*Bipartite states* are one of the basic objects in Quantum Information Theory and will be defined in what follows:

## Pure States

### Definition

Let H = H_{A} ⊗ H_{B} be a Hilbert space defined as a tensor product of two Hilbert spaces H_{A} and H_{B}. We call some pure state ∣*ψ*⟩_{AB} on the composite system *A* ∪ *B* **bipartite**, if it is written with respect to the partition *A**B*, 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 H_{A} and H_{B} respectively.

### Schmidt Theorem (Schmidt Decomposition)

There is a statement in linear algebra, according to which for every ∣*ψ*⟩_{AB} there exist bases ∣*u*_{i}⟩_{A} and ∣*v*_{j}⟩_{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(*d**i**m*(H_{A}), *d**i**m*(H_{B})) 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 *ρ*_{AB} be a mixed state on a composite system *A* ∪ *B*. Then we say that *ρ*_{AB} is a **bipartite mixed state** on H_{A} ⊗ H_{B} 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(*d**i**m*(H_{A})^{2}, *d**i**m*(H_{B})^{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*⟩*l**e*_{A1…An} belonging to a Hilbert space H = H_{1} ⊗ … ⊗ H_{n} we can define the **generalized Schmidt Decomposition**

∣*ψ*⟩*l**e*_{A1…An} = ∑_{i = 1}^{min{dA1, …, dAn}}*a*_{i}∣*e*_{A1}⟩*l**e* ⊗ …⟩*l**e*∣*e*_{An}⟩*l**e*.

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.