It is pointed out that separability problem for arbitrary multi-partite states can be fully solved by a finite size, elementary recursive algorithm. In the worse case scenario, the underlying numerical procedure, may grow doubly exponentially with the state's rank. Nevertheless, we argue that for generic states, analysis of concurrence matrices essentially reduces the task of solving separability problem in
m×n dimensions to solving a set of linear equations in about
(Dmn+D−1) variables, where
D decreases with
mn and for large
mn it should not exceed 4. Moreover, the same method is also applicable to multipartite states where it is at least equally efficient.
We show how the separability problem is dual to that of decomposing any given matrix into a conic combination of rank-one partial isometries, thus offering a duality approach different to the positive maps characterization problem. Several inmediate consequences are analyzed: (i) a sufficient criterion for separability for bipartite quantum systems, (ii) a complete solution to the separability problem for pure states also of bipartite systems independent of the classical Schmidt decomposition method and (iii) a natural generalization of these results to multipartite systems.
Abstract We study separability in arbitrary multipartite quantum systems based on principal base matrices. Necessary conditions are presented for different kinds of separable states. These conditions can give a complete classification of multipartite quantum states. While the usual Bloch representation of a density matrix uses three types of generators, the representation with principal base matrices has one uniform type of generator which simplifies computation. In this paper, we take advantage of this simplicity to derive useful and operational criteria to detect multipartite separability. We first obtain criteria on detecting 1 − 3 separable, 2 − 2 separable, 1 − 1 − 2 separable and fully separable four-partite quantum states. We then study k-separability for multipartite quantum states in arbitrary dimensions. Detailed examples are given to show that our criteria are able to detect more entanglement states than some existing criteria.
We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary
U s.t
UAlU∗=Bl,
l∈{1,...,p}, where
Al and
Bl are
nxn complex matrices. We observe that the problem is simplest when
U is diagonal, where we see that the `paths' in the graph defined by non-zero elements of
Al and
Bl determine the solution. Inspired by this we generalize this to the case when
U is block-diagonal to identify a form refered to as the `Solution-form' using `paths' determined by non-zero sub-matrices of
Al,Bl which are non-zero multiples of Unitary. When not in Solution form we find an equivalent problem to solve by diagonalizing a Hermitian or a Normal matrix related to the sub-matrices. The problem is solved in a maximum of
n steps. The same idea can be extended to solve the Simultaneous Unitary Equivalence (S
.U
.Eq) problem where we solve for
U,V in
UAlV∗=Bl,
Al,Bl being
mxn Complex rectangular matrices. Here we work with the 'paths' in the related bi-graph to define the Solution-form. The algorithms have a complexity of
O(pn4). This work finds application in Quantum Evolution, Quantum gate design and Simulation. The salient features of each step of the algorithm can be retained as Canonical features to classify a given collection of complex matrices up to Unitary Similarity.