Entanglement is one of the properties of quantum mechanics which caused Einstein and others to dislike the theory. In 1935, Einstein, Podolsky, and Rosen formulated the EPR paradox, demonstrating that entanglement makes quantum mechanics a nonlocal theory. Einstein famously derided entanglement as "spooky action at a distance."
On the other hand, quantum mechanics was highly successful in producing correct experimental predictions, and the phenomenon of "spooky action" could in fact be observed. Some suggested the existence of unknown microscopic parameters, known as "hidden variables", that were deterministic and obeyed the locality principle, but gave rise to quantum mechanical behavior in the bulk. However, in 1964 Bell showed that the effects of quantum entanglement could be experimentally distinguished from the effects of a broad class of local hiddenvariable theories. Subsequent experiments verified the quantum mechanical predictions, and entanglement has now become accepted as a bona fide physical phenomenon. The "Bell inequalities" are described in greater detail in the article EPR paradox.
Entanglement obeys the letter if not the spirit of relativity. Although two entangled systems can interact across large spatial separations, no useful information can be transmitted in this way, so causality cannot be violated through entanglement. This occurs for two subtle reasons: (i) quantum mechanical measurements yield probabilistic results, and (ii) the no cloning theorem forbids the statistical inspection of entangled quantum states.
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. This process is known as quantum teleportation. Despite its name, quantum teleportation can not be used to transmit information faster than light, because a classical information channel is involved.
Though an area of active research, some of the essential properties of entanglement are now understood, and it is the basis for emerging technologies such as quantum computing and quantum cryptography. In the following article, we will briefly survey the mathematical formulation of entanglement.
The following discussion builds on the theoretical framework developed in the articles braket notation and mathematical formulation of quantum mechanics.
Consider two systems A and B, with respective Hilbert spaces H_{A} and H_{B}. The Hilbert space of the composite system is H_{A} × H_{B}. If the first system is in state ψ⟩_{A} and the second in state φ⟩_{B}, the state of the composite system is
This is called a pure state.
Pick observables (and corresponding Hermitian operators) Ω_{A} acting on H_{A}, and Ω_{B} acting on H_{B}. According to the spectral theorem, we can find a basis {i⟩_{A}} for H_{A} composed of eigenvectors of Ω_{A}, and a basis {j⟩_{B}} for H_{B} composed of eigenvectors of Ω_{B}. We can then write the above pure state as
for some choice of complex coefficients a_{i} and b_{j}. This is not the most general state of H_{A}×H_{B}, which has the form
If such a state cannot be factored into the form of a separable state, it is known as an entangled state.
For example, given two basis vectors {0⟩_{A}, 1⟩_{A}} of H_{A} and two basis vectors {0⟩_{B}, 1⟩_{B}} of H_{B}, the following is an entangled state:
If the composite system is in this state, neither system A nor system B have a definite state. Instead, their states are superposed with one another. In this sense, the systems are "entangled".
Now suppose Alice is an observer for system A, and Bob is an observer for system B. If Alice performs the measurement Ω_{A}, there are two possible outcomes, occurring with equal probability:
If the former occurs, any subsequent measurement of Ω_{B} performed by Bob always returns 1. If the latter occurs, Bob's measurement always returns 0. Thus, system B has been altered by Alice performing her measurement on system A., 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. (There is a possible loophole: if Bob could make multiple duplicate copies of the state he receives, he could obtain information by collecting statistics. This loophole is closed by the no cloning theorem, which forbids the creation of duplicate states.) Causality is thus preserved, as we claimed above.
Quantifying entanglement is an important step towards better understanding the phenomenon. The method of density matrices provides us with a formal measure of entanglement. Let the state of the composite system be Ψ⟩. The projection operator[?] for this state is denoted
We define the density matrix of system A, a linear operator in the Hilbert space of system A, as the trace of ρ_{T} over the basis of system B:
For example, the density matrix of A for the entangled state discussed above is
and the density matrix of A for the pure state discussed above is
This is simply the projection operator of ψ⟩_{A}. Note that the density matrix of the composite system, ρ_{T}, also takes this form. This is unsurprising, since we assumed that the state of the composite system is pure.
Given a general density matrix ρ, we can calculate the quantity
where k is Boltzmann's constant, and the trace is taken over the space H in which ρ acts. It turns out that S is precisely the entropy of the system corresponding to H.
The entropy of any pure state is zero, which is unsurprising since there is no uncertainty about the state of the system. The entropy of any of the two subsystems of the the entangled state discussed above is kln 2 (which can be shown to be the maximum entropy for a onelevel system). If the overall system is pure, the entropy of its subsystems can be used to measure its degree of entanglement with the other subsystems.
It can also be shown that unitary operators acting on a state (such as the time evolution operator obtained from the Schrödinger equation) leave the entropy unchanged. This associates the reversibility of a process with its resulting entropy change, which is a deep result linking quantum mechanics to information theory and thermodynamics.
The language of density matrices is also used to describe quantum ensembles, or a collection of identical quantum systems.
Consider a "blackbox" 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 called a pure ensemble.
However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state z+⟩ (spins aligned in the positive z direction), and the other with state y⟩ (spins aligned in the negative y direction.) Generally, there can be any number of populations, each corresponding to a different state. This is a mixed ensemble.
We can describe an ensemble as a collection of populations with weights w_{i} and corresponding states α_{i}⟩. The density matrix of the ensemble is defined as
All the above results for density matrices and the quantum entropy remain valid with this definition. Motivated by this, as well as the manyworlds interpretation, many physicists now believe that all mixed ensembles can be explained as entangled quantum states.
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