Existence of the Schmidt decomposition for tripartite systems

Arun Kumar Pati

For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain condition. If the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis, then the Schmidt decomposition for a tripartite system exists. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis.

Asher Peres

Spandan Das, Goutam Paul

Pati (Physics Letters A, 2000) derived a sufficient condition for the existence of Schmidt decomposition in tripartite Hilbert spaces. In this paper, we show that the condition is erroneous by demonstrating some counter-examples. Moreover, we suitably modify the condition and provide a correctness proof. We also show for the first time how this can be generalized to $n$-partite Hilbert spaces for any $n\geq 2$. Finally, we prove that this condition is also a necessary condition.

박정수

Yat Wong, Changchun Zhong, Liang Jiang

Marcio F. Cornelio, A. F. R. de Toledo Piza

We present a method to find the decompositions of tripartite entangled pure states with one qubit which are smaller than two successive Schmidt decompositions. We use this method to get a classification of the entangled states according to their decompositions. Furthermore, we also use this method to classify the entangled states that can be interconverted through stochastic local operations and classical communication. More general tripartite systems are briefly discussed.