Entanglement

'''Entanglement''' is a property of [[quantum mechanics|composite quantum system]] where the [[joint state]] cannot be written as a product of states of its component systems. [[Schrodinger|Erwin Schrödinger]] coined the word 'entanglement' as a translation for the German ''Verschränkung'' in a letter to [[Einstein|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 space]]s $H\_A$ and $H\_B$. The Hilbert space of the composite system is the [[tensor product]] : $H\_A\; \backslash otimes\; H\_B.$ If the first system is in state $\backslash scriptstyle|\; \backslash psi\; \backslash rangle\_A$ and the second in state $\backslash scriptstyle|\; \backslash phi\; \backslash rangle\_B$, the state of the composite system is : $|\backslash psi\backslash rangle\_A\; \backslash otimes\; |\backslash phi\backslash rangle\_B.$ States of the composite system which can be represented in this form are called ''[[separable state]]s'', or (in the simplest case) ''[[product state]]s''. Not all states are separable states (and thus product states). Fix a [[basis (linear algebra)|basis]] $\backslash scriptstyle\; \backslash \{|i\; \backslash rangle\_A\backslash \}$ for $H\_A$ and a basis $\backslash scriptstyle\; \backslash \{|j\; \backslash rangle\_B\backslash \}$ for $H\_B$. The most general state in $H\_A\; \backslash otimes\; H\_B$ is of the form: : $|\backslash psi\backslash rangle\_\{AB\}\; =\; \backslash sum\_\{i,j\}\; c\_\{ij\}\; |i\backslash rangle\_A\; \backslash otimes\; |j\backslash rangle\_B$. This state is separable if there exist $c^A\_i,c^B\_j$ so that $\backslash scriptstyle\; c\_\{ij\}=\; c^A\_ic^B\_j,$ yielding $\backslash scriptstyle\; |\backslash psi\backslash rangle\_A\; =\; \backslash sum\_\{i\}\; c^A\_\{i\}\; |i\backslash rangle\_A$ and $\backslash scriptstyle\; |\backslash phi\backslash rangle\_B\; =\; \backslash sum\_\{j\}\; c^B\_\{j\}\; |j\backslash rangle\_B.$ It is inseparable if for all $c^A\_i,c^B\_j$ we have $\backslash scriptstyle\; c\_\{ij\}\; \backslash neq\; c^A\_ic^B\_j.$ If a state is inseparable, it is called an ''entangled state''. For example, given two basis vectors $\backslash scriptstyle\; \backslash \{|0\backslash rangle\_A,\; |1\backslash rangle\_A\backslash \}$ of $H\_A$ and two basis vectors $\backslash scriptstyle\; \backslash \{|0\backslash rangle\_B,\; |1\backslash rangle\_B\backslash \}$ of $H\_B$, the following is an entangled state: : $\backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\; \backslash left\; (\; |0\backslash rangle\_A\; \backslash otimes\; |0\backslash rangle\_B\; +\; |1\backslash rangle\_A\; \backslash otimes\; |1\backslash rangle\_B\; \backslash right\; ).$ Which is often simply written as: : $\backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\; \backslash left\; (\; |00\backslash rangle\; +\; |11\backslash rangle\; \backslash right\; ).$ 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 state]]s are : $|\backslash Phi^\backslash pm\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_A\; \backslash otimes\; |0\backslash rangle\_B\; \backslash pm\; |1\backslash rangle\_A\; \backslash otimes\; |1\backslash rangle\_B)$ : $|\backslash Psi^\backslash pm\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_A\; \backslash otimes\; |1\backslash rangle\_B\; \backslash pm\; |1\backslash rangle\_A\; \backslash otimes\; |0\backslash rangle\_B)$. 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 [[Greenberger–Horne–Zeilinger state|GHZ state]] is : $|\backslash mathrm\{GHZ\}\backslash rangle\; =\; \backslash frac\{|0\backslash rangle^\{\backslash otimes\; M\}\; +\; |1\backslash rangle^\{\backslash otimes\; M\}\}\{\backslash sqrt\{2\}\},$ which reduces to the Bell state $|\backslash Phi^+\backslash rangle$ for $M=2$. The traditional GHZ state was defined for $M=3$. GHZ states are occasionally extended to ''[[qudit]]s'', 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 ([http://pra.aps.org/abstract/PRA/v47/i6/p5138_1 Masahiro Kitagawa and Masahito Ueda]). For two [[boson]]ic modes, a [[NOON state]] is : $|\backslash psi\_\backslash text\{NOON\}\; \backslash rangle\; =\; \backslash frac\{|N\; \backslash rangle\_a\; |0\backslash rangle\_b\; +\; |\{0\}\backslash rangle\_a\; |\{N\}\backslash rangle\_b\}\{\backslash sqrt\{2\}\},\; \backslash ,$ This is like a Bell state $|\backslash Phi^+\backslash rangle$ 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. [http://prl.aps.org/abstract/PRL/v71/i9/p1355_1 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. == See also == * [[Bell's Theorem]] * [[Bell Test Experiments]] * [[Entanglement Swapping]] * [[Separable State]] * [[Entanglement Distillation]] [[Category:Handbook of Quantum Information]]