# Entanglement

Entanglement is a property of composite quantum system where the joint state cannot be written as a product of states of its component systems.

Erwin Schrödinger coined the word 'entanglement' as a translation for the German Verschränkung in a letter to Albert Einstein "to describe the correlations between two particles that interact and then separate, as in the EPR experiment.".

### Definition

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

HA ⊗ HB.

If the first system is in state $\scriptstyle| \psi \rangle_A$ and the second in state $\scriptstyle| \phi \rangle_B$, the state of the composite system is

ψA ⊗ ∣ϕB.

States of the composite system which can be represented in this form are called separable states, or (in the simplest case) product states.

Not all states are separable states (and thus product states). Fix a basis $\{|i \rangle_A\}$ for HA and a basis $\{|j \rangle_B\}$ for HB. The most general state in HA ⊗ HB is of the form:

ψAB = ∑i, jcijiA ⊗ ∣jB.

This state is separable if there exist ciA, cjB so that $c_{ij}= c^A_ic^B_j,$ yielding $|\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A$ and $|\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.$ It is inseparable if for all ciA, cjB we have $c_{ij} \neq c^A_ic^B_j.$ If a state is inseparable, it is called an entangled state.

For example, given two basis vectors $\{|0\rangle_A, |1\rangle_A\}$ of HA and two basis vectors $\{|0\rangle_B, |1\rangle_B\}$ of HB, the following is an entangled state:

$\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B \right ).$

Which is often simply written as:

$\tfrac{1}{\sqrt{2}} \left ( |00\rangle + |11\rangle \right ).$

If the composite system is in this state, it is impossible to attribute to either system A or system B a definite pure state. Another way to say this is that while the von Neumann entropy of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled".

### Canonical Entangled States

There are several canonical entangled states that appear often in theory and experiments.

For two qubits, the Bell states are

$|\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B)$
$|\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B)$.

These four pure states are all maximally entangled (according to the entropy of entanglement) and form an orthonormal basis (linear algebra) of the Hilbert space of the two qubits. They play a fundamental role in Bell's theorem.

For M>2 qubits, the GHZ state is

$|\mathrm{GHZ}\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}},$

which reduces to the Bell state ∣Φ +  for M = 2. The traditional GHZ state was defined for M = 3. GHZ states are occasionally extended to qudits, i.e. systems of d rather than 2 dimensions.

Also for M>2 qubits, there are spin squeezed states. Spin squeezed states are a class of states satisfying certain restrictions on the uncertainty of spin measurements, and are necessarily entangled (Masahiro Kitagawa and Masahito Ueda).

For two bosonic modes, a NOON state is

$|\psi_\text{NOON} \rangle = \frac{|N \rangle_a |0\rangle_b + |{0}\rangle_a |{N}\rangle_b}{\sqrt{2}}, \,$

This is like a Bell state ∣Φ +  except the basis kets 0 and 1 have been replaced with "the N photons are in one mode" and "the N photons are in the other mode".

Finally, there also exist twin Fock states for bosonic modes, which can be created by feeding a Fock state into two arms leading to a beam-splitter. They are the sum of multiple of NOON states, and can used to achieve the Heisenberg limit. Phys. Rev. Lett. 71, 1355 (1993): Interferometric detection of optical phase shifts at the Heisenberg limit

For the appropriately chosen measure of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled. The partially entangled states are generally easier to prepare experimentally.