billions of personal devices , ranging from in - car gps to mobile phones and fitness bracelets , connect us to the cloud .these ubiquitous interconnections of the physical and the digital world are opening up a host of new opportunities for predictive mobility models .each user of these devices produces rich information that can help us to capture their daily mobility routine .this core knowledge , when obtained from massive number of individuals , impacts a wide range of areas such as health monitoring  , ubiquitous computing  , disaster response  or smart traffic management  .  in the age of ubiquitous computing ,recent contributions to mobility modeling have flourished in computer science  , transportation engineering  , geographic information sciences  , and complexity sciences  .while these findings have enhanced our level of understanding of mobility modeling we need further work to tackle the problem of individual predictability .human mobility has been studied using a multitude of proxies ( for example call detail records ( cdr ) , gps , wifi , travel surveys ) , and a variety of techniques have been suggested for predictive models , such as markov chains , naive bayes , artificial neural networks , time series analysis .studies report varying results for the predictive power of these models , with accuracy as high as 93% and as low as under 40% . in this paper we set out to uncover the reasons behind these differences in performance by a thorough investigation of the factors that may influence an estimation of mobility predictability .the key contributions of this paper are :  1 .  we describe the factors that have influenced the various ranges when estimating predictability .these include : ( a ) does the analysis concern the upper limit of predictability , or actual next - place prediction ?( b ) what is the specific formulation of the prediction problem ?e.g. is the goal to predict the next location , or is the goal to identify location in the next time - bin ?( c ) what is the spatial resolution ?e.g. is the analysis based on gps vs. cdr data ?( d ) what is the temporal resolution e.g. minutes , hours ? 2 .we quantify the amount of explorations and locations visited only once , and show that these are key limiting factors in the accuracy of predictions for individual mobility .3 .  we measure the predictive power of a number of contextual features ( e.g. social proximity , time , call / sms ) .4 .  we study the problem of predictability of human mobility using a novel , longitudinal , high - precision location dataset for more than 400 users .the rest of the paper is organized as follows .we first provide an overview of related work in the field of human mobility prediction .next , we introduce the dataset and describe the preprocessing steps . in the subsequent sectionwe describe the baseline models , and compare their performances .finally we introduce the exploration prediction problem and report the performance of the predictive models .in a seminal paper song _et al_.  investigate the limits of predictability of human mobility , using call detail records ( cdr ) as proxy for human movement .in their analysis , the authors discretize location into a sequence of places , and estimate an upper limit for the predictive performance using fano s inequality on the temporal entropy of visits .their results show that for a majority of users , this upper bound is surprisingly high ( 93% ) .this framework has been further explored to refine the upper limit . specifically , lin et al . study the effects of spatial and temporal resolution on the predictability limit , smith et al . consider the spatial reachability constraints when selecting the next place to visit , and obtain a tighter upper bound of 81 - 85% , and lu et al . analyze the predictability of the population of haiti after the earthquake in 2010 , and show that the upper limit of predictability remains as high as 85% .  the work described above focuses on the upper limit of predictability based on estimating the entropy of trajectorieswhen the topic is actual prediction performance , some of the most studied models are markov chains , where the probability of the next location is assumed to depend only on the current location .markov chains have been applied to a variety of data sets .lu et al . applied markov chain models to cdr - based locations in cote divore , with a prediction goal of estimating the last location of the day at the prefecture ( county ) level . under these conditionsthe models perform extremely well , reaching an accuracy of over 90% . in  authors apply the markov models to wifi traces at darthmouth campus and find that the best performing model is order 2 and has a median accuracy of about .finally , bapierre et al . applied a variable - order markov chain to the reality mining  and geolife  datasets .another frequently used category of models is naive bayes , where the probability of next location is factorized as independent probabilities for a number of context variables .gao et al . applied this approach to the nokia data challenge dataset  using time and location features , and obtained an accuracy of approximately 50% .do et al . applied the same technique but used a larger number of features including also sms , calls and bluetooth proximity , and obtained an accuracy of approximately 60% . in a subsequent paper the same authors then explore a kernel density estimation approach for improving performance .a number of more complex methods have also been explored in the literature , including non - linear time series  , principal component analysis  , gaussian mixtures  and dynamic bayesian networks  .while recent work on predictability has resulted in richer methods and incorporated interesting new features such as social contacts , they have not deeply characterized the intrinsic characteristics of human mobility that form the basis for the limitations in predicting the next visited location . in this paperwe focus on that aspect , showing that in weeks , individuals visit on average unique locations , of which of them are visited only once . despite most of the trips being among of their recurrent locations ; the occurrence of an exploration can be predicted with at best of accuracy .separating the two types of visited locations and improving the ways to predict an exploration would advance the methods in this area .in this study we analyze a dataset from the copenhagen network study  .the project has collected mobile sensing data from smartphones for more than 800 students at the technical university of denmark ( dtu ) .the data sources include gps location , bluetooth , sms , phone contacts , wifi , and facebook friendships .data collection was approved by the danish data protection agency , and informed consent has been obtained for all study all participants .  for this studywe focus on the location data , which is collected by the smartphone with frequency of one sample every 15 minutes .each location sample contains a timestamp , a latitude and longitude , and an accuracy value .the location is determined by the best available provider , either gps or wifi , with a median accuracy of meters ; more than 90% of the samples are reported to have an accuracy better than meters .for individual participants , there may be periods missing data .these periods can occur for various reasons , for example due to a drained battery , the phone being switched off , the location probe being disabled , or due to software issues . since we are interested in reconstructing mobility histories without large gaps , we select the longest period that has at least one sample in 90% of the 15-minutes time - bins for each participant .moreover we consider only participants that have at least 3 months long period of such contiguous data .we are left with 454 users , with data collection periods of data ranging from three months to one year .[ fig : periods_lengths ] shows the distribution of period lengths .the data is mainly concentrated in denmark where the study takes place , but because students use the phones during travel , the dataset spans several other countries as well .[ fig : locmap ] shows a map of the locations in the world ( left pane ) and in denmark ( right pane ) .in this work we are interested in the location prediction task .this task can be broadly stated as follows : given your location history , how well can we predict your future location ? the specific details of how this question is implemented have a profound impact on the prediction accuracy .below we investigate how various factors , e.g. spatial and temporal data resolution play a role in determining the reported accuracy for a single underlying dataset .  because the prediction task can be stated in many different ways , we start the discussion by analyzing different problem formulations . in terms of spatial predictionit is possible to discretize space in grid cells , voronoi cells or define places using a clustering method . in terms of temporal predictionwe could decide to predict a location in the next time - bin , or within a time horizon , or as the next visited place . in this paperwe select two of the most common problem formulations : _ next - cell _ and _ next - place_. in the _ next - cell _ formulation we discretize space into grid cells , and we predict the cell in the next time - bin . in the _ next - place _ formulation we detect visits to places and we predict the next visited place .the following sections provide details on the two alternative formulations , and show how each formulation affects the prediction task .in the first problem formulation , we convert geographical coordinates _( lon , lat ) _ into discrete symbols by placing a uniform grid on the map and retrieving the grid cell i d associated with the coordinates . specifically , we start by considering a grid of approximate size 50 meters x 50 meters . at each timestep minutes , we convert the current _ ( lon , lat ) _ into a cell i d , therefore producing a sequence of symbols through which we can represent a user s location history .[ fig : cell - process ] illustrates the process .    in this formulation, the problem can be restated as follows : given your past cell sequence up to time , which cell will you visit at time ? before trying to perform any prediction at all , following the process suggested in , we calculate the theoretical upper limit for the predictability of the cells sequence . fig .[ fig : predictability ] shows how the maximum predictability for the grid cell formulation is peaked at around 0.95 .we now consider different baseline strategies for next grid cell prediction .for each of the strategies , we perform prediction in an online manner , by training the algorithm on the data up to timestep , and predicting cell at timestep .we measure the accuracy as number of correct predictions over the number of total predictions .we first consider the _ toploc _ strategy , where at each timestep we predict the most frequent symbol in the history so far .given the highly stationary nature of most human mobility trajectories , we expect this simple strategy to achieve a relatively high accuracy .[ fig : timebin - accuracy ] top panel shows the distribution of accuracies for all the users .the accuracy of the _ toploc _ is indeed reasonable , peaking at around 0.4 .we now consider the markov chain model . in this model ,the prediction of next state depends only on the current state .the transition probabilities between locations are estimated based on past transitions in the location history . for making a prediction, we then consider the transition that has the highest probability among all possible transitions from the current cell .if the current state has never been seen before , then we have no information about the transition probability to other states . in this casewe fall back and predict the most frequent state .again we fit the model in an online manner , updating at each step the transition probabilities and then making a prediction for the next timestep .[ fig : timebin - accuracy ] middle panel shows the distribution of accuracies for all the users .the accuracy of the _ markov _ model is much higher than toploc , peaking at around 0.7 .  considering the highly stationary nature of typical trajectories , we hypothesize that a significant part of the markov prediction power in this formulation comes from self - transitions , that is , the model predicting the user to remain in the same state as in the previous time - bin .to test this hypothesis , we consider the _ stationary _ strategy : at each step we predict that the user will remain in the current cell .[ fig : timebin - accuracy ] bottom panel shows that the distribution of accuracies for _ stationary _ closely matches the one for _markov_. furthermore fig .[ fig : accuracy - correlation ] shows how the two are very strongly correlated ( pearson s , ) .this strongly suggests that , in this formulation , most of the markov model power comes from self - transitions , as suspected .we now investigate another issue related to this problem formulation .intuitively , we expect that the size of our spatial units will influence the accuracy of prediction . predicting a user s location with the precision of few metersis intuitively much more difficult that predicting with precision of several kilometers . in order to examine the effect of spatial resolution, we also consider results for cell size 500 meters and 5000 meters , and apply the _ markov _ model .[ fig : cell - accuracy - spatial ] compares the accuracy for different spatial resolutions . as expected the accuracy dramatically improves as the spatial size increases .finally we investigate the effect of temporal resolution within this problem formulation .our findings above suggest that using a very fine - grained temporal resolution will increase the number of self - transitions , thus driving up the accuracy of the prediction that is mainly able to capture stationarity .we achieve this by discretizing the location at 50 meters cell size , but varying the temporal time binning to 15 minutes , 30 minutes and 60 minutes , and then running the _ markov _ model for each scenario .[ fig : cell - accuracy - temporal ] compares the accuracy for different temporal resolutions .as expected , the accuracy is decreased as the time - bins grow larger due fewer self - transitions .we now consider an alternative problem formulation . instead of predicting the cell in the next timestep , we want to predict only when we observe a transition between places , eliminating the possiblity of self - transitions . in order to do so ,we convert the raw gps locations into a sequence of stops at places .a large amount of literature has been dedicated to the problem of place detection , such as methods based on wifi fingerprint  , grid clustering  , and kernel density estimation  .  in this paperwe consider the following process , based on density - based clustering approaches such as .each user is treated individually .we define a `` stop '' as sequence of location - observations where the user has been approximately stationary , that is , the distance between position at time and is less than a threshold meters , roughly corresponding to the gps accuracy .this produces a sequence of stops , each one with a centroid calculated as the median of the locations coordinates , and a duration equal to the time between the last location and the first location sample . in order to filter out the short stops during commute, we consider only stops with duration greater than 15 minutes .the average number of stops per user per day is 2.89 with standard deviation 0.89 .we are now interested in grouping stops into places , where a `` place '' is a group of spatially related stops representing a self - contained area such as a building . in order to do so, we apply the dbscan  clustering to the stops in the geographical coordinate space , using the haversine distance .we set as parameter the grouping distance meters , and _min_pts _ = 2 .this distance threshold is set to produce places of the approximate size of a large building .the result of the clustering is an assignment of a cluster label to each stop , where the label represents the place that the stop belongs to .finally , in order avoid artifacts due to missing samples or noise , we merge multiple consecutive stops at the same place into one .this process converts the raw location history into a sequence of stops at places .[ fig : stops - extraction ] and [ fig : dbscan ] illustrate the complete process of stop detection .are examined sequentially , and are grouped into a stop as long as they are within a distance threshold . in the example , and are assigned to the first stop but does not , since it too far away .subsequently , and are assigned to stop 2 . ]    as example result of this process , let us consider the stops and places extracted for a user .the sequence of stops at places can be represented as a weekly schedule capturing the user s movement patterns . in fig .[ fig : schedule ] each row represents a week from monday to sunday ; each place is encoded as a different color . inspecting this visualization it immediately possible to spot the periodic patters characterizing human mobility , such as evening returns to the home location , and morning trips to class .we can also spot many irregularities however , that deviate from the normal schedule : small stops , new explorations , and day - by - day variability .finally we can also see a large change in routine starting week 20 , where the home location changes .each user can be characterized by a similar plot .the prediction task can now be re - formulated as follows : given a sequence of stops up to step , can we predict your next stop at step ?notice that a key difference from the cell grid formulation is that in this case there are ( by definition ) no self - transitions ; we are interested in the place changes only .  as before ,we start by investigating the upper predictability limit bound . fig .[ fig : predictability ] shows how the maximum predictability for the stops formulation is peaked at 0.68 , significantly lower than what we observe in the grid cells formulation .we now apply the two prediction strategy _ toploc _ and _ markov _ to this new formulation .the two models remain conceptually the same , but instead of trying to predict the grid cell at each step , they try to predict the next stop ( note that in this formulation we can not use the _ stationary _ strategy , as by construction we are interested in transitions to new places ) . in this casewe also fit each user separately , and we perform the prediction in an online manner . fig .[ fig : stops - toploc - markov ] shows the accuracy for both models .it is evident that the accuracy for these models ( around 0.3 for toploc and 0.4 for markov ) is significantly lower in the next - place formulation , indicating that this problem formulation presents a more difficult task .we have investigated how the details of the problem formulation strongly impact the reported accuracy for location prediction tasks .we now focus on next - place prediction and study the influence of different contextual features on the prediction task .we formalize the problem as follows . at each step , we want to compute the most probable next location given the current location . we may also want to include other context variables , such as time of the day , day of the week , call activity , or distance from home for example .in other words we want to compute , where is the next location , and , , are the variables representing different contexts . for this purposewe use a logistic regression model , and we study the usefulness of various predictor features . the goal of the model is not to suggest a new state - of - the - art method , but rather to evaluate the importance of individual contextual features . specifically , we consider the current location , the time metadata ( hour of the day , day of the week , hour of the week , weekend ) , a ` home ' binary indication , distance from home , call and sms activity , and bluetooth proximity . table [ tab : features ] provides a summary of the features ..description of the features used for the logistic regression models . [ cols="<,<",options="header " , ]   we model each user separately since we want to perform next - place prediction at the individual level . as before ,we perform an online prediction where we fit the data up to step , and we predict the next location at step . for each user, we fit a logistic regression model using all the individual features separately , and then a model with all features .[ fig : stops - acc ] shows the accuracy for each of the models , averaged by user .the markov chain model baseline is highlighted in red . using the current location and time features, the logistic regression model outperforms the markov chain based model . even using the current locationonly ( which is conceptually very similar to a markov chain model ) , the logistic regression shows stronger performance , likely due to the explicit optimization of the model .it is also interesting that other context variable such as call and sms data have little predictive power in this model formulation .the most complex model that considers all features is practically identical in performance to the model using only current location and hour of the week .although the logistic regression model does improve the accuracy over the markov model , the absolute value of accuracy is remains low ( below 45% ) .we therefore investigate the possible reasons of this difficulty in prediction .it is well known that the majority of individuals tend to spend most of the times at very few places such as home and work , and only sporadically visit other places .this phenomenon has been described using concepts such as preferential return  , heavy - tailed stay times and return rate based on the number of visits  . for the location prediction tasks ,the consequence is that the target classes are very unbalanced , which implies that most records belong to very few classes and most classes are represented by only few records . to illustrate this issue, we consider the extreme case of places visited only once .[ fig : visited_once ] shows that surprisingly this fraction is quite large ( 0.7 ) .this fact is , in large part , the central reason behind the difficulty of the prediction task .    as we shall see below, another central challenge is not just that our population visits a large number of different places , but also that many new places are discovered over time .we consider a stop at a location as `` exploration '' if this place has not been seen in the location history so far for a given user .in other words , this place is being visited for the first time by the user . to express this formally , we consider the sequence of stops for each user .we consider a stop as _ return _ if has been seen before in the location history , that is there exists a stop for .otherwise we consider stop an _ exploration _ , that is the place is visited for the first time at step . for example given a location sequence a b a c b c , the target variable exploration would have values 1 1 0 1 0 0 .we can then estimate the probability of exploration as fraction of explorations over the number of stops . to our surprise , this probability is particularly large : between 0.2 and 0.25 ( fig .[ fig : p_exploration ] ) .this implies that most users discover a new place every 4 or 5 stops .the fact that a large fraction of stop - locations have never been seen before poses a challenge for the prediction task , since by construction any model that tries to predict a place from an alphabet of previously visited places will be unable to predict new , unseen symbols .moreover , another consequence of this frequent exploration is that the pool of possible places constantly grows over time and , given the longitudinal nature of our dataset , ends up being very large .[ fig : exploration_week ] shows how the average number of new place explored per week remains approximately constant around 4 , and consequently the total number of places keeps growing to hundreds of places ( fig .[ fig : exploration_cumulated ] ) .this is a problem for the prediction task , as the number of possible places that the classifier needs to choose from increases constantly .in fact if we measure the relation between the number of unique places per user and the performance of the best performing logistic regression model using pearson s correlation coefficient , we find a quite strong negative correlation ( , ) .on the other hand we find no significant correlation for accuracy with period length or number of stops .these facts suggest that the exploration phenomenon is a key reason for the relatively low accuracy of mobility prediction tasks at high spatial resolution .given the importance of exploration , we now consider a novel task in mobility prediction : exploration prediction .   the exploration prediction task can be stated as follows : given a user s location history up to step , will the stop at step an exploration or a return ?the first question is : what should be the baseline model for the exploration prediction task ?surprisingly , most literature on human mobility prediction has focused on next location prediction but has overlooked the exploration prediction problem , and to the best of our knowledge no suitable solution has been proposed for this task . we therefore suggest , as a reasonable baseline , random guessing with probability equal to our prior knowledge of the fraction of explorations : .for our main model we use as before the logistic regression model with the same features constructed for the next place prediction model .we also add two additional features : _ explore_now _ and _ explore_before _ , which capture if the current stop or the previous stops were explorations , respectively .the intuition for these is that multiple explorations may occur in a row , and therefore the current exploration may increase the likelihood for an exploration at the next stop .as before , we fit each individual separately , and we perform an online prediction , that fits based on the data up to step , and predicts exploration at step .we fit one logistic regression model for each of the single features , and a more complex model with all the features at once .  measuring the performance of these modelsrequires a few considerations . in this case , the classification problem is imbalanced , that is the number of positive cases ( exploration ) is much smaller than negative cases ( return ) , as shown in fig . [fig : p_exploration ] .this implies that accuracy is therefore not a good metric , since a classifier predicting always return ( the most probable class ) would have good performance , but would not be useful .instead we employ the score , which is the harmonic mean of precision ( the fraction of correctly predicted explorations over all predicted explorations ) and recall ( the fraction of correctly predicted explorations over all true explorations ) .[ fig : exploration - f1 ] shows the results of the exploration prediction .score for all models . ]  as we would expect , the model with the most complete set of features outperforms the others . among the single feature models ,perhaps not surprisingly , the current location feature has the best performance .this finding can be explained by the role of some places as `` gateways '' for exploration such as public transport hubs ( e.g. central train station ) .the individual features that perform also well are the time - related ones , in agreement with the intuition that exploration tends to happen to at specific times of the day or week .the _ explore_now _ and _ explore_before _ also perform well , suggesting an element of burstiness in the exploratory behavior .if we consider our best performing model , we find that it has average precision of 0.3 and recall of 0.65 .overall the performance of this model is far from perfect , showing that the exploration prediction problem is a challenging one .in this paper we first show that when interpreting results of predictive performance there are a number of factors that must be taken into consideration . the problem formulation is the central factor what should be taken into account wheninterpreting predictability results , since e.g. predicting the next time - bin is a very different ( and much easier ) task than predicting the next transition .we show that the most challenging problem is the next - place prediction , which is arguably the most useful task for practical applications such as travel recommendations .another issue to be taken into account is the spatial resolution of the prediction , here we show how more coarse spatial precision results in an easier task . similarly the time resolution also has an effect on the predictive power .we suggest that the factors described in this paper should be taken in consideration as context when comparing results from prediction models .other than the factors discussed above , we believe that one further reason for performance differences could be the demographics of the dataset .the population under study is here composed by students that have no single workplace but tend to change multiple classes per week , even multiple times per day .moreover a younger population may have a more irregular schedule and more exploratory behavior .certainly more work is needed to conclusively link demographics and predictability . for future directions , we suggest considering demographic factors when trying to characterize human mobility , as it has been done , for example , by linking changes in mobility patterns with unemployment status  .we also discussed the issue of exploration , and we show how frequently new places are discovered . based on that , we show that the mechanism of exploration is an important part of human mobility and plays a role in next - place prediction . because any model that tries to predict a next place from a set of visited place will fail when an exploration occurs .this problem has rarely been addressed in mobility prediction literature , which almost always assumes that the next place can be determined from the past history . providing a full solution for nextexplored place prediction is beyond the scope of this work , and here we simply aim to stress the fact that the prediction of explorations is very different from the predictions to returns to known places. some previous work on next - place prediction using social information  or nearby points of interest  may be the starting point for investigating this problem .  in this sense , we raise the question on whether the simple location history is enough for accurate next - place prediction .as we have discussed , there are indeed a lot of regularities both in the sequence of visits , and in the daily and weekly temporal patterns of visitation .however there are a lot of `` exceptions to the rules '' , where schedules change , plans are canceled , and people run late .we speculate that other channels such as email , social media , calendar , class schedule may be needed for achieving a satisfying accuracy in the prediction task .this work is funded in part by the high resolution networks project ( the villum foundation ) , as well as social fabric ( university of copenhagen ) .lane nd , mohammod m , lin m , yang x , lu h , ali s , et al .bewell : a smartphone application to monitor , model and promote wellbeing . in : 5th international icst conference on pervasive computing technologies for healthcare ; 2011 .p. 2326 .quercia d , lathia n , calabrese f , di lorenzo g , crowcroft j. recommending social events from mobile phone location data . in : data mining ( icdm ) , 2010ieee 10th international conference on .ieee ; 2010 .p. 971976 .aalto l , gthlin n , korhonen j , ojala t. bluetooth and wap push based location - aware mobile advertising system . in : proceedings of the 2nd international conference on mobile systems , applications , and services .acm ; 2004 .p. 4958 .balmer m , meister k , rieser m , nagel k , axhausen kw , axhausen kw , et al .agent - based simulation of travel demand : structure and computational performance of matsim - t .eth , eidgenssische technische hochschule zrich , ivt institut fr verkehrsplanung und transportsysteme ; 2008 .schneider cm , rudloff c , bauer d , gonzlez mc .daily travel behavior : lessons from a week - long survey for the extraction of human mobility motifs related information . in : proceedings of the 2nd acm sigkdd international workshop on urban computing .acm ; 2013 .smith g , wieser r , goulding j , barrack d. a refined limit on the predictability of human mobility . in : pervasive computing and communications ( percom ) ,2014 ieee international conference on .ieee ; 2014 .p. 8894 .cho e , myers sa , leskovec j. friendship and mobility : user movement in location - based social networks . in : proceedings of the 17th acmsigkdd international conference on knowledge discovery and data mining .acm ; 2011 .p. 10821090 .kang jh , welbourne w , stewart b , borriello g. extracting places from traces of locations . in : proceedings of the 2nd acm international workshop on wireless mobile applications and services on wlan hotspots .acm ; 2004 .p. 110118 .zheng vw , zheng y , xie x , yang q. collaborative location and activity recommendations with gps history data . in : proceedings of the 19th international conference on world wide web .acm ; 2010 .p. 10291038 .montoliu r , gatica - perez d. discovering human places of interest from multimodal mobile phone data . in : proceedings of the 9th international conference on mobile and ubiquitous multimedia .acm ; 2010 .

Abstract: predictive models for human mobility have important applications in many fields such as traffic control , ubiquitous computing and contextual advertisement . the predictive performance of models in literature varies quite broadly , from as high as 93% to as low as under 40% . in this work we investigate which factors influence the accuracy of next - place prediction , using a high - precision location dataset of more than 400 users for periods between 3 months and one year . we show that it is easier to achieve high accuracy when predicting the time - bin location than when predicting the next place . moreover we demonstrate how the temporal and spatial resolution of the data can have strong influence on the accuracy of prediction . finally we uncover that the exploration of new locations is an important factor in human mobility , and we measure that on average 20 - 25% of transitions are to new places , and approx . 70% of locations are visited only once . we discuss how these mechanisms are important factors limiting our ability to predict human mobility .


given the ubiquity of friendships in social interactions and complex social systems , the value of any particular friendship can be difficult to quantify .but some things are known : friendships are useful for finding new jobs , as in the `` weak ties '' or granovetter effect  ; they are useful for marketing because personal attributes including fixed variables like age , ethnicity and location , as well as more fluid variables like preferences and opinions exhibit homophily  and thus correlate across friendship ties ; and , they are useful for social filtering or search  , i.e. , your friends are good at predicting what information you will like .but , how valuable are friendships within complex collaborative environments , particularly those found online ?on the one hand , teams composed of friends may perform better as a result of their extensive collaboration history , i.e. , friendships may increase awareness and understanding of others goals and motivations  or they may increase commitment to group objectives  , thereby facilitating more successful collaboration .friendships may also reduce within - group conflict  , yielding similar results . on the other hand, friendships could detract from performance because friends spend less time focusing on group objectives and more time socializing  .friendship may even be irrelevant for effective collaboration .if a competition s outcome depends mainly on team coordination , teams composed of experts may coordinate effectively regardless of within - team friendships : highly practiced individuals may simply know from experience how to work well with other experts and thus naturally anticipate or adapt to their teammates actions ( e.g. , by fitting into established and effective team roles ) . in such a case , friendships do not matter and teams composed of skilled strangers will perform best  .thus , the extent to which expertise and friendship matter to success is an open question in complex collaborative environments , particularly those found online .  here , we focus on the topic of friendships , collaboration and competition in online game environments , and specifically in the first - person shooter ( fps ) genre , in which teams of players compete against each other in non - persistent virtual worlds .we are also interested in the variety of social dynamics naturally observed in this complex environment , and the extent to which they vary with demographic and psychometric variables .there are many studies in cscw on the design of , and human action within , multi - user dungeons  , massively multiplayer online games ( mmogs )  ( among others ) , social virtual worlds  , and non - persistent game sites  .work on friendships in online gaming has focused on how friendships and social relationships are a motivation to play games  or how playing games builds offline relationships and social capital of the players  .work on guilds and parties  in massively multiplayer online role - playing games ( mmorpgs ) suggests that teams are important for individual satisfaction and performance , and anecdotal evidence suggests they are conducive to the formation of friendships .however , the impact of friendships themselves on team or individual performance has not been studied quantitatively ( but see  for some qualitative insights ) . and, work on first - person shooter type games has generally focused on qualitative analysis of user actions and in - game communication  .most work on the utility of friendships in teams is found in the management literature .several such studies argue that friendships play a significant role in mitigating certain types of within - group conflicts ; thus , teams of friends perform better because internal conflict handicaps performance  .others have focused on designing effective teams in business or educational environments  . in these settings , friendships seem to improve group cohesion , satisfaction , and some measures of performance .thus , we expect friendships to play a significant role in the collaboration effectiveness in virtual environments , including those found in the team - based competitive environments of the fps genre .finally , the increasing cscw interest in `` serious games ''  as a model for creating effective work spaces makes the study of real online games , and particularly those that allow collaboration with both friends and strangers , a valuable laboratory by which to understand the dynamics and determinants of collaboration and sociality in virtual environments .  here, we study these questions in the context of the popular online fps game _ halo : reach_. we combine two large and novel data sets : one composed of rich online behavioral samples from the events within _ reach _ competitions and one composed of rich demographic , play style , psychometric and friendship variables collected from players of _ reach _ through an anonymous online survey .this combination of in - game behavioral data with survey data on the same participants has been used productively to study human behavior in mmorpgs  and online parlor games  , but this study is the first time it has been applied to the fps genre or to the question of friendship , collaboration and competition .the survey asked participants to label their online and offline friendships with other _reach _ players , and these data allow us to test the hypothesis that friendships have a direct and significant impact on the results of these competitions .features within the game allow us to control for the impact of expertise .an in - game `` matchmaking '' system draws groups of friends and individual players from the general population to create competitions ; thus , players collaborate with and compete against a wide variety of teammates and competitors . finally , we note that the sheer size of _ reach _ , which is played by millions of individuals , provides a wealth of data by which to investigate these and other questions about collaboration and competition .we find that friendships significantly improve both individual and team performance in these complex collaborative environments .friendships are not as important to success as the raw expertise of teammates ( friendship does not compensate for being a bad player ) , but individuals themselves perform better the more friends they play with , independent of their team s performance .our survey shows that _ reach _ players are also highly social , tending to prefer team - oriented play and experiencing a strong sense of group cohesiveness and coordination .furthermore , older players ( 24 or older ; 30% of players ) tend to be more socially oriented and exhibit greater proclivity for pro - social behaviors .above and beyond performance , friendships also reshape the style of play within a competition , and players structure their choices around opportunities to play repeatedly with friends . as a result, friendships can be accurately inferred directly from behavioral time seres data .these results illustrate the strong role that friendships play in complex collaborative and competitive virtual environments .they also shed light on social dynamics within fps games and suggest algorithms designed to account for the benefits of friendship in predicting or designing competitive environments ._ halo : reach _ is a multiplayer online first person shooter game played by more than ten million people worldwide .it was publicly released by bungie inc ., a former subdivision of microsoft game studios , on 14 september 2010 and _ reach _ players have now generated more than 1 billion games .players choose from among seven game types and numerous subtypes , which are played on more than 33 terrain maps with 74 weapons .games can be played alone , with or against other players via the xbox live online system .both individual game and player summaries are available through the halo reach stats api . through this interface, we collected the player details of all individuals who participated in our online survey and each of their full game histories , which yielded 2,445,617 complete games .  among other information ,each game file includes the sequence of the scoring events at the per - second resolution and a list of players by team .scoring events are annotated with the name of the player generating the event ( a unique xbox live gamertag ) , the number of points scored and the player giving up the points ( if applicable ) .  unlike professional sports , teams in _ reach _ do not persist across competitions . instead ,each time a competition is created , individuals or small `` parties '' of players ( typically friends ) are grouped into teams by an in - game `` matchmaking '' algorithm ; when the competition is complete , the players ( or a subset ) may choose to play together again as a party , or may reenter the matchmaking process to find new teammates or competitors .the result is that players routinely collaborate with or compete against different strangers in successive games . _s matchmaking system uses an algorithm called trueskill  to ensure equal total `` skill '' for both teams , which allows us to control for this variable in our subsequent analysis .  among the sampled games ,there are three basic types : _ campaign games _ , a sequence of story - driven , non - competitive , player - versus - environment ( pve ) maps that many players complete prior to trying other types of games ; _ firefight games _ ( also pve ) , in which a team of human - controlled players battle successive waves of computer - controlled enemies ; and _ competitive games _ , a player - versus - player ( pvp ) game type , in which teams of equal size ( typically 2 , 4 , 6 or 8 players ) compete to either be the first to reach some fixed number of points or have the largest score after a fixed length of time .( the precise number of players per team , points required to win and length of a game depends on the game subtype . ) in competitive games , players avatars move through the game map simultaneously , in real - time , navigating complex terrain , acquiring avatar modifications and encountering opponents .points are scored by dealing sufficient damage to eliminate an opposing avatar and for each such success , a team gains a single point .eliminated players must then wait several seconds before their avatar is placed back into the game at one of several specified `` spawn '' locations .  ]in addition to the in - game behavioral data available through the api , we collected data on 1191 unique _ reach _ players via an anonymous online survey .the survey design incorporated extensive feedback from expert _ reach _ players , including ourselves , to better frame questions in appropriate language and to focus on relevant social dynamics .we advertised the survey through halo - centric online forums and other community websites . while this did result in a biased sample ( see below ) , the bias was towards participants who were active and therefore had more in - game data to draw upon .participants supplied their xbox live gamertag , which we used to download their _ reach _ player details and their entire _reach _ game history up to the date of survey participation . within the survey ,an initial screening was conducted based on participant age : participants who indicated their age was under 18 were routed through an additional email - based parental consent step .the player detail files contain only summary statistics about _ reach _ game play and do not contain demographic or social network information .the survey solicited these missing demographic variables ( age , sex , location , highest education level , primary language ) , along with preferred play style ( lone wolf , team leader , or team support ) , and psychometric responses like measures of group cohesion  , entativity  , and conflict  . in answering these questions ,respondents were instructed to consider their `` primary halo group , '' defined as the group of friends with whom they primarily play _ reach _ at the time of the survey .finally , respondents indicated their relationship ( online friend , offline friend or not a friend ) with every other unique gametag that appeared in their game history .because this list could include hundreds or even thousands of gamertags , it was sorted by the frequency of co - plays , which placed the most likely true positives at the top .the interpretation of the meaning of an online and offline friend was left to the participant , although we suggested online friends were people `` you play with regularly and would say you know at least casually . ''this is conceptually distinct from any feature available on halo or xbox for indicating friendships .as is typical with many online games  and especially first person shooters  , our survey respondents were mostly young men .the median age of our participants was 20 , which is considerably lower than the mean age of 37 for video game players overall  .of the 1191 respondents , an overwhelming majority ( 94.9% ) reported their sex as male ( compared with 80.8% male in mmo players  and 58% reported by the industry overall  ) .the reported age distribution of the respondents is relatively smooth , with the exception of a large spike at 18 ( see figure [ fig : age_dist ] ) , presumably caused by individuals younger than 18 misreporting their age so as to avoid the parental consent requirement .this contrasts with the reported age distribution of mmo games , in which the plurality of players are in their 30s rather than their 20s  .however , the age distribution also exhibits a long right tail , with 12.8% being at least 30 years old . in agreement with the age distribution ,the most frequent response indicated some college education .although 35 countries were represented in the respondents , nearly three - quarters of respondents were from the united states , with another 14% from canada and the united kingdom .an overwhelming majority94% of our respondents indicated english was their first language .our respondents were very active players : they reported playing video games for an average of 23.3 hours per week . although this number may appear high to non - gamers , it is slightly lower than the 25.9 hours per week reported by mmo players and the 27.5 hours per week reported by the industry in 2007 for video game players in general  .we also asked participants to report their preferred playing style : team leader or team support ( both collaborative styles ) or `` lone wolf '' ( a style that does not coordinate actions with the team ) .the survey did not provide definitions of these labels to respondents ; however , feedback from our expert players indicated that their meanings should have been commonly understood within the _ reach _ community .although the popular stereotype of fps games is non - collaborative , we find that 78.6% of players prefer playing as a team , in either the leading or supporting roles .thus , _ reach _ players are in fact strongly motivated by the collaborative aspects of the game  .when asked how often they played as a leader , participants reported typically playing their preferred role , although interestingly the lone wolves report playing a leader much more often than we would naively expect , given the anti - social nature of the lone wolf style ( fig .[ fig : ldrfreq ] ) .  ]we find some support for the popular perception that younger players prefer the non - collaborative lone wolf role  . in our sample ,people who preferred the team support role were indeed older ( 23.5 years ) than those who preferred to play as lone wolves ( 21.6 years ; ) .the modest size of the difference , however , suggests that while the effect is real , it is only a very weak tendency , with many younger players preferring collaborative roles and many older players preferring non - collaborative ones .interestingly , people who preferred to play as leaders were also significantly younger ( 21 years ) than those who played team support ( ) .  using our expertiseas _ reach _ players , we modified three classic psychometric surveys intended to assess group dynamics : entativity ( how much a group is like a single entity ) , cohesion ( how tightly - knit a group is ) , and conflict ( how much internal conflict a group has ) .past work on collaboration and team performance suggests that low - levels of conflict within a group of friends should correlate with better performance  .the entativity portion of the survey contained 14 questions , which had reasonable variability ( sd=0.976 on a 5-point scale ) and good internal consistency ( cohen s ) .we therefore averaged the responses to create a single entativity metric .the average perceived entativity was 3.93 out of a 5-point scale , indicating respondents typically felt their teams were good at acting as one .the portion of the survey on cohesiveness contained 7 questions , which exhibited similar variability ( sd=0.972 ) and internal reliability ( ) , so again we averaged the responses to create a single metric . in agreement with the perceived entativity , players typically felt their teams were very cohesive ( 4.1 out of 5 ) . finally , the portion of the survey on group conflict consisted of 8 questions with slightly lower variability ( sd=0.92 ) and less internal reliability ( ) , which appears to be related to the presence of reverse - scored questions .nonetheless , we felt the consistency was sufficient and averaged the scores . here ,perceived conflict was low ( 1.77 ) , but somewhat higher than might be expected given the high entativity and cohesion ratings .  ]advertising the anonymous survey on halo - oriented community websites likely produced a biased sample of respondents relative to the overall population of _ reach _ players . to characterize how different our respondent population was from the overall player population , we downloaded a uniformly random sample of the first 400,000,000 _ reach _ games from which we extracted 963,000 unique gamertags .we then downloaded each of their player detail files and estimated the background distributions of various player summary statistics . within this population, the average proportion of wins exhibited by a player is almost exactly 1/2 , indicating a fairly unbiased sample of all players ( fig .[ fig : wins_963 ] ) .  compared to the background distribution ,our respondents were much better players , winning many more of their games ( fig .[ fig : wins_963 ] ) than the typical _ reach _ player .in addition to being more successful players , our respondents also typically invested almost 10 times more actual time playing _ reach _ than the typical player ( fig .[ fig : playtime_963 ] ) .that is , our respondents were serious halo players , likely devoting a large share of their overall time spent playing video games to this particular game .  ]although popular fps stereotypes suggest that demographic variables like player sex may correlate with game performance , we find that males and females behave and perform similarly .for instance , males and females played roughly the same number of games ( female , male ) , and the same amount of time per game ( female , male ; total seconds played divided by total games played ) .however , women died slightly more often than men ( female deaths / game , male deaths / game , , ) . )was corrected for the number of tests done .] of course , because of the relatively small number of female players ( 5.1% or 61 respondents ) , there may be other small but real effects that we are unable to detect .we also observed interesting differences in game play by player age .congruent with the popular stereotypes of fps gamers , the older a respondent was the more kills he had per game ( , , ) , as shown in figure [ fig : by_age ] .we also see older players were significantly more capable in the lone wolf role , a result that runs contrary to the suggestion that young players prefer the lone wolf role because they are more capable in the role popularly attributed to faster reflexes while older players must coordinate in teams in order to effectively compete  .one surprising age - correlated behavioral difference is in the number of betrayals ( killing one s own teammate , which results in a loss of a point to the team and a longer respawn time for the offending player ) , as shown in figure [ fig : by_age ] .younger players ( age 18 ) showed a disproportionate amount of this type of team disloyalty relative to the older players ( , , ) , with the former group exhibiting a betrayal rate about 40% higher than the latter groups .because betrayals result in a penalty against the team s overall score , this suggests younger players exhibit significantly increased anti - social behavior while older players generally exhibit a more pro - social or cooperative orientation ., title="fig : " ] + , title="fig : " ]   unlike purely survey - based studies , the access granted through the api to the detailed in - game data allows us to pair each survey response with rich independent behavioral data on the same individuals . through this pairing ,we may quantitatively verify reported variables on in - game behaviors .if the survey responses are accurate , they should predict corresponding patterns within the behavioral data , and allow us to subsequently predict survey responses from behavioral data alone .for instance , participants who report preferring the lone wolf role do indeed have significantly fewer `` assists '' to teammates ( in which two or more team members collaborate to score a point ) than participants who prefer to be team leaders ( ) or team support ( ) .we also find meaningful differences in the reported group dynamic psychometrics .for instance , the more a player sees their group as an entity , the more assists they have per game ( , ) .surprisingly , the opposite is true for number of kills : the more entitative groups have significantly fewer kills per game ( , ) , perhaps because these groups travel as a pack and thus share their kills among their number .there are no significant differences in the number of deaths per game , however , suggesting perhaps that such groups are no better at warding off attacks than less entitative groups .we observed a similar effect for perceptions of cohesiveness more cohesive groups had significantly fewer kills per game ( , ) , and there were no differences in the number of deaths per game .we did not , however , see the same relationship between cohesiveness and assists that we did for entativity and assists , suggesting the measures may be tapping different aspects of the players subjective perceptions of their groups .a player going out of their way to assist a teammate , and thereby potentially giving up opportunities to score their own points , may require a greater sense of unity than mere closeness or camaraderie .studies of team dynamics in the management literature  suggest that groups with greater levels of internal conflict should exhibit different game play dynamics than groups with lower levels .surprisingly , however , we found no strong or notable relationships between responses about group conflict and game play .there are several possible explanations for this pattern , but we do not explore them here .before we consider the question of whether ground - truth friendship labels can be predicted purely from in - game behavioral data , we briefly discuss a few additional results from the survey data regarding friendships .  of the 1191 respondents ,597 played at least one game with another survey respondent .this overlap allows us to test whether respondents perceptions of their online and offline friendships are reciprocal  , i.e. , when one player labels another as a friend , that friend also labels the player as a friend . for online friendships ,the reciprocity is 36.9% ; that is , when one of two survey respondents indicated the other was an online friend , about a third of the time did they agree they were friends .the agreement on offline friendships was higher ( 60.9% ) .while both of these values may seem low , both are in fact much higher than the rates observed in other online social networks , e.g. , twitter ( 22% )  . for a social network where friendship ties represent a significant mutual investment of time , it is not clear why we do not see rates approaching 100% , although there are a few possible reasons .for instance , we notice that there are a disproportionate number of respondents who list exactly 29 online friends , and this could indicate a number of respondents did not realize the list of teammates was scrollable . if this is the case , online and offline friends who fell `` under the fold '' may have been missed .another explanation may be variability in the interpretation of the term `` online friend '' or actual differences in friendships of _ reach _ players .this is a question for future research .  ]if players choose to structure their online activities in _ halo : reach _ around opportunities to play with friends , then the friendship labels respondents provided to us should be predictable from in - game behavioral data alone .logistically , friends often synchronize their play times , at the same time of day or day of the week ; this leads each player s gamertag to appear in the other player s game files in highly non - random ways . within _ reach _, if friends are to play together , they must use the in - game `` party '' mechanism , in which one player `` joins '' another player to form a larger unit .once they have formed this unit , the players are automatically placed together by the matchmaking algorithm into the same new competition and almost always on the same team ( though splitting a party can happen , which leads to interesting in - game behavior ; see below . )thus , players who are friends will tend to appear together in a sequence of games and , within each sequence , they will typically be on the same side of a competition .we find significant differences in the frequency of playing together when a participant labeled a co - player as an online friend , offline friend , both , or neither .pairs of players who are both online and offline friends typically played the most together , followed by online but not offline friends , then offline but not online friends and finally strangers ( fig .[ fig : friends_mutual ] ) .that is , friends of both types play many more games together than do strangers .  additionally , the party mechanism implies that when friends play together , their gamertags should appear together in a series of games , and therefore these series should be a good predictor of friendship . we define a _ series _ to be a sequence of consecutive games played together with no more than a one - hour gap between consecutive games .( other gap sizes yield similar results to those reported here , indicating the robustness of this feature . ) for each pair of players , we measured the number of series greater than length and the length of the longest series .figure [ fig : nseries ] shows that online friends were significantly more likely to have played series of games together , as were offline friends .the greatest number of such series was played by pairs who were both online and offline friends .similarly , the longest series for online and offline friend pairs is significantly longer than the longest series for non - friend pairs .  to illustrate this point ,consider the average longest series . for strangers ,the average was only 1.25 games in a row , indicating a strong tendency for these groups to dissolve quickly .however , for pairs of players in which either labeled the other as an online friend , the average was 10.20 and for those labeled as both online and offline friends , it was 13.15 games ( corresponding to roughly two hours of clock time ) .that is , friends play many more games together in a sequence ( on average , greater than 8 times more ) than do strangers , and the structure of the party mechanism in _ reach _ implies that this behavior must be actively selected for by players .these patterns suggest that certain purely behavioral features may be highly accurate predictors of unknown friendship labels . recovering this kind of hidden informationhas been done with a range of other electronic data on social interactions , including mobile phone call records  , geographic coincidences  and email  , but not previously interactions within online games .  ]  given the number of competitions two players played and the number of those that were played together , the length of the longest series , and how many series of different length the pair played together ( here : , although other values could also be used ) , we try to predict whether either player indicated the other was an online friend ( for those cases where multiple teammates responded to the survey ) .we fit the data to a logistic regression model , and obtain an accuracy of 98.7% , representing an auc of 0.96 .the corresponding roc curve can be seen in figure [ fig : pred_friends_on ] , and the ranking of the features can be seen in table [ tab : pred_friends_on ] ..parameters in a logistic regression predicting whether a player is labeled an online friend [ cols=">,>,>,>,>",options="header " , ]   that being said , our results do suggest some interesting clues . for instance, we observed more assists and fewer betrayals of teammates when there were more friends on the team , in agreement with research showing friendships decrease within - group conflict  .alternatively , the large amount of time friends spend playing together may improve the division of specialized roles and improve communication within the team , leading to a better coordinated effort  .overall , friendships not only improve performance , but they also reshape the style of play within competitions .that is , players compete differently when they play with friends than when they play with strangers . for instance , groups of friends who are split across teams have to choose whether to compete against their friends ( cooperate with their teammates ) or defect against the teammates ( cooperate with their friends ) .when this happens , friends tend to defect against their teammates , illustrated by a nearly double betrayal rate when two friends are on the opposing team .that is , friendship ties dominate teammate ties , and players sacrifice their own competitive success to help their friends .similarly , betrayal rates ( both by individuals and by teams ) decrease as the number of friends on the team increase , illustrating a significant pro - social effect as compared to teams of strangers .although our logistic regression models found that friendship variables make marginal improvements over skill variables in our ability to predict success , this is understandable . in a contextin which skill matters , a team of unskilled friends is sure to do more poorly than a team of skilled strangers .the important point is that even among skilled players , friendship ties provide a lift in both individual and team performance .to conclude , we found a number of interesting relationships between in - game behavior within _ halo : reach _ and player demographic , group covariates and friendship ties .most generally , we found that friendships strongly influence the social dynamics and collaboration within complex virtual environments like the fps game _ halo : reach_. friendships lead to improved individual and team performance , increased pro - social behavior , and likely increased long - term appeal of and engagement in the game .the impact of friendships is sufficiently strong , with players actively structuring their in - game activities around opportunities to play with friends , that hidden or unknown friendship ties can in fact be correctly inferred directly from behavioral time series using common sense heuristics .that is , it should be possible to automatically identify pairs of individuals who would call each other friends , if asked , merely by watching the way they interact with each other online .collaborative systems , including online fps games like _ reach _ but also a wide variety of other systems , that can infer and account for such friendship ties could thus enable a wide range of better outcomes by providing fewer mismatches between individuals or by suggesting individuals to interact with in the future . given the increasing ubiquity of social gaming and _ ad hoc _ competition in general , the importance of such algorithms is likely significant .we thank chris schenk for help implementing the web survey , nick yee and thore graepel for helpful discussions , and the halo community for their support and participation .this project was funded in part by a grant from the james s. mcdonnell foundation .y. xu , x. cao , a. sellen , r. herbrich , and t. graepel .sociable killers : understanding social relationships in an online first - person shooter game . in _ proc . computer supported cooperative work _ , pages 197206 .acm , 2011 .

Abstract: how important are friendships in determining success by individuals and teams in complex collaborative environments ? by combining a novel data set containing the dynamics of millions of _ ad hoc _ teams from the popular multiplayer online first person shooter _ halo : reach _ with survey data on player demographics , play style , psychometrics and friendships derived from an anonymous online survey , we investigate the impact of friendship on collaborative and competitive performance . in addition to finding significant differences in player behavior across these variables , we find that friendships exert a strong influence , leading to both improved individual and team performance even after controlling for the overall expertise of the team and increased pro - social behaviors . players also structure their in - game activities around social opportunities , and as a result hidden friendship ties can be accurately inferred directly from behavioral time series . virtual environments that enable such friendship effects will thus likely see improved collaboration and competition .   [ data mining ]


coherent superposition states are an important aspect of quantum systems , one which is believed to make possible their use for more efficient realization of energy and information manipulation tasks . in the last few years , much attention has been given for the development , analysis , and application of a resource theory for quantum coherence ( rtc ) .for a recent review , see ref .it is certainly worthwhile recognizing the existence of several levels of incoherent operations ( io ) in the rtc . naturally , depending on its properties , different coherence functions may or may not be a coherence monotone under a specific element of a given set of io .this kind of observation points for the need of further investigations concerning the similarities and differences among the properties enjoyed by varied quantumness functions .here we shall pay special attention to the geometric quantum coherence function which is obtained using hilbert - schmidt s distance ( hsd ) . although the possible non - contractivity under general quantum operations of the hsd demands care when of its application for quantumnesses quantification , it is a fact that the hsd has been a handy tool for a variety of investigations in quantum information science .in addition to that , recently it was shown that the hsd leads to an observable upper bound for the robustness of coherence , which is a full quantum coherence monotone .besides , because of its friendliness for analytical calculations , the hsd can be a useful tool for starting on quantumnesses quantification , as happened , for instance , with quantum entanglement and quantum discord .our main goal in this article is obtaining the hilbert - schmidt coherence ( hsc ) of multipartite qudit states and to use it to analyze the distribution and manipulation of quantum coherence in this kind of scenario .we shall then deal with two relevant - associated topics .we will look at the possibility of controlling the local and/or non - local coherences of a system by manipulating its local populations . and we will investigate the issue of a possible non - monotonic behavior of quantum distance measures under tensor products .the non - monotonicity under tensor products ( nmutp ) of a distance measure , investigated recently for the trace distance , is the possibility for an inversion in the dissimilarity relation between pairs of states when we consider their copies .that is to say , for four density matrices , it can happen that once there is no known analytical expression for the trace distance coherence for general states of systems with dimension greater or equal to three , and with the aim of unveiling the possible implications of the nmutp in quantum information science , here we shall analyze it using the hsd and look for its possible implications in coherence quantification .the remainder of the article is structured as follows . in sec .[ sec : hsd_nqudits ] we make use of bloch s representation to write the hilbert - schmidt distance between two generic -qudit states as an euclidean distance between the associated rescaled bloch s vectors . in sec .[ sec : hsc_nqudits ] we use this result to obtain an analytical formula for the hilbert - schmidt coherence of -qudit states .we instantiate this general result by looking at one - qubit ( sec .[ sec : qubit ] ) , one - qutrit ( sec .[ sec : qutrit ] ) , two - qubit ( sec . [ sec:2qubits ] ) , and two copies of one - qubit ( sec .[ sec : rho_rho ] ) states . after writing , in sec .[ sec:2qubits ] , the total hsc of two - qubit states in terms of its local and nonlocal parts , in sec .[ sec : rho_rho ] we study the possibility for controlling these quantities by manipulating the local populations of two copies of one - qubit states .the contrasting behaviors of hsc , -norm coherence , and relative entropy of coherence in this regard are noticed .the non - monotonicity of the hsd under tensor products is investigated in sec .[ sec : nmutp ] , where we report the first example of an important implication of this kind of property of a quantum distance function .we summarize our findings and discuss related open questions in sec .[ sec : conclusion ] .let , the identity matrix , and be a basis for .we can use these operators to form a basis for , with , and to write any -qudit state as : if the matrices satisfy the algebraic relations then the unit trace of requires . besides , the other parameters , which are the components of the so called bloch s vector , can be obtained as follows :   now let s regard the hilbert - schmidt distance ( hsd ) , between two generic -qudit states .using we shall have thus , if we define the components of the _ rescaled bloch s vector _ as with the analogous holding for , the hsd between and can be written as the euclidean distance between the vectors and :   we observe that if the generators are hermitian , then the distinguishability of the two states gains a nice physical significance in terms of differences between the corresponding pairs of observables mean values .notice that for this kind of generator we have , e.g. , , with . for a generic two - qudit density matrix , ref . showed an optimized way to compute numerically the components of the associated bloch s vector ( the fortran code produced for that purpose can be accessed freely in ) .incoherent states , , are represented by density matrices which are diagonal in some reference , orthonormal , basis .that is to say  with being a probability distribution .the hilbert - schmidt coherence ( hsc ) of a state is defined as its minimum hilbert - schmidt distance ( hsd ) to incoherent states :   for computing the hsc , one convenient choice for the generators is the generalized gell mann matrices : these three sets of matrices are named the _ diagonal _ , _ symmetric _ , and _ antisymmetric _ sets , respectively .the particular cases with and give the well known pauli and gell mann matrices , respectively .  if the reference basis is chosen to be the -qubit _ computational basis _ , with , then the presence of any non - null component of the bloch s vector corresponding to generators from the symmetric and/or antisymmetric sets implies in the presence of off diagonal elements in the density matrix .thus , the more general -qudit incoherent state is obtained utilizing the identity matrix and the generators from the diagonal set : above we used the notation .  as seen in eq .( [ eq : hsd_nqudits ] ) , the hsd is given in terms of the sum of positive terms , which are equal to the square of the difference between corresponding components of the rescaled vectors .therefore , the minimum in eq .( [ eq : hsc ] ) is obtained if we set for all and .thus we see that the optimal incoherent state under the hsd is   we shall define the _ coherence vector _ , , as the vector obtained using the components of the rescaled bloch s vector in eq .( [ eq : nbv ] ) corresponding to one or more generators from the symmetric and/or antisymmetric sets .hence the hsc of a -qudit state can be cast as the euclidean norm of this dimension coherence vector : in the next sub - sections , we exemplify the application of this general result to some particular cases .in this example , , and .so , the one - qubit hsc is given by in this section , and in the remainder of this article , when there is no chance for confusion , we omit the density matrix in the expression for the mean values .   here , , and .so ,    now , , and where the local part of the coherence was set to with being the one - qubit reduced states ( their hsc read as in eq .( [ eq : c1qb ] ) ) . the non - local part of the coherence may be cast as    as a special case of the class of states regarded in the last sub - section , let us look to two copies of one - qubit states : . for this kind of statewe have and   considering that the local populations are equal to with , we notice that although we can not control the local hsc by manipulating , we can change the non - local hsc by modifying the difference between those populations : .the result above is to be compared with the one for the -norm coherence ( l1c ) , which , in contrast to the hsc , is a coherence monotone under the incoherent operations regarded in ref .the l1c is given by the sum of the absolute values of the off - diagonal elements of a density matrix .although is present in the off - diagonal elements of , it cancels out when we perform that sum , and we get with .so , we can not control the non - local l1c by tuning local populations .because of its operational interpretation , the relative entropy of coherence ( rec ) may be considered to be one of the main coherence quantifiers introduced up today .the rec is defined and given by : with being the density matrix obtained by setting the off - diagonal elements of to zero and is von neumann s entropy . using the additivity of von neumann s entropy under tensor products , i.e. , and the fact that , we see that the rec is also additive under tensor products , i.e. , .so , we shall be able to change the rec of the two copies of only if we can change its own rec .an interesting result we report here in this direction is that , contrary to what happens with the hsc and l1c , we can modify the rec of a qubit by controlling its populations . for a qubit state , , with the form of the equation for is an indicative that the term , related to the populations , does not cancel out .we verify that this is indeed the case via an example , which is shown in fig .[ fig : rec ] .   for a fixed value of the -norm coherence if we look at the bloch s sphere in the space , a constant value of ( or of ) will be associated with points in the lateral surface of a cylinder of physical states within the sphere . in this cylinder , for running from to , different values of shall lead to different values of .the uppermost and downmost circles of that cylinder surface touch the sphere surface , and in these circles lie pure states for which the bloch s vector norm , shown in the upper ( red ) curve , is equal to one . ]one can verify that , similarly to what happens with the trace distance ( td ) , the hilbert - schmidt distance ( hsd ) does not suffer from the non - monotonicity under tensor product ( nmutp ) issue for general pure states or for one - qubit collinear states .actually , one can show that this holds true for all -norm distances . for general qudit states , the probability for random quartets of states to presentthe nmutp behavior of hsd decays with increasing the system dimension , also similar to what was seen for the td . for completeness ,we show such probabilities in fig .[ fig : nmutp ] .  .]  regarding the hilbert - schmidt coherence ( hsc ) , once we see that the nmutp behavior of the hsd can be revealed on its associated hsc if we manipulate accordingly ; what we will do by using an example .we will consider and , what gives us and so in order to invert the dissimilarity relation , we fix and change , which can vary from zero up to . as , we can use e.g. to get , and consequently   these last two inequalities are a manifestation of the nmutp of hsd : in the last equation we used the fact that , under the hsd , .in this article , we started obtaining a closed formula for the hilbert - schmidt distance ( hsd ) between two generic -qudit states in terms of the euclidean distance between the corresponding normalized bloch s vectors .we used this result to derive an analytical expression for the hilbert - schmidt quantum coherence ( hsc ) of -qudit systems .this formula was then exemplified in simple cases and we applied it to investigated issues related to the distribution and transformation of quantum coherence in some composite systems .  after splitting the total hsc of two - qubit states into its local and non - local parts , we analyzed the possibility of controlling these coherences by tuning the local populations of two copies of a generic one - qubit state .related to that matter , we observed some interesting contrasting behaviors of the hsc , -norm coherence ( l1c ) , and relative entropy of coherence ( rec ) .we noticed that it is not possible to change the local hsc and l1c by tuning local populations .however , contrary the what was seen for the l1c , we can control the hsc by doing that kind of operation .the rec of two copies of a qubit state can be changed only if the local rec is changed , what we showed being possible by the control of local populations .this raises yet another bold difference among these coherence functions . in future works, it could be fruitful to investigate this kind of issue considering another coherence quantifiers .we have identified the first important implication of the non - monotonicity under tensor product ( nmutp ) property of a quantum distance measure .we showed that the nmutp of hsd , when applied to some states , can indicate that although a configuration is more coherent than another state , two independent copies of the first state possess less hsc than two copies of the last .we emphasize that we succeeded in doing that at this moment mainly because we used the hsd .this unexpected result , obtained for the hsd , calls for further investigations of the nmutp issue regarding other quantum distance measures and also considering other of its possible consequences .this work was supported by the brazilian funding agencies : conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) , processes 441875/2014 - 9 and 303496/2014 - 2 , and instituto nacional de cincia e tecnologiade informao quntica ( inct - iq ) , process 2008/57856 - 6 .                  c. napoli , t. r. bromley , m. cianciaruso , m. piani , n. johnston , and g. adesso , robustness of coherence : an operational and observable measure of quantum coherence , phys .116 , 150502 ( 2016 ) .b. dakic , y. o. lipp , x. ma , m. ringbauer , s. kropatschek , s. barz , t. paterek , v. vedral , a. zeilinger , c. brukner , and p. walther , quantum discord as resource for remote state preparation , nat .phys . 8 , 666 ( 2012 ) .

Abstract: using bloch s parametrization for qudits ( -level quantum systems ) , we write the hilbert - schmidt distance ( hsd ) between two generic -qudit states as an euclidean distance between two vectors of observables mean values in , where is the dimension for qudit . then , applying the generalized gell mann s matrices to generate , we use that result to obtain the hilbert - schmidt quantum coherence ( hsc ) of -qudit systems . as examples , we consider in details one - qubit , one - qutrit , two - qubit , and two copies of one - qubit states . in this last case , the possibility for controlling local and non - local coherences by tuning local populations is studied and the contrasting behaviors of hsc , -norm coherence , and relative entropy of coherence in this regard are noticed . we also analyze the non - monotonicity of hsd under tensor products and report the first instance of a consequence ( for coherence quantification ) of this kind of property of a quantum distance measure .