Bipartite states and Schmidt decomposition

''[[Bipartite]] states'' are one of the basic objects in Quantum Information Theory and will be defined in what follows: =Pure States = == Definition == Let $\backslash mathcal\{H\}=\backslash mathcal\{H\}\_A\backslash otimes\backslash mathcal\{H\}\_B$ be a Hilbert space defined as a [[tensor product]] of two Hilbert spaces $\backslash mathcal\{H\}\_A$ and $\backslash mathcal\{H\}\_B$. We call some [[pure state]] $|\backslash psi\backslash rangle\_\{AB\}$ on the composite system $A\backslash cup\; B$ '''bipartite''', if it is written with respect to the partition $AB$, which means $|\backslash psi\backslash rangle\_\{AB\}\; =\; \backslash sum\backslash limits\_\{ij\}\backslash chi\_\{ij\}|i\backslash rangle\_A\backslash otimes\; |j\backslash rangle\_B$, where $|i\backslash rangle\_A$ and $|j\backslash rangle\_B$ are bases in $\backslash mathcal\{H\}\_A$ and $\backslash mathcal\{H\}\_B$ respectively. == Schmidt Theorem (Schmidt Decomposition) == There is a statement in linear algebra, according to which for every $|\backslash psi\backslash rangle\_\{AB\}$ there exist bases $|u\_i\backslash rangle\_A$ and $|v\_j\backslash rangle\_B$ such that $|\backslash psi\backslash rangle\_\{AB\}\; =\; \backslash sum\backslash limits\_\{i=1\}^\{n\}\backslash sqrt\{\backslash tilde\{\backslash chi\_\{i\}\}\}|u\_i\backslash rangle\_A\backslash otimes\; |v\_i\backslash rangle\_B$, where $n=\backslash min\backslash left(dim(\backslash mathcal\{H\}\_A),dim(\backslash mathcal\{H\}\_B)\backslash right)$ and $\backslash ;\; \backslash sum\_\{i=1\}^\{n\}\backslash tilde\{\backslash chi\_\{i\}\}\; =\; 1$. The Schmidt coefficients $\backslash ;\; \backslash sqrt\{\backslash tilde\{\backslash chi\_\{i\}\}\}\backslash ge\; 0$ are the square roots of the eigenvalues of the two partial traces of $\backslash ;\; \backslash varrho\_\{AB\}=|\backslash psi\backslash rangle\_\{AB\}\backslash langle\; \backslash psi|$, $\backslash ;\; \backslash varrho\_A\; =\; Tr\_B[\backslash varrho\_\{AB\}]\; =\backslash sum\backslash limits\_\{i=1\}^\{n\}\backslash tilde\{\backslash chi\_\{i\}\}|u\_i\backslash rangle\_A\; \backslash langle\; u\_i|$ and $\backslash ;\; \backslash varrho\_B\; =\; Tr\_A[\backslash varrho\_\{AB\}]\; =\backslash sum\backslash limits\_\{i=1\}^\{n\}\backslash tilde\{\backslash chi\_\{i\}\}|v\_i\backslash rangle\_B\; \backslash langle\; v\_i|,$ and the ones which are non-zero have the same multiplicity. The Schmidt Decomposition is useful for the separability characterization of pure states: # The state $\backslash ;\; |\backslash psi\backslash rangle\_\{AB\}$ is separable if and only if there is only one non-zero Schmidt coefficient $\backslash ;\; \backslash tilde\{\backslash chi\_\{i\}\}=1$, $\backslash ;\; \backslash tilde\{\backslash chi\_\{j\}\}=0\; \backslash quad\; \backslash forall\; j\; \backslash neq\; i$; # If more than one Schmidt coefficients are non-zero, then the state is entangled; # If all the Schmidt coefficients are non-zero and equal, then the state is said to be ''maximally entangled''. = Mixed States = Operators on a finite dimensional Hilbert spaces form a normed vector space. Considering operators as vectors is helpful for the definition of a mixed bipartite state. == Definition == Let $\backslash rho\_\{AB\}$ be a mixed state on a composite system $A\backslash cup\; B$. Then we say that $\backslash rho\_\{AB\}$ is a '''bipartite [[mixed state]]''' on $\backslash mathcal\{H\}\_A\backslash otimes\; \backslash mathcal\{H\}\_B$ and write $\backslash rho\_\{AB\}\; =\; \backslash sum\backslash limits\_\{ij\}\backslash lambda\_\{ij\}G^A\_i\backslash otimes\; G^B\_j\; .$ == Schmidt Decomposition == The Decomposition can be also written for operators: $\backslash rho\_\{AB\}\; =\; \backslash sum\backslash limits\_\{i\}^\{\backslash tilde\{\backslash lambda\}\}\backslash lambda\_\{i\}\backslash tilde\{G\}^A\_i\backslash otimes\; \backslash tilde\{G\}^B\_i,$ where $\backslash tilde\{\backslash lambda\}=\backslash max\backslash left(dim(\backslash mathcal\{H\}\_A)^2,dim(\backslash mathcal\{H\}\_B)^2\backslash right)$ are Schmidt numbers, which can be connected to the [[separability]] question of a bipartite state. = Generalization to multipartite states = Since the interest in entanglement theory is also shifting to the multipartite case, i.e. to systems composed of $\backslash ;\; n>2$ subsystems, the question of a ''generalized'' Schmidt Decomposition arises naturally. '''Definition:''' For a pure state $\backslash ;\; |\backslash psi\backslash ranglele\_\{A\_1\; \backslash ldots\; A\_n\}$ belonging to a Hilbert space $\backslash ;\; \backslash mathcal\{H\}=\; \backslash mathcal\{H\}\_1\; \backslash otimes\; \backslash ldots\; \backslash otimes\; \backslash mathcal\{H\}\_n$ we can define the '''generalized Schmidt Decomposition''' :$\backslash ;\; |\backslash psi\backslash ranglele\_\{A\_1\; \backslash ldots\; A\_n\}\; =\; \backslash sum\_\{i=1\}^\{min\backslash \{d\_\{A\_1\},\backslash ldots,d\_\{A\_n\}\backslash \}\}\; a\_i\; |e\_\{A\_1\}\backslash ranglele\; \backslash otimes\; \backslash ldots\; \backslash ranglele\; |e\_\{A\_n\}\backslash ranglele\; .$ In the multipartite setting, pure states admit a generalized Schmidt Decomposition only if, tracing out any subsystem, the rest is in a [[Multipartite entanglement| fully separable]] state. [[Category:Handbook of Quantum Information]]