# Quantum entanglement

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. This leads to correlations between observable physical properties of the systems. 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. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. However, classical information cannot be transmitted through entanglement faster than the speed of light.

Quantum entanglement is the basis for emerging technologies such as quantum computing and quantum cryptography, and has been used for experiments in quantum teleportation. At the same time, it produces some of the more theoretically and philosophically disturbing aspects of the theory, as one can show that the correlations predicted by quantum mechanics are inconsistent with the seemingly obvious principle of local realism, which is that information about the state of a system should only be mediated by interactions in its immediate surroundings. Different views of what is actually occurring in the process of quantum entanglement give rise to different interpretations of quantum mechanics.

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## Background

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 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. One apparent way to explain quantum entanglement is an approach known as "hidden variable theory", in which unknown deterministic microscopic parameters would cause the correlations. However, in 1964 Bell showed that such a theory could not be "local", the quantum entanglement predicted by quantum mechanics being experimentally distinguishable from a broad class of local hidden-variable theories. Results of subsequent experiments have overwhelmingly supported quantum mechanics. It is known that there are a number of "loopholes" in these experiments, but these are generally considered to be of minor importance.

Entanglement produces some interesting interactions with the principle of relativity that states that information cannot be transferred faster than the speed of light. 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 cannot be used to transmit information faster than light, because a classical information channel is involved.

## Formalism

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

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

[itex] H_A \otimes H_B [itex]

If the first system is in state [itex] | \psi \rangle_A[itex] and the second in state [itex] | \phi \rangle_B[itex], the state of the composite system is

[itex]|\psi\rangle_A \otimes |\phi\rangle_B,[itex]

which is often also written as

[itex]|\psi\rangle_A \; |\phi\rangle_B.[itex]

States of the composite system which can be represented in this form are called separable states.

Pick observables (and corresponding Hermitian operators) ΩA acting on HA, and ΩB acting on HB. According to the spectral theorem, we can find a basis {|i⟩A} for HA composed of eigenvectors of ΩA, and a basis {|j⟩B} for HB composed of eigenvectors of ΩB. We can then write the above pure state as

[itex]\left( \sum_i a_i |i\rangle_A \right) \left( \sum_j b_j |j\rangle_B \right)[itex],

for some choice of complex coefficients ai and bj. This is not the most general state of [itex] H_A \otimes H_B[itex], which has the form

[itex]\sum_{i,j} c_{ij} |i\rangle_A \; |j\rangle_B[itex].

If such a state is not separable, it is known as an entangled state.

For example, given two basis vectors {|0⟩A, |1⟩A} of HA and two basis vectors {|0⟩B, |1⟩B} of HB, the following is an entangled state:

[itex]{1 \over \sqrt{2}} \bigg( \mid0\rangle_A \otimes \mid 1\rangle_B - \mid 1\rangle_A \otimes \mid 0\rangle_B \bigg)[itex].

If the composite system is in this state, it is impossible to attribute to either system A or system B a definite pure 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 ΩA, there are two possible outcomes, occurring with equal probability:

1. Alice measures 0, and the state of the system collapses to |0⟩A |1⟩B
2. Alice measures 1, and the state of the system collapses to |1⟩A|0⟩B.

If the former occurs, any subsequent measurement of ΩB 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.

## Entropy

Quantifying entanglement is an important step towards better understanding the phenomenon of entropy. The method of density matrices provides us with a formal measure of entanglement.

Consider as above systems A and B each with a Hilbert space HA, HB. Let the state of the composite system be

[itex] |\Psi \rangle \in H_A \otimes H_B. [itex]

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

[itex]\rho_T = |\Psi\rangle \; \langle\Psi|[itex].

which is the projection operator onto this state. The state of A is the partial trace of ρT over the basis of system B:

[itex]\rho_A \equiv \sum_j \langle j|_B \left( |\Psi\rangle \langle\Psi| \right) |j\rangle_B = \hbox{Tr}_B \; \rho_T [itex].

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

[itex]\rho_A = (1/2) \bigg( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \bigg)[itex]

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

[itex]\rho_A = |\psi\rangle_A \langle\psi|_A [itex].

This is simply the projection operator of |ψ⟩A. Note that the density matrix of the composite system, ρT, 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

[itex]S = - k \hbox{Tr} \left( \rho \ln{\rho} \right)[itex]

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 entangled state discussed above is kln 2 (which can be shown to be the maximum entropy for a one-level 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.

## Ensembles

The language of density matrices is also used to describe quantum ensembles, or a collection of identical quantum systems.

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 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 wi and corresponding states |αi⟩. The density matrix of the ensemble is defined as

[itex]\rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|[itex].

All the above results for density matrices and the quantum entropy remain valid with this definition. Motivated by this, as well as the many-worlds interpretation, many physicists now believe that all mixed ensembles can be explained as entangled quantum states.

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

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