Bipartite states and Schmidt decomposition

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

Contents

Pure States

Definition

Let \mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B be a Hilbert space defined as a tensor product of two Hilbert spaces \mathcal{H}_A and \mathcal{H}_B. We call some pure state |\psi\rang_{AB} on the composite system A\cup B bipartite, if it is written with respect to the partition AB, which means
|\psi\rang_{AB} = \sum\limits_{ij}\chi_{ij}|i\rang_A\otimes |j\rang_B ,
where |i\rang_A and |j\rang_B are bases in \mathcal{H}_A and \mathcal{H}_B respectively.


Schmidt Theorem (Schmidt Decomposition)

There is a statement in linear algebra, according to which for every |\psi\rang_{AB} there exist bases |u_i\rang_A and |v_j\rang_B such that
|\psi\rang_{AB} = \sum\limits_{i=1}^{n}\tilde{\chi_{i}}|u_i\rang_A\otimes |v_i\rang_B ,
where n=\min\left(dim(\mathcal{H}_A),dim(\mathcal{H}_B)\right) 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\rang_{AB}\langle \psi|, \; \varrho_A = Tr_B[\varrho_{AB}] =\sum\limits_{i=1}^{n}\tilde{\chi_{i}}|u_i\rang_A \langle u_i| and \; \varrho_B = Tr_A[\varrho_{AB}] =\sum\limits_{i=1}^{n}\tilde{\chi_{i}}|v_i\rang_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 \; |\psi\rang_{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;
  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\cup B. Then we say that ρAB is a bipartite mixed state on \mathcal{H}_A\otimes \mathcal{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 \tilde{\lambda}=\max\left(dim(\mathcal{H}_A)^2,dim(\mathcal{H}_B)^2\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 \; n>2 subsystems, the question of a generalized Schmidt Decomposition arises naturally.

Definition: For a pure state \; |\psi\rangle_{A_1 \ldots A_n} belonging to a Hilbert space \; \mathcal{H}= \mathcal{H}_1 \otimes \ldots \otimes \mathcal{H}_n we can define the generalized Schmidt Decomposition

\; |\psi\rangle_{A_1 \ldots A_n} = \sum_{i=1}^{min\{d_{A_1},\ldots,d_{A_n}\}} a_i |e_{A_1}\rangle \otimes \ldots \rangle |e_{A_n}\rangle .

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.

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