# Joint Schmidt-type decomposition for two bipartite pure quantum states

## TL;DR

This paper introduces two novel joint Schmidt-type decompositions for bipartite pure states, extending the classical Schmidt decomposition to rank-1 operators with non-orthonormal bases and complex coefficients.

## Contribution

It presents new methods for decomposing bipartite pure states, generalizing the Schmidt decomposition to rank-1 operators with non-orthogonal bases and complex coefficients.

## Key findings

- Derived purity-type formulas for one-party reductions.
- Established relations between Schmidt decomposition and Bloch representation.
- Extended the applicability of Schmidt decomposition to rank-1 operators.

## Abstract

It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulae for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.