Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence

The generic real (b=1) and complex (b=2) two-qubit states are 9-dimensional and 15-dimensional in nature, respectively. The total volumes of the spaces they occupy with respect to the Hilbert-Schmidt and Bures metrics are obtainable as special cases of formulas of Zyczkowski and Sommers. We claim that if one could determine certain metric-independent 3-dimensional "eigenvalue-parameterized separability functions" (EPSFs), then these formulas could be readily modified so as to yield the Hilbert-Schmidt and Bures volumes occupied by only the separable two-qubit states (and hence associated separability probabilities). Motivated by analogous earlier analyses of "diagonal-entry-parameterized separability functions", we further explore the possibility that such 3-dimensional EPSFs might, in turn, be expressible as univariate functions of some special relevant variable--which we hypothesize to be the maximal concurrence (0 < C <1) over spectral orbits. Extensive numerical results we obtain are rather closely supportive of this hypothesis. Both the real and complex estimated EPSFs exhibit clearly pronounced jumps of magnitude roughly 50% at C=1/2, as well as a number of additional matching discontinuities.Comment: 12 pages, 7 figures, new abstract, revised for J. Phys.

Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems

We seek to derive the probability--expressed in terms of the Hilbert-Schmidt (Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres-Horodecki test on the partial transposes (PT's) of the associated 4 x 4 density matrices). But the full implementation of the test--requiring that the determinant of the PT be nonnegative for separability to hold--appears to be, at least presently, computationally intractable. So, we have previously implemented--using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)--the weaker implied test of nonnegativity of the six 2 x 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of 1024/{135 pi^2} =0.76854$. Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 x 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of 22/35 = 0.628571. Then, we conclude that a still further improved upper bound of 1129/2100 = 0.537619 can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 x 3 principal minors. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESF's that comes close to reproducing our numerical estimate of the true separability function.Comment: 16 pages, 9 figures, a few inadvertent misstatements made near the end are correcte

Zircon as a tracer of plumbing processes in an active magmatic system: insights from mingled magmas of the 2010 dome collapse, Montserrat, Lesser Antilles Arc, Caribbean

This project has received funding from the European Union's Hori-zon 2020 research and innovation programme under the Marie Skodowska-Curie grant agreement No. 749611 (JHS) . We also ac-knowledge funding from the Natural Environment Research Council Isotope Geoscience Facilities Steering Committee, grant IP-1746-1117 (JB) . The work has been financially supported by the Spanish grant CGL 2017-84469-P (JHS) . Funding for open access charge: Universidad de Granada/CBUA. Alexander Varychev at the University of Heidelberg, Germany, and Jeremy Rushton at the British Geological Survey, U.K., are thanked for help with analytical work. Jose Luis Macias is thanked for editorial handling. We are obliged to two anonymous reviewers for the time and effort they took to help us improve the clarity and rigour of our interpretations. 19Soufriere Hills Volcano, Montserrat, erupted from 1995 to 2010, with activity including dome growth, destructive pyroclastic density currents and Vulcanian explosions. Monitoring data, such as gas emissions, show the system is still in a state of unrest. The recent eruptions provide an opportunity to study, in real time, a complex subduction-related subvolcanic transaustal melt-mush reservoir, its magma fluxes, and the timing of crystal and melt storage prior to eruptive paroxysms. How and when mush destabilisation occurs prior to volcanic eruptions continues to be a question of intense debate. Evidence of mafic magma intrusion, a potential eruptive trigger, is preserved in enclaves with quenched and diffuse margins that are mingled with crystal-rich andesite. Here, in this first study of Soufriere Hills Volcano zircon, we report zircon ages and compositions for mafic-intermediate enclaves and host andesites from the most recent dome collapse in 2010 to place temporal constraints on magma reservoir processes. Zircon U-238-Th-230 ages disequilibrium crystallisation ages ranging between c. 2-250 ka constrain the longevity of the magmatic plumbing system. Uniform Hf isotopes, epsilon Hf 11.3 +/- 12 to 14.6 +/- 1.5, indicate invariant compositions that are typical for island arc magma sources. Zircon trace element concentrations and Ti-in-zircon crystallisation temperatures indicate crystallisation in isolated, small-volume, lenses with variable fractions of melt of heterogeneous compositions. We suggest amalgamation of assorted crystal cargoes from these lenses occurred prior to eruption during mush destabilisation triggered by mafic magma recharge. Zircon textures, on the other hand, shed light on recent centimetre-scale magma mingling immediately prior to eruption. Euhedral-subhedral zircon is preferentially preserved in or near quenched contacts of the least-evolved enclave and host andesite. By contrast, reheating of the andesite by the mafic magma recharge in the presence of zircon-undersaturated melts promoted zircon resorption. This led to the formation of subhedral-anhedral corroded zircon that is typical in the host andesite mush. Zircon thus reveals processes ranging from 100,000s of years of andesite storage to short-term partial destruction in response to transient heating and magma mixing events.European Commission 749611Natural Environment Research Council Isotope Geoscience Facilities Steering Committee IP-1746-1117Spanish Government CGL 2017-84469-PUniversidad de Granada/CBU

Separability and Fourier representations of density matrices

Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for $d$-dimensional spaces, and the resulting set of unitary matrices $S(d)$ is a basis for $d\times d$ matrices. If $N=d_{1}\times d_{2}\times...\times d_{b}$ and H^{[ N]}=\bigotimes H^{% [ d_{k}]}, we give a sufficient condition for separability of a density matrix $\rho$ relative to the $H^{[ d_{k}]}$ in terms of the $L_{1}$ norm of the spin coefficients of $\rho >.$ Since the spin representation depends on the form of the tensor product, the theory applies to both full and partial separability on a given space $H^{[ N]}$% . It follows from this result that for a prescribed form of separability, there is always a neighborhood of the normalized identity in which every density matrix is separable. We also show that for every prime $p$ and $n>1$ the generalized Werner density matrix $W^{[ p^{n}]}(s)$ is fully separable if and only if $s\leq (1+p^{n-1}) ^{-1}$

High-Temperature Expansions of Bures and Fisher Information Priors

For certain infinite and finite-dimensional thermal systems, we obtain --- incorporating quantum-theoretic considerations into Bayesian thermostatistical investigations of Lavenda --- high-temperature expansions of priors over inverse temperature beta induced by volume elements ("quantum Jeffreys' priors) of Bures metrics. Similarly to Lavenda's results based on volume elements (Jeffreys' priors) of (classical) Fisher information metrics, we find that in the limit beta -> 0, the quantum-theoretic priors either conform to Jeffreys' rule for variables over [0,infinity], by being proportional to 1/beta, or to the Bayes-Laplace principle of insufficient reason, by being constant. Whether a system adheres to one rule or to the other appears to depend upon its number of degrees of freedom.Comment: Six pages, LaTeX. The title has been shortened (and then further modified), at the suggestion of a colleague. Other minor change

Supergoop Dynamics

We initiate a systematic study of the dynamics of multi-particle systems with supersymmetric Van der Waals and electron-monopole type interactions. The static interaction allows a complex continuum of ground state configurations, while the Lorentz interaction tends to counteract this configurational fluidity by magnetic trapping, thus producing an exotic low temperature phase of matter aptly named supergoop. Such systems arise naturally in $\mathcal{N}=2$ gauge theories as monopole-dyon mixtures, and in string theory as collections of particles or black holes obtained by wrapping D-branes on internal space cycles. After discussing the general system and its relation to quiver quantum mechanics, we focus on the case of three particles. We give an exhaustive enumeration of the classical and quantum ground states of a probe in an arbitrary background with two fixed centers. We uncover a hidden conserved charge and show that the dynamics of the probe is classically integrable. In contrast, the dynamics of one heavy and two light particles moving on a line shows a nontrivial transition to chaos, which we exhibit by studying the Poincar\'e sections. Finally we explore the complex dynamics of a probe particle in a background with a large number of centers, observing hints of ergodicity breaking. We conclude by discussing possible implications in a holographic context.Comment: 35 pages,11 figures. v2: updated references to include a previous proof of classical integrability, exchanged a figure for a prettier versio

Holographic Vitrification

We establish the existence of stable and metastable stationary black hole bound states at finite temperature and chemical potentials in global and planar four-dimensional asymptotically anti-de Sitter space. We determine a number of features of their holographic duals and argue they represent structural glasses. We map out their thermodynamic landscape in the probe approximation, and show their relaxation dynamics exhibits logarithmic aging, with aging rates determined by the distribution of barriers.Comment: 100 pages, 25 figure