The separability problem, a long-standing open challenge in mathematics and quantum information theory. In this work, we propose and investigate a conjecture as a sub-question of the separability problem. Specifically, we investigate whether the form is separable for any eight-dimensional unit vector . This is actually the so-called absolute separability problem. For studying the conjecture, we classify the vectors in a finitely-dimensional bipartite Hilbert space, and construct their equivalence classes in terms of the so-called Schmidt rank, under both product general linear and product linear groups. Then we construct seven equivalence classes of , using which we partially prove the conjecture. Our results highlight the the complexity of the difference between bipartite and tripartite spaces.
The separability problem, a long-standing open challenge in mathematics and quantum information theory. In this work, we propose and investigate a conjecture as a sub-question of the separability problem. Specifically, we investigate whether the form is separable for any eight-dimensional unit vector . This is actually the so-called absolute separability problem. For studying the conjecture, we classify the vectors in a finitely-dimensional bipartite Hilbert space, and construct their equivalence classes in terms of the so-called Schmidt rank, under both product general linear and product linear groups. Then we construct seven equivalence classes of , using which we partially prove the conjecture. Our results highlight the the complexity of the difference between bipartite and tripartite spaces.
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC (PPT) matrix can be decomposed into a sum of s+1 SPC (PPT) matrices of which the first s are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC (PPT) matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC (PPT) matrices to the case of weakly irreducible SPC (PPT) matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).
It is well know that the problem of the description of separable states of finite dimensional quantum system is central for the quantum information theory. The aim of the present paper is to point out that the conjecture of An Min Wang [1] concerning the separability of pure states is valid for pure states of bipartite quantum systems. A general condition for separability of a quantum state could be in principle obtained from the measure of entanglement. For a bipartite pure state the natural measure of entanglement is given by the von Neumann entropy of the reduced density matrices. Then, a bipartite state is separable if and only if this measure of entanglement is equal to zero, i.e., if and only if the reduced states are pure states. The purity of any state, described by a density matrix, can be verified directly using the Bloch vector, associated with that state. One of the restrictions is on the value of the squared Euclidean norm of the Bloch vector. Exactly on this restriction is based the An Min Wang criterion for the separability of a bipartite quantum state. The other restrictions concern the symmetric product of the Bloch vector with itself. We shall prove that the equations of the second set of restrictions follows from the An Min Wang restrictions and from the equations fulfilled by the Fano parameters of any pure bipartite state. In order to prove the An Min Wang conjecture we shall use three different parametrizations for the pure states of bipartite quantum systems: the generalized Bloch vector parametrization [2-10]; the Fano parametrization [3-10]; the Schmidt parametrization [11,12], and the relations between them.
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We deal mainly with the bipartite finite dimensional case and with two types of matrices, one of them being the PPT matrices. We proved that many results holds for both types. If these matrices have specific Hermitian Schmidt decompositions then the matrices are separable in a very strong sense. We proved that both types have what we call split decompositions. We defined the notion of weak irreducible matrix, based on the concept of irreducible state defined recently. These split decomposition theorems together with the notion of weak irreducible matrix, imply that these matrices are weak irreducible or a sum of weak irreducible matrices of the same type. The separability problem for these types of matrices can be reduced to the set of weak irreducible matrices of the same type. We also provided a complete description of weak irreducible matrices of both types. Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable, we found sharp inequalites providing separability for both types.
We study separability properties in a five-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states that can be written as linear combinations of permutation operators, or equivalently, commute with unitaries of the form $U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}U.$ We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.
This thesis explores the applications of matrix spaces in quantum information and computational complexity, specifically in the areas of quantum state/channel discrimination, entanglement transformation and isomorphism testing.We achieve the following contributions:• We derive a necessary condition which determines whether a set of orthogonal bipartite states can be distinguished by positive-partial-transpose (PPT) operations, in the many-copy scenario.We reduce the discrimination task to the following problem:Decide whether there exist a nonzero bipartite matrix with positive partial-transpose, such that all its eigenvectors with nonzero eigenvalues is orthogonal to a given bipartite vector space.As applications, we reprove and extend a variety of existing results, including the local indistinguishability of the bipartite maximally entangled state and its orthogonal complement [Yu, Duan and Ying, IEEE transaction on Information Theory, 2014] and the maximum dimension of non-positive-partialtranspose subspaces [Johnston, Physical Review A, 2013].• We show that determining the parallel distinguishability of quantum channels is equivalent to determining whether the orthogonal complement of a given matrix space contains nonzero positive semidefinite matrices.Our characterization immediately implies a necessary condition to decide the parallel distinguishability.We further prove that our condition is also sufficient for two large families of quantum channels, which leads to an alternating proof for the parallel distinguishability of vi Abstract unitary operations [Acín, Physical Review Letters, 2001].We also present an illustrative example showing our necessary condition cannot determine the parallel distinguishability.• We systematically study the tripartite-to-bipartite entanglement transformations by stochastic local operations and classical communication (SLOCC).Such a problem is equivalent to computing the maximal rank of a matrix space [Chitambar, Duan and Shi, Physical Review A, 2010].We analyze the SLOCC convertibility in both the finite-copy and asymptotic regimes.In particular, we derive explicit formulas which compute asymptotic entanglement transformation rates for two families of tripartite states by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants.Significantly, we show that the problem of deciding the asymptotic SLOCC convertibility of tripartite pure states to the bipartite maximally entangled states and the non-commutative symbolic determinant identity testing problem is algorithmically equivalent, which builds a new connection between problems in algebraic complexity and problems in asymptotic SLOCC entanglement transformations.• We devise algorithms which test isometry between alternating matrix spaces over finite field.Algorithms for such a problem in time polynomial in the underlying vector space size resolves the believed bottleneck case of the group isomorphism problem, i.e. testing isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order.Our approach is to view it as a linear algebraic analog of the graph isomorphism problem.We devise an average-case efficient algorithm for the alternating matrix space isometry problem over a key range of parameters.in a random model of alternating matrix spaces in vein of the Erdős-Rényi model of random graphs.To devise our algorithm, we developed linear algebraic individualization and refinement techniques, which are crucial in the first average-case efficient algorithm for graph isomorphism, devised by Babai, Erdős, and Selkow in the 1970s [Babai, Erdős and Selkow, SIAM Journal on Computing, 1980].We also adapt Luks' dynamic programming technique for graph isomorphism [Luks, STOC, 1999] to slightly improve the worst-case time complexity of the alternating matrix space isometry problem.Very little of the work in this thesis would have been possible without the support of my supervisors, collaborators, colleagues, friends and family.First and foremost, I would like to thank my supervisor, Runyao Duan.Without his guidance and insight, I would not have complete my PhD study.I am greatly indebted to him for leading me into the quantum world.Dating back to 2013 in Wuhan University, when I was still a third-year undergraduate student, I was so inspired by his talk about quantum computation.He encouraged me to investigate the parallel distinguishability of quantum channels, which becomes the departure point of my research career.Furthermore, I am grateful for Runyao's spiritual support for my life.He is not only my academic supervisor but also a spiritual mentor, sharing his experience and scope in both research and life.Thanks to my co-supervisor, Youming Qiao, my research interests have been developed and extended.I would like to thank him for his inspirational discussions and conversations from which I really benefit a lot.I have learned about many interesting topics from him, especially for introducing me to the beauty of computational complexity theory.
In matrix theory, a well established relation (AB)T=BTAT holds for any two matrices A and B for which the product AB is defined. Here T denote the usual transposition. In this work, we explore the possibility of deriving the matrix equality (AB)Γ=AΓBΓ for any 4×4 matrices A and B, where Γ denote the partial transposition. We found that, in general, (AB)Γ=AΓBΓ holds for 4×4 matrices A and B but there exist particular set of 4×4 matrices for which (AB)Γ=AΓBΓ holds. We have exploited this matrix equality to investigate the separability problem. Since it is possible to decompose the density matrices ρ into two positive semi-definite matrices A and B so we are able to derive the separability condition for ρ when ρΓ=(AB)Γ=AΓBΓ holds. Due to the non-uniqueness property of the decomposition of the density matrix into two positive semi-definte matrices A and B, there is a possibility to generalise the matrix equality for density matrices lives in higher dimension. These results may help in studying the separability problem for higher dimensional and multipartite system.