Quantum entanglement occurs when particles such as photons, electrons, molecules as large as buckyballs,^[1][2] and even small diamonds^[3][4] 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,^[5] momentum, spin, polarization, etc.

According to the Copenhagen interpretation of quantum mechanics, their shared state is indefinite until measured.^[6] Quantum entanglement is a form of quantum superposition. 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^[7] be found to have taken the appropriately correlated value (e.g., counterclockwise spin). Thus, there is a correlation between the results of measurements performed on entangled pairs, and this correlation is observed even though the entangled pair may have been separated by arbitrarily large distances.^[8]

This behavior is consistent with quantum mechanical theory and has been demonstrated experimentally, and it is accepted by the physics community. However there is some debate^[9] about a possible underlying mechanism 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 a paper of Albert Einstein, Boris Podolsky and Nathan Rosen in 1935; the EPR paradox,^[10] and several papers by Erwin Schrödinger shortly thereafter.^[11][12] Although these first studies focused on the counterintuitive properties of entanglement, with the aim of criticizing quantum mechanics, eventually entanglement was verified experimentally^[13], 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[link]

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.^[10] In this study, they formulated the EPR paradox, a thought experiment that attempted to show that quantum mechanical theory was incomplete. They wrote:

We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.^[10]

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 Schrödinger 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".^[14] 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:^[11]

I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.

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 relativity.^[15] Einstein later famously derided entanglement as "spukhafte Fernwirkung"^[16] or "spooky action at a distance".

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 Bell demonstrated precisely how one of their key assumptions, the principle of locality, 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.^[17] His inequality is experimentally testable, and there have been numerous relevant experiments, starting with the pioneering work of Freedman and Clauser in 1972^[18] and Aspect's experiments in 1982.^[19] 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 BB84 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.

David Kaiser of MIT mentioned in his book, How the Hippies Saved Physics, that the possibilities of instantaneous long-range communication derived from Bell's theorem stirred interest among hippies, psychics, and even the CIA, with the counter-cultural playing a critical role in its development toward practical use.^[20]

Concept[link]

Quantum systems 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 superposition and share a single quantum state until a measurement is made.^[21]

An example of entanglement occurs when subatomic particles decay 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 spin: 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 Bob, 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 in this example experiment, the correlation seen with aligned measurements (i.e., up and down only) 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.^{[clarification needed]}

One might imagine that using a die instead of a coin could solve the problem, but the fundamental issue about measuring spin in different directions is that these measurements cannot 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 cannot show Bell-like correlations,^[22] 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,^[23][24] and that when measurements of the entangled particles are made in moving, relativistic reference frames in which each respective measurement occurs before the other, the measurement results remain correlated.^[25][26]

In a very recent experiment, "delayed-choice entanglement swapping" has been used to decide whether two particles were entangled or not after they had already been measured.^[27]

Entanglement, non-locality and hidden variables[link]

There is much confusion about the meaning of entanglement, non-locality and hidden variables 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?"^[28] The answer to the latter question involves the issue of locality, 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 force) 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 that do not produce such correlations. A well-known example of this is the Werner state that is entangled for certain values of Failed to parse (Missing texvc executable; please see math/README to configure.): p_{sym} , but can always be described using local hidden variables.^[29] 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 to quantum teleportation and to superdense coding, whereas non-locality is defined according to experimental statistics and is much more involved with the foundations and interpretations of quantum mechanics.

Methods of creating entanglement[link]

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.^[30] Other methods include the use of a fiber coupler to confine and mix photons, the use of quantum dots to trap electrons until decay occurs, the use of the Hong-Ou-Mandel effect, etc. In the earliest tests of Bell's theorem, the entangled particles were generated using atomic cascades.

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

Applications of entanglement[link]

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 coding and quantum teleportation.^[31] Not all researchers agree that entanglement is vital to the functioning of a quantum computer.^[32] Entanglement is used in some protocols of quantum cryptography.^[33][34]

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

Quantum mechanical framework[link]

The following subsections are for those with a good working knowledge of the formal, mathematical description of quantum mechanics, including familiarity with the formalism and theoretical framework developed in the articles: bra-ket notation and mathematical formulation of quantum mechanics.

Pure states[link]

Consider two noninteracting systems Failed to parse (Missing texvc executable; please see math/README to configure.): A

and Failed to parse (Missing texvc executable; please see math/README to configure.): B

, with respective Hilbert spaces Failed to parse (Missing texvc executable; please see math/README to configure.): H_A

and Failed to parse (Missing texvc executable; please see math/README to configure.): H_B

. The Hilbert space of the composite system is the tensor product

Failed to parse (Missing texvc executable; please see math/README to configure.): H_A \otimes H_B .

If the first system is in state Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle| \psi \rangle_A

and the second in state Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle| \phi \rangle_B

, the state of the composite system is

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle_A \otimes |\phi\rangle_B.

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

Not all states are separable states (and thus product states). Fix a basis Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \{|i \rangle_A\}

for Failed to parse (Missing texvc executable; please see math/README to configure.): H_A
and a basis Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \{|j \rangle_B\}
for Failed to parse (Missing texvc executable; please see math/README to configure.): H_B

. The most general state in Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle H_A \otimes H_B

is of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B

.

This state is separable if Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle c_{ij}= c^A_ic^B_j,

yielding Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A
and Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.
It is inseparable if Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle c_{ij} \neq c^A_ic^B_j.
If a state is inseparable, it is called an entangled state.

For example, given two basis vectors Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \{|0\rangle_A, |1\rangle_A\}

of Failed to parse (Missing texvc executable; please see math/README to configure.): H_A
and two basis vectors Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \{|0\rangle_B, |1\rangle_B\}
of Failed to parse (Missing texvc executable; please see math/README to configure.): H_B

, the following is an entangled state:

Failed to parse (Missing texvc executable; please see math/README to configure.): {1 \over \sqrt{2}} \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 Failed to parse (Missing texvc executable; please see math/README to configure.): A

or system Failed to parse (Missing texvc executable; please see math/README to configure.): B
a definite pure state. Another way to say this is that while the von Neumann entropy 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.[35] 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 Failed to parse (Missing texvc executable; please see math/README to configure.):  H_A \otimes H_B 
space, but which cannot be separated into pure states of each Failed to parse (Missing texvc executable; please see math/README to configure.):  H_A 
and Failed to parse (Missing texvc executable; please see math/README to configure.):  H_B 

).

Now suppose Alice is an observer for system Failed to parse (Missing texvc executable; please see math/README to configure.): A , and Bob is an observer for system Failed to parse (Missing texvc executable; please see math/README to configure.): B . If in the entangled state given above Alice makes a measurement in the Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \{|0\rangle, |1\rangle\}

eigenbasis of A, there are two possible outcomes, occurring with equal probability:[36]

1. Alice measures 0, and the state of the system collapses to Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle |0\rangle_A |1\rangle_B

.

1. Alice measures 1, and the state of the system collapses to Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle |1\rangle_A |0\rangle_B

.

If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system B has been altered by Alice performing a local measurement on system A. This remains true even if the systems A and B are spatially separated. This is the foundation of the EPR paradox.

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see no-communication theorem.

Ensembles[link]

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,

where the Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

as representing an ensemble where Failed to parse (Missing texvc executable; please see math/README to configure.): w_i
is the proportion of the ensemble whose states are Failed to parse (Missing texvc executable; please see math/README to configure.): |\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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho = \sum_i p_i \rho_i^A \otimes \rho_i^B,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_i^A 's and Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_i^B '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 Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_i^A

and Failed to parse (Missing texvc executable; please see math/README to configure.): \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. The general bipartite case has been shown to be NP-hard.[37] For the Failed to parse (Missing texvc executable; please see math/README to configure.):  2 \times 2
and Failed to parse (Missing texvc executable; please see math/README to configure.):  2 \times 3
cases, a necessary and sufficient criterion for separability is given by the famous Positive Partial Transpose (PPT) condition.[38]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): |\mathbf{z}+\rangle

with spins aligned in the positive Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{z}
direction, and the other with state Failed to parse (Missing texvc executable; please see math/README to configure.): |\mathbf{y}-\rangle
with spins aligned in the negative Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{y}
direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

Reduced density matrices[link]

The idea of a reduced density matrix was introduced by Paul Dirac in 1930.^[39] Consider as above systems Failed to parse (Missing texvc executable; please see math/README to configure.): A

and Failed to parse (Missing texvc executable; please see math/README to configure.): B
each with a Hilbert space Failed to parse (Missing texvc executable; please see math/README to configure.): H_A

, Failed to parse (Missing texvc executable; please see math/README to configure.): H_B . Let the state of the composite system be

Failed to parse (Missing texvc executable; please see math/README to configure.): |\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 Failed to parse (Missing texvc executable; please see math/README to configure.): A . However, it still is possible to associate a density matrix. Let

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_T = |\Psi\rangle \; \langle\Psi|

.

which is the projection operator onto this state. The state of Failed to parse (Missing texvc executable; please see math/README to configure.): A

is the partial trace of Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_T
over the basis of system Failed to parse (Missing texvc executable; please see math/README to configure.): B

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j \langle j|_B \left( |\Psi\rangle \langle\Psi| \right) |j\rangle_B = \hbox{Tr}_B \; \rho_T

.

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_A

is sometimes called the reduced density matrix of Failed to parse (Missing texvc executable; please see math/README to configure.): \rho
on subsystem A. Colloquially, we "trace out" system B to obtain the reduced density matrix on A.

For example, the reduced density matrix of Failed to parse (Missing texvc executable; please see math/README to configure.): A

for the entangled state

Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B ) / \sqrt{2}

discussed above is

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): A

for the pure product state Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle_A \otimes |\phi\rangle_B

discussed above is

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_A = |\psi\rangle_A \langle\psi|_A .

In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure. Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one dimensional AKLT spin chain:^[40] the ground state can be divided into a block and an 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 chains, where it has full rank. It was proved that in the thermodynamic limit, the spectrum of the reduced density matrix of a large block of spins is an exact geometric sequence^[41] in this case.

Entropy[link]

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[link]

In classical information theory, the Shannon entropy, Failed to parse (Missing texvc executable; please see math/README to configure.): H

is associated to a probability distribution,Failed to parse (Missing texvc executable; please see math/README to configure.): p_1, \cdots, p_n

, in the following way:^[42]

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right),

.

In general, one uses the Borel functional calculus to calculate Failed to parse (Missing texvc executable; please see math/README to configure.): \; \log \rho . If ρ acts on a finite dimensional Hilbert space and has eigenvalues Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda_1, \cdots, \lambda_n , the Shannon entropy is recovered:

Failed to parse (Missing texvc executable; please see math/README to configure.): S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i

.

Since an event of probability 0 should not contribute to the entropy, and given that Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{p \to 0} p \log p \;=\; 0 , the convention is adopted that Failed to parse (Missing texvc executable; please see math/README to configure.): 0 \log 0 \; = 0 . This extends to the infinite dimensional case as well: if ρ has spectral resolution Failed to parse (Missing texvc executable; please see math/README to configure.): \rho = \int \lambda d P_{\lambda} , assume the same convention when calculating

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho \log_2 \rho = \int \lambda \log_2 \lambda d P_{\lambda} .

As in statistical mechanics, the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is Failed to parse (Missing texvc executable; please see math/README to configure.): \log 2

(which can be shown to be the maximum entropy for Failed to parse (Missing texvc executable; please see math/README to configure.): 2 \times 2
mixed states).

As a measure of entanglement[link]

Entropy provides one tool which can be used to quantify entanglement, although other entanglement measures exist.^[43] 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho \in H \otimes H

is said to be a maximally entangled state if the reduced state of ρ is the diagonal matrix

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} \frac{1}{n}& \; & \; \\ \; & \ddots & \; \\ \; & \; & \frac{1}{n}\end{bmatrix}.

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^{[citation needed]} (comparing the two definitions, we note that, in the present context, it is customary to set the Boltzmann constant Failed to parse (Missing texvc executable; please see math/README to configure.): k = 1 ). For example, by properties of the Borel functional calculus, we see that for any unitary operator U,

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

Failed to parse (Missing texvc executable; please see math/README to configure.): U(t) \; = \exp \left(\frac{-i H t }{\hbar}\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. Rényi entropy also can be used as a measure of entanglement.

See also[link]

References[link]

Further reading[link]

External links[link]

• The original EPR paper
• Quantum Entanglement at Stanford Encyclopedia of Philosophy
• How to entangle photons experimentally
• A creative interpretation of Quantum Entanglement
• Albert's chest: entanglement for lay persons
• How Quantum Entanglement Works
• Explanatory video by Scientific American magazine
• Two Diamonds Linked by Strange Quantum Entanglement
• Entanglement experiment with photon pairs - interactive
• Multiple entanglement and quantum repeating
• Quantum Entanglement and Bell's Theorem at MathPages
• Audio - Cain/Gay (2009) Astronomy Cast Entanglement
• Recorded research seminars at Imperial College relating to quantum entanglement
• Quantum Entanglement and Decoherence: 3rd International Conference on Quantum Information (ICQI)
• Ion trapping quantum information processing
• IEEE Spectrum On-line: The trap technique
• Was Einstein Wrong?: A Quantum Threat to Special Relativity
• Citizendium: Entanglement

This article contains Japanese text. Without proper rendering support, you may see question marks, boxes, or other symbols instead of kanji and kana.

Michio Kaku (加来 道雄, Kaku Michio^?, born January 24, 1947) is an American theoretical physicist, the Henry Semat Professor of Theoretical Physics in the City College of New York of City University of New York, a co-founder of string field theory,^[1] a futurist, and a "communicator" and "popularizer" of science. He has written several books about physics and related topics; he has made frequent appearances on radio, television, and film; and he writes extensive online blogs and articles. He has written two New York Times best sellers, Physics of the Impossible (2008) and Physics of the Future (2011). He has hosted several TV specials for BBC-TV, the Discovery Channel, and the Science Channel.

Early life and education[link]

Kaku was born in San Jose, California to Japanese immigrant parents. His grandfather came to the United States to take part in the clean-up operation after the 1906 San Francisco Earthquake^{[citation needed]}. His father was born in California but was educated in Japan and spoke little English. Both his parents were put in the Tule Lake War Relocation Center, where they met and where his two brothers were born.

At Cubberley High School in Palo Alto, Kaku assembled an atom smasher in his parents' garage for a science fair project.^[2] At the National Science Fair in Albuquerque, New Mexico, he attracted the attention of physicist Edward Teller, who took Kaku as a protégé, awarding him the Hertz Engineering Scholarship. Kaku graduated summa cum laude from Harvard University in 1968 and was first in his physics class. He attended the Berkeley Radiation Laboratory at the University of California, Berkeley and received a Ph.D. in 1972, and in 1972 he held a lectureship at Princeton University.

During the Vietnam War, Kaku completed his U.S. Army basic training at Fort Benning, Georgia and his advanced individual training at Fort Lewis, Washington.^[3] However, the Vietnam War ended before he was deployed as an infantryman.

Academic career[link]

Kaku became a visiting professor at the Institute for Advanced Study in Princeton,^[4] and New York University.^[5] He currently holds the Henry Semat Chair and Professorship in theoretical physics at the City College of New York.^[6]

Kaku has had over 70 articles published in physics journals such as Physical Review, covering topics such as superstring theory, supergravity, supersymmetry, and hadronic physics.^[7] In 1974, along with Prof. Keiji Kikkawa of Osaka University, he authored the first papers describing string theory in a field form.^[8][9]

Kaku is the author of several textbooks on string theory and quantum field theory.

Popular science[link]

Kaku is most widely known as a popularizer of science.^[10] He has written books and appeared on many television programs as well as film. He also hosts a weekly radio program.

Books[link]

Kaku is the author of various popular science books.

• Hyperspace (1994)
• Beyond Einstein (with Jennifer Thompson) (1995)
• Visions: How Science Will Revolutionize the 21st Century^[11] (1998)
• Einstein's Cosmos (2004)
• Parallel Worlds (2004)
• Physics of the Impossible (2008)
• Physics of the Future (2011)

Hyperspace was a best-seller and was voted one of the best science books of the year by both The New York Times^[10] and The Washington Post. Parallel Worlds was a finalist for the Samuel Johnson Prize for non-fiction in the UK.^[12]

Radio[link]

Kaku is the host of the weekly, one-hour radio program Explorations, produced by the Pacifica Foundation's WBAI in New York. "Explorations" is syndicated to community and independent radio stations and makes previous broadcasts available on the program's website. Kaku defines the show as dealing with the general topics of science, war, peace, and the environment.

In April 2006, Kaku began broadcasting Science Fantastic on 90 commercial radio stations, the only nationally syndicated science program on commercial radio in the United States. It is syndicated by Talk Radio Network and now reaches 130 radio stations and America's Talk on XM. The program is formatted as a live listener call-in show, focusing on "futurology," which he defines as the future of science.^{[citation needed]} Featured guests include Nobel laureates and top researchers on the topics of string theory, time travel, black holes, gene therapy, aging, space travel, artificial intelligence, and SETI. When Kaku is busy filming for television, Science Fantastic goes on hiatus, sometimes for several months. Kaku is also a frequent guest on many programs, where he is outspoken in all areas and issues he considers of importance, such as the program Coast to Coast AM, where on 30 November 2007, he reaffirmed his belief that there is a 100% probability of extraterrestrial life in the universe.^[13]

Kaku has appeared on The Opie and Anthony Show a number of times, discussing popular fiction such as Back to The Future, Lost, and the theories behind time-travel that these and other fictional entertainment focus on.

Television and film[link]

Kaku has appeared in many forms of media and on many programs and networks, including Good Morning America, The Screen Savers, Larry King Live, 60 Minutes, Nightline, 20/20, Naked Science, CNN, ABC News, CBS News, NBC News, Al Jazeera English, Fox News Channel, The History Channel, Conan, The Science Channel, The Discovery Channel, TLC, Countdown with Keith Olbermann, The Colbert Report, The Art Bell Show and its successor, Coast To Coast AM, BBC World News America, The Opie & Anthony Show, The Covino & Rich Show, Head Rush, The Late Show with David Letterman, and Real Time with Bill Maher.

In 1999, Kaku was one of the scientists profiled in the feature-length film Me and Isaac Newton, directed by Michael Apted. It played theatrically in the United States, was later broadcast on national TV, and won several film awards.^{[citation needed]}

In 2005, Kaku appeared in the short documentary Obsessed & Scientific. The film is about the possibility of time travel and the people who dream about it. It screened at the Montreal World Film Festival and a feature film expansion is in development talks. Kaku also appeared in the ABC documentary UFOs: Seeing Is Believing, in which he suggested that while he believes it is extremely unlikely that extraterrestrials have ever actually visited Earth, we must keep our minds open to the possible existence of civilizations a million years ahead of us in technology, where entirely new avenues of physics open up. He also discussed the future of interstellar exploration and alien life in the Discovery Channel special Alien Planet as one of the multiple speakers who co-hosted the show, and Einstein's Theory of Relativity on The History Channel.

In February 2006, Kaku appeared as presenter in the BBC-TV four-part documentary Time which seeks to explore the mysterious nature of time. Part one of the series concerns personal time, and how we perceive and measure the passing of time. The second in the series deals with cheating time, exploring possibilities of extending the lifespan of organisms. The geological time covered in part three explores the ages of the earth and the sun. Part four covers the topics of cosmological time, the beginning of time and the events that occurred at the instant of the big bang.

On January 28, 2007, Kaku hosted the Discovery Channel series 2057. This three-hour program discussed how medicine, the city, and energy could change over the next 50 years.

In 2008, Kaku hosted the three-hour BBC-TV documentary Visions of the Future, on the future of computers, medicine, and quantum physics, and he appeared in several episodes of the History Channel's Universe series.

On December 1, 2009, he began hosting a 12-episode weekly TV series for the Science Channel at 10 pm, called Sci Fi Science: Physics of the Impossible, based on his best-selling book. Each 30-minute episode discusses the scientific basis behind imaginative schemes, such as time travel, parallel universes, warp drive, star ships, light sabers, force fields, teleportation, invisibility, death stars, and even superpowers and flying saucers. Each episode includes interviews with the world's top scientists working on prototypes of these technologies, interviews with science fiction fans, clips from science fiction movies, and special effects and computer graphics. Although these inventions are impossible today, the series discusses when these technologies might become feasible in the future.^[14]

In 2010, he began to appear in a series on the website Gametrailers.com called Science of Games, discussing the scientific aspects of various popular video games such as Mass Effect 2 and Star Wars: The Force Unleashed.

Kaku is popular in mainstream media because of his knowledge and his accessible approach to presenting complex subjects in science. While his technical writings are confined to theoretical physics, his public speaking and media appearances cover a broad range of topics, from the Kardashev scale to more esoteric subjects such as wormholes and time travel. In January 2007, Kaku visited Oman. While there, he talked at length to select members of that country's decision makers. In an interview with local media, Dr Kaku elaborated on his vision of mankind's future. Kaku considers climate change and terrorism as serious threats in man's evolution from a Type 0 civilization to Type 1.^[15]

He is pictured in Symphony of Science's newest song (as of September 6, 2011), 'The Quantum World.'

On October 11, 2010, Michio Kaku appeared in the BBC program "What Happened Before the Big Bang" (along with Laura Mersini-Houghton, Andrei Linde, Roger Penrose, Lee Smolin, Neil Turok, and other notable cosmologists and physicists), where he propounded his theory of the universe created out of nothing.^[16]

Social policy advocacy[link]

Kaku has publicly stated his concerns over matters including the anthropogenic cause of global warming, nuclear armament, nuclear power and the general misuse of science.^[17] He was critical of the Cassini–Huygens space probe because of the 72 pounds (33 kg) of plutonium contained in the craft for use by its radioisotope thermoelectric generator. Conscious of the possibility of casualties if the probe's fuel were dispersed into the environment during a malfunction and crash as the probe was making a 'sling-shot' maneuver around earth, Kaku publicly criticized NASA's risk assessment.^[18] He has also spoken on the dangers of space junk and called for more and better monitoring. Kaku is generally a vigorous supporter of the exploration of outer space, believing that the ultimate destiny of the human race may lie in extrasolar planets; but he is critical of some of the cost-ineffective missions and methods of NASA.

Kaku credits his anti-nuclear war position to programs he heard on the Pacifica Radio network, during his student years in California. It was during this period that he made the decision to turn away from a career developing the next generation of nuclear weapons in association with Edward Teller and focused on research, teaching, writing and media.^{[citation needed]} Kaku joined with others such as Helen Caldicott, Jonathan Schell, Peace Action and was instrumental in building a global anti-nuclear weapons movement that arose in the 1980s, during the administration of U.S. President Ronald Reagan.

Kaku was a board member of Peace Action and on the board of radio station WBAI-FM in New York City where he originated his long running program, Explorations, that focused on the issues of science, war, peace and the environment.

His remark from an interview in support of SETI, "We could be in the middle of an intergalactic conversation...and we wouldn't even know.", is used in the third Symphony of Science installment "Our Place in the Cosmos".

Personal life[link]

Kaku is married to Shizue Kaku and has two daughters.^[19]

Bibliography[link]

References[link]

External links[link]

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• Official website
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