Partial traces and the geometry of entanglement: Sufficient conditions for the separability of Gaussian states

The notion of partial trace of a density operator is essential for the understanding of the entanglement and separability properties of quantum states. In this paper, we investigate these notions putting an emphasis on the geometrical properties of the covariance ellipsoids of the reduced states. We thereafter focus on Gaussian states and we give new and easily numerically implementable sufficient conditions for the separability of all Gaussian states. Unlike the positive partial transposition criterion, none of these conditions is however necessary.

Detection Power of Separability Criteria Based on a Correlation Tensor: A Case Study

Detection power of separability criteria based on a correlation tensor is tested within a family of generalized isotropic states in [Formula: see text]. For [Formula: see text] all these criteria are weaker than the positive partial transposition (PPT) criterion. Interestingly, our analysis supports the recent conjecture that a criterion based on symmetrically informationally complete positive operator-valued measure (SIC-POVMs) is stronger than realignment criterion.

Case Report: Surgical Treatment of Severe Facial Wounds and Proptosis in a Dog Due to a Traffic Accident

Although facial wounds caused by traffic accidents in dogs are common, the surgical management of severe facial injuries involving the soft tissue, bone, dentition, nose and orbit are challenging. A 2 year-old Korean Jindo dog was diagnosed with severe skin defects of the face and proptosis caused by a vehicular accident. Along the left lateral maxilla, severe injury involving the overlying skin and platysma muscle occurred, to the extent that the middle part of the sphincter colli profundus pars intermedia muscle was exposed. Repair surgeries of the skin defects and globe displacement were performed using a local subdermal plexus rotation flap and a partial transposition of the dorsal rectus muscle combined with small intestinal submucosa (SIS) instead of enucleation as the first attempt. SIS was used to sustain the torn medial region. In this case, the surgery resulted in good cosmetic and functional outcome in the dog, despite the atypical complexities upon presentation.

Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable.

Separability of diagonal symmetric states: a quadratic conic optimization problem

We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS inCd⊗Cd(symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states ford≤4. Furthermore, ford≥5, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus onN-partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.

Nongeneric positive partial transpose states of rank five in 3×3 dimensions

In [Formula: see text] dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. These rank four states are completely understood. We say that they have rank [Formula: see text] since both a state [Formula: see text] and its partial transpose [Formula: see text] have rank four. The next problem is to understand the extremal PPT states of rank [Formula: see text]. We call two states [Formula: see text]-equivalent if they are related by a product transformation. A generic rank [Formula: see text] PPT state [Formula: see text] is extremal, and both [Formula: see text] and [Formula: see text] have six product vectors in their ranges, and no product vectors in their kernels. The three numbers [Formula: see text] are [Formula: see text]-invariants that help us classify the state. There is no analytical understanding of such states. We have studied numerically a few types of nongeneric rank five PPT states, in particular, states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to [Formula: see text]-equivalence, where they appear as boundary states on one single five-dimensional face on the set of normalized PPT states. The interior of the face consists of rank [Formula: see text] states with four common product vectors in their kernels, it is a simplex of separable states surrounded by entangled PPT states. We say that a state [Formula: see text] is [Formula: see text]-symmetric if [Formula: see text] and [Formula: see text] are [Formula: see text]-equivalent, and is genuinely [Formula: see text]-symmetric if it is [Formula: see text]-equivalent to a state [Formula: see text] with [Formula: see text]. Genuine [Formula: see text]-symmetry implies a special form of [Formula: see text]-symmetry. We have produced numerically, by a special method, a random sample of rank [Formula: see text] [Formula: see text]-symmetric states. About 50 of these are of type [Formula: see text], among those all are extremal and about half are genuinely [Formula: see text]-symmetric. All these genuinely [Formula: see text]-symmetric states can be transformed to have a circulant form. We find however that this is not a generic property of genuinely [Formula: see text]-symmetric states. The remaining [Formula: see text]-symmetric states found in the search have product vectors in their kernels, and they inspired us to study such states without regard to [Formula: see text]-symmetry.