A fundamental property of quantum systems, first described by Erwin Schrödinger, in which two quantum systems can become correlated in a way which is impossible in classical physics. The two systems retain this correlation such that, under certain circumstances, subsequent action on one system can then have implications for the outcome of a measurement on the other. Explicitly verified by experiment, its applications include quantum computing and secure telecommunications via quantum cryptography. Entanglement also explains the destruction of the interference pattern in the double-slit experiment when either of the paths is monitored, an effect often explained using the Heisenberg uncertainty principle. Two quantum systems are said to be entangled if the wave function describing the combination of those two systems cannot be simply factored into a product of two wave functions corresponding to two separate systems. The classic display of entanglement involves a single light particle, or photon, which is fired into a special crystal in which it is split into two identical daughter photons, both having half the frequency of the parent (a process called parametric down-conversion). Via their shared parentage, the two daughters remain entangled no matter what their separation; if one is horizontally polarized, for example, then the other has to be vertically polarized. Quantum mechanics guarantees that these properties remain indeterminate until actually measured, but if experimenters examine one photon and find that it has a vertical polarization, they can then be certain that a measurement of the other photon will show it to have horizontal polarization. In this limited sense, detecting one particle of an entangled pair has an effect on the other, even though the two may be widely spaced.
Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin-up, the other one will always be observed to be spin-down and vice versa, this despite the fact that it is impossible to predict, according to quantum mechanics, which set of measurements will be observed.
Quantum entanglement is closely concerned with the emerging technologies of quantum computing and quantum cryptography, and has been used to experimentally realize quantum teleportation. The correlations predicted by quantum mechanics, and observed in experiment, reject the principle of local realism, which is that information about the state of a system should only be mediated by interactions in its immediate surroundings.
Background
Entanglement is one of the properties of quantum mechanics which caused Einstein and others to dislike the theory.
On the other hand, quantum mechanics has been highly successful in producing correct experimental predictions, and the strong correlations associated with the phenomenon of quantum entanglement have in fact been observed.
Observations on entangled states naively appear to conflict with the property of Einsteinian relativity that information cannot be transferred faster than the speed of light.
Although no information can be transmitted through entanglement alone, it is possible to transmit information using a set of entangled states used in conjunction with a classical information channel.
Pure States
The following discussion builds on the theoretical framework developed in the articles bra-ket notation and mathematical formulation of quantum mechanics. The Hilbert space of the composite system is the tensor product
If the first system is in state and the second in state , the state of the composite system is
which is often also written as
States of the composite system which can be represented in this form are called separable states, or product states.
Not all states are product states. The most general state in is of the form
.If a state is not separable, it is called an entangled state.
For example, given two basis vectors of HA and two basis vectors of HB, the following is an entangled state:
.If the composite system is in this state, it is impossible to attribute to either system A or system B a definite pure state.
Now suppose Alice is an observer for system A, and Bob is an observer for system B. If Alice makes a measurement in the {|0>, |1>} eigenbasis of A, there are two possible outcomes, occurring with equal probability:
Alice measures 0, and the state of the system collapses to Alice measures 1, and the state of the system collapses to . Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system.In some formal mathematical settings, it is noted that the correct setting for pure states in quantum mechanics is projective Hilbert space endowed with the Fubini-Study metric.
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 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:
,
where the wi's sum up to 1 (in the infinite dimensional case, we would take the closure of such states in the trace norm). When there is less than total information about the state of a quantum system we need density matrices to represent the state (see experiment discussed below).
Following the definition in previous section, for a bipartite composite system, mixed states are just density matrices on . Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as
,where 's and 's are they themselves states on the subsystems A and B respectively. In other words, a state is separable if it is probability distribution over uncorrelated states, or product states. In general, finding out whether or not a mixed state is entangled is considered difficult. For example, it could produce two populations of electrons: one with state (spins aligned in the positive direction), and the other with state (spins aligned in the negative direction.) Generally, there can be any number of populations, each corresponding to a different state.
Reduced Density Matrices
Consider as above systems A and B each with a Hilbert space HA, HB. Let the state of the composite system be
As indicated above, in general there is no way to associate a pure state to the component system A.
which is the projection operator onto this state. The state of A is the partial trace of ρT over the basis of system B:
.For example, the density matrix of A for the entangled state discussed above is
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 discussed above is
In general, a bipartite pure state ρ is entangled if and only if one, therefore both, of its reduced states are mixed states.
Entropy
In this section we briefly discuss entropy of a mixed state and how it can be viewed as a measure of entanglement.
Definition
In classical information theory, to a probability distribution , one can associate the Shannon entropy:
where the logarithm is taken in base 2.
Since one can think of a mixed state ρ as a probability distribution over an ensemble, this leads naturally to the definition of the von Neumann entropy:
where the logarithm is again taken in base 2. This extends to the infinite dimensional case as well: if ρ has spectral resolution , then we assume the same convention when calculating
As in statistical mechanics, one can say that the more uncertainty (number of microstates)the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is log2 (which can be shown to be the maximum entropy for mixed states).
As a measure of entanglement
Entropy provides one tool which can be used to quantify entanglement (although other entanglement measures exist).
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. Therefore, a bipartite pure state
is said to be a maximally entangled state if there exists some local bases on H such that the reduced state of ρ is the diagonal matrix
For mixed states, the reduced von Neumann entropy is not the unique entanglement measure.
Applications of entanglement
Entanglement has many applications in quantum information theory. Mixed state entanglement can be viewed as a resource for quantum communication. Among the most well known such applications of entanglement are superdense coding and quantum state teleportation.
Quantum computers use entanglement and superposition.
The Reeh-Schlieder theorem of quantum field theory is sometimes seen as the QFT analogue of quantum entanglement.
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