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, :$\backslash rho\; =\; \backslash sum\_i\; p\_i\; \backslash rho\_i^A\; \backslash otimes\; \backslash rho\_i^B$. == Pure states == For pure states the above definition can be represented as follows. Consider two quantum systems $A$ and $B$, with respective Hilbert spaces $\backslash mathcal\{H\}\_A$ and $\backslash mathcal\{H\}\_B$. The Hilbert space of the composite system is the tensor product $\backslash mathcal\{H\}\_A\backslash otimes\backslash mathcal\{H\}\_B$. If the state $|\backslash Psi\backslash rangle\_\{AB\}$of the composite system can be represented in the form :$|\backslash Psi\backslash rangle\_\{AB\}=|\backslash psi\backslash rangle\_A\; \backslash otimes\; |\backslash phi\backslash rangle\_B$, where $|\backslash psi\backslash rangle\_A\backslash in\backslash mathcal\{H\}\_A$ and $|\backslash phi\backslash rangle\_B\backslash in\backslash mathcal\{H\}\_B$ 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]]