Sensitivity of estimates of the volume of the set of separable states to the choice of "uniform" distributions

Uniform Entanglement Frames

We present several criteria for genuine multipartite entanglement from universal uncertainty relations based on majorization theory. Under non-negative Schur-concave functions, the vector-type uncertainty relation generates a family of infinitely many detectors to check genuine multipartite entanglement. We also introduce the concept of $k$-separable circles via geometric distance for probability vectors, which include at most $(k-1)$-separable states. The entanglement witness is also generalized to a universal entanglement witness which is able to detect the $k$-separable states more accurate…

Characterizing the boundary of the set of absolutely separable states and their generation via noisy environments

We characterize the boundary of the convex compact set of absolutely separable states, referred as {\bf AS}, that cannot be transformed to entangled states by global unitary operators, in $2\otimes d$ Hilbert space. However, we show that the absolutely separable states of rank-$(2d-1)$ are extreme points of such sets. We then discuss conditions to examine if a given full-rank absolutely separable state is an interior point or a boundary point of {\bf AS}. Moreover, we construct two-qubit absolutely separable states which are boundary points but not extreme points of {\bf AS} and prove the exis…

On the structure of the body of states with positive partial transpose

We show that the convex set of separable mixed states of the 2 x 2 system is a body of constant height. This fact is used to prove that the probability to find a random state to be separable equals 2 times the probability to find a random boundary state to be separable, provided the random states are generated uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.

Geometric Multiaxial Representation of N-qubit Mixed Symmetric Separable States

Study of an N qubit mixed symmetric separable states is a long standing challenging problem as there exist no unique separability criterion. In this regard, we take up the N-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer elegant mathematical analysis since the dimension of the Hilbert space reduces from 2^N to N + 1. Since there exists a one to one correspondence between spin-j system and an N-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of spin density matrix. Further, we use g…

Factorization and entanglement in general XYZ spin arrays in non-uniform transverse fields

We determine the conditions for the existence of a pair of degenerate parity breaking separable eigenstates in general arrays of arbitrary spins connected through $XYZ$ couplings of arbitrary range and placed in a transverse field, not necessarily uniform. Sufficient conditions under which they are ground states are also provided. It is then shown that in finite chains, the associated definite parity states, which represent the actual ground state in the immediate vicinity of separability, can exhibit entanglement between any two spins regardless of the coupling range or separation, with the r…

可分集的半定扩展复杂度及其在近似解纠缠器中的应用

Semidefinite extension complexity of the separable set, with applications to approximate disentanglers

本文证明可分量子态集合的半定扩展复杂度具有定量下界，通过结合伪密度定理与块正算子，改进了HNW的拟多项式界，并给出Fawzi定理的定量版本，结果由Lean形式化验证。