Related papers: The positive partial transpose conjecture for n=3
We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein {\it et al.} [Phys. Rev. A \textbf{73}, 012320 (2006)]. The conjecture states…
The convex set of quantum states of a composite $K \times K$ system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an…
In order to compute the Schmidt decomposition of $A\in M_k\otimes M_m$, we must consider an associated self-adjoint map. Here, we show that if $A$ is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC)…
We show that pure states of multipartite quantum systems are multiseparable (i.e. give separable density matrices on tracing any party) if and only if they have a generalized Schmidt decomposition. Implications of this result for the…
The $p_3$-PPT criterion is an experimentally viable relaxation of the well-known positive partial transposition (PPT) criterion for the certification of quantum entanglement. Recently, it has been generalized to various families of…
Positivity of the density operator reflects itself in terms of sequences of inequalities on observable moments. Uncertainty relations for non-commuting observables form a subset of these inequalities. In addition, criterion of positivity…
We study bipartite entangled states in arbitrary dimensions and obtain different bounds for the entanglement measures in terms of teleportation fidelity. We find that there is a simple relation between negativity and teleportation fidelity…
It is quite common to use the generalized probabilistic theories (GPTs) as generic models to reconstruct quantum theory from a few basic principles and to gain a better understanding of the probabilistic or information theoretic foundations…
In this paper we report on new results relating to a conjecture regarding properties of $n\times n$, $n\leq 6$, positive definite matrices. The conjecture has been proven for $n\leq 4$ using computer-assisted sum of squares (SoS) methods…
We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > \epsilon$ for some…
Given a PPT state $A=\sum_{i=1}^nA_i\otimes B_i \in M_k\otimes M_k$ and a vector $v\in\Im(A)\subset\mathbb{C}^k\otimes\mathbb{C}^k$ with tensor rank $k$, we provide an algorithm that checks whether the positive map $G_A:M_k\rightarrow M_k$,…
Let $\alpha(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $\alpha(2n+1) = (2n+1) \alpha(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents…
Equivalence between Positive Partial Transpose (PPT) entanglement and bound entanglement is a long-standing open problem in quantum information theory. So far limited progress has been made, even on the seemingly simple case of Werner…
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…
We show that positivity of {\it every} partial transpose of $N$-partite quantum states implies new inequalities on Bell correlations which are stronger than standard Bell inequalities by a factor of $2^{(N-1)/2}$. A violation of the…
Recently, in [1], the author proved that many results that are true for PPT matrices also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called SPC in [1]…
We introduce and study bipartite quantum states that are invariant under the local action of the cyclic sign group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. Their important semi-definite…
We show that partial transposition of any $2\otimes n$ state can have at most $(n-1)$ number of negative eigenvalues. This extends a decade old result of $2\otimes 2$ case by Sanpera et al [Phys. Rev. A {\bf 58}, 826 (1998)]. We then apply…
We investigate the canonical forms of positive partial transposition (PPT) density matrices in ${\cal C}^2 \otimes {\cal C}^M \otimes {\cal C}^N$ composite quantum systems with rank $N$. A general expression for these PPT states are…
We study extremality in various sets of states that have positive partial transposes. One of the tools we use for this purpose is the recently formulated criterion allowing to judge if a given state is extremal in the set of PPT states.…