Quantum entanglement

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

Quantum mechanics
$\hat{H} \| \psi_n(t) \rangle = i \hbar \frac{\partial}{\partial t} \| \psi_n(t) \rangle$ $\frac { 1 }{ { c }^{ 2 } } \frac { { \partial }^{ 2 }{ \phi }_{n} }{ { \partial t }^{ 2 } } -{ { \nabla }^{ 2 }{ \phi }_{n} }+{ \left( \frac { mc }{ \hbar } \right) }^{ 2 }{ \phi }_{n}=0$ Manifold Dynamics: Schrödinger and Klein-Gordon equations
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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 must be described 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 are 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, as to be expected due to their entanglement. However, this behavior gives rise to paradoxical effects: any measurement of a property of a particle can be seen as acting on that particle (e.g., by collapsing a number of superposed states) and will change the original quantum property by some unknown amount; and in the case of entangled particles, such a measurement will 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] and several papers by Erwin Schrödinger shortly thereafter,^[2]^[3] describing what came to be known as the EPR paradox. Einstein and others considered such behavior to be impossible, as it violated the local realist view of causality (Einstein referring 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 one-hundredth of a percent of the travel time of light 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,^[11]^[12]^[13]^[14] electrons, molecules the size of buckyballs,^[15]^[16] and even small diamonds.^[17]^[18]^[19]^[20] 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."^[21] 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.^[22] Einstein later famously derided entanglement as "spukhafte Fernwirkung"^[23] 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 produced relatively little other published work. So, despite the interest, the weak point in EPR's argument was not discovered until 1964, when John Stewart Bell proved that one of their key assumptions, the principle of locality, which underlies the kind of hidden variables interpretation hoped for by EPR, was mathematically inconsistent with the predictions of quantum theory. 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.^[24] His inequality is experimentally testable, and there have been numerous relevant experiments, starting with the pioneering work of Stuart Freedman and John Clauser in 1972^[25] and Alain Aspect's experiments in 1982.^[26] They have all shown agreement with quantum mechanics rather than the principle of local realism. However, the issue is not finally settled, as 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 Charles H. Bennett and Gilles Brassard^[27] and E91 by Artur Ekert.^[28] Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

Concept

Meaning of entanglement

An entangled system is defined to be one whose quantum state cannot be factored as a product of states of its local constituents, that is to say, they are not individual particles but are an inseparable whole. If entangled, one constituent cannot be fully described without considering the other(s). Note that the state of a composite system is always expressible as a sum, or superposition, of products of states of local constituents; it is entangled if this sum necessarily has more than one term.

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. Entanglement is broken when the entangled particles decohere through interaction with the environment; for example, when a measurement is made.^[29]

As an example of entanglement: a subatomic particle decays into an entangled pair of other particles. The 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-½ 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.)

The special property of entanglement can be better observed if we separate the said two particles. Let's put one of them in the White House in Washington and the other in UC Berkeley (think about this as a thought experiment, not an actual one). Now, if we measure a particular characteristic of one of these particles (say, for example, spin), get a result, and then measure the other particle using the same criterion (spin along the same axis), we find that the result of the measurement of the second particle will match (in a complementary sense) the result of the measurement of the first particle, in that they will be opposite in their values.

The above result may or may not be perceived as surprising. A classical system would display the same property, and a hidden variable theory (see below) would certainly be required to do so, based on conservation of angular momentum in classical and quantum mechanics alike. The difference is that a classical system has definite values for all the observables all along while the quantum system does not. In a sense to be discussed below, the quantum system considered here seems to acquire a probability distribution for the outcome of a measurement of the spin along any axis of the other particle upon measurement of the first particle. This probability distribution is in general different from what it would be without measurement of the first particle. This may certainly be perceived as surprising in the case of spatially separated entangled particles.

Paradox

The paradox 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 result could have been communicated to the other particle (assuming that information cannot travel faster than light) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. 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 too is measured.

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements spacelike, hence, a message connecting the events would have to travel faster than light. According to the principles of special relativity, it is not possible for any information to travel between two such measuring events. It is not even possible to unambiguously say which of the measurements came first. For two spacelike separated events $x 1$ and $x 2$ there are inertial systems in which $x 1$ is first and others in which $x 2$ is first. 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.

In conclusion, the result shows that it is quite impossible for the measurement of one particle's properties to determine the other's properties, even though it seems that way at first, as the speed of light is the universal speed limit, and nothing can travel faster than that speed.

Hidden variables theory

A possible resolution to the 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 of the paradox, 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 widely separated detectors attempting to detect the particle in two different places must instantaneously 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 that Bell's inequality is not satisfied. However, all experiments have loophole problems.^[30]^[31] 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.^[32]^[33]

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,^[34] and thus entanglement is a fundamentally non-classical phenomenon.

Other types of experiments

In a 2012 experiment, "delayed-choice entanglement swapping" was used to decide whether two particles were entangled or not after they had been measured.^[35]

In a 2013 experiment, entanglement swapping has been used to create entanglement of 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".^[36] What this means is that two particles can be entangled even if they are distanced from each other in time. Two entangled particles will thus show the property of entanglement even if they are measured in two different times.

In three independent experiments it was shown that classically-communicated separable quantum states can be used to carry entangled states.^[37]

In August 2014, researcher Gabriela Barreto Lemos and team were able to "take pictures" of objects using photons that have not interacted with the subjects, but were entangled with photons that did interact with such objects. Lemos, from the University of Vienna, is confident that this new quantum imaging technique could find application where low light imaging is imperative, in fields like biological or medical imaging.^[38]

Mystery of time

There are physicists who say that time is an emergent phenomenon that is a side effect of quantum entanglement.^[39]^[40] In other words, time is an entanglement phenomenon, which places all equal clock readings (of correctly prepared clocks - or of any objects usable as clocks) into the same history. This was first understood by physicist Don Page and William Wootters in 1983.^[41] The Wheeler–DeWitt equation that combines general relativity and quantum mechanics — by leaving out time altogether — was introduced in the 1960s and it was taken up again in 1983, when the theorists Don Page and William Wootters made a solution based on the quantum phenomenon of entanglement. Page and Wootters argued that entanglement can be used to measure time.^[42]

In 2013, at the Istituto Nazionale di Ricerca Metrologica (INRIM) in Turin, Italy, researchers performed the first experimental test of Page and Wootters' ideas. Their result has been interpreted to confirm that time is an emergent phenomenon for internal observers but absent for external observers of the universe just as the Wheeler-DeWitt equation predicts.^[42]

Source for the arrow of time

Physicist Seth Lloyd says that quantum uncertainty gives rise to entanglement, the putative source of the arrow of time. According to Lloyd; "The arrow of time is an arrow of increasing correlations."^[43] The approach to entanglement would be from the perspective of the causal arrow of time, with the assumption that the cause of the measurement of one particle determines the effect of the result of the other particle's measurement.

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. Entanglement is also mathematically well-defined (see below). 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?"^[44] The latter question involves the issue of locality, i.e. does the agent of change have to be in physical contact (at least via some intermediary such as a field force), for a change to occur in some other thing? The 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.^[45] In particular, in the Many-worlds interpretation, the underlying description is fully local.^[46] 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.^[47] In short, entanglement of a two-party state is necessary but not sufficient for that state to be non-local. Moreover, it was shown that, for arbitrary number of party, there exist states that are genuinely entangled but admits a fully local strategy. It is important to recognize 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 $H A$ and $H B$ . The Hilbert space of the composite system is the tensor product

H_A \otimes H_B.

If the first system is in state $\scriptstyle| \psi \rangle_A$ 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 $H A$ and a basis $\scriptstyle \{|j \rangle_B\}$ for $H B$ . The most general state in $H A \otimes H B$ 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 vectors $\scriptstyle [c^A_i],[c^B_j]$ 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 any vectors $\scriptstyle [c^A_i],[c^B_j]$ at least for one pair of coordinates $\scriptstyle c^A_i,c^B_j$ 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 $H A$ and two basis vectors $\scriptstyle \{|0\rangle_B, |1\rangle_B\}$ of $H B$ , 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.^[48] 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 $H A \otimes H B$ space, but which cannot be separated into pure states of each $H A$ and $H B$ ).

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:^[49]

Alice measures 0, and the state of the system collapses to $\scriptstyle |0\rangle_A |1\rangle_B$ .
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 w_i are positive-valued probabilities (they sum up to 1), the vectors $α i$ are unit vectors, 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 $w i$ is the proportion of the ensemble whose states are $|\alpha_i\rangle$ . When a mixed state has rank 1, it therefore describes a pure ensemble. When there is less than total information about the state of a quantum system we need density matrices to represent the state.

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.

Following the definition above, for a bipartite composite system, mixed states are just density matrices on $H A \otimes H B$ . That is, it has the general form

\rho =\sum_{i} w_i\left[\sum_{j} \bar{c}_{ij} (|\alpha_{ij}\rangle\otimes|\beta_{ij}\rangle)\right]\otimes \left[\sum_k c_{ik} (\langle\alpha_{ik}|\otimes\langle\beta_{ik}|)\right]

where the w_i are positively valued probabilities, $\sum_j |c_{ij}|^2=1$ , and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as^[50]^:131–132

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

where the $w i$ are positively valued probabilities and the $\rho_i^A$ 's and $\rho_i^B$ 's are themselves mixed states (density operators) 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. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that $\rho_i^A$ and $\rho_i^B$ are themselves 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.^[51] For the $2 \times 2$ and $2 \times 3$ cases, a necessary and sufficient criterion for separability is given by the famous Positive Partial Transpose (PPT) condition.^[52]

Reduced density matrices

The idea of a reduced density matrix was introduced by Paul Dirac in 1930.^[53] Consider as above systems $A$ and $B$ each with a Hilbert space $H A, H B$ . Let the state of the composite system be

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

As indicated above, in general there is no way to associate a pure state to the component system $A$ . However, it still is possible to associate a density matrix. Let

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

which is the projection operator onto this state. The state of $A$ is the partial trace of $ρ T$ 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.

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

For example, the 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.

Two applications that use them

Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional AKLT spin chain:^[54] 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^[55] 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:^[56]

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 a non-polynomial function such as $log 2 (ρ)$ . If the nonnegative operator $ρ$ acts on a finite-dimensional Hilbert space and has eigenvalues $\lambda_1, \cdots, \lambda_n$ , $log 2 (ρ)$ turns out to be nothing more than the operator with the same eigenvectors, but the eigenvalues $\log_2(\lambda_1), \cdots, \log_2(\lambda_n)$ . The Shannon entropy is then:

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 $2 \times 2$ mixed states).

As a measure of entanglement

Entropy provides one tool which can be used to quantify entanglement, although other entanglement measures exist.^[57] 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 $ρ \in H A \otimes H B$ 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 only reasonable entanglement measure.

As an aside, the information-theoretic definition is closely related to entropy in the sense of statistical mechanics (comparing the two definitions, we note that, in the present context, it is customary to set the Boltzmann constant $k = 1$ ). For example, by properties of the Borel functional calculus, we see that for any unitary operator $U$ ,

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

Indeed, without this 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. Here the entropy is unchanged.

The reversibility of a process is associated with the resulting entropy change, i.e., a process is reversible if, and only if, it leaves the entropy of the system invariant. Therefore, the march of the arrow of time towards thermodynamic equilibrium is simply the growing spread of quantum entanglement.^[58] This provides a connection between quantum information theory and thermodynamics.

Rényi entropy also can be used as a measure of entanglement.

Entanglement measures

Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned, entanglement entropy is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature ^[57] and no single one is standard.

Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult (NP-hard) to compute.^[59]

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.^[60]

Most researchers believe that entanglement is necessary to realize quantum computing (although this is disputed by some^[61]).

Entanglement is used in some protocols of quantum cryptography.^[62]^[63] 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.^[64]

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.^[65] Spin squeezed states are a class of squeezed coherent states satisfying certain restrictions on the uncertainty of spin measurements, and are necessarily entangled.^[66] Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement.^[67]

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.^[68]

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.^[69] 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.

Testing a system for entanglement

Systems which contain no entanglement are said to be separable. For 2-Qubit and Qubit-Qutrit systems (2 × 2 and 2 × 3 respectively) the simple Peres–Horodecki criterion provides both a necessary and a sufficient criterion for separability, and thus for detecting entanglement. However, for the general case, the criterion is merely a sufficient one for separability, as the problem becomes NP-hard.^[70]^[71] A numerical approach to the problem is suggested by Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement".^[72] Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached. An implementation of the algorithm (including a built in Peres-Horodecki criterion testing) is brought in the "StateSeparator" web-app.

Naturally entangled systems

The electron shell of multi-electron atoms always consists of entangled electrons. The correct ionization energy can be calculated only by consideration of electron entanglement.^[73]

It has been shown by femtosecond transition spectroscopy, that in the photosystem of plants, entangled photons exist. An efficient conversion of the photon energy into chemical energy is possible only due to this entanglement.^[74]^[75]

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External links

Quantum mechanics

Background	Introduction Classical mechanics Old quantum theory Glossary History

Fundamentals	Bra–ket notation Complementarity Density matrix Entanglement Hamiltonian Interference Decoherence Nonlocality Quantum state Superposition Tunnelling Scattering theory Symmetry in quantum mechanics Uncertainty Wave function

Formulations	Formulations Heisenberg Interaction Matrix mechanics Schrödinger Path integral formulation Phase space

Equations	Dirac Klein–Gordon Pauli Rydberg Schrödinger

Interpretations	Interpretations Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden variables Many-worlds Objective collapse Quantum logic Relational Stochastic Transactional

Experiments	Afshar Bell's inequality Cold Atom Laboratory Davisson–Germer Delayed choice quantum eraser Double-slit Franck–Hertz experiment Mach-Zehnder inter. Elitzur–Vaidman Popper Quantum eraser Schrödinger's cat Stern–Gerlach Wheeler's delayed choice

Applications	Quantum biology Quantum chemistry Quantum chaos Quantum cryptography Quantum electronics Quantum gravity Quantum machine Quantum mind Quantum optics Quantum field theory Quantum statistical mechanics Quantum information science Quantum technology Fractional quantum mechanics Relativistic quantum mechanics

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