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Quantum entanglement

Quantum entanglement

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Spontaneous parametric down-conversion process can split photons into type II photon pairs with mutually perpendicular polarization.

Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently – instead, a quantum state may be given for the system as a whole.

Measurements of physical properties such as position, momentum, spin, polarization, etc. performed on entangled particles are found to be appropriately correlated. For example, if a pair of particles is generated in such a way that their total spin is known to be zero, and one particle is found to have clockwise spin on a certain axis, then the spin of the other particle, measured on the same axis, will be found to be counterclockwise. Because of the nature of quantum measurement, however, this behavior gives rise to effects that can appear paradoxical: any measurement of a property of a particle can be seen as acting on that particle (e.g. by collapsing a number of superimposed states); and in the case of entangled particles, such action must be on the entangled system as a whole. It thus appears that one particle of an entangled pair "knows" what measurement has been performed on the other, and with what outcome, even though there is no known means for such information to be communicated between the particles, which at the time of measurement may be separated by arbitrarily large distances.

Such phenomena were the subject of a 1935 paper by Albert Einstein, Boris Podolsky and Nathan Rosen,[1] describing what came to be known as the EPR paradox, and several papers by Erwin Schrödinger shortly thereafter.[2][3] Einstein and others considered such behavior to be impossible, as it violated the local realist view of causality (Einstein referred to it as "spooky action at a distance"),[4] and argued that the accepted formulation of quantum mechanics must therefore be incomplete. Later, however, the counterintuitive predictions of quantum mechanics were verified experimentally.[5] Experiments have been performed involving measuring the polarization or spin of entangled particles in different directions, which – by producing violations of Bell's inequality – demonstrate statistically that the local realist view cannot be correct. This has been shown to occur even when the measurements are performed more quickly than light could travel between the sites of measurement: there is no lightspeed or slower influence that can pass between the entangled particles.[6] Recent experiments have measured entangled particles within less than one part in 10,000 of the light travel time between them.[7] According to the formalism of quantum theory, the effect of measurement happens instantly.[8][9] It is not possible, however, to use this effect to transmit classical information at faster-than-light speeds[10] (see Faster-than-light → Quantum mechanics).

Quantum entanglement is an area of extremely active research by the physics community, and its effects have been demonstrated experimentally with photons, electrons, molecules the size of buckyballs,[11][12] and even small diamonds.[13][14] Research is also focused on the utilization of entanglement effects in communication and computation.

History

May 4, 1935 New York Times article headline regarding the imminent EPR paper.

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.[1] In this study, they formulated the EPR paradox (Einstein, Podolsky, Rosen 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."[1]

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."[15] 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:[2] "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."

Like 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.[16] Einstein later famously derided entanglement as "spukhafte Fernwirkung"[17] 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 proved that one of their key assumptions, the principle of locality, was not consistent with the hidden variables interpretation of quantum theory that EPR purported to establish. Specifically, he demonstrated an upper limit, seen in 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.[18] His inequality is experimentally testable, and there have been numerous relevant experiments, starting with the pioneering work of Freedman and Clauser in 1972[19] and Aspect's experiments in 1982.[20] 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-culture playing a critical role in its development toward practical use.[21]

Concept

Meaning of entanglement

Quantum systems can become entangled through various types of interactions. (For some ways in which entanglement may be achieved for experimental purposes, see the section below on methods.) An entangled system has a quantum state which cannot be factored out into the product of states of its local constituents (e.g. individual particles). If entangled, one constituent cannot be fully described without considering the other(s). Like the quantum states of individual particles, the state of an entangled system is expressible as a sum, or superposition, of basis states, which are eigenstates of some observable(s). Entanglement is broken when the entangled particles decohere through interaction with the environment; for example, when a measurement is made.[22]

An example of entanglement occurs when a subatomic particle decays into a pair of other particles. These decay events obey the various conservation laws, and as a result, the measurement outcomes of one daughter particle must be highly correlated with the measurement outcomes of the other daughter particle (so that the total momenta, angular momenta, energy, and so forth remains roughly the same before and after this process). For instance, a spin-zero particle could decay into a pair of spin-1/2 particles. Since the total spin before and after this decay must be zero (conservation of angular momentum), whenever the first particle is measured to be spin up on some axis, the other (when measured on the same axis) is always found to be spin down. (This is called the spin anti-correlated case; and if the prior probabilities for measuring each spin are equal, the pair is said to be in the singlet state.)

Apparent paradox

The seeming paradox here is that a measurement made on either of the particles apparently collapses the state of the entire entangled system – and does so instantaneously, before any information about the measurement could have reached the other particle (assuming that information cannot travel faster than light). In the quantum formalism, the result of a spin measurement on one of the particles is a collapse into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is taken to be random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it is measured. The distance and timing of the measurements can be chosen so as to make the interval between the two measurements spacelike, i.e. from any of the two measuring events to the other a message would have to travel faster than light. Then, according to the principles of special relativity, it is not in fact possible for any information to travel between two such measuring events – it is not even possible to say which of the measurements came first, as this would depend on the inertial system of the observer. Therefore the correlation between the two measurements cannot appropriately be explained as one measurement determining the other: different observers would disagree about the role of cause and effect.

The hidden variables theory

A possible resolution to the apparent paradox might be to assume that the state of the particles contains some hidden variables, whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. It was originally believed by Einstein and others (see the previous section) that this was the only way out, and therefore that the accepted quantum mechanical description (with a random measurement outcome) must be incomplete. (In fact similar paradoxes can arise even without entanglement: the position of a single particle is spread out over space, and two detectors attempting to detect the particle at different positions must attain appropriate correlation, so that they do not both detect the particle.)

Violations of Bell's inequality

The hidden variables theory fails, however, when we consider measurements of the spin of entangled particles along different axes (for example, along any of three axes which make angles of 120 degrees). If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the local realist or hidden variables view were correct, the results would always satisfy Bell's inequality. A number of experiments have shown in practice, however, that Bell's inequality is not satisfied. This tends to confirm that the original formulation of quantum mechanics is indeed correct, in spite of its apparently paradoxical nature. Even when measurements of the entangled particles are made in moving relativistic reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.[23][24]

The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are incompatible in the sense that these measurements' maximum simultaneous precision is constrained by the uncertainty principle. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating correlations,[25] and thus entanglement is a fundamentally non-classical phenomenon.

Other types of experiment

In a 2012 experiment, "delayed-choice entanglement swapping" was used to decide whether two particles were entangled or not after they had already been measured.[26]

In a 2013 experiment, entanglement swapping has been used to create entanglement between photons that never coexisted in time, thus demonstrating that "the nonlocality of quantum mechanics, as manifested by entanglement, does not apply only to particles with spacelike separation, but also to particles with timelike [i.e., temporal] separation".[27]

In three independent experiments it was shown that classically-communicated separable quantum states can be used carry entangled states.[28]

Special Theory of Relativity

Another theory explains quantum entanglement using special relativity.[29] According to this theory, faster-than-light communication between entangled systems can be achieved because the time dilation of special relativity allows time to stand still in light's point of view. For example, in the case of two entangled photons, a measurement made on one photon at present time would determine the state of the photon for both the present and past at the same moment. This leads to the instantaneous determination of the state of the other photon. Corresponding logic is applied to explain entangled systems, i.e. electron and positron, that travel below the speed of light.

Non-locality and hidden variables

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, which has critical implications for the interpretations of quantum mechanics. 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?"[30] 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.

Bell's theorem and related results rule out a local realistic explanation for quantum mechanics (one which obeys the principle of locality while also ascribing definite values to quantum observables). However, in other interpretations, the experiments that demonstrate the apparent non-locality can also be described in local terms: If each distant observer regards the other as a quantum system, communication between the two must then be treated as a measurement process, and this communication is strictly local.[31] In particular, in the many worlds interpretation, the underlying description is fully local.[32] More generally, the question of locality in quantum physics is extraordinarily subtle and sometimes hinges on precisely how it is defined.

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 p_{sym}, but can always be described using local hidden variables.[33] 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.

Quantum mechanical framework

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

Consider two noninteracting systems A and B, with respective Hilbert spaces HA and HB. The Hilbert space of the composite system is the tensor product

 H_A \otimes H_B.

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

|\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 \scriptstyle \{|i \rangle_A\} for HA and a basis \scriptstyle \{|j \rangle_B\} for HB. The most general state in HAHB is of the form

|\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B.

This state is separable if there exist c^A_i,c^B_j so that \scriptstyle c_{ij}= c^A_ic^B_j, yielding \scriptstyle |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A and \scriptstyle |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B. It is inseparable if for all c^A_i,c^B_j we have \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 \scriptstyle \{|0\rangle_A, |1\rangle_A\} of HA and two basis vectors \scriptstyle \{|0\rangle_B, |1\rangle_B\} of HB, the following is an entangled state:

\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right ).

If the composite system is in this state, it is impossible to attribute to either system A or system 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.[34] 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 HAHB space, but which cannot be separated into pure states of each HA and HB).

Now suppose Alice is an observer for system A, and Bob is an observer for system B. If in the entangled state given above Alice makes a measurement in the \scriptstyle \{|0\rangle, |1\rangle\} eigenbasis of A, there are two possible outcomes, occurring with equal probability:[35]

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

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-semidefinite 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 positive valued wi 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 ρ as representing an ensemble where wi 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 HAHB. Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as[36]:131–132

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

where pi are positive valued probabilities, \rho_i^A's and \rho_i^B's are themselves states on the subsystems A and B respectively. In other words, a state is separable if it is a 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. The general bipartite case has been shown to be NP-hard.[37] For the 2 × 2 and 2 × 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 |\mathbf{z}+\rangle with spins aligned in the positive z direction, and the other with state |\mathbf{y}-\rangle with spins aligned in the negative 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

The idea of a reduced density matrix was introduced by Paul Dirac in 1930.[39] Consider as above systems A and B each with a Hilbert space HA, HB. 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 ρT over the basis of system B:

\rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j \langle j|_B \left( |\Psi\rangle \langle\Psi| \right) |j\rangle_B = \hbox{Tr}_B \; \rho_T.

ρA is sometimes called the reduced density matrix of ρ on subsystem A. Colloquially, we "trace out" system B to obtain the reduced density matrix on A.

For example, the reduced density matrix of A for the entangled state

\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right),

discussed above is

\rho_A = \tfrac{1}{2} \left ( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \right )

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 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

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

The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state. When the eigenvalue has value .5, von Neumann entropy is at a maximum, corresponding to maximum entanglement.

In classical information theory, the Shannon entropy, H is associated to a probability distribution,p_1, \cdots, p_n, in the following way:[42]

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{Tr} \left( \rho \log_2 {\rho} \right).

In general, one uses the Borel functional calculus to calculate log(ρ). If ρ acts on a finite-dimensional Hilbert space and has eigenvalues \lambda_1, \cdots, \lambda_n, the Shannon entropy is recovered:

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

 \lim_{p \to 0} p \log p = 0,

the convention 0 log(0) = 0 is adopted. This extends to the infinite-dimensional case as well: if ρ has spectral resolution

 \rho = \int \lambda d P_{\lambda},

assume the same convention when calculating

 \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 log(2) (which can be shown to be the maximum entropy for 2 × 2 mixed states).

As a measure of entanglement

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

\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 k = 1). For example, by properties of the Borel functional calculus, we see that for any unitary operator U,

S(\rho) = S \left (U \rho U^* \right).

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{-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.

Therefore the march of the arrow of time towards thermodynamic equilibrium is simply the growing spread of quantum entanglement.[44]

Quantum field theory

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

Applications

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.[45]

Most researchers believe that entanglement is necessary to realize quantum computing (although this is disputed by some[46]).

Entanglement is used in some protocols of quantum cryptography.[47][48] This is because the "shared noise" of entanglement makes for an excellent one-time pad. Moreover, since measurement of either member of an entangled pair destroys the entanglement they share, entanglement-based quantum cryptography allows the sender and receiver to more easily detect the presence of an interceptor.

In interferometry, entanglement is necessary for surpassing the standard quantum limit and achieving the Heisenberg limit.

Entangled states

There are several canonical entangled states that appear often in theory and experiments.

For two qubits, the Bell states are

|\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B)
|\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B).

These four pure states are all maximally entangled (according to the entropy of entanglement) and form an orthonormal basis (linear algebra) of the Hilbert space of the two qubits. They play a fundamental role in Bell's theorem.

For M>2 qubits, the GHZ state is

|\mathrm{GHZ}\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}},

which reduces to the Bell state |\Phi^+\rangle for M=2. The traditional GHZ state was defined for M=3. GHZ states are occasionally extended to qudits, i.e. systems of d rather than 2 dimensions.

Also for M>2 qubits, there are spin squeezed states.[49] Spin squeezed states are a class of states satisfying certain restrictions on the uncertainty of spin measurements, and are necessarily entangled.[50]

For two bosonic modes, a NOON state is

|\psi_\text{NOON} \rangle = \frac{|N \rangle_a |0\rangle_b + |{0}\rangle_a |{N}\rangle_b}{\sqrt{2}}, \,

This is like a Bell state |\Phi^+\rangle except the basis kets 0 and 1 have been replaced with "the N photons are in one mode" and "the N photons are in the other mode".

Finally, there also exist twin Fock states for bosonic modes, which can be created by feeding a Fock state into two arms leading to a beam-splitter. They are the sum of multiple of NOON states, and can used to achieve the Heisenberg limit.[51]

For the appropriately chosen measure of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled. The partially entangled states are generally easier to prepare experimentally.

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.[52] 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.

Author:Bling King
Published:Jun 29th 2014
Modified:Jun 29th 2014
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There is no such thing as time
Posted by Bling King

    

     Upon further ponderance I have come to the conclusion that time does not exist except in the law of physics. I have come to this conclusion through the observation of how things change and why they change at the pace in which they change. To me it seems that every change that takes place  in the universe is not dictated by time but rather physics. It is the law of physics that dictates the rate and speed at which all things change. For example if you have a car  that is traveling at 100 miles an hour the speed at  which the car travels is all dictated by physical changes and therfor controlled by the law of physics..Therfor it seems that for any change to take place all you need is physics and the law of physics that governs the physical changes. Time does not need be a factor and bears no relavance. As long as we have the law of physics everything will happen in accordance with those laws.

The composition of time
Posted by Bling King

   

    Time has 3 components. A front a middle and a rear. In the front time has what appears to be something of perspectual perspectualness that will move things forward at a set forth proponent. This part of time is easy to see and witness. However it is not easy to predict at which point time will make forward momentum happen. It would seem that this forward momentum is always in inactment but I would disagree with this. To me it seems more as if time interacts with things on its own accord leaving somethings unchanged for long standing periods of time. An example of this would be how time occasionally interacts with the speed of light. The speed of light remains constant but occasionally time will manifest itself into the equation and make modifications of the speed that light travels. For instance light will move forward forthwittingly at a billion miles a second but if it encounters any kind of resistance then time will inject itself and change the speed at which it was moving. Which leads me to the assumption that in order for time to inject itself into any equation a proponent has to take place that makes a physical change that would cause time to interject itself. If no physical change takes place than time has also not been a factor.

    The middle proponent of time is the area in which time is manipulating  the change that takes...Read More

👄What turns me on
Posted by Bling King

    I get turned on by some funny stuff. I'm not really into like full blown kinkiness or at least I wouldn't consider myself to be a kinky person but I do have a few fetishes. Some of them are a little out of the ordinary. For instance I have this one fetish about being tied up  and thrown in the ocean and then rescued by a mermaid. I think this fantasy comes from when I was a kid and I used to dream of mermaids and always wanted to meet one. Well one day its gonna happen. Now don't go telling me mermaids don't exist. You don't know cause they are in fact real and as soon as I meet one I will prove it to you. As far as some of my other turn ons  I guess what really gets me excited is people who  tell other people to shut the fuck up. I love when a woman just looks at a man and tells him to shut his mouth. To me thats a big turn on because the woman seems assertive like a dominatrix or something. If she will be assertive in a conversation she will be assertive in the bedroom or so I  would like to believe.

Time is a dialectable derelict
Posted by Bling King

To fathom the fortrighteousness of time one has to contemplate the personification of forthwittial forthwittil. Time forthwittingly will only listen to the commands of its on inner personification to which there is no directional direction or so it would seem but on further inquisitories I have come to realize that there is a forthwittingly forthwittal of which time has pronounced and those commands seem to speak to the nature of to which time corresponds. To review these pronouncements for your own bemusement look at time as if you had it captured it  in a bottle. What would happen? We know on the inside of the bottle time would force the inner workings of the bottle to correspond to times diabolical commands. Causing everything to change to times everlescent rules. however on the outside of the bottle things would not change, everything would stay in constant neutrality or would it? The question remains if there was no time would things still be allowed to happen and if so at what pace and what would dictate the pace at which things would change. There seems to be no rule in place for the dictation of the pace change which takes place. So it would seem that time has decided that factor somehow within itself. There could be a correlation at which things change and the pace being dictated by physics and the amount the physical world can be allowed to change within its own accord of set boundaries. To actually find...Read More

Free from time constraints
Posted by Bling King

 

 

 

There was a time when time did not matter. The thing that was an utmost relevance now was of no matter. The diffrence it made seemed miniscule and now it is constantly dictating everything that takes place before me. What is this thing that controls and makes everything manifest itself to its constraints and why and how does it do this. Time is nothing but the utmost miracle before us. Something that has always had to exist for anything ever to take place. There is no changing its course there is no variance in its absolute everlasting existance. To control time would be the utmost  crown jewel of all accomplishments if indeed it could ever be controlled. The only way I ever see time being manipulated to change its values is to speed up everything that time has interacted with. In order to do such a thing you would have to understand the nature of the objects in question and how they are effected by time. For instance a speeding car will slow down in time without constant force being distrubuted by the engine. To slow down the car one only has to take their foot off the accelarator and gradually time will do the rest but if you could freeze time at the speed at which the car was traveling then time would not  exist because the...Read More

the truth about time
Posted by Bling King

        I have looked at time many times and I have noticed a few components. There is a precise proponent that ushers in a manifestation. Whenever something new is going to happen you can look at that event which is about to take place and precisely predict exactly when it has started. Once you realize a manifestation has taken place you can precisely predict its out come. If you know that a manifestation has started to take place then you will know you are being guided through the realm precisely by the forces of an enlightenment. Throughout time this manifestation will remain constant starting with a beginning and an end and ending in a preconcieved enlightenment. Sometimes an enlightenment can take weeks and some times an enlightenment can take centuries. It depends on how many times that enlightenment has been benounced to the realm. 

 

nothing
Posted by Bling King

I suspect a suffcient of sufficence of suffiacantel suffiance of suffiance of absurdity of absurdanace. In all actual actuality there is an  actual actuality of actualityness in retrospect to the retorospective respect in which every person who has an intellectual intellect can see that the world is a prominance of prominance in which the order will reside as long as the order is maintained. Once that order is relinquished chaos will ensue. For chaos to be a calamity there only needs to be a perspectual perspective of perspectance that escalates the chaos to that height. What would cause that is a person or persons in the realm of the realmatical realmatics looking beyond thier own existance to the existance of there forfathers to see what has become of thier existance. If you look at your own existance for what it is you will see that it is neither logical nor illogical for it makes all the sense of a sensimatical sensematic. As long as you have a reason for your own existance then it is fruitful for you to exist. Once that reason or reasons are gone you will no longer care whether it is you live or die. In the realm in which we live is a prospectus prospectant of prospectantin which all will ensue. To change the prospectus prospectus you need to look to the realm and see what the prospectus prospectant is and manifest it to your own liking. My...Read More

The conclusive conclusion
Posted by Bling King

In all actual reality the realm is manifested of certain procedural procedures that come forth frequently to forthrightous forthrightenous. In the place of predicament I have found that I can properly place things in the procedural sequence unbenowst to people of the realm. In order to conflict the conflictions you have to equate the equation of equationalness in to proper equations. Very simple but also very tedious. You do this by equating the equation into percise preciseness. An example of an equation would be a placement of perdicament of a certain event in which you wish it to be. The next manifestation I could manifest is a manifestual manifestation of manifests of a sequance of certainal circumstances. Put together a sequence by asking the sequence in order to manifest itself and then tell the manifestations to happen in frequence in which they will unfold.

The Unattainable future
Posted by Bling King

     If the future is a grain of sand and its falling through an hour glass nothing in the world can stop it. It will eniquivaocalby blind as to where its going when it comes to its rest it has befallen its fate and will remain where it lay for an eternity knowing nothing about itself or it's surroundings. I am that grain of sand. Nothing ever can change my destiny for only time here makes a diffrence.. To benounce the future is the only way to change ones fortune. The time it takes to make an equivical change remains the utmost mystery of the universe.

🤯In the eyes of myself
Posted by Bling King

 

 

There where three men. All who seemed frightened. They stood on the edge of the canyon looking on as a fourth man tumbled to his death. We could have saved him said one of the men. He should have saved himself said another. The third man just look at them bewildered and brought a handgun to his own head and pulled the trigger. Blood spattered. The two men watched as he slumped to the ground. The first man screamed and the second threw himself to the side of the man on the ground. Why?!! he screamed. It was the only sound heard. Sobbing he looked at the man standing and said you did this! You and your frigging righteous speech about the lives we leave and the sacrifice we must make. Your the devil. I am not the devil said the standing man only the truth. The truth about what? The other man screamed. Your life he said and he jumped.

The man heard a ringing and he sat up slowly. It was over the dream but his thoughts where still on the side of the canyon. How did this happen. How did it all just fade away? The dream came and went in an instant leaving his mind boggled and his eyes heavy. I knew I was there thought the man but how? It was all to familiar the...Read More

The story Elijah and Ellen
Posted by Bling King

The story of Elijah and Ellan. This is the story of Elijah and Ellan. Ellan is a beutiful temptress and Elijah is a dutiful servant of Ellan's. Together the pair fell in love and soon became a duo of in excessible excession. They frolicked in the sun under the rare occurance of rain they took shelter in the arms of each other. One day while hiding from the glares of the sun under an oak tree that provided an abundance of shade they looked into each others souls and realized there where no people suited for each other then the two of them where suited for each other. They basked in the notion that they where the most two compatible souls on the planet. As they where thinking this a giant unforseen acclamaited acclamation occurred. The planet began to tremble and shake beneath them and the stars came out. The sun hid amongst the clouds and everything from start to finish began to take shape. There where huge explosions and giant surges of wind and rain. The two began to run for their shelter knowing at the exact moment the trembling and violent agressions of unacclaimated weather started that they most likely wouldn't make it to see another sunrise. The planet was exploding with molten lava and the tempertures where unbearable as for the two of them could remember they had never seen a winter climate and didn't expect they ever would. The planet had been warming out of...Read More

today was a day of dismal despair
Posted by Bling King

Things have gone down hill drastically now for a very long time. We seem to be some what defeated but yet i know we still have some power and prominance. We are fighting an up hill battle and there is no way forward from here from what i can see. We are trudging along a path that goes nowhere.

⚔️The Greatest Warrior of All Time
Posted by Bling King

 

 

Today i conquered and beat all adverseries there where to beat. Tomorrow new adversaries will arise. I will be ready, there is never a shortage of enemies who wish to dethrone me from the top of the world. I didn't get here by being passive and yeilding to the oppostion. I got here by defeating them both mentally and physically and in entiriety.

In a time of desilute despair
Posted by Bling King

     There was a time when I was in desilute despair. The only thing I had was me myself and I to fall back on. I looked at the person who was my opponent and I knew one of  us was going to die and I was going to do everytrhing I could to make dam sure it wasn't me. I pulled my six shooter from its holster and aimed at the guy looking at me  about 30 yards away. He also went for his gun and in lightning speed he was laid sprawled out on the dirt bleeding and moaning. I had heard a shot but new that it had come from my own gun. He never even got a shot off. I was unscathed and again undeafeted. Anybody who ever tried to kill me was dead and their where over 30 who had tried and failed to kill yours truly.

Gravity
Posted by Bling King

Gravity is the force of nature that pulls cellestrial bodies toward one another. The cause of gravity is the enertia of a bodies movement through space and time. This happens by an object preconcievably traveling through the cosmos at an alarming rate of acceleration. The faster an object travels the more enertia it will build up and then will therefore have a greater ability to move. the more it moves the more other objects will cling to it. the way this can be proved is by taking an object and hurtling it towards another object the two objects would collide do to the enertia pulling them towards each other. Thy would not stay on their current trajectory but their paths would alter towards one another in a greater force than their initial gravitational pull. the best test to accomodate this theory would be tow baseballs flying through the air at speeds over one hundred miles an hour. The baseballs would not interject themselves with one another normally but at this speed would do so do to the balls enertia pulling them towards one another.

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