Bell's inequality.

Observations pertaining to entangled states
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, the current state of belief is that no
useful information can be transmitted in this way, meaning that
causality
cannot be violated through entanglement. This is the statement of
the no
communication theorem.

Even if information can not be transmitted
through entanglement alone, it is believed 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 may
still not permit information to be transmitted faster than light,
because a
classical information channel is required to complete the
process.

In addition experiments are underway to see if
entanglement is the result of retrocasuailty.

## Pure States

The following discussion builds on the
theoretical framework developed in the articles bra-ket
notation and
mathematical formulation 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

- H_A \otimes H_B .

If the first system is in state | \psi \rangle_A
and the second in state | \phi \rangle_B, the state of the
composite system is

- |\psi\rangle_A \otimes |\phi\rangle_B,

which is often also written as

- |\psi\rangle_A \; |\phi\rangle_B.

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

Not all states are product states. Fix a basis
\ for H_A and a basis \ for H_B. The most general state in H_A
\otimes H_B is of the form

- \sum_ c_ |i\rangle_A \otimes |j\rangle_B.

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:

- \bigg( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \bigg).

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:

- Alice measures 0, and the state of the system collapses to |0\rangle_A |1\rangle_B
- Alice measures 1, and the state of the system collapses to |1\rangle_A |0\rangle_B.

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.

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 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.

## 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 a 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:

- \rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,

where the w_i'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 \rho as representing an ensemble where
w_i is the proportion of the ensemble whose states are
|\alpha_i\rangle. 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 H_A \otimes H_B. Extending the definition of
separability from the pure case, we say that a mixed state is
separable if it can be written as

- \rho = \sum_i p_i \rho_i^A \otimes \rho_i^B ,

where \rho_i^A's and \rho_i^B'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 \rho_i^A and \rho_i^B 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 2
\times 2 and 2 \times 3 cases, a necessary and sufficient criterion
for separability is given by the famous Positive
Partial Transpose (PPT) condition.

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 |\mathbf+\rangle with
spins
aligned in the positive \mathbf direction, and the other with state
|\mathbf-\rangle with spins aligned in the negative \mathbf
direction. Generally, this is a mixed ensemble, as there can be any
number of populations, each corresponding to a different
state.

## 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

- |\Psi \rangle \in H_A \otimes H_B.

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

- \rho_T = |\Psi\rangle \; \langle\Psi|.

which is the projection
operator onto this state. The state of A is the partial
trace of \rho_T over the basis of system B:

- \rho_A \ \stackrel\ \sum_j \langle j|_B \left( |\Psi\rangle \langle\Psi| \right) |j\rangle_B = \hbox_B \; \rho_T .

\rho_A is sometimes called the reduced density
matrix of \rho 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

- \rho_A = (1/2) \bigg( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \bigg)

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 |\psi\rangle_A \otimes |\phi\rangle_B discussed above
is

- \rho_A = |\psi\rangle_A \langle\psi|_A .

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 p_1, \cdots, p_n, one can associate the Shannon
entropy:

- H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i,

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

- S(\rho) = - \hbox \left( \rho \log \right),

where the logarithm is again taken in base 2. In
general, to calculate \; \log \rho, one would use the Borel
functional calculus. If ρ acts on a finite dimensional Hilbert
space and has eigenvalues \lambda_1, \cdots, \lambda_n, then we
recover the Shannon entropy:

- S(\rho) = - \hbox \left( \rho \log \right) = - \sum_i \lambda_i \log \lambda_i.

Since an event of probability 0 should not
contribute to the entropy, we adopt the convention that 0 \log 0 \;
= 0. This extends to the infinite dimensional case as well: if ρ
has spectral
resolution \rho = \int \lambda d P_, then we assume the same
convention when calculating

- \rho \log \rho = \int \lambda \log \lambda d P_ .

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 \log 2
(which can be shown to be the maximum entropy for 2 \times 2 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 . Therefore, a bipartite pure state

- \rho \in H \otimes H

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

- \begin \frac& \; & \; \\ \; & \ddots & \; \\ \; & \; & \frac\end.

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,

- S(\rho) \; = S(U \rho 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.

- U(t) \; = \exp \left(\frac\right)

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.

## 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 best 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.Other uses:

- Quantum computers use entanglement and superposition.

- The Reeh-Schlieder theorem of quantum field theory is sometimes seen as the QFT analogue of quantum entanglement.

## See also

## 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.

- I. Bengtsson and K. Zyczkowski, "Geometry of Quantum States. An Introduction to Quantum Entanglement", Cambridge University Press, Cambridge, 2006.

## External links

- Multiple entanglement and quantum repeating
- How to entangle photons experimentally
- Quantum Entanglement
- Recorded research seminars at Imperial College relating to quantum entanglement
- How Quantum Entanglement Works
- Quantum Entanglement and Decoherence: 3rd International Conference on Quantum Information (ICQI
- The original EPR paper
- Ion trapping quantum information processing
- IEEE Spectrum On-line: The trap technique

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