Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of N-representability

We recently presented a constructive solution to the N-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2,q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2,q)-positivity conditions for q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation

Structure of Fermionic Density Matrices: Complete N-representability Conditions

We present a constructive solution to the N-representability problem---a full characterization of the conditions for constraining the two-electron reduced density matrix (2-RDM) to represent an N-electron density matrix. Previously known conditions, while rigorous, were incomplete. Here we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompositions of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems

GE PETtrace RF power failures related to poor power quality

Introduction Anyone who has ever overseen the installation of a new cyclotron is aware of the importance of addressing the numerous vendor-supplied site specifications prior to its arrival. If the site is not adequately prepared, the facility may face project cost overruns, poor cyclotron performance and unintended maintenance costs. Once a facility has identified the space, providing sufficient power is the next step. Every cyclotron vendor will provide you with a set of power specifications, but meeting these specifications can be difficult, especially when the cyclotron is placed in an existing structure. The cyclotron is an interesting collection of power supplies providing power to sensitive electronic circuitry. It is not sufficient to just provide enough power; you must also provide quality power. It is hoped that our efforts to resolve our poor power quality problems will assist others as they replace aging cyclotrons in existing institutions whose power quality has degraded over the years. The University of Iowa Hospitals and Clinics completed installation of a GE PETtrace 800 cyclotron in November 2011. Four months prior to installation, GE service personnel arrived to do a power assessment. The result was that we met their specifications, but with reservations. We could easily provide the quantity of power required, but the specification also states that GE recommends that primary power remain at 480 VAC ± 5%. GE service personnel attached a power quality analyzer to the cyclotron main power panel and determined that we did have some events of 7 to 8 % sag, but they were in-frequent, perhaps once or twice a week lasting 20 to 50 msec. Sags were confirmed to be the result of large non-linear loads elsewhere in the hospital. If these occurred during a run, they may shut down the cyclotron, specifically the RF power supply. Further investigation revealed the presence of harmonics on our power. Harmonics are the multiples of 60Hz power that are reflected back into your facility’s power grid from large motor drivers. Commercial air handler, water pump and fan motors often use variable frequency drives (VFDs) for proportional control to meet the changing facility demands. This pro-vides a significant on-going cost savings, but may play havoc with power quality throughout the institution. Harmonic distortion is often quantified as a total harmonic distortion (THD) percentage. Though not specifically mentioned in the site-specifications, our experience here will show that it is important not to overlook harmonic distortion. Its effects can be varied, erratic and wide-spread throughout the cyclotron system. When asked, GE service referred us to IEEE standards for electrical systems and equipment which states that THD is recommended to be below 5 % for most applications, but below 3 % for sensitive settings including airports and hos-pitals1. Mitigation of voltage sag and harmonic distortion is an expensive and complex topic. It is recommended that you consult with your cyclotron vendor to determine if there exists a field-tested solution. Additionally, you should consult a power systems specialist to do an audit of your building’s power system. Material and Methods Characterization of Power Quality: This was accomplished using a Hioki 3197 Power Quality Analyzer and a couple Dranetz PX-5 Power Xplorers. Each monitoring cycle logged data for about a week, which seemed to be about the limit for these units when logging both THD and surge/sag events down to the duration of a single 60Hz cycle. Analysis of the circuit diagrams and communication with GE engineers indicated that the main power contactors to the cyclotron RF system were dropping power to protect the system. The feedback for this shutoff is a detection signal from the front-end EHT (high-voltage generation) circuit that is set at a level to be representative of the 5% AC deviation specification. RF Power System Contactors: Every time the contactors of the RF power distribution system are energized/de-energized, some arching occurs at the contact surfaces. This arching pits the contactor surfaces such that over time the contactor surfaces become irregular and potentially resistive. Since the RF protection circuit triggered by the EHT circuit is downstream from the contactors, it is not so hard to envision why the system becomes more sensitive over time2. Additionally, the harmonic distortion also exists on the AC voltage energizing the contactors. As a result, they may not actuate as smoothly (de-pendent of degree of harmonic distortion) and further hasten the normal rate of pitting of contactor surfaces. Results and Conclusion Within weeks of installation, we began to get RF power shutoffs. They were infrequent at first, but soon began to occur numerous times a week, then numerous times a day. At approximately 3 months post installation, it was often difficult to get through a standard 30 to 45 minute bombardment to make F-18 for our daily patient FDG doses. We limped along for over a year until the University was willing to invest in a solution to address our power problems. Periodic Power Analyses: These analyses, per-formed over the next year, indicated that our power quality worsened in the winter and re-turned to functional levels in the summer. The instance of voltage sag remained approximately the same throughout the year (a few short sags per week), but the THD was down to 6 % in the summer and nearly 10% in the winter. This result, combined with RF shutdown tracking and lack of correlation between observed power sags and RF shutdowns, led us to the conclusion that our very high harmonic distortion combined with small power fluctuations (< 5 %) were the culprit. Mitigation Planning: There are a number of power conditioning technologies, but imposing the need to remove both voltage sag as well as harmonic distortion, quickly narrows the field. What remains are the following options: 1) UPS line conditioner with batteries, 2) UPS line conditioner with flywheel or 3) motor-generator power isolator. Battery maintenance costs ruled out the UPS battery line conditioner. Of the remaining two, if you have the space, the motor-generator is the simplest and cheapest (favored by forward military hospital units). But for the space constrained user, like us, the UPS flywheel line conditioner became the preferred option. Additionally, it was identified in a power audit that the THD was only 4% at the transformers connected directly to the local power utility company supply (upstream of load effect and harmonic distortion sources). This was to be expected as load effects and harmonic distortion are worse if your tie-in point to the building power grid is at the same level or downstream of their sources. Additionally, a test was performed during a hospital backup generator test, wherein the suspected primary offenders (large motors and VFDs) were diverted to backup. As a result, the THD measured at the cyclotron primary power panel dropped by 2.5 %. Working with University electricians, an outside power consultant, GE engineering and University Hospital Radiology Engineering, a two phase plan was created. Phase 1: With a repurposed utility transformer, the cyclotron and PET cameras got their own dedicated transformer connected to the main utility power feed. We also replaced the old contactors in the RF power distribution system. Since installation, the measured THD has remained at 4.5 to 5 % year round and the sag incidence and magnitude are slightly improved. Phase 2: With a quote from GE for a flywheel UPS we should be able to fully condition the power entering our facility, removing the load effect voltage sags as well as the harmonic distortion. One year of operation after Phase 1 implementation, it has been decided that Phase 1 was all that was required. We haven’t had a single new instance of RF shutdown since

Compatibility Relations between the Reduced and Global Density Matrixes

It is a hard and important problem to find the criterion of the set of positive-definite matrixes which can be written as reduced density operators of a multi-partite quantum state. This problem is closely related to the study of many-body quantum entanglement which is one of the focuses of current quantum information theory. We give several results on the necessary compatibility relations between a set of reduced density matrixes, including: (i) compatibility conditions for the one-party reduced density matrixes of any $N_A\times N_B$ dimensional bi-partite mixed quantum state, (ii) compatibility conditions for the one-party and two-party reduced density matrixes of any $N_A\times N_B\times N_C$ dimensional tri-partite mixed quantum state, and (iii) compatibility conditions for the one-party reduced matrixes of any $M$-partite pure quantum state with the dimension $N^{\otimes M}$.Comment: 14 page

Practical experience implementing the Comecer ALCEO Metal solid targetry system

Introduction The Comecer ALCEO Metal system is intended to be a comprehensive solid targetry system, capable of all steps necessary to produce copper isotopes (60Cu, 61Cu & 64Cu) from enriched nickel: plating, transfer to/from cyclotron, irradiation, and dissolution/purification. To develop plating and chemistry methods, we plate natural nickel, and irradiate with deuterons to produce 61Cu. This alleviates the need for expensive enriched nickel isotopes, but gives a lower activity yield. We report a few issues with the ALCEO system, and some of our modifications. Material and Methods BRIEF DESCRIPTION OF SYSTEM: The ALCEO system uses cylindrical shuttles (dia 28 mm, height 35 mm) comprised of an Al body with a Pt well, onto which the Ni is plated. Shuttles are transferred pneumatically from the hot cell to the irradiation module, on the end of the cyclotron beamline. The plating and dissolution are both done at the electrochemical cell, located in the hot cell. This cell is connected by capillary tubes to the electrolytic solution reservoir (for plating), or the acid reservoir (for dissolution). These tubes form a recirculation loop, through which the fluid is propelled by an inline micropump throughout plating and dissolution. The platinum well is 16 mm in diameter, while an O-ring is used to plate only the center (6 mm in diameter). A constant DC voltage is applied. PLATING: We dissolve natural nickel nitrate (99.999% pure) into an electrolytic solution comprised of deionized water, ammonium hydroxide and ammonium chloride (pH = 9.3). We use 30–100 mg of natNi in a 10mg/mL solution. We have varied the ALCEO electrochemical cell voltage between 2.25–3 VDC, and tried to maintain a low pump flow rate between 1–2 mL/min. The electrochemical cell uses a fixed metal tube as the anode (~3mm above the plating surface). This tube also delivers the electrolytic plating solution to the Pt surface, forming part of the recirculating loop. The Pt surface is in contact with a gold-plated cathode. Due to issues discussed below, we have built a custom plating rig for the ALCEO shuttles, which does not use the pump/recirculation loop, but leaves the reservoir of electrolytic solution in place, atop the plating surface. A graphite rod is used for the anode, rotating offcenter ~2mm above the plating surface. We use the same size O-ring to plate only the center 6 mm of the Pt surface, and apply a constant DC voltage. TRANSFER TO/FROM CYCLOTRON: The pneumatic transfer tube is 50 ft long between the hot cell and the cyclotron vault, and has a rise of 14 ft from under the floor to the ceiling of the cyclotron vault. IRRADIATION: The ALCEO irradiation module holds the plating surface orthogonal to the beam path. The module has a 10-mil-thick (0.010”) Al front foil, supported by a hex-grid. Once we realized the thickness, we replaced this with a 1-mil-thick Al foil, followed by a 1-mil-thick Havar foil. The foil is cooled by a flow of helium, while the shuttle and grid are cooled by a flow of water. The helium and water are cooled in heat ex-changers by chilled water. As initially plumbed, the chilled water flowed through the heat ex-changers in series, cooling the helium first, then the water. After initial runs, we plumbed the heat exchangers in parallel, teeing the chilled water to the supply of each heat exchanger, and teeing the returns together. Irradiation is performed with a PETtrace 800 accelerating deuterons to 8.4 MeV on target. The beam current limit is 20 μA for the ALCEO Metal target. A set point of 19 μA is used to avoid the system tripping off. DISSOLUTION/PURIFICATION: The ALCEO system circulates 5 mL of 6M HCl, while heating the shuttle to 100 ⁰C for 40 min. This solution is loaded onto a column containing 10 g of 200–400 mesh chromatographic resin in chloride form. A separation is performed yielding three solutions: The column is washed with 40 mL of 6M HCl to obtain the Recovered Nickel Solution, then 20 mL of 4M HCl to obtain the Cobalt Solution, then 10 mL of 0.5M HCl to obtain the Cop-per Product Solution. Results and Conclusion PLATING: Using the ALCEO method, the platings obtained had a tendency to mound, (up to 0.75 mm thick for 50 mg) giving a lower density of 3–4 g/cm3. This was attributed to the anode tube being fixed in place over the center of the plating surface. Using the custom rig, almost no mounding was observed, (0.25mm thick for 50 mg) giving a density of 7 g/cm3, closer to nickel’s nominal density of 8.9 g/cm3. FIGS. 1 and 2 attempt to show the mounding from the ALCEO method, and relative flatness from the custom rig. Both methods give a rough, or “fuzzy” plated surface. FIG. 2 shows that the custom rig exaggerates this “fuzziness”. Using the ALCEO method, a slower pump speed (~1 mL/min) gives a smoother plating surface, but the pump has a tendency to lock up at this lower set point, stalling the plating. Using the ALCEO method, slight increases in the voltage (2.65 V instead of 2.25 V), can form thin stalagmites of nickel, electrically connecting the anode and the plating surface, ending the chance for a useable plating. Using the custom rig, no stalagmites are seen, adjusting the voltage from 2.3 to 3.0 V. This leads to the potential for a faster plating. Both the ALCEO method and the custom rig have obtained plating efficiencies of 95 %. TRANSFER TO/FROM CYCLOTRON: The shuttle typically transfers without issue in < 15 seconds. Once or twice it has remained in the transfer tube, but been retrieved by cycling the com-pressed air/vacuum a couple of times. IRRADIATION: During initial testing, the temperature of the return water rose rapidly during irradiation. This was attributed to the chilled water already gaining heat from the helium heat exchanger. Once the parallel chilled water plumbing was implemented, the water temperature rose much more slowly. Initial testing with the 10-mil Al foils gave a very poor activity yield. The 1-mil Al foil ruptured under the 20 psi helium pressure before beam was applied. The 1-mil Havar foil produced 1.57 mCi of 61Cu at EOB, giving an activity yield of 0.308 mCi/μAh (results summarized in TABLE 1). This compares to yields of 1.4 obtained by [1], and 0.29 obtained by [2] for deuterons on natNi. DISSOLUTION/PURIFICATION: The dissolution in 6M HCl is close to 100% efficient by weight. After purification, the three solutions were assayed by dose calibrator as summarized in TABLE 2. The purification is very efficient at removing the starting material, and long-lived Co isotopes from the Copper Product Solution, as seen with the 57-hour EOB measurements. However, much of the desired 61Cu is removed as well, with only 32% remaining in the product. The lack of nuclide impurities in the Copper Product Solution was confirmed by gamma spectroscopy using a HP-Ge detector

Further exploration of C-11 HP target on PETtrace

Introduction At WTTC 14 we presented data on the target yields of our GE PETtrace C-11 HP target in comparison to the target yields we had been getting on the MC17 prior to its decommissioning1. Discussion with other attendees alerted us to the fact that the target may be too “thin”, allowing the beam to spread out and interact with the walls, which could result in a lower target yield. Additionally, a GE service engineer indicated that we could be striking the top of the target with some of the beam, due both to target thinning and the “banana” effect from the magnetic fringe fields. Experiments were carried out to determine the potential magnitude of this effect and the efficacy of potential solutions. Material and Methods All experiments were performed on a GE PET-trace cyclotron. The first set of experiments was performed on the C-11 HP target in its natural mounting state (no aids). The change is gas pressure as a function of beam current was measured, from 5 to 70 microamps for three different gas fill pressures: 210, 230 and 250 PSI. The second set of experiments was performed after mechanically lifting the back end of the target with a box, changing the target angle from 23.9 degrees past horizontal to 25.2 degrees past horizontal. While this change in angle does not seem drastic, it did pick up all the slack in the target mount due the sagging of the target from its longer length than other GE targets. The change in gas pressure as a function of beam current was measured, from 5 to 80 microamps for four different gas fill pressures: 190, 210, 230 and 250 PSI. (Note that the box is a temporary solution and the target will sag over time without a more permanent solution for supporting the back end of the target.) Results and Conclusion The graphical results of pressure rise as a function of beam current are shown in FIGURE 1. Note that measurements were stopped when the pressure approached 470 PSI, based on advice from GE engineers. There is some flattening out for the 190-PSI data, even with the increase in angle as an attempt to counteract the banana effect (note that GE’s recommended fill pressure is 187 PSI). Increases in the fill pressure helped in keeping the target thick, but with the tradeoff that less beam can be put onto the target before reaching the maximum specified pressure. Final-ly, using a lifting mechanism to raise the back of the target also helped to prevent thinning, as seen in the r-squared values for the linear fit, shown in TABLE 1. The data presented indicate that a shorter target that can withstand higher pressures could be beneficial for the PETtrace cyclotron, allowing the beam to fully stop before striking the walls, be it through target thinning or the “banana” effect while still allowing the user to run high beam currents

Djinn Lite: a tool for customised gene transcript modelling, annotation-data enrichment and exploration

BACKGROUND: There is an ever increasing rate of data made available on genetic variation, transcriptomes and proteomes. Similarly, a growing variety of bioinformatic programs are becoming available from many diverse sources, designed to identify a myriad of sequence patterns considered to have potential biological importance within inter-genic regions, genes, transcripts, and proteins. However, biologists require easy to use, uncomplicated tools to integrate this information, visualise and print gene annotations. Integrating this information usually requires considerable informatics skills, and comprehensive knowledge of the data format to make full use of this information. Tools are needed to explore gene model variants by allowing users the ability to create alternative transcript models using novel combinations of exons not necessarily represented in current database deposits of mRNA/cDNA sequences. RESULTS: Djinn Lite is designed to be an intuitive program for storing and visually exploring of custom annotations relating to a eukaryotic gene sequence and its modelled gene products. In particular, it is helpful in developing hypothesis regarding alternate splicing of transcripts by allowing the construction of model transcripts and inspection of their resulting translations. It facilitates the ability to view a gene and its gene products in one synchronised graphical view, allowing one to drill down into sequence related data. Colour highlighting of selected sequences and added annotations further supports exploration, visualisation of sequence regions and motifs known or predicted to be biologically significant. CONCLUSION: Gene annotating remains an ongoing and challengingtask that will continue as gene structures, gene transcription repertoires, disease loci, protein products and their interactions become moreprecisely defined. Djinn Lite offers an accessible interface to help accumulate, enrich, and individualise sequence annotations relating to a gene, its transcripts and translations. The mechanism of transcript definition and creation, and subsequent navigation and exploration of features, are very intuitive and demand only a short learning curve. Ultimately, Djinn Lite can form the basis for providing valuable clues to plan new experiments, providing storage of sequences and annotations for dedication to customised projects. The application is appropriate for Windows 98-ME-2000-XP-2003 operating systems

Extending Torelli map to toroidal compactifications of Siegel space

It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regular for $g\ge9$; this disproves a 1973 conjecture of Namikawa.Comment: To appear in Inventiones Mathematica

The class of the locus of intermediate Jacobians of cubic threefolds

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressions for the classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces.Comment: v2: new section 9 on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds. Final version to appear in Invent. Mat

Superfluidity of a perfect quantum crystal

In recent years, experimental data were published which point to the possibility of the existence of superfluidity in solid helium. To investigate this phenomenon theoretically we employ a hierarchy of equations for reduced density matrices which describes a quantum system that is in thermodynamic equilibrium below the Bose-Einstein condensation point, the hierarchy being obtained earlier by the author. It is shown that the hierarchy admits solutions relevant to a perfect crystal (immobile) in which there is a frictionless flow of atoms, which testifies to the possibility of superfluidity in ideal solids. The solutions are studied with the help of the bifurcation method and some their peculiarities are found out. Various physical aspects of the problem, among them experimental ones, are discussed as well.Comment: 24 pages with 2 figures, version accepted for publication in Eur.Phys.J.