text:
Journal topic
Atmos. Chem. Phys., 19, 1013–1026, 2019
https://doi.org/10.5194/acp-19-1013-2019
Atmos. Chem. Phys., 19, 1013–1026, 2019
https://doi.org/10.5194/acp-19-1013-2019

Research article 25 Jan 2019

Research article | 25 Jan 2019

# Speciated and total emission factors of particulate organics from burning western US wildland fuels and their dependence on combustion efficiency

Speciated and total emission factors of particulate organics from burning western US wildland fuels and their dependence on combustion efficiency
Coty N. Jen1,a, Lindsay E. Hatch2, Vanessa Selimovic3, Robert J. Yokelson3, Robert Weber1, Arantza E. Fernandez4, Nathan M. Kreisberg4, Kelley C. Barsanti2, and Allen H. Goldstein1,5 Coty N. Jen et al.
• 1Department of Environmental Science, Policy, and Management, University of California, Berkeley, CA 94720, USA
• 2Department of Chemical and Environmental Engineering and College of Engineering, Center for Environmental Research and Technology, University of California, Riverside, CA 92507, USA
• 3Department of Chemistry, University of Montana, Missoula, MT 59812, USA
• 4Aerosol Dynamics Inc., Berkeley, CA 94710, USA
• 5Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, USA
• anow at: Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Correspondence: Coty N. Jen (cotyj@andrew.cmu.edu)

Abstract

Western US wildlands experience frequent and large-scale wildfires which are predicted to increase in the future. As a result, wildfire smoke emissions are expected to play an increasing role in atmospheric chemistry while negatively impacting regional air quality and human health. Understanding the impacts of smoke on the environment is informed by identifying and quantifying the chemical compounds that are emitted during wildfires and by providing empirical relationships that describe how the amount and composition of the emissions change based upon different fire conditions and fuels. This study examined particulate organic compounds emitted from burning common western US wildland fuels at the US Forest Service Fire Science Laboratory. Thousands of intermediate and semi-volatile organic compounds (I/SVOCs) were separated and quantified into fire-integrated emission factors (EFs) using a thermal desorption, two-dimensional gas chromatograph with online derivatization coupled to an electron ionization/vacuum ultraviolet high-resolution time-of-flight mass spectrometer (TD-GC × GC-EI/VUV-HRToFMS). Mass spectra, EFs as a function of modified combustion efficiency (MCE), fuel source, and other defining characteristics for the separated compounds are provided in the accompanying mass spectral library. Results show that EFs for total organic carbon (OC), chemical families of I/SVOCs, and most individual I/SVOCs span 2–5 orders of magnitude, with higher EFs at smoldering conditions (low MCE) than flaming. Logarithmic fits applied to the observations showed that log (EFs) for particulate organic compounds were inversely proportional to MCE. These measurements and relationships provide useful estimates of EFs for OC, elemental carbon (EC), organic chemical families, and individual I/SVOCs as a function of fire conditions.

1 Introduction

Wildfires in the western US have become larger and more frequent, and this trend is expected to continue in the coming decades (Dennison et al., 2014; Miller et al., 2009). This is due to historical wildfire suppression, leading to high fuel loading and climate changes that include longer springs and summers, earlier snowmelts, and prolonged droughts (Dennison et al., 2014; Jolly et al., 2015; Spracklen et al., 2009; Westerling et al., 2006). Smoke emissions from wildfires primarily contain carbon dioxide (CO2), carbon monoxide (CO), and thousands of organic compounds in the gas and particle phases. These organic compounds can significantly influence atmospheric chemistry, cloud formation, regional visibility, and human health. Thus, increased occurrences and magnitudes of wildfires will likely lead to greater smoke impacts on regional and global environments.

The extent to which smoke will adversely impact human health and the environment depends, in part, on the chemical composition and amount of emissions produced. In general, biomass burning, which includes wildfires, is the main global source of fine carbonaceous aerosol particles ( 75 %) in the atmosphere (Andreae and Merlet, 2001; Bond and Bergstrom, 2006; IPCC, 2014; Park et al., 2007). Individual and categorized organic emissions from wildfires have previously been identified and quantified (Akagi et al., 2011; Andreae and Merlet, 2001; Hatch et al., 2015; Kim et al., 2013; Koss et al., 2018; Liu et al., 2017; Mazzoleni et al., 2007; Naeher et al., 2007; Oros et al., 2006; Oros and Simoneit, 2001a; Simoneit, 2002; Stockwell et al., 2015; Yokelson et al., 2013). The bulk of previous studies on speciated organic compound emissions focused on gas-phase volatile organic compounds (VOCs). Particle-phase results are typically reported as total organic carbon (OC) or particulate matter (PM) with aerodynamic diameters less than 10 or 2.5 µm (PM10 and PM2.5). These types of measurements provide no chemical specificity of the particle phase and thus limit the ability to predict how smoke will age in the atmosphere and impact the environment.

Several studies have examined specific particle-phase organic compounds in smoke such as toxic retene and other polycyclic aromatic hydrocarbons (PAHs) (e.g., Jayarathne et al., 2018; Kim et al., 2013; Naeher et al., 2007; Sullivan et al., 2014) or abundant tracer compounds, like levoglucosan and vanillic acid (Simoneit et al., 1999). Out of the likely thousands of unique compounds, roughly 400 known particle-phase organic compounds and their amounts produced per mass of dry fuel burned (a quantity known as the emission factor, EF) have been published and organized by wildland type (Oros et al., 2006, 2001a, b). These compounds span many chemical families (i.e., functionalities), like sugars and methoxyphenols, and provide key insights into how different wildland burns lead to different organic particulate composition and EFs.

New advances in instrumentation, such as two-dimensional gas chromatography or electrospray ionization coupled to high-resolution mass spectrometers (Isaacman et al., 2011; Laskin et al., 2009), now allow for unprecedented levels of molecular speciation of atmospheric aerosol particles that can further identify and quantify the thousands of previously unreported biomass burning compounds. Nevertheless, current fire and atmospheric chemistry models that predict the amount of smoke produced, its atmospheric transformation/transportation, and its physiochemical properties (e.g., French et al., 2011; Reinhardt et al., 1997; Wiedinmyer et al., 2011) do not model the thousands of organic compounds emitted from fires, due primarily to limited computational resources. To address this deficiency, Alvarado et al. (2015) used measured volatility distribution bins of organic compounds from fresh smoke and modeled their atmospheric aging by assuming shifts in volatility distribution via reactions with ozone and hydroxl radicals. Though this approach does better predict secondary organic aerosol particle formation, it still does not consider the wide variety of chemical compounds found in smoke, thus limiting its ability to predict physiochemical properties of aged smoke particles and their impacts on the environment. Therefore, a better or estimable representation of the chemical composition in smoke particles within models requires condensing the information from molecular-level speciation into useable relationships that correlate typical particle composition to a measurable burn variable.

The purpose of this study is to (1) identify, classify, and quantify organic compounds in smoke particles produced during laboratory burns and (2) provide scalable EFs of individual compounds and their chemical families from various fuels as a function of fire conditions. A selection of fuels and fuel combinations commonly consumed in western US wildland fires were burned at the US Forest Service Fire Sciences Laboratory (FSL) in Missoula, MT, during the NOAA Fire Influence on Regional and Global Environments Experiment (FIREX) campaign in 2016 (Selimovic et al., 2018). Regression models representing EFs as a function of fire conditions are provided for groupings of organic compounds that vary in chemical complexity, from generalized organic carbon and total particulate organic compounds to specific chemical families and individual compounds. In addition, a mass spectral database, compatible with the National Institute of Standards and Technology (NIST) Mass Spectral Search program, containing the mass spectra, retention indices, and identities/compound classifications for all the separated compounds observed from the various burns, is included. This database will be a valuable resource for the community for identifying specific chemicals in air masses impacted by biomass burning plumes and understanding the dominant source materials burned, fire characteristics, and atmospheric transformations.

2 Materials and methods

A total of 34 different fuels were combusted in 75 “stack” burns during the 2016 FIREX campaign at FSL (Selimovic et al., 2018). Most fuels were representative of common biomass components found in the western US wildlands. Non-western US wildland fuels were also burned and are used to demonstrate the applicability of the reported regression models across a wider range of fuels. A detailed description of the FSL combustion room can be found elsewhere (Christian et al., 2004; Stockwell et al., 2014) with pertinent details described here. The 12.5 m × 12.5 m × 22 m combustion room contained a fuel bed on the floor. Fuels were placed in the fuel bed and ignited by resistance-heated coils. Above the fuel bed was a 3.6 m inverted funnel connected to a 1.6 m diameter exhaust stack that vented through the roof of the combustion room. The room was kept at positive pressure to provide a constant air flow that diluted and carried the smoke up the stack. A platform was located 17 m above the fuel bed and allowed instrument sampling access into the stack (see Fig. S1 in the Supplement). The samples studied here were collected from the platform and thus represent fresh emissions.

Smoke from the stack was pulled through a custom-built sampler known as DEFCON, Direct Emission Fire CONcentrator (see diagram in Fig. S2). DEFCON's inlet was a 20.3 cm × 1.3 cm OD stainless steel tube that reached 15.2 cm into the stack. 10.3 LPM of smoke was pulled through the inlet, with two 150 ccm flows branching off from the main sample flow. These low-flow channels went to two parallel flows consisting of a Teflon filter followed by sorbent tube for gas-phase sample collection; analysis of those samples will be described in future publications. The remaining 10 LPM was passed through a 1.0 µm cutoff cyclone before being sampled onto a 10 cm quartz fiber filter (Pallflex Tissuquartz). Total residence time was  2 s. One quartz fiber filter collected both particles and likely low volatility gases for the duration of each fire which lasted  5–50 min. These filters were analyzed for this study. A few fires were terminated “early” when a small amount of fuel and smoldering combustion remained. Prior to collection, filters were baked at 550 C for 12 h and packed in similarly baked aluminum foil inside Mylar bags. The flows were monitored to ensure constant flow rates. Flow paths within DEFCON were passivated with Inertium® (Advanced Materials Components Express, Lemont, PA) which has been shown to reduce losses of oxygenated organics (Williams et al., 2006). After each burn, the inlet of DEFCON was replaced with a clean tube and the remainder of the system was purged with clean air. A background filter sample was collected each morning prior to the burns to estimate background contributions from sampling components and room air.

Table 1List of fuels analyzed, with the number of compounds separated and quantified from each burn. Note conifer fuel type refers to a realistic mixture of a coniferous ecosystem unless otherwise noted.

A total of 29 fire-integrated smoke filter samples, including one from each specific fuel (with some replicates), were selected and analyzed using a thermal desorption, two-dimensional gas chromatograph with online derivatization coupled to an electron ionization/vacuum ultraviolet ionization high-resolution time-of-flight mass spectrometer (TD-GC × GC-EI/VUV-HRToFMS) (Isaacman et al., 2012; Worton et al., 2017). A list of analyzed fuels is given in Table 1. Chromatograms from replicate burns showed minor variation; thus the remaining 46 burns were not analyzed in detail. Punched samples of each filter (0.21–1.64 cm2) were thermally desorbed at 320 C under a helium flow using a thermal desorption system (TDS3 and TDSA2, Gerstel). Desorbed samples were then mixed with a gaseous derivatization agent, MSFTA (N-methyl-N-(trimethylsilyl)trifluoroacetamide). MSFTA replaces the hydrogen in polar hydroxyl, amino, and thiol groups with the trimethylsilyl group, creating a less polar and thus elutable compound. Derivatized samples then were focused on a quartz wool glass liner at 30 C (cooled injection system, CIS4, Gerstel) before rapid heating to 320 C for injection into the gas chromatograph (GC, Agilent 7890). GC × GC separation was achieved with a 60 m × 0.25 mm × 0.25 µm semi-nonpolar capillary column (Rxi-5Sil MS, Restek) followed by a medium-polarity, second-dimension column (1 m × 0.25 mm × 0.25 µm, Rtx-200MS, Restek). A dual-stage thermal modulator (Zoex), consisting of a guard column (1 m × 0.25 mm, Rxi, Restek), was used to cryogenically focus the effluent from the first column prior to heated injection onto the second column (modulation period of 2.3 s). The main GC × GC oven ramped at 3.5 C min−1 from 40 to 320 C and was held at the final temperature for 5 min and the secondary oven ramped at the same rate from 90 to 330 C and was held for 40 min. Separated compounds were then ionized either by traditional EI (70 eV) or VUV light (10.5 eV). The HRToFMS (ToFWerk) was used to detect the ions and was operated with a resolution of 4000 and transfer line and ionizer chamber temperatures of 270 C. VUV light was provided by the Advanced Light Source, beamline 9.0.2, at Lawrence Berkeley National Laboratories. During the VUV experiments, the HRToFMS operated at a lower ionizer chamber temperature of 170 C to further reduce molecular fragmentation (Isaacman et al., 2012).

Punches from the filter samples were also analyzed for organic and elemental carbon (OC and EC, respectively) using a Sunset Model 5 Lab OCEC Aerosol Analyzer following the NIOSH870 protocol in the Air Quality Research Center at the University of California, Davis. Thermal pyrolysis (charring) was corrected using laser transmittance. OC and EC were also measured on the background filters.

## 2.1 Emission factor calculations

The mass loadings for all separated compounds measured by TD-GC × GC-EI/VUV-HRToFMS were determined using a set of calibration curves. Full details of the data conversion to mass loadings and emission factors with associated uncertainties are provided in the Supplement with important steps outlined here. The TD-GC × GC-EI/VUV-HRToFMS responses to a wide range of standard compounds commonly found in biomass burning samples were measured at varying mass loadings to create calibration curves. Measured peaks from the filters were calibrated using a standard compound that exhibited similar first- and second-dimension retention times and compound classification. For example, a sampled compound classified as sugar was quantified using the nearest sugar standard compound in the chromatogram. Unknown compounds were matched to the nearest eluting standard compound, similar to the approach taken by Zhang et al. (2018). The mass loadings of all observed compounds were then background subtracted; however, the mass on the background filter for all compounds was negligible. The compound's emission factor (EFcompound) was then calculated by normalizing mass loadings by background-corrected sampled CO2 mass. This ratio was then multiplied by the corresponding EF${}_{{\mathrm{CO}}_{\mathrm{2}}}$, as given in Selimovic et al. (2018). EFs for OC and EC were calculated similarly, using background-corrected OC and EC mass loadings.

Figure 1(a) Measured organic and elemental carbon (OC and EC, respectively) as a function of modified combustion efficiency (MCE) and (b) OC∕EC as a function of MCE. Symbols indicate the different fuel types.

3 Results and discussion

## 3.1 Emission factors of organic and elemental carbon

OC and EC EFs were first related to the fire-integrated modified combustion efficiency (MCE). MCE reflects the mix of combustion processes in the fire and is defined as background-corrected values of ${\mathrm{CO}}_{\mathrm{2}}/\left({\mathrm{CO}}_{\mathrm{2}}+\mathrm{CO}\right)$ (Akagi et al., 2011; Ward and Radke, 1993). MCE values near 1 indicate almost pure flaming, while values near 0.8 are almost pure smoldering, with 0.9 representing a roughly equal mix of these processes. Figure 1a shows the EFs of OC and EC (EFOC and EFEC) as a function of MCE across a variety of fuel types (see Table 1). Decreasing MCE (more smoldering) results in increased OC and decreased EC emissions across all studied fuel types. These observed trends are in general agreement with previous studies (e.g., Christian et al., 2003; Hosseini et al., 2013). EFs for OC and EC generally follow a logarithmic relationship such that log(EFOC) is inversely proportional to MCE (slope of −9.506) and log(EFEC) is directly proportional to MCE (slope 5.441). Comparison of the slopes suggests that decreasing MCE of a fire will produce an increasing amount of OC compared to EC. This is further confirmed by examining the ratio of OC to EC (OC∕EC) with MCE. Figure 1b illustrates how OC∕EC sharply increases with more smoldering fire conditions (slope of −16.555). This trend also follows a similar inversely proportional logarithmic relationship as EFOC vs. MCE but with even stronger correlation (R2=0.85 compared to 0.66, respectively). Note that values for Douglas fir rotten log (burn 31), peat (burn 55), rice straw (burn 60), and Engelmann spruce duff (burn 26) fires are not shown due to measured EC at background levels. In addition, significant losses ( 40 %) of organic compounds were only observed for the Douglas fir rotten log burn and were determined by comparing GC × GC chromatograms taken  1 month after collection at FSL and  2 years after collection prior to OC∕EC analysis. The combined results clearly show that flaming combustion produces slightly more particulate EC compared to OC, whereas smoldering combustion emits levels of OC 1–2 orders of magnitude higher compared to EC.

Figure 2Two-dimensional chromatogram of smoke collected from burning lodgepole pine (burn 63). The first dimension separates compounds by their volatility and the second dimension by their polarity. Each point ( 800 in total) represents a separated compound with the colors signifying the compound's classification. Size of a point approximately scales with its emission factor (see Sect. S3).

## 3.2 Identification and quantification of I/SVOCs

Filter samples were analyzed for intermediate and semi-volatile organic compounds (I/SVOCs) using the TD-GC × GC EI-VUV-HRToFMS. Between 100 and 850 peaks (i.e., unique compounds) were separated in each fire-integrated chromatogram, with fewer peaks observed for more flaming fires such as from shrub fuels (see Table 1). An example two-dimensional chromatogram of a lodgepole pine burn (burn 63) is shown in Fig. 2. Based on the GC × GC configuration, all compounds elute between dodecane (C 106µg m−3 and n-alkane retention index, RI, of 1200) and hexatriacontane (C 10−1µg m−3, RI = 3600) and thus are classified as I/SVOCs, with a small fraction classified as low-volatility organic compounds (Donahue et al., 2009). In total, approximately 3000 unique compounds were separated across the 29 analyzed burns (see Table 1). From those compounds, 149 compounds were identified using a combination of matching authentic standards, RI, EI mass spectrum (via NIST Mass Spectral database, 2014 version), and VUV parent and fragment mass ions. True positive identification requires analysis of a standard compound on the instrument; however comparing the NIST match to parent mass determined from VUV mass spectrum analysis can also provide a level of identification (Worton et al., 2017). Identified compounds account for 4 %–37 % of the total observed organic mass (mean of 20 % with a standard deviation of 9 %). A table of these identified compounds with their identifying methods (e.g., standard matching, previous literature, or NIST Mass Spectral database), RI, five most abundant mass ions from the EI mass spectra, and fuel source(s) is given in Table S1 in the Supplement.

To help reduce the chemical complexity from the 3000 observed compounds, each separated compound was sorted into a chemical family. This was achieved using a combination of parent ion mass (VUV), fragment ion mass spectra (VUV and EI), RI, and second-dimension retention time to estimate the compound's functionality. More details on the classification process and examples within each category can be found in the Supplement. The chemical families were broadly named and include non-cyclic aliphatic/oxygenated, sugars, PAHs/methylated + oxygenated, resin acids/diterpenoids, sterols/ triterpenoids, organic nitrogen, oxygenated aromatic heterocycles, oxygenated cyclic alkanes, methoxyphenols, substituted phenols, and substituted benzoic acids. Almost 400 compounds, including the identified and most frequently observed compounds in the analyzed burns, were grouped into these families. The remainder of the compounds, which were both uncategorizable and unidentifiable, were placed into the unknown category. Figure 2 illustrates the chemical families (indicated by color) of all the separated compounds emitted from an example lodgepole pine burn.

Despite many compounds remaining unknown, their defining traits such as mass spectra or retention index (i.e., volatility) can be compared to atmospheric samples to help the community better define the composition of biomass-burning-derived, particle-phase organic compounds. As such, all  3000 observed compounds have been compiled into a publicly available mass spectral database and first reported here as the University of California, Berkeley, Goldstein Library of Organic Biogenic and Environmental Spectra (UCB-GLOBES) for FIREX (see Supplement). This spectral library is compatible with NIST Mass Spectral Search and contains mass spectra, n-alkane RI, potential compound identification or chemical families, EFs as a function of fire conditions, and fuel sources of all unique compounds detected from the 29 analyzed burns.

Figure 3(a) Contributions of unknown mass to the total observed mass for the 29 analyzed burns. (b) Mass fractions for each chemical family compared to total classified mass. Fuels are grouped by type, and numbers after fuel name indicate the burn number during the FIREX campaign.

## 3.3 Average observed I/SVOC composition

The masses of observed I/SVOCs from each chemical family were summed over each fire-integrated sample and normalized to either the total observed I/SVOC mass or total classified I/SVOC mass. Figure 3a illustrates mass fractions of the unidentified and unclassified (unknown) compounds out of the total observed mass from the 29 analyzed burns. Mass fractions for each chemical family out of the total observed mass are given in Table S5. Unknowns represent  35 %–90 % of I/SVOCs mass emitted during the analyzed burns, with woody debris (rotten logs) exhibiting the highest mass fraction of unknowns ( 90 %). Since the compounds that make up the unknown mass fraction varied between burns, differences in the mass fractions between fuel types are not indicative of higher emissions of any particular compound. However, notably the two woody debris burns showed similar unknown compounds (i.e., 99 % of the unknown mass was of compounds found in both burns) but occurred under two different fire conditions (burn 13 at MCE = 0.98 and burn 31 at MCE = 0.78). In both cases, the unknown mass fractions were similar at 87 %–89 %. This observation provides some indication that fuel type plays a larger role than MCE in determining the unknown organic mass fraction in smoke particles.

Given that the unknown compounds typically varied between burns, the mass of each classified chemical family was normalized to the total observed classified mass (i.e., excluding the unknown mass) in order to better compare classified compounds between burns. These results are shown in Fig. 3b. Conifers, coniferous litter, and wood exhibited the highest fraction of sugars (38 %, 29 %, and 44 %, respectively) compared to other fuels (between 6 %–30 %). Furthermore, levoglucosan was the largest single contributor to the sugars for these burns and ranged from 10 % to 40 % of the total sugars. These observations are consistent with previous studies that have shown high levoglucosan emissions from cellulose-rich wood samples (Mazzoleni et al., 2007; Simoneit et al., 1999). Coniferous fuels also emitted higher amounts of resin acids/diterpenoids (7 %, 16 %, 7 %, and 3 % for conifers, coniferous litter, coniferous duff, and woody debris respectively), as previously observed (Hays et al., 2002; Oros and Simoneit, 2001a; Schauer et al., 2001). Peat (from Indonesia) emitted the largest fraction of aliphatic compounds (52 %) compared to other fuels, in agreement with previous observations (George et al., 2016; Iinuma et al., 2007; Jayarathne et al., 2018). Manzanita burns produced the highest amounts of substituted phenols (34 % of total classified mass compared to 1 %–4 % for other fuels), mostly as hydroquinone (Hatch et al., 2018; Jen et al., 2018). Organic nitrogen compounds, most of which were nitro-organics, also contributed significantly (up to 43 %) to the total observed classified mass for all fuels. These compounds tend to absorb light (Laskin et al., 2015) and may contribute to observed brown carbon light absorption from these burns (Selimovic et al., 2018). However, it should be noted that the instrument is not as sensitive to this class of compounds. Thus, the EF uncertainty is high (factor of 2) for compounds that are not positively identified with a standard but are categorized as organic nitrogen.

Figure 3b also provides some evidence that fuels within the same type generally show similar mass fractions of chemical families. For example, conifers, which consist of a mixture of coniferous ecosystem fuel components (e.g., canopy, duff, litter, and twigs), exhibit relatively similar mass fractions, with sugars accounting for 30 %–50 %, 4 %–28 % non-cyclic aliphatic, 13 %–30 % organic nitrogen, 2 %–20 % resin acid/diterpenoids, and 1 %–4 % PAH/methyl + oxy across the MCE range of 0.90–0.95. Coniferous duff (MCE = 0.85–0.89) exhibited a lower sugar fraction (11 %–31 %) but higher non-cyclic aliphatics (15 %–38 %) than the conifers. Burning grasses (MCE = 0.90–0.95) produced roughly equal amounts of sugars and organic nitrogen compounds (30 %) and higher amounts of oxygenated cyclic compounds (3 %–11 %), like lactones, than the coniferous fuels ( 1 %). Shrubs (MCE = 0.92–0.98) exhibited the largest ranges in chemical family mass fractions (e.g., 0 %–42 % organic nitrogen compounds and 2 %–43 % substituted phenols), suggesting that plants in this fuel type are less similar to each other than coniferous fuels. This may be due to a wider range of plant chemical composition for shrubs than for the other fuel types. Overall, the I/SVOC mass fractions tend to be more similar for fuels within a fuel type with the most variation exhibited for fuel mixtures and shrubs.

Figure 4Summed emission factors (EFs) within a chemical family for each burn as a function of modified combustion efficiency (MCE). Each panel depicts a different family with (a) total observed I/SVOC EF, (b) unknowns, (c), sugars, (d) polycyclic aromatic hydrocarbons (PAHs, including methylated and oxygenated forms), (e) methoxyphenols, and (f) sterols/triterpenoids. Dashed lines represent a log fit of the form log(EF) inversely proportional to MCE. The dotted lines represented a factor of 2 above and below the model. Symbols denote different fuel types.

## 3.4 EFs as a function of fire conditions (MCE)

Unlike the dependence of chemical family mass fractions on fuel type, EFs for each chemical family showed a correlation with MCE across all fuels examined and to a much lesser extent than fuel type. Figure 4 presents EFs for total observed organic compounds and five of the chemical families (unknowns, sugars, PAHs/methyl/oxy, methoxyphenols, and sterols/triterpenoids, with others given in Fig. S4) as a function of MCE. Fuels not found in the western US are also included in these figures to demonstrate that their EFs generally follow the trend with MCE. The notable exception is peat, a semi-fossilized fuel (Stockwell et al., 2016), whose EFs for all chemical families are roughly an order of magnitude lower than other fuels at similar MCE values, except non-cyclic aliphatic/oxygenated EF, which is approximately equal to burns at similar MCE (see Fig. S4). In general, EFs measured by the TD-GC × GC-EI/VUV-HRToFMS agree with previous literature (Hays et al., 2002; McDonald et al., 2000; Oros et al., 2006; Oros and Simoneit, 2001a, b). For example, Oros and Simoneit (2001a) provided the sum of carboxylic acids and alkanes–alkanes–alkanols for conifer burns at  1 g kg−1, which is within our reported range of 0.4–2.3 g kg−1 (MCE = 0.90–0.95) for non-cyclic aliphatic/oxygenated emitted from burning conifers. Other chemical family EFs presented here for conifers, including PAHs (0.4 g kg−1), diterpenoids (1–3 g kg−1), and methoxyphenols (1 g kg−1), are also in good agreement with those published in Oros and Simoneit (2001a). Hays et al. (2002) reported EFs for unknown compounds from ponderosa pine of  20 g kg−1, higher than 11 g kg−1 (MCE = 0.94) for the ponderosa pine burn studied here. This may be due to more compounds being classified here than in Hays et al. (2002) or differences in MCE between the studies. In contrast to previous work, EFs for chemical families reported here were measured over a wider range of fire conditions and fuel types and show a clear relationship with MCE.

Figure 5Emission factors (EFs) of (a) levoglucosan (sugar), (b) fluoranthene (PAH), (c) acetovanillone (methoxyphenol), and (d) coniferyl aldehyde (methoxyphenol) for various fuel burns as a function of MCE. Dashed lines indicate a log fit of the form log(EF) inversely proportional to MCE. Dotted lines show a factor of 2 above and below the model. Note that peat (open pentagon) is not included in any of the fits. Different symbols represent fuel categories.

Chemical family EFs span  3 orders of magnitude, and therefore logarithmic fits (given as a dashed line in the semi-log graphs of Fig. 4) were applied to all the measurements excluding peat. Slopes range between −9.425 for sterols/triterpenoids and −14.637 for PAHs/methyl + oxy (fitted slopes, intercepts, and their errors for all chemical families are provided in Table S6). Three decimal places are provided for both the slope and intercept in order to reproduce the regression line. The R2 values for the sugars (Fig. 4) and resin acids/diterpenoids (Fig. S4) are noticeably lower at 0.32 and 0.31, respectively, than the other chemical families (R2∼0.4). This is primarily due to the high mass fraction of sugars and resin acids/diterpenoids found in conifers, as stated above, and suggests that high emissions of both types of compounds are indicative of burning conifers. Also, coniferous litter emitted high amounts of resin acids/diterpenoids. Removing conifers from the semi-logarithmic model for sugars yields a log10(EF${}_{\mathrm{sugars}}\right)=-\mathrm{10.299}$(MCE) + 9.361 with R2=0.66, and removing conifers and coniferous litter for resin acids/diterpenoids results in log10(EF${}_{\mathrm{resin}}\right)=-\mathrm{19.598}$(MCE) + 16.360 with R2=0.77. These R2 values are then more similar to those of other chemical families. In general, these results indicate that MCE can be used to estimate EFs for various chemicals across a broad range of fuels, including those not found in the western US wildlands except peat, with a minimal dependence on fuel type.

The goodness-of-fit for the multi-fuel regression models can be evaluated by comparing the predicted EFs to those measured for the various fuel types in this study and others (Liu et al., 2017). As evident in Fig. 4a, the predicted total I/SVOCs to observed EFs are between 0.7 and 11 times higher for shrubs, 0.90 and 0.97 for grasses, 0.22 and 0.74 for conifers, 0.63 and 3.0 for coniferous duff, and 0.28 and 0.85 for woody debris. The model is also compared to previously reported EFs from wildfires. Specifically, Liu et al. (2017) reported MCE values from three different California wildfires and total organic aerosol (OA) particle EFs, which are the most equivalent to total I/SVOC EFs measured here (though at MCE values of <0.8, we observed higher IVOC mass loadings in our chromatograms which would likely not be included in the OA EFs at lower particle mass loadings; May et al., 2013). Liu et al. (2017) measured MCE values of 0.935, 0.877, and 0.923, with OA EFs of 23.3, 30.9, and 18.8 g kg−1, respectively. The model given in Fig. 4a predicts total I/SVOC EFs of 8, 35, and 10 g kg−1 for those MCE values. The predicted EFs are within a factor of 2–3, consistent with measured total I/SVOC EFs reported here. No other previous experiments report chemical family EFs from wildfires with corresponding MCE values; thus the accuracy of applying the chemical family regressions cannot be evaluated at this time. Without this information, uncertainty in using the reported regression models in predicting EFs of various chemical families is estimated to be a factor of 3. However, this uncertainty in EFs is minor when compared to uncertainties in estimating the amount of fuel burned in large-scale carbon emission fire models (French et al., 2011; Urbanski et al., 2011), which is primarily due to high spatial and temporal variations in fuel loadings and lack of observational data. Thus, these regressions can be used to approximate EFs of various chemical families for a wide range of fuels and fuel mixtures from measured MCE values.

Figure 5 shows the fire-integrated EFs of four specific compounds, levoglucosan (sugar), fluoranthene (PAH), acetovanillone (methoxyphenol), and coniferyl aldehyde (methoxyphenol), as a function of MCE. Acetovanillone and coniferyl aldehyde, both methoxyphenols, have been reported previously as tracers for lignin pyrolysis and levoglucosan (and more broadly sugars) from cellulose (Hawthorne et al., 1989; Oros and Simoneit, 2001a; Schauer et al., 2001; Simoneit, 2002). In addition, fluoranthene and other PAHs are known carcinogenic compounds (Boffetta et al., 1997; Kim et al., 2013). EFs for levoglucosan, the most widely reported particulate tracer compound for biomass burning (Mazzoleni et al., 2007; Simoneit et al., 1999; Sullivan et al., 2014), range between  0.004 and 1 g kg−1 from this study. Hosseini et al. (2013) reported EFlevo for chaparral ecosystems at 0.02–0.1 g kg−1, similar to EFlevo for shrubs measured (0.004–0.1) in this study. Schuaer et al. (2001) provided average EFlevo for pine trees at 1.4 g kg−1, roughly a factor of 2 higher than the average 0.6 g kg−1 EFlevo for conifers of this study. Oros and Simoneit (2001a) examined levoglucosan emissions from various types of pine trees, with an average EFlevo of 0.02 g kg−1, a factor of 30 lower than reported here. Many reasons could explain this difference, such as different smoke sampling and filter extraction procedures and different MCE conditions during sampling. Regardless, the levoglucosan EFs reported in this study generally fall within the ranges measured by previous groups.

EFs for the compounds shown in Fig. 5 span 2–5 orders of magnitude across fire conditions and fuels, including fuels found outside of the western US. Similar to the chemical family EFs, peat displays significantly lower EFs (factor of  10) than the other fuels. Consequently, applied logarithmic fits, given as dashed lines in Fig. 5, exclude peat. Slopes of these fits range from −6.455 to −19.443 for the four displayed compounds. Figure 5a also shows that the R2 value for levoglucosan is the lowest (0.34) compared to the other compounds (0.40–0.63). This poorer correlation with MCE is similar to that seen for the sugar EFs in Fig. 4c, which shows that EFs from burning conifers were higher than the model predicted. Removing the conifers' levoglucosan EFs results in log(EF${}_{\mathrm{levo}}\right)=-\mathrm{9.547}$(MCE) + 8.041 and improves the correlation (R2=0.43). Nonetheless, these measurements suggest compound EFs do depend to some extent on fuel type in addition to MCE. However, the spread of measured EFs around the logarithmic fit in Fig. 5 indicates a factor of 3 uncertainty in estimating EFs from MCE.

In addition to the well-known biomass burning particulate compounds shown in Fig. 5, these measurements provide useful models to estimate EFs for hundreds of previously unreported compounds (not shown in Fig. 5). These commonly detected compounds (i.e., found in >10 burns and occurring in almost all fuel types) exhibited EFs that were inversely proportional to MCE. Regression parameters for compounds not displayed here are provided in the UCB-GLOBES FIREX mass spectral library. Many of these compounds still remain unidentified or unknown (see Fig. 3) but are now quantified as a function of fire conditions. Future work can be done to identify these compounds and ultimately, with the use of these regressions, estimate their contribution to I/SVOC mass in fresh smoke and model how they chemically transform in the atmosphere.

4 Conclusions

Smoke produced from burning a wide variety fuels, primarily from the western US wildlands, was collected onto quartz fiber filters at the Fire Science Laboratory and analyzed for elemental and organic carbon. The organic carbon fraction was further separated, identified, classified, and quantified using the TD-GC × GC-EI/VUV-HRToFMS with online derivatization. Each separated compound's mass spectrum, n-alkane retention index, chemical family, EF vs. MCE relationship, and fuel source are reported here in a publicly available mass spectral library (UCB-GLOBES FIREX) for future comparisons and identification of biomass burning organic compounds in atmospheric samples. Between 10 % and 65 % of the I/SVOC mass for each burn could be specifically identified or placed into a chemical family. Fuels within the same type tended to exhibit similar mass fractions, regardless of fire condition (as quantitated modified combustion efficiency, MCE). For example, similar unknown compounds accounted for  90 % of the total observed mass for the two woody debris burns (MCE = 0.78–0.98). Conifers exhibited similar sugar and resin acid/diterpenoid mass fractions (out of total classified mass) of 30 %–50 % and 2 %–20 %, respectively (MCE = 0.90–0.95). Burns of coniferous duff (MCE = 0.85–0.89) emitted higher classified mass fractions of methoxyphenols (6 %–18 %) than conifers. Peat, a semi-fossilized fuel, displayed a high classified mass fraction of non-cyclic aliphatic/oxy compounds (52 %). Shrubs showed the widest range in mass fractions, indicating fuels in this type were the most dissimilar.

Unlike mass fractions which depend primarily on fuel type, measured emission factors (EFs), classified into either organic carbon, chemical families, or specific compounds, primarily depended on fire conditions (MCE). Regardless of classification, EFs spanned 2–5 orders of magnitude from smoldering to flaming conditions. EFs were shown to follow an inversely proportional relationship to MCE across the wide variety of all fuels studied. However, peat EFs for chemical families (except non-cyclic aliphatic compounds) and specific compounds were approximately a factor of 10 lower than fuels at similar MCE values. This is likely due to significant differences in the fuel structure of peat. Furthermore, conifers exhibited higher sugar (factor of 5) and levoglucosan (factor of 3) emissions compared to other fuels within the same MCE range. This indicates that fuel type and specific fuels play some role in the EFs, though more minor compared to MCE. This is particularly true for nitrogen species and fuel-specific tracer compounds, i.e. compounds that are only emitted from a particular fuel, which will be discussed in a forthcoming paper. However, in general, EFs for these particulate compounds primarily depend on MCE and can be estimated from the fire conditions.

To provide modelers with useful relationships in estimating particle-phase I/SVOC emissions, logarithmic fits were applied to the measured EFs as a function of MCE. These regression models can be used to approximate EFs of I/SVOCs or their chemical families from average MCE of real wildfires, for which fuel loadings, fuel types, and fuel mixtures are often unknown. For example, comparison with Liu et al. (2017) shows the estimated particulate organics from the regression model to be within a factor of 2–3 of those measured in that study. The comparison between predicted and previously measured EFs is affected by methodology, concentration regime, and the definitions of I/SVOC. Regardless, these regression models provide approximate EFs (within a factor of 3) of numerous chemical families and organic species as solely a function of fire conditions across a wide variety of fuels. These regressions will allow modelers and other experimentalists to better define the chemical composition of smoke particles emitted from wildland burns in the western US and potentially other parts of the world.

Data availability
Data availability.

UCB-GLOBES can be downloaded from the Goldstein website at https://nature.berkeley.edu/ahg/data/MSLibrary/ (last access: 29 May 2018). The specific library for FIREX is “FSL_FIREX2016_vX.msp”, where “X” is the version number. The library contains information on all separated compounds observed during the FSL FIREX campaign in 2016 and will be periodically updated as compounds are matched across other campaigns. Observed emission factors for all of the observed compounds for each of the analyzed burns can be accessed for free through the NOAA FIREX data archives (http://esrl.noaa.gov/csd/groups/csd7/measurements/2016firex/FireLab/DataDownload/, last access: 29 May 2018).

Supplement
Supplement.

Author contributions
Author contributions.

CNJ, LEH, VS, RJY, NMK, KCB, and AHG formulated the science question and designed the experimental setup. CNJ, LEH, VS, RJY, and AEF collected the data at FSL. CNJ and AHG analyzed the I/SVOC data. VS and RJY analyzed the CO and CO2 data. RW organized the ALS campaign. CNJ wrote the manuscript, with all authors contributing comments.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

This work was supported by NOAA (NA16OAR4310107, NA16OAR4310103, and NA16OAR4310100) to UCB, UCR, and UM. Coty N. Jen acknowledges support from NSF PFS (AGS-1524211). Indonesian peat sampling was supported by NASA grant NNX13AP46G. Authors thank the staff at FSL and organizers of FIREX. The Advanced Light Source provided the VUV light and is supported by DOE. Special thanks are given to Bruce Rude and Kevin Wilson at LBNL for their assistance during the beamline campaign.

Edited by: Nga Lee Ng
Reviewed by: two anonymous referees

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## Introduction to quantum computing with Q# – Part 12, Bell’s inequality

After a short multi-part detour into the world of quantum cryptography, in this part 12 of the series, we are going to return to some of the foundational concepts of quantum mechanics, and look at the programmatic verification of Bell’s inequality.

### Where were we last time?

In part 5 we looked in detail at the phenomenon of entanglement. In that blog post, which I recommend you have a look at again before proceeding here, we introduced the famous EPR thought experiment and discussed how Bohr and others responded to this paradox. We also explored the problem of non-locality which appears to be a consequence of EPR. In fact, for Albert Einstein it was the key argument proving that quantum mechanics is incomplete. Einstein famously objected to any violation of local realism and therefore suggested that quantum mechanics is incomplete. Namely, he suggested to seek for a theory and ontology based on the “hidden variables” concept, which the mathematical formalism of quantum mechanics doesn’t take into account, and which, in his view, pre-determined the quantum correlations between the particles making up an EPR pair.

Finally, we briefly mentioned that John S. Bell, in his 1964 “On The Einstein Podolsky Rosen Paradox” formukated a test that could be used to determine whether Einstein was correct in his search for a “hidden variables” approach, or whether the nature itself prohibits such theories. And this is exactly where we will pick up in this post.

### Bell’s theorem

Bell introduced a mathematical framework for inequalities that will be satisfied if one relies on the existence of hidden variables when analyzing an entangled state (later generalized by Clauser, Horne, Shimony, and Holt). He derived his inequality for the singlet state – one of the Bell states we discussed in this series already:

$$\ket{11} \rightarrow \frac{1}{\sqrt{2}}(\ket{01} – \ket{10}) = \ket{\Psi^-}$$

Bell imagined the following set up, based on Bohm’s EPR experiment variant, which we already covered in part 5. Two entangled particles in a singlet state $\ket{\Psi^-}$ fly off in the opposite directions, where they hit detectors (e.g.
Stern-Gerlach spin measuring apparatuses) configured to be measuring at angles $\theta_A$ and $\theta_B$. Obviously if $\theta_A = \theta_B$, then the original conditions from Bohm’s EPR experiment are recovered and the measured values $A(\theta_A)$ and $B(\theta_B)$ are always perfectly correlated – opposite to each other (because we started with $\ket{\Psi^-}$). However, if the two detectors are at different angles, Bell realized that the quantum mechanically predicted correlation between the obtained measurements can be written as:

$$\braket{A(\theta)B(\phi)} = -cos(\theta_A – \theta_B)$$

He then followed to derive the inequality that gets violated by the predictions of quantum mechanics, but not by hidden variable theories assuming locality.

The original inequality that Bell derived was as follows:

$$|P(\vec{a},\vec{b}) – P(\vec{a},\vec{c})| – P(\vec{b},\vec{c}) \leq 1$$

where $P_{xy}$ refers to the average value of the product of the spins of the measured particles (qubits), $a$ and $b$ are unit vectors corresponding to the two detectors and $c$ is any other unit vector. A step-by-step derivation of the inequality is beyond our scope here, and there exist various excellent resources for that already, including Bell’s original paper and the briefly mentioned CHSH paper. John Presskill has a very accessible example using classical coins in his lecture notes, and Maccone builds upon this to provide a simple proof himself. It is also worth noting that Bell’s inequality has now been generalized to apply to a class of problems, and is available in numerous variants, hence it is being referred to in plural: inequalities.

That said, the above mentioned $\braket{A(\theta)B(\phi)}$ correlation is important, because it tells us that:

• $P(\vec{a},\vec{b}) = -cos(\theta_a – \theta_b)$
• $P(\vec{a},\vec{c}) = -cos(\theta_a – \theta_c)$
• $P(\vec{b},\vec{c}) = -cos(\theta_b – \theta_c)$

We know that cosine is a periodic function, with its values oscillating between $1$ and $-1$ so it is relatively easy to deduce from the above that there will be some angles, for which Bell’s inequalities would not be violated, and some for which the violation will be maximal. Namely, when:

• $\theta_a = 0^\circ$
• $\theta_b = \frac{\pi}{3} = 60^\circ$
• $\theta_c = \frac{\pi}{2} = 120^\circ$

Then:

• $P(\vec{a},\vec{b}) = -0.5$
• $P(\vec{a},\vec{c}) = 0.5$
• $P(\vec{b},\vec{c}) = -0.5$

And substituting these values into the inequality:

$$1.5 \nleq 1$$

Which of course is a flagrant violation of the Bell’s inequality. As such, the Bell’s theorem states:

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

Blanchard writes in The Message of Quantum Science:

Standard quantum mechanics is the only probabilistic theory where the probabilities are postulated ab initio and are not a consequence of hidden deterministic processes at a deeper level. Such processes are called hidden variables and reflect the ignorance of the observer. However, the majority of physicists today believe that the probabilities in quantum mechanics are not attributable to the ignorance of hypothetical hidden variables, but are of a fundamentally different nature. This is corroborated by Bell’s inequalities, which hold in any theory with (local) hidden variables, and which were experimentally found to be violated.

Bell’s theorem turned out to be a spectacular success and was experimentally confirmed numerous times. The first one to achieve an empirical confirmation of Bell’s inequalities was Alain Aspect at the École supérieure d’optique in Orsay, the results of which were published in 1982. Aspect wrote:

(…) our experiment yields the strongest violation of Bell’s inequalities ever achieved, and excellent agreement with quantum mechanics. Since it is a straightforward transposition of the ideal Einstein-Podolsky-Rosen-Bohm scheme, the experimental procedure is very simple, and needs no auxiliary measurements as in previous experiments with single-channel polarizers. We are thus led to the rejection of realistic local theories if we accept the assumption that there
is no bias in the detected samples: Experiments support this natural assumption.

Aspect’s experiment left two major loopholes, which have since been (mostly) closed by other experimental setups that followed. At this point is clear – with profound consequences – any local realist theory, as championed by Einstein, is wrong. This also has dramatic epistemological consequences about the very nature of reality. Elegance and Enigma dedicated a chapter to survey various physicists about the implications of Bell’s theorem. Tim Mauldin wrote there:

Assuming we can accept what we seem to see, namely, that every experiment has a unique outcome (contrary to the many-worlds view) and that the correlations between experiments performed at spacelike separation violate Bell’s inequality, then we can conclude that nature is nonlocal. That is, in some way certain events at spacelike separation are physically connected to each other. Einstein’s dream of a perfectly local physics, in which the occurrences in any small region of the spacetime depend only on what happens in that region, cannot be fulfilled. It is an open question what the implications of this fact are for the relativistic account of spacetime.

A lot of authors point to Bell’s theorem as a direct confirmation of non-locality in the fabric of nature and David Wallace even hinted at the superluminal interactions. He wrote:

The violations of Bell’s inequalities seem to tell us is that the dynamics of the microworld allows interactions that are faster than light (or slower than light but backward in time, I guess, if that really means anything). If the only interactions in the world are subluminal, Bell’s inequalities would be satisfied; they’re not, so systems can interact superluminally.

However, the view adopted in this post series, namely the reality-without-realism concept, is a lot more subtle, and more in-line with Bohr’s spirt of Copenhagen, who also advocated locality, albeit much differently than Einstein. This is nicely expressed by David Mermin, who, referring to Asher Peres’s famous statement “unperformed tests have no outcomes”, provides the following viewpoint:

So for me, nonlocality is too unsubtle a conclusion to draw from the violation of Bell inequalities. My preference is for conclusions that focus on the impropriety of seeking explanations for what might have happened but didn’t. Evolution has hard-wired us to demand such explanations, since it was crucial for our ancestors to anticipate the consequences of all possible contingencies in their (classical) struggles.

Finally, this view is corroberrated by David Griffiths, who provided a very on-point and rather sharp-witted summary of the Bell’s theorem and the EPR paradox:

It is a curious twist of fathe that the EPR paradox, which assumed locality in order to prove realism, led finally to the demis of locality and left the issue of realism undecided – the outcome (as Bell put it) einstein would have liked least. Most physicists today consider that if they can’t have local realism, there’s not much point in realism at all, and for this reason nonlocal hidden variable theories occupy a rather peripheral niche.

### Bell’s inequalities in Q#

While it took almost two decades until Bell’s original idea was verified experminetally, we can now easily test for Bell’s inequalities using a quantum computer. Just like there are many variants of these inequalities, we could choose a number of ways to approach this problem in Q#.

As we already discussed, quantum mechanics predicts that maximal violations of Bell’s inequalities will happen for the following angles:

• $\theta = \frac{\pi}{3} = 60^\circ$ angle between $\vec{a}$ and $\vec{b}$
• $\phi = \frac{\pi}{3} = 60^\circ$ angle between $\vec{b}$ and $\vec{c}$
• ($\theta + \phi) = \frac{2\pi}{3} = 120^\circ$ angle between $\vec{a}$ and $\vec{c}$

A very simple quantum computational model to validate Bell’s inequalities was summarized by Diego Garcia-Martin and German Sierra from Universidad Autonoma de Madrid in their paper Five Experimental Tests on the 5-qubit IBM Quantum Computer. We will use their approach here, with some small modifications. We can measure $P(\vec{a},\vec{b})$, $P(\vec{a},\vec{c})$ and $P(\vec{b},\vec{b})$ by running 3 separate quantum circuits for each of the cases.

We shall begin with the singlet state $\ket{\Psi^-}$. We already know from the earlier posts, that we can obtain it from a pair of qubits in the state $\ket{11}$, running the $H$ gate on the first qubit, and then a $CNOT$ gate over both of them. Thus, given qubits $q1, q2$, the following Q# operation creates $\ket{\Psi^-}$:

We know that we need to create an experimental setup that will cater for the three different cases, corresponding to $P(\vec{a},\vec{b})$, $P(\vec{a},\vec{c})$ and $P(\vec{b},\vec{c})$. Therefore, we will define three separate Q# operations for them.

The first operation will be called $BellsInequalityAB$, will correspond to $P(\vec{a},\vec{b})$ and will use $\theta = \frac{\pi}{3} = 60^\circ$ angle between $\vec{a}$ and $\vec{b}$. The circuit representing the operation is shown below.

We already wrote the Q# code for the initial part of the circuit – setting up of $\ket{\Psi^-}$ state. What follows is transformation for measurement purposes. Since Q# doesn’t support measurement in an arbitrary basis, but instead only allows measurements in one of the Pauli bases, we will apply an $R_z$ rotation of $\frac{\pi}{3}$ radians. On the other hand, the circuit defines an $H$ gate followed by a standard basis $PauliZ$ measurement to achieve measurement along the X basis, but that can be expressed in the code by a single operation – measurement in $PauliX$ basis – instead. The Q# code for the circuit is shown next.

The return is a tuple of two classical bits – a pair of $00$, $10$, $01$ or $11$. Pauli X basis measurement is done using the $MResetX$ built-in operation.

Our second Bell’s inequality operation will be called $BellsInequalityAC$, will correspond to $P(\vec{a},\vec{c})$ and will use $\theta = \frac{2\pi}{3} = 120^\circ$ angle between $\vec{a}$ and $\vec{c}$. The circuit is presented next.

The Q# code is almost identical to the previous case – with the only exception being the wider angle.

The final circuit will allow us to measure the results for $P(\vec{b},\vec{c})$ and – as should be apparent at this point – will depend on $\theta = \frac{2\pi}{3} = 60^\circ$ angle between $\vec{b}$ and $\vec{c}$. The circuit is shown next – it is, unurprisngly, similar to the previous two variants, with the notable difference of the $R_z$ rotation gate being now applies to both qubits.

The corresponding Q# code is:

At this point we have all the necessary Q# operations ready – what we are still missing, is the code to orchestrate their execution using a reasonable sample size, as well as some code that will output the results in an accessible format. We will, as usually, set up a small self contained Q# program for that purpose. The entry point is shown below:

This snippet hints at the existence of a helper $Run$ operation, which we are yet to see, that takes in two input parameters – a string identifier for the operation for visualization purposes as well as a delegate pointing to one of our three Bell inequality component operations. Such setup will allow us to reuse the output/presentation code for each of the cases. One other interesting tidbit to mention is that to make the final calculation for the Bell’s inequality, we make use of the built-in $AbsD$ function, which is part of the $Microsoft.Quantum.Math$ namespace and returns the absolute value of a double-precision floating-point number.

The $Run$ operation is shown next, completing our code.

Ultimately, since we did all the heavy Q# parts already, this should be straight forward to read. The main purpose here is to provide a standardized way of dealing with and outputting the results of the other three operations. The sample size is fixed to 4096 runs. Since all three Bell’s inequality operations we wrote earlier return a tuple pair corresponding to the two classical bits measured, we use a 4 element mutable array to keep a running total of the obtained combinations. The final $P_{xy}$ is calculated as the sum of the probabilities of measuring $00$ or $11$ minus the probabilities of finding $01$ or $10$.

### Experimental results

When running our program now, we should see results similar to the ones below.

This is very encouraging – because it aligns perfectly with the predictions of quantum mechanics. Remember, that we expected the maximum violation to be at 1.5, because:

• $P(\vec{a},\vec{b}) = cos(60^\circ) = -0.5$
• $P(\vec{a},\vec{c}) = cos(120^\circ) = 0.5$
• $P(\vec{b},\vec{c}) = cos(60^\circ) = -0.5$

The experimentally obtained results for a trial run of 4096 repetitions are:

• $P(\vec{a},\vec{b}) = -0,491943359375$
• $P(\vec{a},\vec{c}) = 0,498291015625$
• $P(\vec{b},\vec{c}) = -0,508056640625$

This fully confirms the nonlocal nature of quantum phenomena, and excludes the possibility of local hidden variable theories. Of course at this point we are only running this in a local simulator, not on real quantum hardware, but it’s nevertheless excellent to see that we did not make any mistakes in our reasoning. In fact, taking a step back, and considering how difficult it was to experimentally confirm the Bell’s theorem, it is quite mind boggling that we can simply write less than a 100 lines of Q# and quickly get the answer to the question that in the first half of the 20th century has troubled some of the greatest mind in physics. Moreover, very soon we’ll be able to run these in Azure on the actual quantum hardware too.

At this point it’s worth mentioning that other quantum cloud providers have already allowed similar Bell’s inequality verification to be done on their computers. The aforementioned paper from Garcia-Martin and Sierra documents their findings from running the tests on the 5-qubit IBM hardware. Their experimental results were:

• $P(\vec{a},\vec{b}) = −0.392±0.014$
• $P(\vec{a},\vec{c}) = 0.401±0.009$
• $P(\vec{b},\vec{c}) = −0.389±0.012$

This still produces the inequality violation, albeit not as large as on the Q# local simulator:

$$1.182 ± 0.020 \nleq 1$$

Their conclusion, in that case (back in 2018), was that volatility and inefficiency of the quantum circuits were still too high for scientifc use cases:

Overall, the results obtained in these experiments, although moderately good in most cases, are still far from optimum.

However, the fidelity of quantum hardware has dramatically increased in the last years so I am confident that we will see real life problem solving and useful physical experiments being carried out on quantum computers in the cloud very soon. I am excited and curious to soon be able to see the results on Azure Quantum.

### Summary

Bell’s theorem is one of the most puzzling aspects of the nature of reality – Henry Stapp even pronounced it to be “the most profoud discovery in science”. It has forced physicists to effectively choose to give up either locality or realism, or to reach into exotic theories of Bohmian or Everrettian type. With regards to quantum computing, I continuously marvel at the fact that we can use quantum computing to experimentally verify these problems without much efforts – and doing that in Q# is a true pleasure.

This concludes part 12 of this series – last part in 2020 – and we’ll continue after New Year’s! Happy Holidays!

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# Weil restriction

In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.

## Definition

Let L/k be a finite extension of fields, and X a variety defined over L. The functor $\mathrm{Res}_{L/k}X$ from k-schemesop to sets is defined by

$\mathrm{Res}_{L/k}X(S) = X(S \times_k L)$

(In particular, the k-rational points of $\mathrm{Res}_{L/k}X$ are the L-rational points of X.) The variety that represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists.

From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L $\to$ Spec k and is right adjoint to fiber product, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed topoi, and the hypotheses on X can be weakened to e.g. stacks. This comes at the cost of having less control over the behavior of the restriction of scalars.

## Properties

For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties. The dimension of the resulting variety is multiplied by the degree of the extension.

Under appropriate hypotheses (e.g., flat, proper, finitely presented), any morphism $T \to S$ of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability.

## Examples and applications

1) Let L be a finite extension of k of degree s. Then $\mathrm{Res}_{L/k}$(Spec L) = Spec(k) and $Res_{L/k}\mathbb{A}^1$ is an s-dimensional affine space $\mathbb{A}^s$ over Spec k.

2) If X is an affine L-variety, defined by

$X = \text{Spec} L[x_1, \dots, x_n]/(f_1,\dots,f_m)$

we can write $\mathrm{Res}_{L/k}X$ as Spec $k[y_{i,j}]/(g_{l,r})$, where yi,j ($1 \leq i \leq n, 1 \leq j \leq s$) are new variables, and gl,r ($1 \leq l \leq m, 1 \leq r \leq s$) are polynomials in $y_{i,j}$ given by taking a k-basis $e_1, \dots, e_s$ of L and setting $x_i = y_{i,1}e_1 + \dots + y_{i,s}e_s$ and $f_t = g_{t,1}e_1 + \dots + g_{t,s}e_s$.

3) Restriction of scalars over a finite extension of fields takes group schemes to group schemes.

In particular:

4) The torus

$\mathbb{S} := \mathrm{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_m$

where Gm denotes the multiplicative group, plays a significant role in Hodge theory, since the Tannakian category of real Hodge structures is equivalent to the category of representations of S. The real points have a Lie group structure isomorphic to $\mathbb{C}^\times$. See Mumford–Tate group.

5) The Weil restriction of a (commutative) group variety is again a (commutative) group variety, if L is separable over k. Aleksander Momot applied restriction of scalars on group varieties and obtained numerous generalizations of classical results from transcendence theory.

6) Restriction of scalars on abelian varieties (e.g. elliptic curves) yields abelian varieties, if L is separable over k. James Milne used this to reduce the Birch and Swinnerton-Dyer conjecture for abelian varieties over all number fields to the same conjecture over the rationals.

7) In elliptic curve cryptography, the Weil descent attack uses the Weil restriction to transform a discrete logarithm problem on an elliptic curve over a finite extension field L/K, into a discrete log problem on the Jacobian variety of a hyperelliptic curve over the base field K, that is potentially easier to solve because of K's smaller size.

## Weil restrictions vs. Greenberg transforms

Restriction of scalars is similar to the Greenberg transform, but does not generalize it, since the ring of Witt vectors on a commutative algebra A is not in general an A-algebra.

## References

The original reference is Section 1.3 of Weil's 1959-1960 Lectures, published as:

• Andre Weil. "Adeles and Algebraic Groups", Progress in Math. 23, Birkhäuser 1982. Notes of Lectures given 1959-1960.

Other references:

• Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud. "Néron models", Springer-Verlag, Berlin 1990.
• James S. Milne. "On the arithmetic of abelian varieties", Invent. Math. 17 (1972) 177-190.


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In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every nonzero vector of V. According to that sign, the quadratic form is called positive definite or negative definite.

A semidefinite (or semi-definite) quadratic form is defined in the same way, except that "positive" and "negative" are replaced by "not negative" and "not positive", respectively. An indefinite quadratic form is one that takes on both positive and negative values.

More generally, the definition applies to a vector space over an ordered field.[1]

## Associated symmetric bilinear form

Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space.[2] A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations:

$\, Q(x) = B(x,x)$
$\, B(x,y) = B(y,x) = \tfrac{1}{2} (Q(x + y) - Q(x) - Q(y))$

## Example

As an example, let $V=\mathbb{R}^2$, and consider the quadratic form

$Q(x)=c_1{x_1}^2+c_2{x_2}^2 \,$

where x = (x1, x2) and c1 and c2 are constants. If c1 > 0 and c2 > 0, the quadratic form Q is positive definite. If one of the constants is positive and the other is zero, then Q is positive semidefinite. If c1 > 0 and c2 < 0, then Q is indefinite.


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# Draw an Ogive for the Data Given Beelow and from the Graph Determine: - ICSE Class 10 - Mathematics

ConceptMeasures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data

#### Question

Draw an ogive for the data given beelow and from the graph determine:

(1) the median marks

(2) the number of students who obtained more than 75% marks

 Marks No.of students 0-9 5 10-19 9 20-29 16 30-39 22 40-49 26 50-59 18 60-69 11 70-79 6 80-89 4 90-99 3

#### Solution

 Marks No.of students Cumulative frequency 05-9.5 5 5 9.5-19.5 9 14 19.5-29.5 16 30 29.5-39.5 22 52 39.5-49.5 26 78 49.5-59.5 18 96 59.5-69.5 11 107 69.5-79.5 6 113 79.5-89.5 4 117 89.5-99.5 3 120

Total number of terms = 120

∴  Median=120/2=60^(th)term

Through mark 60, draw a parallel line to x-axis whicch meets the  curve at A. from A draw  a perpendicular to x-axis meeting it at B.

The value of point B the median =43

Total marks=100

75% of total marks =75/100xx100=75

The number of students getting more than 75% marks = 120-111 = 9

Is there an error in this question or solution?

#### APPEARS IN

Selina Solution for Selina ICSE Concise Mathematics for Class 10 (2018-2019) (2017 to Current)
Chapter 24: Measure of Central Tendency(Mean, Median, Quartiles and Mode)
24E | Q: 3
Solution Draw an Ogive for the Data Given Beelow and from the Graph Determine: Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data.
S


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The construction of solar thermal power plants with several thousand $m2$ of collector area requires quality control measures for components, subsystems, and the entire collector rows. While quality control has a significant potential to increase the solar field efficiency, the main objective is to assure high-quality standards for the whole solar field. Quality control, assembly documentation, and performance measurements are required by the investors. Based on previous R&D work in collector development and prototype qualification, measurement systems have been developed for use in solar field construction and operation supervision. In particular, close-range photogrammetry can be used to measure the geometry of collector steel structures. The measurement system consists of a digital camera, which moves around the structure automatically while shooting photos of the concentrator structure from various positions. The photos are evaluated with photogrammetry software to check the assembly quality. The whole measurement and evaluation procedure is computer controlled and is fast enough to be integrated in a solar collector production line. This paper deals with the required measurement accuracy and shows ways to reach, maintain, and control this accuracy in the rough environment of an on-site production line.

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# How to obtain a convergent solution iteratively for a linear system of equations

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is:

$$\underbrace{\begin{bmatrix} r_{11} & r_{12} & r_{13} & \cdots & r_{1j} \\ r_{21} & r_{22} & r_{23} & \cdots & r_{2j} \\ r_{31} & r_{32} & r_{33} & \cdots & r_{3j} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ r_{i1} & r_{i2} & r_{i3} & \cdots & r_{ij} \end{bmatrix}}_\textit{R} \underbrace{\begin{bmatrix} a_{1} & 0 & 0 & \cdots & 0 \\ 0 & a_{2} & 0 & \cdots & 0 \\ 0 & 0 & a_{3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & a_{j} \end{bmatrix}}_\textit{A} + \underbrace{\begin{bmatrix} b_{1} & b_{2} & b_{3} & \cdots & b_{j} \\ b_{1} & b_{2} & b_{3} & \cdots & b_{j} \\ b_{1} & b_{2} & b_{3} & \cdots & b_{j} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ b_{1} & b_{2} & b_{3} & \cdots & b_{j} \end{bmatrix}}_\textit{B} = \underbrace{\begin{bmatrix} c_{1} & c_{1} & c_{1} & \cdots & c_{1} \\ c_{2} & c_{2} & c_{2} & \cdots & c_{2} \\ c_{3} & c_{3} & c_{3} & \cdots & c_{3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c_{i} & c_{i} & c_{i} & \cdots & c_{i} \end{bmatrix}}_\textit{C}\\ R_{i, j}A_{j, j} + B_{i, j} = C_{i, j}$$

Now matrix R is fully known (input), matrices A, B, C are unknown. I am working on an iterative procedure in which I provide initial guesses for A and B, then calculate C. The iteration is to be carried out till I get a converged value for C (output I require). So far no luck in choosing the initial guess too.

To illustrate the problem I have given an example for a $2 \times 2$ matrix:

$$\begin{bmatrix} r_{11} & r_{12} \\ r_{21} & r_{22} \\ \end{bmatrix} \begin{bmatrix} a_{1} & 0 \\ 0 & a_{2} \\ \end{bmatrix} + \begin{bmatrix} b_{1} & b_{2} \\ b_{1} & b_{2} \\ \end{bmatrix} = \begin{bmatrix} c_{1} & c_{1} \\ c_{2} & c_{2} \\ \end{bmatrix}\\$$ this gives us:

$$a_{1}r_{11} + b_{1} = a_{2}r_{12} + b_{2} = c_{1}\\ a_{1}r_{21} + b_{1} = a_{2}r_{22} + b_{2} = c_{2}$$

From these equations I get:

$$a_{2} = a_{1}\frac{(r_{11} - r_{21})}{(r_{12} - r_{22})}; \quad b_{2}=b_{1} + a_{1}\frac{(r_{11} - r_{21})(r_{11} - r_{12})}{(r_{12} - r_{22})}$$

This helps in the iterative method, but I would like to generalize it for a bigger matrix. Can this be solved with Mathematica?

UPDATE

I have added two reproducible Python codes that should be helpful for a $2 \times 2$ matrix:

### Code 1

import numpy as np
R = np.matrix([[2.5, 2.9], [2.3, 2.7]])
m = R.shape[0]
n = R.shape[1]
A = np.matrix(np.zeros(shape=(n,n)))
B = np.matrix(np.zeros(shape=(m,n)))
A[0,0] = 1
B[0,0] = 1
k = 0
for i in range (1, 10000):
A[0,0] = A[0, 0] - 0.0000001
B[0,0] = B[0, 0] - 0.0000001
for j in range(1, n):
A[j, j] = A[(j-1), (j-1)] * ((R[0, 0] - R[1, 0]) / (R[0, 1] - R[1, 1]))
B[0, j] = B[0, 0] + A[0, 0] * (((R[0, 0] - R[1, 0]) / (R[0, 1] - R[1, 1])) * (R[0, 0] - R[0, 1]))
B[1, 0] = B[0, 0]
B[1, j] = B[0, 0] + A[0, 0] * (((R[0, 0] - R[1, 0]) / (R[0, 1] - R[1, 1])) * (R[0, 0] - R[0, 1]))
C = R * A + B
C_convergence = np.all(np.all(np.diff(C, axis = 1) == 0, axis = 1), axis = 0)
k = k + 1
print k
print C

### Code 2

import numpy as np
R = np.matrix([[2.5, 2.9], [2.3, 2.7]])
m = R.shape[0]
n = R.shape[1]
A = np.matrix(np.zeros(shape=(n,n)))
B = np.matrix(np.zeros(shape=(m,n)))
A[0,0] = 1
B[0,0] = 1
k = 0
C_convergence = False
k = 0
while C_convergence == False:
A[0,0] = A[0, 0] - 0.0000001
B[0,0] = B[0, 0] - 0.0000001
for j in range(1, n):
A[j, j] = A[(j-1), (j-1)] * ((R[0, 0] - R[1, 0]) / (R[0, 1] - R[1, 1]))
B[0, j] = B[0, 0] + A[0, 0] * (((R[0, 0] - R[1, 0]) / (R[0, 1] - R[1, 1])) * (R[0, 0] - R[0, 1]))
B[1, 0] = B[0, 0]
B[1, j] = B[0, 0] + A[0, 0] * (((R[0, 0] - R[1, 0]) / (R[0, 1] - R[1, 1])) * (R[0, 0] - R[0, 1]))
C = R * A + B
C_convergence = np.all(np.all(np.diff(C, axis = 1) == 0, axis = 1), axis = 0)
k = k + 1
print k
print C

• In the 2x2 example you have 4 equations and 6 values to solve for {a1,a2,b1,b2,c1,c2}. Am I wrong? – Dr. belisarius Feb 20 '15 at 19:02
• Yes, but {a1, a2, b1, b2} are initial guesses and I shall iterate it till I get a converged {c1, c2}. – nxkryptor Feb 20 '15 at 19:14
• Could you please exemplify your iteration process with a numerical example in the 2x2 case? I just get an exact solution with 2 degrees of freedom in one step – Dr. belisarius Feb 20 '15 at 19:39
• I tried in Python which is provided in the update. – nxkryptor Feb 20 '15 at 20:20

## 1 Answer

This can be handled by Solve straightforwardly:

r = RandomInteger[{-10, 10}, {2, 2}];
a = {{a1, 0}, {0, a2}};
b = {{b1, b2}, {b1, b2}};
c = {{c1, c1}, {c2, c2}};
Solve[r.a + b == c, {a1, a2, b1, b2, c1, c2}]

{{a2 -> 18 a1, b2 -> -45 a1 + b1, c1 -> -9 a1 + b1, c2 -> 9 a1 + b1}}

You can interpret this as saying that for any a1 and b1, the other parameters, if calculated this way, will fulfill your equation.


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# Link between the Grassmann algebra and spinors

What is the exact link between spinors and the Grassmann algebra? I'm pretty sure there's one, based on the following:

• The Berezin integral in path integrals is done over the Grassmann algebra of $\Bbb C$
• There's a mapping from the tangent bundle of a supermanifold (which is locally $\Bbb R^n \times$ the Grassmann algebra), and the spinor bundle of a spin manifold
• On the other hand, spinors are defined just as the vector space associated to a representation of the spin group.

I'm not quite sure what the relation is. If we deal with spinors larger than basic Dirac spinors $\Bbb C^2$ (like a product of two spinors), are they automatically described by a member of the Grassmann algebra? Can it be shown that only those transform properly under the spin group?

• An obvious link is provided by the spin-statistics theorem. – Qmechanic Jun 28 '17 at 9:23

## 1 Answer

There is no direct link between the Graßmann algebra and spinors. The reason both are connected in physics is the spin-statistic theorem: The fields/operators associated to fermionic objects, i.e. Dirac spinors, anticommute, which is basically the defining property of the Graßmann algebra.

The algebra that actually controls the spinor representation - whose irreducible representation actually essentially defines what a Dirac spinor is - is the Clifford algebra, not the Graßmann algebra. Quantization links the two, see my and David Bar Moshe's answers to this question. The gist is that the defining relation of the Graßmann algebra is $\theta^2 = 0$, while that of the Clifford algebra is $\theta^2 = \hbar \eta(\theta,\theta)$ for the metric $\eta$. For a "fermionic"/anticommuting classical variable, the $\theta^2$ is the super-Poisson bracket. So precisely like the quantization of bosonic variables deforms the Poisson bracket by factors of order $\hbar$ and higher, the quantization of fermionic variables deforms the anticommutator, and consequently the Graßmann algebra into the Clifford algebra.

• Isn't that link between supermanifolds and spin manifolds also true in the classical case, though? I don't think I've seen anything indicating quantization involved. – Slereah Jun 28 '17 at 9:24
• @Slereah You'll have to be more precise about what link you mean in order for me to answer that. – ACuriousMind Jun 28 '17 at 9:25
• The exact theorem is (from arxiv.org/abs/dg-ga/9704002) : For a spin manifold $(M_1, g)$ with a spinor bundle $\pi_S : S \to M_1$ and a supermanifold $(M_2, g)$, for every $p \in M_1$, there is a canonical isomorphism of $\mathbb Z_2$-graded vector spaces $$\iota : T_p M_1 + S_p^* \to T_p M_2$$ – Slereah Jun 28 '17 at 9:26


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# XOR Power

Logic Level 3

$$A \text{ XOR } B$$ is true if either $$A$$ is true or $$B$$ is true (but not both). ($$XOR$$ stands for "exclusive or".)

Is it possible to make a statement equivalent to $$A \text{ AND } B$$ using only $$XOR$$ as an operation? (Please note: only XOR! NOT isn't allowed.)

×


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# Is this NFA correct?

For the language L = $\{\Sigma^*. 0 .\Sigma^5 . 1. \Sigma^*\}$

The NFA must have 8 states. Also, what would be the upper bound on the number of states of a DFA recognizing L.

-
The NFA looks good, though you should have said that $\Sigma=\{0,1\}$. There is no upper bound on the number of states of a DFA recognizing $L$; did you mean lower bound? –  Brian M. Scott Nov 8 '12 at 7:50

Delete the $\lambda$-edge from $q_1$ to $q_8$. Presently the automaton accepts every string over $\{0,1\}$.
In your question delete the set-brackets in $\{ \Sigma^* \cdots\Sigma^* \}$. In itself, $\Sigma^*$ is a set (of strings), so it is not necessary (in fact wrong) to add another `level' of sets.