The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory

We derive several results in classical Euclidean elementary geometry using the steering ellipsoid formalism from quantum mechanics. This gives a physically motivated derivation of very non-trivial geometric results, some of which are entirely new. We consider a sphere of radius $r$ contained inside another sphere of radius $R$, with the sphere centres separated by distance $d$. When does there exist a nested tetrahedron circumscribed about the smaller sphere and inscribed in the larger? We derive the Grace-Danielsson inequality $d^2 \leq (R+r)(R-3r)$ as the sole necessary and sufficient condition for the existence of a nested tetrahedron. Our method also gives the condition $d^2 \leq R(R-2r)$ for the existence of a nested triangle in the analogous 2-dimensional scenario. These results imply the Euler inequality in 2 and 3 dimensions. Furthermore, we formulate a new inequality that applies to the more general case of ellipses and ellipsoids.Comment: 8 pages, 1 figure. Published versio

Quantum steering ellipsoids

Graphical representations are invaluable for visualising physical systems and processes. In quantum information theory, the Bloch vector representation of a single qubit is ubiquitous, but visualising higher-dimensional quantum systems is far less straightforward. The quantum steering ellipsoid provides a method for geometrically representing the state of two qubits, the most fundamental system for studying quantum correlations. This thesis constitutes a significant development of the steering ellipsoid formalism. As well as offering new insight into the study of two-qubit states, we extend this powerful geometric approach to explore scenarios beyond two qubits. We find necessary and sufficient conditions for when an ellipsoid inside the Bloch ball describes a valid (i.e. positive semidefinite) two-qubit state. Combined with the notion of ellipsoid chirality, this enables a geometric characterisation of entanglement. We find a family of "maximally obese" two-qubit states whose ellipsoids have maximal volume. These states have optimal correlation properties within the set of all two-qubit states with a single maximally mixed marginal. We study a three-qubit scenario and discover that ellipsoid volume obeys an elegant monogamy of steering relationship. From this we can derive the Coffman-Kundu-Wootters (CKW) inequality for concurrence monogamy, providing an intuitive geometric derivation of this classic result. Remarkably, we find that steering ellipsoids offer a fresh perspective on questions beyond quantum state space. Entanglement witnesses are also very naturally represented and classified using the formalism. This gives a physical interpretation to any ellipsoid inside the Bloch ball as a block positive two-qubit operator, which we may then classify further. We can also use steering ellipsoids to derive some highly nontrivial results in classical Euclidean geometry, extending Euler's inequality for the circumradius and inradius of a triangle.Open Acces

The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory

Abstract. We derive several results in classical Euclidean elementary geometry using the steering ellipsoid formalism from quantum mechanics. This gives a physically motivated derivation of very non-trivial geometric results, some of which are entirely new. We consider a sphere of radius r contained inside another sphere of radius R, with the sphere centres separated by distance d. When does there exist a 'nested' tetrahedron circumscribed about the smaller sphere and inscribed in the larger? We derive the Grace-Danielsson inequality d 2 ≤ (R + r)(R − 3r) as the sole necessary and sufficient condition for the existence of a nested tetrahedron. Our method also gives the condition d 2 ≤ R(R − 2r) for the existence of a nested triangle in the analogous two-dimensional scenario. These results imply the Euler inequality in two and three dimensions. Furthermore, we formulate a new inequality that applies to the more general case of ellipses and ellipsoids. Mathematics Subject Classification. 51M04 · 51M16 · 51P05 · 81P40

Quantum steering ellipsoids, extremal physical states and monogamy

A Corrigendum for this article has been published in 2015 New J. Phys. 17 019501Any two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman-Kundu- Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry. Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Eulers inequality for the circumradius and inradius of a triangle.The EPSRC and the ARC Centre of Excellence grant no. CE110001027. DJ is funded by the Royal Society. TR would like to thank the Leverhulme Trust. SJ acknowledges EPSRC grant EP/ K022512/1

Proposal for demonstration of long-range cluster state entanglement in the presence of photon loss

Photonic cluster states are a crucial resource for optical quantum computing. Recently a quantum dot single photon source has been demonstrated to produce strings of photons in a linear cluster state, but high photon loss rates make it impossible to characterize the entanglement generated by conventional methods. We present a benchmarking method for such sources that can be used to demonstrate useful long-range entanglement with currently available collection/detection efficiencies below 1%. Measurement of the polarization state of single photons in different bases can provide an estimate for the three-qubit correlation function hZXZi. This value constrains correlations spanning more than three qubits, which in turn provide a lower bound for the localizable entanglement between any two qubits in the large state produced by the source. Finite localizable entanglement can be established by demonstrating hZXZi > 2/3 . This result enables photonic experiments demonstrating computationally useful entanglement with currently available technology

Physical properties of blood are altered in young and lean women with polycystic ovary syndrome

Classic features of polycystic ovary syndrome (PCOS) include derangement of metabolic and cardiovascular health, and vascular dysfunction is commonly reported. These comorbidities indicate impaired blood flow; however, other than limited reports of increased plasma viscosity, surprisingly little is known regarding the physical properties of blood in PCOS. We aimed to investigate whether haemorheology was impaired in women with PCOS. We thus measured a comprehensive haemorheological profile, in a case-control design, of lean women with PCOS and age-matched healthy controls. A clinical examination determined similar cardiovascular risk for the two groups. Whole blood and plasma viscosity was measured using a cone-plate viscometer. The magnitude and rate of red blood cell (RBC) aggregation was determined using a light-transmission aggregometer, and the degree of RBC deformability was measured via laser-diffraction ektacytometry. Plasma viscosity was significantly increased in women with PCOS. Blood viscosity was also increased for PCOS at lower-to-moderate shear rates in both native and standardised haematocrit samples. The magnitude of RBC aggregation–a primary determinant of low-shear blood viscosity–was significantly increased in PCOS at native and 0.4 L·L(-1) haematocrit. No difference was detected between PCOS and CON groups for RBC deformability measurements. A novel measure indicating the effectiveness of oxygen transport by RBC (i.e., the haematocrit-to-viscosity ratio; HVR) was decreased at all shear rates in women with PCOS. In a group of young and lean women with PCOS with an unremarkable cardiovascular risk profile based on clinical data, significant haemorheological impairment was observed. The degree of haemorheological derangement observed in the present study reflects that of overt chronic disease, and provides an avenue for future therapeutic intervention in PCOS

A Comparison between Four Analytical Methods for the Measurement of Fe(II) at Nanomolar Concentrations in Coastal Seawater

Dissolved Fe(II) in seawater is deemed an important micronutrient for microbial organisms, but its analysis is challenging due to its transient nature. We conducted a series of Fe(II) method comparison experiments, where spikes of 5 to 31 nM Fe(II) were added to manipulated seawaters with varying dissolved oxygen (37 to 156 μM) concentrations. The observed Fe(II) concentrations from four analytical methods were compared: spectrophotometry with ferrozine, stripping voltammetry, and flow injection analysis using luminol (with, and without, a pre-concentration column). Direct comparisons between the different methods were undertaken from the derived apparent Fe(II) oxidation rate constant (k1). Whilst the two luminol based methods produced the most similar concentrations throughout the experiments, k1 was still subject to a 20–30% discrepancy between them. Contributing factors may have included uncertainty in the calibration curves, and different responses to interferences from Co(II) and humic/fulvic organic material. The difference in measured Fe(II) concentrations between the luminol and ferrozine methods, from 10 min–2 h after the Fe(II) spikes were added, was always relatively large in absolute terms (>4 nM) and relative to the spike added (>20% of the initial Fe(II) concentration). k1 derived from ferrozine observed Fe(II) concentrations was 3–80%, and 4–16%, of that derived from luminol observed Fe(II) with, and without, pre-concentration respectively. The poorest comparability of k1 was found after humic/fulvic material was added to raise dissolved organic carbon to 120 μM. A luminol method without pre-concentration then observed Fe(II) to fall below the detection limit (<0.49 nM) within 10 min of a 17 nM Fe(II) spike addition, yet other methods still observed Fe(II) concentrations of 2.7 to 3.7 nM 30 min later. k1 also diverged accordingly with the ferrozine derived value 4% of that derived from luminol without pre-concentration. These apparent inconsistencies suggest that some inter-dataset differences in measured Fe(II) oxidation rates in natural waters may be attributable to differences in the analytical methods used rather than arising solely from substantial shifts in Fe(II) speciation

Pathways and timescales of Southern Ocean hydrothermal iron and manganese transport

Scarcity of iron and manganese limits the efficiency of the biological carbon pump over large areas of the Southern Ocean. The importance of hydrothermal vents as a source of these micronutrients to the euphotic zone of the Southern Ocean is debated. Here we present full depth profiles of dissolved and total dissolvable trace metals in the remote eastern Pacific sector of the Southern Ocean (55–60° S, 89.1° W), providing evidence of enrichment of iron and manganese at depths of 2000–4000 m. These enhanced micronutrient concentrations were co-located with 3He enrichment, an indicator of hydrothermal fluid originating from ocean ridges. Modelled water trajectories revealed the understudied South East Pacific Rise and the Pacific Antarctic Ridge as likely source regions. Additionally, the trajectories demonstrate pathways for these Southern Ocean hydrothermal ridge-derived trace metals to reach the Southern Ocean surface mixed layer within two decades, potentially supporting a regular supply of micronutrients to fuel Southern Ocean primary production

Sublithospheric diamond ages and the supercontinent cycle.

Subduction related to the ancient supercontinent cycle is poorly constrained by mantle samples. Sublithospheric diamond crystallization records the release of melts from subducting oceanic lithosphere at 300-700 km depths1,2 and is especially suited to tracking the timing and effects of deep mantle processes on supercontinents. Here we show that four isotope systems (Rb-Sr, Sm-Nd, U-Pb and Re-Os) applied to Fe-sulfide and CaSiO3 inclusions within 13 sublithospheric diamonds from Juína (Brazil) and Kankan (Guinea) give broadly overlapping crystallization ages from around 450 to 650 million years ago. The intracratonic location of the diamond deposits on Gondwana and the ages, initial isotopic ratios, and trace element content of the inclusions indicate formation from a peri-Gondwanan subduction system. Preservation of these Neoproterozoic-Palaeozoic sublithospheric diamonds beneath Gondwana until its Cretaceous breakup, coupled with majorite geobarometry3,4, suggests that they accreted to and were retained in the lithospheric keel for more than 300 Myr during supercontinent migration. We propose that this process of lithosphere growth-with diamonds attached to the supercontinent keel by the diapiric uprise of depleted buoyant material and pieces of slab crust-could have enhanced supercontinent stability