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 non-local 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.

In 1964, Bell? invented an experimental test, based on the EPR paradox, that could distinguish quantum entanglement from the effects of a broad class of hidden variable theories. Subsequent experiments verified the quantum mechanical approach. Since then, entanglement has been accepted as a bona fide physical phenomenon.

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, because of the probabilistic nature of quantum mechanical processes. Causality? is thus preserved.

Though an area of active research, the essential properties of entanglement are now understood, and it is the basis for emerging technologies such as quantum computing and [quantum cryptography]?.

The following discussion builds on the theoretical framework developed in the articles bra-ket notation and mathematical formulation of quantum mechanics.

Consider two systems A and B, with respective Hilbert spaces *H _{A}* and

- |ψ>
_{A}|φ>_{B}

This is known as a *pure state*.

Pick observables (and corresponding [Hermitian operators]?) Ω_{A} acting on *H _{A}*, and Ω

- (Σ
_{i}a_{i}|i>_{A})(Σ_{j}b_{j}|j>_{B})

for some choice of complex coefficients a_{i} and b_{j}. This is not the most general state of *H _{A}* ×

- Σ
_{i,j}c_{ij}|i>_{A}|j>_{B}

If such a state cannot be factored into the form of a pure state, it is known as a mixed, or *entangled* state.

Choose two basis vectors {|0>_{A}, |1>_{A}} of *H _{A}*, and two basis vectors {|0>

- 2
^{-1/2}( |0>_{A}|1>_{B}- |1>_{A}|0>_{B})

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:

- Alice measures λ
_{0}, and the state of the system collapses to |0>_{A}|1>_{B}. - Alice measures λ
_{1}, and the state of the system collapses to |1>_{A}|0>_{B}.

If the former occurs, any subsequent measurement of
Ω_{B} performed by Bob always returns μ_{1}. If the latter occurs, Bob's measurement always returns μ_{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.

Because the outcome of Alice's measurement is random and cannot be controlled by Alice, no information can be transmitted from Alice to Bob in this manner. Causality is thus preserved, as we claimed above.

The method of *density matrices* provides us with a formal measure of entanglement. Let the state of the composite system be |Ψ>, which can be either pure or entangled. The [projection operator]? for
this state is denoted

- ρ
_{T}= |Ψ> <Ψ|

The density matrix of system A is a linear operator in the Hilbert space of system A, defined as the trace? of ρ_{T} over the basis of system B:

- ρ
_{A}- = Σ
_{j}<j|_{B}(|Ψ> <Ψ|) |j>_{B} - = Tr
_{B}ρ_{T}

- = Σ

For example, the density matrix of A for the entangled state discussed above is

- ρ
_{A}= 1/2 ( |0>_{A}<0|_{A}+ |1>_{A}<1|_{A})

and the density matrix of A for the pure state discussed above is

- ρ
_{A}= |ψ>_{A}<ψ|_{A}

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

- S = -
*k*Tr ( ρ ln ρ )

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 the entangled state discussed above is *k*ln 2 (which can be shown to be the maximum entropy for a composition of two two-level systems.) Generally, the more entangled a system, the larger its entropy.

It can also be shown that [unitary operators]? acting on a state (such as the time evolution operator obtained from Schrodinger's 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.