Bipartite states are one of the basic objects in Quantum Information Theory and will be defined in what follows:
Pure States
Definition
Let H = HA ⊗ HB be a Hilbert space defined as a tensor product of two Hilbert spaces HA and HB. We call some pure state ∣ψ⟩AB on the composite system A ∪ B bipartite, if it is written with respect to the partition AB, which means $|\psi\rangle_{AB} = \sum\limits_{ij}\chi_{ij}|i\rangle_A\otimes |j\rangle_B$, where ∣i⟩A and ∣j⟩B are bases in HA and HB respectively.
Schmidt Theorem (Schmidt Decomposition)
There is a statement in linear algebra, according to which for every ∣ψ⟩AB there exist bases ∣ui⟩A and ∣vj⟩B such that $|\psi\rangle_{AB} = \sum\limits_{i=1}^{n}\sqrt{\tilde{\chi_{i}}}|u_i\rangle_A\otimes |v_i\rangle_B$, where n = min(dim(HA), dim(HB)) and $\; \sum_{i=1}^{n}\tilde{\chi_{i}} = 1$.
The Schmidt coefficients $\; \sqrt{\tilde{\chi_{i}}}\ge 0$ are the square roots of the eigenvalues of the two partial traces of $\; \varrho_{AB}=|\psi\rangle_{AB}\langle \psi|$, $\; \varrho_A = Tr_B[\varrho_{AB}] =\sum\limits_{i=1}^{n}\tilde{\chi_{i}}|u_i\rangle_A \langle u_i|$ and $\; \varrho_B = Tr_A[\varrho_{AB}] =\sum\limits_{i=1}^{n}\tilde{\chi_{i}}|v_i\rangle_B \langle v_i|,$ 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 ∣ψ⟩AB is separable if and only if there is only one non-zero Schmidt coefficient $\; \tilde{\chi_{i}}=1$, $\; \tilde{\chi_{j}}=0 \quad \forall j \neq i$;
- 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 ρAB be a mixed state on a composite system A ∪ B. Then we say that ρAB is a bipartite mixed state on HA ⊗ HB and write $\rho_{AB} = \sum\limits_{ij}\lambda_{ij}G^A_i\otimes G^B_j .$
Schmidt Decomposition
The Decomposition can be also written for operators: $\rho_{AB} = \sum\limits_{i}^{\tilde{\lambda}}\lambda_{i}\tilde{G}^A_i\otimes \tilde{G}^B_i,$ where λ̃ = max(dim(HA)2, dim(HB)2) 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 $\; n>2$ subsystems, the question of a generalized Schmidt Decomposition arises naturally.
Definition: For a pure state ∣ψ⟩leA1…An belonging to a Hilbert space H = H1 ⊗ … ⊗ Hn we can define the generalized Schmidt Decomposition
∣ψ⟩leA1…An = ∑i = 1min{dA1, …, dAn}ai∣eA1⟩le ⊗ …⟩le∣eAn⟩le.
In the multipartite setting, pure states admit a generalized Schmidt Decomposition only if, tracing out any subsystem, the rest is in a fully separable state.