Quantum entanglement

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Quantum entanglement

Quantum entanglement occurs when electrons, molecules even as large as "buckyballs",[1][2] photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description (state), which is indefinite in terms of important factors such as position,[3] momentum, spin, polarization, etc. According to the Copenhagen interpretation of quantum mechanics, their shared state is indefinite until measured.[4] 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[5] be found to have taken the complementary value (e.g., counterclockwise spin). Thus, there is a correlation between the results of measurements performed on entangled pairs, and this occurs even though the entangled pair may have been separated by arbitrarily large distances.[6]

This behavior is theoretically coherent and has been demonstrated experimentally, and it is accepted by the physics community. However there is some debate[7] 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 the EPR paradox paper of Albert Einstein, Boris Podolsky and Nathan Rosen in 1935,[8] and several papers by Erwin Schrödinger shortly thereafter.[9][10] Although these first studies focused on the counterintuitive properties of entanglement, with the aim of criticizing quantum mechanics, eventually entanglement was verified experimentally[11], and recognized as a valid, fundamental feature of quantum mechanics; the focus of the research has now changed to its utilization as a resource for communication and computation.

History

The counterintuitive predictions of quantum mechanics about strongly correlated systems were first discussed by Albert Einstein in 1935, in a joint paper with Boris Podolsky and Nathan Rosen.[8] 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.[8]

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".[12] 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:[9]

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.[13] Einstein later famously derided entanglement as "spukhafte Fernwirkung"[14] 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.[15] His inequality is experimentally testable, and there have been numerous relevant experiments, starting with the pioneering work of Freedman and Clauser in 1972[16] and Aspect's experiments in 1982.[17] 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.

Concept

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.[18]

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

However, with quantum entanglement, if Alice and Bob measure the spin of their particles in directions other than just up or down, with the directions chosen to form a Bell's inequality, they can now observe a correlation that is fundamentally stronger than anything that is achievable in classical physics. Here, the classical simulation of the experiment breaks down because there are no "directions" other than heads or tails to be measured in the coins.[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 can't have definite values at the same time―they are incompatible. In classical physics this does not make sense, since any number of properties can be measured simultaneously with arbitrary accuracy. Bell's theorem implies, and it has been proven mathematically, that compatible measurements can't show Bell-like correlations,[19] 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.[20][21] In another experiment, the measurements of the entangled particles were made in moving, relativistic reference frames in which each respective measurement occurred before the other, and the measurement results remained correlated.[22][23]

Entanglement, 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. 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?"[24] 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 which do not produce such correlations. A well-known example of this is the Werner state that is entangled for certain values of psym, but can always be described using local hidden variables.[25] In short, entanglement of a two-party state is necessary but not sufficient for that state to be non-local. It is important to recognise that entanglement is more commonly viewed as an algebraic concept, noted for being a precedent to non-locality as well as quantum teleportation and 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

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.[26] Other methods include the use of a fiber coupler to confine and mix photons, use of quantum dots to trap electrons until decay occurs, 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's also possible to create entanglement between quantum systems that never directly interacted, through the use of entanglement swapping.

Applications of entanglement

Entanglement has many applications in quantum information theory. With the aid of entanglement, otherwise impossible tasks may be achieved. Among the best-known applications of entanglement are superdense coding and quantum teleportation.[27] Not all researchers agree that entanglement is vital to the functioning of a quantum computer.[28] Entanglement is used in some protocols of quantum cryptography.[29][30]

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

Quantum mechanical framework

The following subsections are for those with a good working knowledge of quantum mechanics, including familiarity with the 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 $\scriptstyle H_A \otimes H_B$ is of the form

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

This state is separable if $\scriptstyle c_{ij}= c^A_ic^B_j,$ 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 $\scriptstyle c_{ij} \neq c^A_ic^B_j.$ 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:

${1 \over \sqrt{2}} \bigg( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \bigg)$.

If the composite system is in this state, it is impossible to attribute to either system 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.[31] 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 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:[32]

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 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 wi'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 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 $H_A \otimes H_B$. Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as

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

where $\rho_i^A$'s and $\rho_i^B$'s are themselves states on the subsystems A and B respectively. In other words, a state is separable if it is probability distribution over uncorrelated states, or product states. We can assume without loss of generality that $\rho_i^A$ and $\rho_i^B$ are pure ensembles. A state is then said to be entangled if it is not separable. In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be NP-hard.[33] 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.[34]

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

Reduced density matrices

The idea of a reduced density matrix was introduced by Paul Dirac in 1930.[35] 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 $\scriptstyle ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B ) / \sqrt{2}$ discussed above is

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

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of A for the pure product state $|\psi\rangle_A \otimes |\phi\rangle_B$ discussed above is

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

In general, a bipartite pure state ρ is entangled if and only if one, meaning both, of its reduced states are mixed states. Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is one dimensional AKLT spin chain:[36] the ground state can be divided into a block and environment. The reduced density matrix of the block is proportional to a projector to a degenerate ground state of another Hamiltonian.

The reduced density matrix also was evaluated for XY spin chains, where it has full rank. It was proved that in thermodynamic limit, the spectrum of the reduced density matrix of large block of spins is exact geometric sequence [37] in this case.

Entropy

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

Definition

In classical information theory, the Shannon entropy, H is associated to a probability distribution,$p_1, \cdots, p_n$, in the following way:[38]

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

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 $\; \log \rho$. 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 is adopted that $0 \log 0 \; = 0$. 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.[39] 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

$\rho \in H \otimes H$

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,

$S(\rho) \; = S(U \rho U^*).$

Indeed, without the above property, the von Neumann entropy would not be well-defined. In particular, U could be the time evolution operator of the system, i.e.

$U(t) \; = \exp \left(\frac{-i H t }{\hbar}\right)$

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.

References

Specific references

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General references

• Bengtsson I, Zyczkowski K (2006). "Geometry of Quantum States". An Introduction to Quantum Entanglement. Cambridge: Cambridge University Press.
• Steward EG (2008). Quantum Mechanics: Its Early Development and the Road to Entanglement. Imperial College Press. ISBN 978-1860949784.
• Horodecki R, Horodecki P, Horodecki M, Horodecki K (2009). "Quantum entanglement". Rev. Mod. Phys. 81 (2): 865–942. arXiv:quant-ph/0702225. Bibcode 2009RvMP...81..865H. doi:10.1103/RevModPhys.81.865.
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