Quantum entanglement

Quantum entanglement is a quantum mechanical phenomenon in which the superposition of states of two or more systems allows the systems to influence one another regardless of their spatial separation.

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]?.

Formalism

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 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

|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_B acting on H_B. By the [spectral theorem]?, we obtain a basis {|i>_A} for H_A with matching eigenvalues {λ_i}, and a basis {|j>_B} for H_B with matching eigenvalues {μ_j}. We can then write the above pure state as

(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 × H_B, which has the form

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>_B, |1>_B} of H_A. An example of an entangled state is

2^-1/2 |0>_A |1>_B - |1>_A|0>_B

Neither system A nor B are in a definite state; instead, their states are superposed with each other. 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:

1. Alice measures λ₀, and the state of the system collapses to |0>_A |1>_B.
2. Alice measures λ₁, 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 μ₁. If the latter occurs, the measurement always returns μ₀. Thus, system B has been altered by Alice performing her measurement on system A. This happens even if the systems A and B are widely separated, which 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.

Entropy

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

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 (up to a proportionality factor) precisely the entropy of the system corresponding to H!

The entropy of any pure state is zero, which is unsurprising as there is no uncertainty about the state of the system. The entropy of the entangled state discussed above is kln 2 (which can be shown to be the maximum entropy for a composition of two two-level systems.) Generally, the more entangled a system is, the larger its entropy.

It can also be shown that [unitary operators]? acting on a state - including 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.