# 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 H = HA ⊗ HB be a Hilbert space defined as a tensor product of two Hilbert spaces HA and HB. We call some pure stateψAB 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 ∣iA and ∣jB are bases in HA and HB respectively.

### Schmidt Theorem (Schmidt Decomposition)

There is a statement in linear algebra, according to which for every ∣ψAB there exist bases ∣uiA and ∣vjB such that$|\psi\rangle_{AB} = \sum\limits_{i=1}^{n}\tilde{\chi_{i}}|u_i\rangle_A\otimes |v_i\rangle_B$,where n = min(dim(HA), dim(HB)) and $\; \sum_{i=1}^{n}\tilde{\chi_{i}}^2 = 1$.

The Schmidt coefficients $\; \tilde{\chi_{i}}$ 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:

1. The state  ∣ψAB is separable if and only if there is only one non-zero Schmidt coefficient $\; \tilde{\chi_{i}}=1$, $\; \tilde{\chi_{j}}=0 \quad \forall j \neq i$;
2. If more than one Schmidt coefficients are non-zero, then the state is entangled;
3. 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  ∣ψleA1…An belonging to a Hilbert space  H = H1 ⊗ … ⊗ Hn we can define the generalized Schmidt Decomposition

∣ψleA1…An = ∑i = 1min{dA1, …, dAn}aieA1⟩le ⊗ …⟩leeAnle.

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.

Category:Handbook of Quantum Information