# Separable and entangled states

An '''entangled state''' is defined as a state that is '''not separable'''. A separable state can be written as a probability distribution over uncorrelated states, product states, :$\rho = \sum_i p_i \rho_i^A \otimes \rho_i^B$. == Pure states == For pure states the above definition can be represented as follows. Consider two quantum systems $A$ and $B$, with respective Hilbert spaces $\mathcal\left\{H\right\}_A$ and $\mathcal\left\{H\right\}_B$. The Hilbert space of the composite system is the tensor product $\mathcal\left\{H\right\}_A\otimes\mathcal\left\{H\right\}_B$. If the state $|\Psi\rangle_\left\{AB\right\}$of the composite system can be represented in the form :$|\Psi\rangle_\left\{AB\right\}=|\psi\rangle_A \otimes |\phi\rangle_B$, where $|\psi\rangle_A\in\mathcal\left\{H\right\}_A$ and $|\phi\rangle_B\in\mathcal\left\{H\right\}_B$ are the states of the systems $A$ and $B$ respectively, then this state is called a ''separable state''. If a state is not separable, it is known as an ''entangled state''. == Ralated papers == * R. F. Werner, ''[http://www.imaph.tu-bs.de/ftp/werner/p26.pdf Quantum states with Einstein-Rosen-Podolsky correlations admitting a hidden-variable model]'', Phys. Rev. A 40, pp. 4277-4281, 1989. * D. Bruss, ''Characterizing Entanglement'', Math. Phys. 43, pp. 4237, 2002. {{arxiv|number=quant-ph/0110078}} {{stub}} Category:Entanglement Category:Handbook of Quantum Information