Fully characterizing the steerability of a quantum state of a bipartite system has remained an open problem ever since the concept of steerability was first defined. In this paper, using our recent geometrical approach to steerability, we suggest a necessary and sufficient condition for a two-qubit state to be steerable with respect to projective measurements. To this end, we define the critical radius of local models and show that a state of two qubits is steerable with respect to projective measurements from Alice's side if and only if her critical radius of local models is less than 1. As an example, we calculate the critical radius of local models for the so-called T-states by proving the optimality of a recently-suggested ansatz for Alice's local hidden state model.

Abstract. Given a squarefree monomial ideal I ⊆ R = k[x 1 , . . . , x n ], we show that α(I), the Waldschmidt constant of I, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of I. By applying results from fractional graph theory, we can then express α(I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing α(I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on α(I), thus verifying a conjecture of Cooper-Embree-Hà-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of P n with few components compared to n, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.

of two-qubit states from the geometry of steering outcomes" (2016 When two qubits, A and B, are in an appropriate state, Alice can remotely steer Bob's system B into different ensembles by making different measurements on A. This famous phenomenon is known as quantum steering, or Einstein-Podolsky-Rosen steering. Importantly, quantum steering establishes the correspondence not only between a measurement on A (made by Alice) and an ensemble of B (owned by Bob) but also between each of Alice's measurement outcomes and an unnormalized conditional state of Bob's system. The unnormalized conditional states of B corresponding to all possible measurement outcomes of Alice are called Alice's steering outcomes. We show that, surprisingly, the four-dimensional geometry of Alice's steering outcomes completely determines both the nonseparability of the two-qubit state and its steerability from her side. Consequently, the problem of classifying two-qubit states into nonseparable and steerable classes is equivalent to geometrically classifying certain four-dimensional skewed double cones.

Abstract. Let K be an arbitrary field. Let a = (a 1 < · · · < an) be a sequence of positive integers. Let C(a) be the affine monomial curve in A n parametrized by t → (t a 1 , ..., t an ). Let I(a) be the defining ideal of C(a) in K[x 1 , ..., xn]. For each positive integer j, let a + j be the sequence (a 1 + j, ..., an + j). In this paper, we prove the conjecture of Herzog and Srinivasan saying that the betti numbers of I(a + j) are eventually periodic in j with period an − a 1 . When j is large enough, we describe the betti table for the closure of C(a + j) in P n .

We address the problem of quantum nonlocality with positive operator valued measures (POVM) in the context of Einstein-Podolsky-Rosen quantum steering. We show that, given a candidate for local hidden state (LHS) ensemble, the problem of determining the steerability of a bipartite quantum state of finite dimension with POVMs can be formulated as a nesting problem of two convex objects. One consequence of this is the strengthening of the theorem that justifies choosing the LHS ensemble based on symmetry of the bipartite state. As a more practical application, we study the classic problem of the steerability of two-qubit Werner states with POVMs. We show strong numerical evidence that these states are unsteerable with POVMs up to a mixing probability of 1 2 within an accuracy of 10 −3 .