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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Non-Boolean quantum amplitude amplification and quantum mean estimation

This paper generalizes the quantum amplitude amplification and amplitude estimation algorithms to work with non-Boolean oracles. The action of a non-Boolean oracle $U_\varphi $ on an eigenstate $\mathinner {|{x}\rangle }$ is to apply a state-dependent phase-shift $\varphi (x)$. Unlike Boolean oracles, the eigenvalues $\exp (i\varphi (x))$ of a non-Boolean oracle are not restricted to be $\pm 1$. Two new oracular algorithms based on such non-Boolean oracles are introduced. The first is the non-Boolean amplitude amplification algorithm, which preferentially amplifies the amplitudes of the eigenstates based on the value of $\varphi (x)$. Starting from a given initial superposition state $\mathinner {|{\psi _0}\rangle }$, the basis states with lower values of $\cos (\varphi )$ are amplified at the expense of the basis states with higher values of $\cos (\varphi )$. The second algorithm is the quantum mean estimation algorithm, which uses quantum phase estimation to estimate the expectation $\mathinner {\langle {\psi _0|U_\varphi |\psi _0}\rangle }$, i.e., the expected value of $\exp (i\varphi (x))$ for a random x sampled by making a measurement on $\mathinner {|{\psi _0}\rangle }$. It is shown that the quantum mean estimation algorithm offers a quadratic speedup over the corresponding classical algorithm. Both algorithms are demonstrated using simulations for a toy example. Potential applications of the algorithms are briefly discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Environment-sensitivity functions for gross primary productivity in light use efficiency models

The sensitivity of photosynthesis to environmental changes is essential for understanding carbon cycle responses to global climate change and for the development of modeling approaches that explains its spatial and temporal variability. We collected a large variety of published sensitivity functions of gross primary productivity (GPP) to different forcing variables to assess the response of GPP to environmental factors. These include the responses of GPP to temperature; vapor pressure deficit, some of which include the response to atmospheric CO 2 concentrations; soil water availability (W); light intensity; and cloudiness. These functions were combined in a full factorial light use efficiency (LUE) model structure, leading to a collection of 5600 distinct LUE models. Each model was optimized against daily GPP and evapotranspiration fluxes from 196 FLUXNET sites and ranked across sites based on a bootstrap approach. The GPP sensitivity to each environmental factor, including CO 2 fertilization, was shown to be significant, and that none of the previously published model structures performed as well as the best model selected. From daily and weekly to monthly scales, the best model's median Nash-Sutcliffe model efficiency across sites was 0.73, 0.79 and 0.82, respectively, but poorer at annual scales (0.23), emphasizing the common limitation of current models in describing the interannual variability of GPP. Although the best global model did not match the local best model at each site, the selection was robust across ecosystem types. The contribution of light saturation and cloudiness to GPP was observed across all biomes (from 23% to 43%). Temperature and W dominates GPP and LUE but responses of GPP to temperature and W are lagged in cold and arid ecosystems, respectively. The findings of this study provide a foundation towards more robust LUE-based estimates of global GPP and may provide a benchmark for other empirical GPP products.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2017

Association of lead, cadmium and mercury with paraoxonase 1 activity and malondialdehyde in a general population in Southern Brazil

Metal exposure is associated with increased oxidative stress (OS), which is considered an underlying mechanism of metal-induced toxicity. Malondialdehyde (MDA) is a final product of lipid peroxidation, and it has been extensively used to evaluate metal-induced OS. Pro-oxidant effects produced by metals can be mitigated by paraoxonase 1 (PON1), an antioxidant enzyme known to prevent cardiovascular disease and atherosclerosis. Among other factors, the Q192R polymorphism and the exposure to heavy metals have been known to alter PON1 activity. Here, we evaluated the association of blood lead (Pb), cadmium (Cd) and mercury (Hg) levels with PON1 activity, and with MDA concentrations in a randomly selected sample of Brazilian adults aged 40 years or older, living in an urban area in Southern Brazil. A total of 889 subjects were evaluated for blood Pb and Cd levels, and 832 were tested for Hg. Geometric mean of blood Pb, Cd and Hg was 1.93 μg/dL, 0.06 μg/L and 1.40 μg/L, respectively. PON1 activity was significantly different among various genotypes: QQ (PON1=121.4 U/mL), QR (PON1=87.5 U/mL), and RR (PON1=55.2 U/mL), p<0.001. PON1 genotypes were associated only with Cd blood levels. Those with QR genotype had Cd concentrations higher (0.07 μg/L) than those with the RR genotype (0.04 μg/L) with p=0.034. However, PON1 activity was not significantly associated with metal concentrations. Cluster analysis showed that men who reported to be current smokers and drinkers with higher blood Pb and Cd levels, had significantly lower PON1 activity than non-smokers or -drinkers, and women with lower Pb and Cd levels. RR genotype carriers had lower PON1 activity than those with the QR genotype, and had higher levels of Pb and Cd compared with other genotype carriers. For blood Hg, no association with PON1 activity or genotype was noted. We found low levels of Pb, Cd and Hg in environmentally exposed Brazilian adults. Cd concentrations were increased in subjects with QR genotype. Those with RR genotype had lower PON1 activity and higher levels of Pb and Cd than other genotype carriers. The results of cluster analysis suggested that smoking status exerts a significant influence on PON1 activity. Other studies with environmentally exposed populations are required to further clarify whether low blood levels of metals influence OS biomarkers. - Highlights: • Low levels of lead, cadmium and mercury were found in a sample of Brazilian adults. • PON1 activity and MDA was not significantly associated with metals concentrations. • Blood cadmium was significantly different according with PON1 genotypes, and RR genotype carriers had lower PON1 activity and higher Pb and Cd levels. • Men, smokers, drinkers, with higher blood Pb and Cd levels had lower PON1 activity. • There was not a trend of an association with PON1 activity or genotype with blood mercury.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Spin Dynamics of Quintet and Triplet States Resulting from Singlet Fission in Oriented Terrylenediimide and Quaterrylenediimide Films

Singlet fission in organic semiconductors provides an important opportunity to study high-spin states in electronically coupled chromophores. Photoexcitation of oriented, crystalline films of N,N-bis(pentadec-8-yl)terrylene-3,4:11,12-bis(dicarboximide) (TDI) and N,N-bis (pentadec-8-yl)quaterrylene-3,4:13,14-bis(dicarboximide) (QDI) result in the formation of the correlated triplet pair quintet state, 5 (T 1 T 1 ) and its subsequent dissociation into two triplet (T-1) excitons. Time-resolved electron paramagnetic resonance (TREPR) spectroscopy and X-ray crystallography are used to show that the orientation dependence of 5 (T 1 T 1 ) spin dynamics is retained as it dissociates into two T 1 excitons, while such information is lost in a randomly oriented sample. In addition, the spin dynamics depend on how the molecules are oriented relative to the external applied magnetic field. Dissociation of 5 (T 1 T 1 ) to form two T 1 excitons is more efficient in TDI and leads to a longer T-1 lifetime than in QDI, making TDI a more viable candidate for photovoltaic applications. Since QDI readily forms a highly oriented film, and the lifetime of its 5 (T 1 T 1 ) state is longer than that of TDI, it may be a good candidate for quantum information science applications that require the generation of a quantum- entangled, four-spin state.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Automated Construction of Artificial Lattice Structures with Designer Electronic States

Manipulating matter with a scanning tunneling microscope (STM) enables the creation of atomically defined artificial structures that host designer quantum states. However, the time-consuming nature of the manipulation process, coupled with the sensitivity of the STM tip, constrains the exploration of diverse configurations and limits the size of the designed features. In this study, we present a reinforcement learning (RL)-based framework for creating artificial structures by spatially manipulating carbon monoxide (CO) molecules on a copper substrate by using the STM tip. The automated workflow combines molecule detection and manipulation, employing deep-learning-based object detection to locate CO molecules and linear assignment algorithms to allocate these molecules to designated target sites. We initially perform molecule maneuvering based on randomized parameter sampling for sample bias, tunneling current set point, and manipulation speed. This data set is then structured into an action trajectory used to train an RL agent. The model is subsequently deployed on the STM for real-time fine-tuning of the manipulation parameters during structure construction. Our approach incorporates path-planning protocols coupled with active drift compensation to enable atomically precise fabrication of structures with significantly reduced human input while realizing larger-scale artificial lattices with the desired electronic properties. Furthermore, using our approach, we demonstrate the automated construction of an extended artificial graphene lattice and confirm the existence of a characteristic Dirac point in its electronic structure. Further challenges regarding the RL-based structural assembly scalability are discussed.

Algorithms↗
DOE OSTI2024

Extending the computational reach of a superconducting qutrit processor

Quantum computing with qudits is an emerging approach that exploits a larger, more connected computational space, providing advantages for many applications, including quantum simulation and quantum error correction. Nonetheless, qudits are typically afflicted by more complex errors and suffer greater noise sensitivity which renders their scaling difficult. In this work, we introduce techniques to tailor arbitrary qudit Markovian noise to stochastic Weyl–Heisenberg channels and mitigate noise that commutes with our Clifford and universal two-qudit gate in generic qudit circuits. We experimentally demonstrate these methods on a superconducting transmon qutrit processor, and benchmark their effectiveness for multipartite qutrit entanglement and random circuit sampling, obtaining up to 3× improvement in our results. To the best of our knowledge, this constitutes the first-ever error mitigation experiment performed on qutrits. Our work shows that despite the intrinsic complexity of manipulating higher-dimensional quantum systems, noise tailoring and error mitigation can significantly extend the computational reach of today’s qudit processors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Flux line relaxation kinetics following current quenches in disordered type-II superconductors

We investigate the relaxation dynamics of magnetic vortex lines in type-II superconductors following rapid changes of the external driving current by means of an elastic line model simulated with Langevin molecular dynamics. A system of flux vortices in a sample with randomly distributed point-like defects is subjected to an external current of appropriate strength for a sufficient period of time so as to be in a moving non-equilibrium steady state. Here, the current is then instantaneously lowered to a value that pertains to either the moving or pinned regime. The ensuing relaxation of the flux lines is studied via one-time observables such as their mean velocity and radius of gyration. We have in addition measured the two-time flux line height autocorrelation function to investigate dynamical scaling and aging behavior in the system, which in particular emerge after quenches into the glassy pinned state.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2019

The stellar halo of isolated central galaxies in the Hyper Suprime-Cam imaging survey

We study the faint stellar halo of isolated central galaxies, by stacking galaxy images in the HSC survey and accounting for the residual sky background sampled with random points. The surface brightness profiles in HSC r band are measured for a wide range of galaxy stellar masses (9.2 < log 10 M * /M ⊙ < 11.4) and out to 120 kpc. Failing to account for the stellar halo below the noise level of individual images will lead to underestimates of the total luminosity by ≤15%. Splitting galaxies according to the concentration parameter of their light distributions, we find that the surface brightness profiles of low-concentration galaxies drop faster between 20 and 100 kpc than those of high-concentration galaxies. Albeit the large galaxy-to-galaxy scatter, we find a strong self-similarity of the stellar halo profiles. They show unified forms once the projected distance is scaled by the halo virial radius. The colour of galaxies is redder in the centre and bluer outside, with high-concentration galaxies having redder and more flattened colour profiles. There are indications of a colour minimum, beyond which the colour of the outer stellar halo turns red again. This colour minimum, however, is very sensitive to the completeness in masking satellite galaxies. We also examine the effect of the extended PSF in the measurement of the stellar halo, which is particularly important for low-mass or low-concentration galaxies. The PSF-corrected surface brightness profile can be measured down to ~31 magarcsec -2 at 3σ significance. PSF also slightly flattens the measured colour profiles.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Intrinsic nonlocality of spin- and polarization-resolved probabilities in strong-field quantum electrodynamics

Spin and polarization are central to precision tests of fundamental physics and for interpreting radiation from astrophysical sources and ultraintense laser-matter experiments. Here, focusing on the fundamental process of nonlinear Compton scattering, we demonstrate that a key assumption underlying current strong-field quantum electrodynamics models, i.e., that emission can be treated as an instantaneous random event sampled from a local differential rate, is inconsistent once emission angles, electron spin, and/or photon polarization are resolved. Namely, even in strictly constant and uniform fields , the resulting fully differential distribution is sign indefinite, yielding negative inferred probabilities. The physical reason is that the photon emission probability builds up over a finite length of the electron trajectory, the formation region, during which the electron direction changes by roughly the same small angle that defines the radiation cone. Therefore, we put forward a new method where we integrate over this formation region analytically to obtain a physically consistent electron spin and photon polarization model. We show that the implementation of our model is compatible with existing Monte Carlo and particle-in-cell workflows. Simulations of a GeV-class electron-laser collision accessible at current petawatt facilities and of emission in a pulsarlike magnetic field are shown to reveal spin and polarization patterns that differ even qualitatively from state-of-the-art local models. In particular, our new model predicts substantial angle-dependent circular photon polarization where the well-known collinear-emission approach yields none, and a pronounced helicity bias in the recoiling electrons absent from current predictions. These findings have direct implications for upcoming strong-field QED experiments and for interpreting polarized radiation from extreme astrophysical environments.

astrophysical electromagnetic fields↗
DOE OSTI2025

Evaluation of Seismic Artificial Intelligence with Uncertainty

Artificial intelligence has transformed the seismic community with deep learning models (DLMs) that are trained to complete specific tasks within workflows. However, there is still a lack of robust evaluation frameworks for evaluating and comparing DLMs. Here, we address this gap by designing an evaluation framework that jointly incorporates two crucial aspects: performance uncertainty and learning efficiency. To target these aspects, we meticulously construct the training, validation, and test splits using a clustering method tailored to seismic data and enact an expansive training design to segregate performance uncertainty arising from stochastic training processes and random data sampling. The framework’s ability to guard against misleading declarations of model superiority is demonstrated through the evaluation of PhaseNet (Zhu and Beroza, 2018), a popular seismic phase picking DLM, under three training approaches. Our framework helps practitioners choose the best model for their problem and set performance expectations by explicitly analyzing model performance with uncertainty at varying budgets of training data.

58 GEOSCIENCES↗
DOE OSTI2020

Site-wide occupancy assessment using camera traps for seven mammalian species at Los Alamos National Laboratory

Los Alamos National Laboratory (LANL or Laboratory) is committed to solving national security challenges through scientific excellence and has been serving the nation and northern New Mexico for over 70 years. Being located on the Pajarito Plateau in the eastern flanks of the Jemez Mountains, the Laboratory is surrounded by a rich diversity of plants and animals. It is common to see many different species of wildlife on Laboratory property; however, sometimes interactions with wildlife can be negative. Vehicle accidents with wildlife have become a common occurrence. With the current and ongoing expansion of the Laboratory on the Pajarito Plateau, it has the potential to further impact wildlife movement including large game species. Local agencies and tribal Pueblos rely on large game species and do not want these species to be restricted from moving across property boundaries. Temporal and spatial aspects of where wildlife occur on the site is a phenomena that is either not well understood in uncommon species or needs periodic reevaluation for common species. Estimating the distribution of multiple species across the landscape provides wildlife biologists with crucial information for monitoring and conserving animal populations in a particular area. Utilizing motion activated wildlife cameras, also known as camera traps, to monitor wildlife populations has become an essential tool for biologists. Camera traps are non-invasive and cost-effective and can document multiple elusive or uncommon wildlife species, such as carnivores, simultaneously. Occupancy modeling provides a flexible framework for the analysis of the distribution for multiple wildlife species. It explicitly recognizes whether a species is spatially common or rare (occupancy = ψ) and if that species is easy or hard to detect (detection probability = p ). Multispecies and multi-season occupancy models can detect trends in species occupancy because individual species may vary in seasonal movements, detection probability, and transition rates between habitats. In this study, we assessed the site as a whole to ascertain when and where medium and large mammal species are present. Understanding wildlife patterns at the Laboratory will better inform future management decisions regarding land use and development strategies. We placed motion activated wildlife cameras in a random systematic sampling design and used these data to create occupancy models. We tested for differences in single-species occupancy and detection probability by season of mammal species captured on 20 camera traps placed across the Laboratory in a 40 mi² (103 km²) area. We focus the interpretation of our findings on seven mammal species found during this study. They are Rocky Mountain elk ( Cervus canadensis nelsoni ; hereafter “elk”), mule deer ( Odocoileus hemionus ; hereafter “deer”), mountain lion ( Puma concolor ; hereafter “lion”), American black bear ( Ursus americanus ; hereafter “bear”), coyote ( Canis latrans ), bobcat ( Lynx rufus ), and gray fox ( Urocyon cinereoargenteus ; hereafter “fox”).

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2019

Computational experiments to characterize bubble formation and movement in waste glass foam layer

Computer simulation experiments are a valuable complement to physical experimentation for systems where there is limited access to quantities of interest. The motivation for this work is to characterize the fluid dynamic and heat transfer processes that affect melt rate of waste slurry during vitrification, which can limit the throughput rate of the melter. Heat transfer to the cold cap from the molten glass below this layer is an important factor determining melt rate. In this study, computational fluid dynamics is used to elucidate the behavior of gas-filled cavities that form underneath the cold cap in a laboratory-scale waste glass melter. The cold cap consists of a reacting feed layer above a foam layer. The cavities are a result of coalesced gases from collapsing primary foam and air from forced convection bubblers. Bubbles in the foam layer move horizontally with the circulation pattern of the melt and escape around the edges of the cold cap to the plenum above. The effects of varying thermophysical properties and foam parameters based on the bounds of high-level waste glass formulations were explored. Using the DAKOTA toolkit, a Latin hypercube space-filling design was used to randomly distribute samples in bins to achieve good coverage over the parameter space with minimal points. Using the results from the set of simulations, a Gaussian process model with a linear trend was used to create response surfaces for several key quantities of interest, including temperatures, temperature gradients, velocities, and foam extent beneath the cold cap.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2021

IAEA Facility-Level Safeguards and Implementation and Advanced Verification Technologies

IAEA Facility-Level Safeguards and Implementation - This presentation is intended to introduce the audience to the application of international safeguards by the International Atomic Energy Agency at nuclear facilities around the world. It covers the State Level Approach, random statistical sampling processes, material balance area structure, non-destructive assay techniques, and inspection areas. Addition of advanced verification technologies for the IAEA

98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL P↗
DOE OSTI2016

Unbiased Ratio Estimator for k-Eigenvalue in Monte Carlo Criticality Calculations

In criticality calculations, the effective multiplication factor (k{sub eff}) is an important parameter for reactor analysis. It can be interpreted as the ratio of the neutron production rate to the loss rate in a reactor core. In Monte Carlo (MC) criticality calculations, k{sub eff} is obtained for each active cycle i as: k-i = P{sub i}/L{sub i} for I = 1 to I where P{sub i} and L{sub i} are the production rate and the loss rate at cycle i, respectively, and they are normalized by the nominal source size for cycle. Since all neutron particles in a cycle will be absorbed or leaks out at the end, the loss rate, L{sub i}, can be set as the source rate, which is the unity. Thus, in MCNP5, k{sub i} is simply calculated with P{sub i} by using three conventional estimators; collision, absorption, and track-length estimators. In domain decomposed MC criticality calculations based on the fission and surface source (FSS) iteration method, k{sub i} is calculated by a ratio of P{sub i} and L{sub i}, where both P{sub i} and L{sub i} are random variables, sampled by using the same particle histories. When the numerator and the denominator are correlated, the variance of the ratio can be smaller than that of the numerator. Then, the final estimate of k{sub eff} is provided as the sample mean of k{sub i}, which will be referred as 'simple' ratio estimator in this paper. In Ref., it is shown that this ratio estimator converges faster and fluctuates less than the conventional estimator. A similar ratio estimator is also used in Serpent 2, as an implicit estimate of k{sub eff}. However, this type of 'simple' ratio estimator is biased, which means that the expectation of the ratio is different from the ratio of the expectation of the numerator and that of the denominator. In this paper, an unbiased ratio estimator which was originally derived in the cytometry analysis is introduced to estimate k{sub eff} and is tested on a continuous-energy UO{sub 2} pin-cell problem. In this test problem, it is shown that the variance of the unbiased ratio estimator is smaller than those of the conventional estimators provided by MCNP5, while the simple ratio estimator can be significantly biased when the number of histories per cycle is too small. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2025

Field Testing of Safeguards Technologies in the Hot Fuel Examination Facility

Recent developments in nuclear fuel reprocessing techniques have yielded more efficient processes and fuel cycle options that strengthen the nuclear industry and production of clean energy. One such area of interest is pyroprocessing of used oxide fuel. However, with these advances in the back end of the nuclear fuel cycle, advances in safeguards instrumentation, measurements, and approaches are needed to ensure special nuclear material (SNM) is accounted for according to regulatory requirements. As a high-level overview of a nominal pyroprocessing approach, used oxide fuel from commercial light water reactors (LWR) is mechanically removed from the metallic cladding. Then the fuel is crushed and randomized representative samples are taken and sent to an analytical lab for analysis. The analytical results of the feed material are used for input accountancy into the rest of the process. The crushed oxide fuel is then moved to the oxide reduction (OR) furnace where it is reduced from an oxide to metallic form. The OR product is distilled to remove salt and then moved to an electrorefiner (ER), where it is immersed in a eutectic mixture of lithium chloride potassium chloride (LiCl-KCl) that typically ranges between 450-550 ?. Within the ER, the usable uranium is electrochemically transported through the molten salt from the anode to the cathode, and then subsequently removed as a relatively pure U product.. A simplified model of pyroprocessing techniques with added emphasis on the safeguards can be seen below in Fig. 1

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2023

Influence of energy density on the microstructure, growth orientation, and anisotropy of magnetic properties in additively manufactured Fe-3.8wt%Si transformer steels

Fe-3.8wt%Si transformer steels were processed using two different additive manufacturing (AM) techniques, laser powder bed fusion (LPBF) and directed energy deposition (DED). While the LPBF processed samples exhibited a strong <001> orientation of the BCC grains along the build axis, the DED processed samples exhibited a randomized texture along the build axis. DED processed samples showed substantially coarser columnar grains as compared to their LPBF counterparts. Here, the columnar grains exhibited a substantial number of low-angle sub-grain boundaries. All samples exhibited very good soft magnetic properties, with saturation magnetization (M s ) values ranging from 205 - 232 emu/gm, and coercivity (H c ) values ranging from 1.2 – 4.2 Oe. The Coercivity (H c ) values were significantly lower when the magnetic field was applied parallel to the build axis, as compared to being perpendicular, which can be rationalized based on the columnar nature of the grains, resulting in a higher number density of grain boundaries in case of the field applied perpendicular to the build axis.

36 MATERIALS SCIENCE↗
DOE OSTI2017

Average subentropy, coherence and entanglement of random mixed quantum states

Compact expressions for the average subentropy and coherence are obtained for random mixed states that are generated via various probability measures. Surprisingly, our results show that the average subentropy of random mixed states approaches the maximum value of the subentropy which is attained for the maximally mixed state as we increase the dimension. In the special case of the random mixed states sampled from the induced measure via partial tracing of random bipartite pure states, we establish the typicality of the relative entropy of coherence for random mixed states invoking the concentration of measure phenomenon. Our results also indicate that mixed quantum states are less useful compared to pure quantum states in higher dimension when we extract quantum coherence as a resource. This is because of the fact that average coherence of random mixed states is bounded uniformly, however, the average coherence of random pure states increases with the increasing dimension. As an important application, we establish the typicality of relative entropy of entanglement and distillable entanglement for a specific class of random bipartite mixed states. In particular, most of the random states in this specific class have relative entropy of entanglement and distillable entanglement equal to some fixed number (to within an arbitrary small error), thereby hugely reducing the complexity of computation of these entanglement measures for this specific class of mixed states.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗