\section{Introduction}
\label{sec:1}
Molecular biology has become a high-volume information science. This rapid transformation has taken place over the past two decades and has been chiefly enabled by two technological advances: (\emph{i}) affordable and accessible high-throughput sequencing platforms, sequence diagnostic platforms and proteomic platforms, and (\emph{ii}) affordable and accessible computing platforms for managing and analyzing these data.
It is estimated that sequence data accumulates at the rate of 100 exabases per day (1 exabase~$=10^{18}$ bases) \cite{Stephens2015Big}. However, the available sequence data are of limited use without understanding their biological implications. Therefore, the development of computational methods that provide clues about functional roles of biological macromolecules is of primary importance.

Many function prediction methods have been developed over the past two decades \cite{Friedberg2006Automated, Rentzsch2009Protein}. Some are based on sequence alignments to proteins for which the function has been experimentally established \cite{Martin2004, Engelhardt2005, Clark2011}, yet others exploit other types of data such as protein structure \cite{Pazos2004, Pal2005}, protein and gene expression data \cite{Huttenhower2006}, macromolecular interactions \cite{Letovsky2003, Nabieva2005}, scientific literature \cite{Camon2005}, or a combination of several data types \cite{Troyanskaya2003, Sokolov2010, Cozzetto2013Protein}.
Typically, each new method is trained and evaluated on different data. Therefore, establishing best practices in method development and evaluating the accuracy of these methods in a standardized and unbiased setting is important. To help choose an appropriate method for a particular task, scientists often form community challenges for evaluating methods \cite{Costello2013}. The scope of these community challenges extends beyond testing methods: they have been successful in invigorating their respective fields of research by building communities and producing  new ideas and collaborations (e.g. \cite{Kryshtafovych2014CASP10}).

In this chapter we discuss a community-wide effort whose goal is to help understand the state of affairs in computational protein function prediction and drive the field forward. We are holding a series of challenges which we named the Critical Assessment of Functional Annotation, or CAFA. CAFA was first held in 2010-2011 (CAFA1) and included 23 groups from 14 countries who entered 54 computational function prediction methods that were assessed for their accuracy. To the best of our knowledge, this was the first large-scale effort to provide insights into the strengths and weaknesses of protein function prediction software in the bioinformatics community. CAFA2 was held in 2013-2014, and more than doubled the number of groups (56) and participating methods (126). Although several repetitions of the CAFA challenge would likely give accurate trajectory of the field, there are valuable lessons already learned from the two CAFA efforts

For further reading on CAFA1, the results were reported in full in \cite{Radivojac2013LargeScale}. As of this time, the results of CAFA2 are still unpublished and will be reported in the near future. The preprint of the paper is available on arXiv \cite{Jiang2015}.

\section{Organization of the CAFA challenge}
We begin our explanation of CAFA by describing the participants. The CAFA challenge generally involves the following groups: the organizers, the assessors, the biocurators, the steering committee, and the predictors (Figure~\ref{fig:cafa_time}A).

\begin{figure}
\sidecaption
\includegraphics[width=\textwidth]{pics/Organization.pdf}
\caption{\textbf{The organizational structure of the CAFA experiment.} (A) Five groups of participants in the experiment together with their main roles. Organizers, assessors and biocurators cannot participate as predictors. (B) Timeline of the experiment.}
\label{fig:cafa_time}
\end{figure}

The main role of the organizers is to run CAFA smoothly and efficiently. They advertise the challenge to recruit predictors, coordinate activities with the assessors, report to the steering committee, establish the set of challenges and types of evaluation, and run the CAFA web site and social networks. The organizers also compile CAFA data and coordinate the publication process.
The assessors develop assessment rules, write and maintain assessment software, collect the submitted prediction data, assess the data, and present the evaluations to the community. The assessors work together with the organizers and the steering committee on standardizing submission formats and developing assessment rules. The biocurators joined the experiment during CAFA2: they provide additional functional annotations that may be particularly interesting for the challenge. The steering committee members are in regular contact with the organizers and assessors. They provide advice and guidance that ensures the quality and integrity of the experiment. Finally, the largest group, the predictors, consists of research groups who develop methods for protein function prediction and submit their predictions for evaluation. The organizers, assessors and biocurators are not allowed to officially evaluate their own methods in CAFA.

CAFA is run as a timed-challenge (Figure~\ref{fig:cafa_time}B). At time $t_0$, a large number of experimentally unannotated proteins are made public by the organizers and the predictors are given several months, until time $t_1$, to upload their predictions to the CAFA server. At time $t_1$ the experiment enters a waiting period of at least several months, during which the experimental annotations are allowed to accumulate in databases such as Swiss-Prot \cite{Bairoch2005} and UniProt-GOA \cite{Huntley2015}. These newly accumulated annotations are collected at time $t_2$ and are expected to provide experimental annotations for a subset of original proteins. The performance of participating methods is then analyzed between time points $t_2$ and $t_3$ and presented to the community at time $t_3$. It is important to mention that unlike some machine learning challenges, CAFA organizers do not provide training data that is required to be used. CAFA, thus, evaluates a combination of biological knowledge, the ability to collect and curate training data and the ability to develop advanced computational methodology.

We have previously described some of the principles that guide us in organizing CAFA \cite{Friedberg2015}. It is important to mention that CAFA is associated with the Automated Function Prediction Special Interest Group (Function-SIG) that is regularly held at the Intelligent Systems for Molecular Biology (ISMB) conference \cite{Wass2014Automated}. These meetings provide a forum for exchanging ideas and communicating research among the participants. Function-SIG also serves as the venue at which CAFA results are initially presented and where the feedback from the community is sought.

\section{The Gene Ontology provides the functional repertoire for CAFA}
\label{sec:go-info}
Computational function prediction methods have been reviewed extensively \cite{Friedberg2006Automated, Rentzsch2009Protein} and are also discussed in the chapter by Cozzetto \& Jones. Briefly, a function prediction method can be described as a classifier: an algorithm that is tasked with correctly assigning biological function to a given protein. This task, however, is arbitrarily difficult unless the function comes from a finite, preferably small, set of functional terms. Thus, given an unannotated protein sequence and a set of available functional terms, a predictor is tasked with associating terms to a protein, giving a score (ideally, a probability) to each association.

The Gene Ontology (GO) \cite{Ashburner2000Gene} is a natural choice when looking for a standardized, controlled vocabulary for functional annotation. GO's high adoption rate in the protein annotation community helped ensure CAFA's attractiveness, as many groups were already developing function prediction methods based on GO, or could  migrate their methods to GO as the ontology of choice. A second consideration is GO's ongoing maintenance: GO is continuously maintained by the Gene Ontology Consortium, edited and expanded based on ongoing discoveries related to the function of biological macromolecules.

One useful characteristic of the basic GO is that its directed acyclic graph structure can be used to quantify the information provided by the annotation; for details on the GO structure see the chapter by Munoz-Torres \textit{et al.} Intuitively, this can be explained as follows: the annotation term ``Nucleic acid binding" is less specific than ``DNA binding" and, therefore, is less informative (or has a lower \textit{information content}). (A more precise definition of information content and its use in GO can be found in \cite{Lord2003Semantic, Schnoes2013Biases}.) The following question arises: if we know that the protein is annotated with the term ``Nucleic acid binding", how can we quantify the additional information provided by the term ``DNA binding" or incorrect information provided by the term ``RNA binding"? The hierarchical nature of GO is therefore important in determining proper metrics for annotation accuracy. The way this is done will be discussed in Section \ref{sec:assessing_quality}.

When annotating a protein with one or more GO terms, the association of each GO term with the protein should be described using an Evidence Code (EC), indicating how the annotation is supported. For example, the Experimental Evidence code (EXP) is used in an annotation to indicate that an experimental assay has been located in the literature, whose results indicate a gene product's function. Other experimental evidence codes include Inferred by Expression Pattern (IEP), Inferred from Genetic Interaction (IGI), and Inferred from Direct Assay (IDA), among others. Computational evidence codes include lines of evidence that were generated by computational analysis, such as orthology (ISO), genomic context (IGC), or identification of key residues (IKR). Evidence codes are not intended to be a measure of trust in the annotation, but rather a measure of provenance for the annotation itself. However, annotations with experimental evidence are regarded as more reliable than computational ones, having a provenance stemming from experimental verification. In CAFA, we treat proteins annotated with experimental evidence codes as a ``gold standard'' for the purpose of assessing predictions, as explained in the next section. The computational evidence codes are treated as predictions.

From the point of view of a computational challenge, it is important to emphasize that the hierarchical nature of the GO graph leads to the property of \emph{consistency} in functional annotation. Consistency means that when annotating a protein with a given GO term, it is automatically annotated with all the ancestors of that term. For example, a valid prediction cannot include ``DNA binding" but exclude ``Nucleic acid binding" from the ontology because DNA binding implies nucleic acid binding. We say that a prediction is not consistent if it includes a child term, but excludes its parent. In fact, the UniProt resource and other databases do not even list these parent terms from a protein's experimental annotation. If a protein is annotated with several terms, a valid complete annotation will automatically include all parent terms of the given terms,  propagated to the root(s) of the ontology. The result is that a protein's annotation can be seen as a consistent sub-graph of GO. Since any computational method effectively chooses one of a vast number of possible consistent sub-graphs as its prediction, the sheer size of the functional repertoire suggests that function prediction is non-trivial.

\section{Comparing the performance of prediction methods}
\label{sec:performance}
In the CAFA challenge, we ask the participants to associate a large number of proteins with GO terms and provide a probability score for each such association. Having associated a set of GO sub-graphs with a given confidence, the next step is to assess how accurate these predictions are. This involves: ($i$) establishing standards of truth and ($ii$) establishing a set of assessment metrics.

\subsection{Establishing standards of truth}
The main challenge to establishing a standard-of-truth set for testing function prediction methods is to find a large set of correctly annotated proteins whose functions were, until recently, unknown. An obvious choice would be to ask experimental scientists to provide these data from their labs. However, scientists prefer to keep the time between discovery and publication as brief as possible, which means that there is only a small window in which new experimental annotations are not widely known and can be used for assessment. Furthermore, each experimental group has its own ``data sequestration window'' making it hard to establish a common time for all data providers to sequester their data. Finally, to establish a good statistical baseline for assessing prediction method performance, a large number of prediction targets are needed, which is problematic since most laboratories research one or only a few proteins each. High-throughput experiments, on the other hand, provide a large number of annotations, but those tend to concentrate only on few functions, and generally provide annotations that have a lower information content  \cite{Schnoes2013Biases}.

Given these constraints, we decided that CAFA would not initially rely on direct communication between the CAFA organizers and experimental scientists to provide new functional data. Instead, CAFA relies primarily on established biocuration activities around the world: we use annotation databases to conduct CAFA as a time-based challenge. To do so, we exploit the following dynamics that occurs in annotation databases: protein annotation databases grow over time. Many proteins that at a given time $t_1$ do not have experimentally-verified annotation, but later, some of proteins may gain experimental annotations, as biocurators add these data into the databases. This subset of proteins that were not experimentally annotated at $t_1$, but gained experimental annotations at $t_2$, are the ones that we use as a test set during assessment (Figure~\ref{fig:cafa_time}B). In CAFA1 we reviewed the growth of Swiss-Prot over time and chose 50,000 \textit{target proteins} that had no experimental annotation in the Molecular Function or Biological Process ontologies of GO. At $t_2$, out of those 50,000 targets we identified 866 \textit{benchmark proteins}; i.e., targets that gained experimental annotation in the Molecular Function and/or Biological Process ontologies. While a benchmark set of 866 proteins constitutes only 1.7\% of the number of original targets, it is large enough set for assessing performance of prediction methods. To conclude, exploiting the history of the Swiss-Prot database enabled its use as the source for standard-of-truth data for CAFA. In CAFA2, we have also considered experimental annotations from UniProt-GOA \cite{Huntley2015} and established 3,681 benchmark proteins out of 100,000 targets (3.7\%).

One criticism of a time-based challenge is that when assessing predictions, we still may not have a full knowledge of a protein's function. A protein may have gained experimental validation for function $f_1$, but it may also have another function, say $f_2$, associated with it, which has not been experimentally validated by the time $t_2$. A method predicting $f_2$ may be judged to have made a false-positive prediction, even though it is correct (only we do not know it yet). This problem, known as the ``incomplete knowledge problem'' or the ``open world problem'' \cite{Dessimoz2013}
is discussed in detail in the chapter by Skunca \textit{et al}. Although the incomplete knowledge problem may impact the accuracy of time-based evaluations, its actual impact in CAFA has not been substantial. There are several reasons for this, including the robustness of the evaluation metrics used in CAFA, and that the newly added terms may be unexpected and more difficult to predict. The influence of incomplete data and conditions under which it can  affect a time-based challenge were investigated and discussed in \cite{Jiang2014Impact}. Another criticism of CAFA is that the experimental functional annotations are not unbiased because some terms have a much higher frequency than others due to artificial considerations. There are two chief reasons for this bias: first, high-throughput assays typically assign shallow terms to proteins, but being high throughput means they can dominate the experimentally-verified annotations in the databases. Second, biomedical research is driven by specific areas of human health, resulting in over-representation of health-related functions \cite{Schnoes2013Biases}. Unfortunately, CAFA1 and CAFA2 could not guarantee unbiased evaluation. However, we will expand the challenge in CAFA3 to collect genome-wide experimental evidence for several biological terms. Such an assessment will result in unbiased evaluation on those specific terms.

\subsection{Assessment metrics}
\label{sec:assessing_quality}
When assessing the prediction quality of different methods, two questions come to mind. First, what makes a good prediction? Second, how can one score and rank prediction methods? There is no simple answer to either of these questions. As GO comprises three ontologies that deal with different aspects of biological function, different methods should be ranked separately with respect to how well they perform in Molecular Function, Biological Process, or the Cellular Component ontologies. Some methods are trained to predict only for a subset of any given GO graph. For example, they may only provide predictions of DNA-binding proteins or of mitochondrial-targeted proteins. Furthermore, some methods are trained only on a single species or a subset of species (say, eukaryotes), or using specific types of data such as protein structure, and it does not make sense to test them on benchmark sets for which they were not trained. To address this issue, CAFA scored methods both in general performance, but also on specific subsets of proteins taken from humans and  model organisms, including \textit{Mus musculus}, \textit{Rattus norvegicus}, \textit{Arabidopsis thaliana}, \textit{Drosophila melanogaster}, \textit{Caenorhabditis elegans}, \textit{Saccharomyces cerevisiae}, \textit{Dictyostelium discoideum}, and \textit{Escherichia coli}. In CAFA2, we extended this evaluation to also assess the methods only on benchmark proteins on which they made predictions; i.e., the methods were not penalized for omitting any benchmark protein.

One way to view function prediction is as an information retrieval problem, where the most relevant functional terms should be correctly retrieved from GO and properly assigned to the amino-acid sequence at hand. Since each term in the ontology implies some or all of its ancestors,\footnote{Some types of edges in Gene Ontology violate the transitivity property (consistency assumption), but they are not frequent.} a function prediction program's task is to assign the best consistent sub-graph of the ontology to each new protein and output a prediction score for this sub-graph and/or each predicted term. An intuitive scoring mechanism for this type of problem is to treat each term independently and provide the precision-recall curve. We chose this evaluation as our main evaluation in CAFA1 and CAFA2.

Let us provide more detail. Consider a single protein on which evaluation is carried out, but keep in mind that CAFA eventually averages all metrics over the set of benchmark proteins. Let now $T$ be a set of experimentally-determined nodes and $P$ a non-empty set of predicted nodes in the ontology for the given protein. Precision ($pr$) and recall ($rc$) are defined as

$$
pr(P,T)=\frac{|P\cap T|}{|P|}; \qquad rc(P,T)=\frac{|P\cap T|}{|T|},
$$

\noindent where $|P|$ is the number of predicted terms, $|T|$ is the number of experimentally-determined terms, and $|P \cap T|$ is the number of terms appearing in both $P$ and $T$; see Figure~\ref{fig:cafa_metrics} for an illustrative example of this measure. Usually, however, methods will associate scores with each predicted term and then a set of terms $P$ will be established by defining a score threshold $t$; i.e., all predicted terms with scores greater than $t$ will constitute the set $P$. By varying the decision threshold $t \in [0, 1]$, the precision and recall of each method can be plotted as a curve $(pr(t), rc(t))_t$, where one axis is the precision and the other the recall; see Figure~\ref{fig:prec-rec-curves} for an illustration of pr-rc curves and \cite{Radivojac2013LargeScale} for pr-rc curves in CAFA1. To compile the precision-recall information into a single number that would allow easy comparison between methods, we used the maximum harmonic mean of precision and recall anywhere on the curve, or the maximum  $F_1$-measure which we call $F_{\max}$
\[
F_{\max} = \underset{t}{\max}\left \{ 2\times\frac{pr(t)\times rc(t)}{pr(t)+rc(t)} \right \},
\]

\noindent where we modified $pr(t)$ and $rc(t)$ to reflect the dependency on $t$. It is worth pointing out that the F-measure used in CAFA places equal emphasis on precision and recall as it is unclear which of the two should be weighted more. One alternative to $F_1$ would be the use of a combined measure that weighs precision over recall, which reflects the preference of many biologists for few answers with a high fraction of correctly predicted terms (high precision) over many answers with a lower fraction of correct predictions (high recall); the rationale for this tradeoff is illustrated in Figure~\ref{fig:prec-rec-curves}. However, preferring precision over recall in a hierarchical setting can steer methods to focus on shallow (less informative) terms in the ontology and thus be of limited use. At the same time, putting more emphasis on recall may lead to overprediction, a situation in which many or most of the predicted terms are incorrect. For this reason, we decided to equally weight precision and recall. Additional metrics within the precision-recall framework have been considered, though not implemented yet.

\begin{figure}[t]
\sidecaption
\includegraphics[width=\textwidth]{pics/metrics_new.pdf}
\caption{\textbf{CAFA assessment metrics.} (A) Red nodes are the predicted terms $P$ for a particular decision threshold in a hypothetical ontology and (B) blue nodes are the true, experimentally determined terms $T$. The circled terms represent the overlap between the predicted subgraph and the true subgraph. There are two nodes (circled) in the intersection of $P$ and $T$, whereas $|P|=5$ and $|T|=3$. This sets the prediction's precision at $\frac{2}{5}=0.4$ and recall at $\frac{2}{3}=0.667$, with $F_1= 2\times \frac{0.4\times 0.667}{0.4+0.667} = 0.5$. The remaining uncertainty ($ru$) is the information content of the uncircled blue node in panel B, while the misinformation ($mi$) is the total information content of the uncircled red nodes in panel A. An information content of any node $v$ is calculated from a representative database as $-\log \Pr(v|\textrm{Pa}(v))$; i.e., the probability that the node is present in a protein's annotation given that all its parents are also present in its annotation.}
\label{fig:cafa_metrics}
\end{figure}

\begin{figure}[t]
\sidecaption
\includegraphics[width=\textwidth]{pics/curves_new.pdf}
\caption{\textbf{Precision-recall curves and remaining uncertainty-misinformation curves.} This figure illustrates the need for multiple assessment metrics, and understanding  the context in which the metrics are used. (A) two pr-rc curves corresponding to two prediction methods $M_1$ and $M_2$. The point on each curve that gives $F_{\max}$ is marked as a circle. Although the two methods have a similar performance according to $F_{\max}$, method $M_1$ achieves its best performance at high recall values, whereas method $M_2$ achieves its best performance at high precision values. (B) two ru-mi curves corresponding to the same two prediction methods  with marked points where the minimum semantic distance is achieved. Although the two methods have similar performance in the pr-rc space, method $M_1$ outperforms $M_2$ in ru-mi space. Note, however, that the performance in ru-mi space depends on the frequencies of occurrence of every term in the database. Thus, two methods may score differently in their $S_{\min}$ when the reference database changes over time, or using a different database.
}

\label{fig:prec-rec-curves}
\end{figure}

Precision and recall are  useful because they are easy to interpret: a precision of $\frac{1}{2}$ means that one half of all predicted terms are correct, where a recall of $\frac{1}{3}$ means that a third of the experimental terms have been recovered by the predictor. Unfortunately, precision-recall curves and $F_1$, while simple and interpretable measures for evaluating ontology-based predictions, are limited because they ignore the hierarchical nature of the ontology and dependencies among terms. They also do not directly capture the information content of the predicted terms.

Assessment metrics that take into account the information-content of the terms were developed in the past \cite{Lord2003Semantic, Lord2003Investigating, Pesquita2009Semantic}, and are also detailed in the chapter by Pesquita. In CAFA2 we used an information-theoretic measure in which each term is assigned a probability that is dependent on the probabilities of its direct parents. These probabilities are calculated from the frequencies of the terms in the database used to generate the CAFA targets. The entire ontology graph, thus, can be seen as a simple Bayesian network \cite{Clark2013Informationtheoretic}. Using this representation, two information-theoretic analogs of precision and recall can be constructed. We refer to these quantities as \emph{misinformation} ($mi$), the information content attributed to the nodes in the predicted graph that are incorrect, and \emph{remaining uncertainty} ($ru$), the information content of all nodes that belong to the true annotation but not the predicted annotation. More formally, if $T$ is a set of experimentally-determined nodes and $P$ a set of predicted nodes in the ontology, then
$$
   ru(P, T)=-\sum_{v\in T-P}\log \Pr(v|\textrm{Pa}(v)); \quad
   mi(P, T)=-\sum_{v\in P-T}\log \Pr(v|\textrm{Pa}(v)),
$$
\noindent where $\textrm{Pa}(v)$ is the set of parent terms of the node $v$ in the ontology (Figure~\ref{fig:cafa_metrics}). A single performance measure to rank methods, the minimum semantic distance $S_{\min}$, is the minimum distance from the origin to the curve $(ru(t), mi(t))_t$. It is defined as
\[
S_{\min} = \underset{t}{\min}\left \{ (ru^k(t) + mi^k(t)) ^{\frac{1}{k}} \right \},
\]

\noindent where $k \geq 1$. We typically choose $k=2$, in which case $S_{\min}$ is the minimum Euclidean distance between the ru-mi curve and the origin of the coordinate system (Figure~\ref{fig:prec-rec-curves}B). The ru-mi plots and $S_{\min}$ metrics compare the true and predicted annotation graphs by adding an additional weighting component to high-information nodes. In that manner, predictions with a higher information content will be assigned larger weights. The semantic distance has been a useful measure in CAFA2 as it properly accounts for term dependencies in the ontology. However, this approach also has limitations in that it relies on an assumed Bayesian network as a generative model of protein function as well as on the available databases of protein functional annotations where term frequencies change over time. While the letter limitation can be remedied by more robust estimation of term frequencies in a large set of organisms, the performance accuracies in this setting are generally less comparable over two different CAFA experiments than in the precision-recall setting.

\section{Discussion}

Critical assessment challenges have been successfully adopted in a number of fields due to several factors. First, the recognition that improvements to methods are indeed necessary. Second, the ability of the community to mobilize enough of its members to engage in a challenge. Mobilizing a community is not a trivial task, as groups have their own research priorities and only a limited amount of resources to achieve them, which may deter them from undertaking a time-consuming and competitive effort a challenge may pose. At the same time, there are quite a few incentives to join a community challenge. Testing one's method objectively by a third party can establish credibility, help point out flaws, and suggest improvements. Engaging with other groups may lead to collaborations and other opportunities. Finally, the promise of doing well in a challenge can be a strong incentive heralding a group's excellence in their field. Since the assessment metrics are crucial to the performance of the teams, large efforts are made to create multiple metrics and to describe exactly what they measure. Good challenge organizers try to be attentive to the requests of the participants, and to have the rules of the challenge evolve based on the needs of the community. An understanding that a challenge's ultimate goal is to improve methodologies and that it takes several rounds of repeating the challenge to see results.

The first two CAFA challenges helped clarify that protein function prediction is a vibrant field, but also one of the most challenging tasks in computational biology. For example, CAFA provided evidence that the available function prediction algorithms outperform a straightforward use of sequence alignments in function transfer. The performance of methods in the Molecular Function category has consistently been reliable and also showed progress over time (unpublished results from CAFA2). On the other hand, the performance in the Biological Process or Cellular Component ontologies has not yet met expectations. One of the reasons for this may be that the terms in these ontologies are less predictable using amino acid sequence data and instead would rely more on high-quality systems data; e.g., see \cite{Costanzo2010}. The challenge has also helped clarify the problems of evaluation, both in terms of evaluating over consistent sub-graphs in the ontology but also in the presence of incomplete and biased molecular data. Finally, although it is still early, some best practices in the field are beginning to emerge. Exploiting multiple types of data is typically advantageous, although we have observed that both machine learning expertise and good biological insights tend to result in strong performance. Overall, while the methods in the Molecular Function ontology seem to be maturing, in part because of the strong signal in sequence data, the methods in the Biological Process and Cellular Component ontologies still appear to be in the early stages of development. With the help of better data over time, we expect significant improvements in these categories in the future CAFA experiments.

Overall, CAFA generated a strong positive response to the call for both challenge rounds, with the number of participants substantially growing between CAFA1 (102 participants) and CAFA2 (147). This indicates that there exists significant interest in developing computational protein function prediction methods, in understanding how well they perform, and in improving their performance. In CAFA2 we preserved the experiment rules, ontologies and metrics we used in CAFA1, but also added new ones to better capture the capabilities of different methods. The CAFA3 experiment will further improve evaluation by facilitating unbiased evaluation for several select functional terms.

More rounds of CAFA are needed to know if computational methods will improve as a direct result of this challenge. But given the community's growth and growing interest, we believe that CAFA is a welcome addition to the community of protein function annotators.

\section{Acknowledgements}
We thank Kymberleigh Pagel and Naihui Zhou for helpful discussions. This work was partially supported by NSF grants DBI-1458359 and DBI-1458477.

\bibliographystyle{spmpsci}


\section{Introduction}

A deeper understanding of the low temperature macroscopic phases of a many-body system
can only be achieved with the aid of a quantum theory.
Of particular interest is the modeling and description of how the system goes
from one phase to another, a process known as ``phase transition''.
In ordinary phase transitions,
which occur at finite temperatures, the
phase change is driven by thermal fluctuations. For example, if we heat a magnet
we arrive at a temperature (Curie temperature) above which it loses its magnetism. In other
words, the magnet initially in the ferromagnetic phase, where all spins are aligned,
changes to the paramagnetic phase, where the magnetic moments are in a disordered state.

However, a phase transition can also occur at or near absolute zero temperature ($T=0$), where
thermal fluctuations are negligible. This process is called
a quantum phase transition (QPT)\cite{sac99} and is attained
varying a tuning parameter in the
Hamiltonian (e.g. an external magnetic field) while keeping the temperature
fixed. When the tuning parameter reaches a particular
value, the so called critical point (CP), the system's Hamiltonian ground state changes
drastically, which reflects in an abrupt modification of the macroscopic properties of the system.
In this scenario, quantum fluctuations, which loosely speaking are governed by the
Heisenberg uncertainty principle, are responsible
for the phase transition.
Although it is impossible to
achieve $T=0$ due to the third law of thermodynamics,
the effects of QPTs can be observed at finite temperatures
whenever the de Broglie wavelength is greater
than the correlation length of thermal fluctuations\cite{sac99}.
Important examples of QPTs are
the paramagnetic-ferromagnetic transition in some metals\cite{row10},
the superconductor-insulator transition\cite{dol10},
and superfluid-Mott insulator transition\cite{gre02}.

The ground state of a many-body system at $T=0$ near a CP is often described by a
non-trivial wave function due to the long-range correlations among the system's constituents.
In Ref.~\refcite{pres00} Preskill argued that quantum entanglement
could be responsible for these correlations and therefore the methods developed by quantum information
theory (QIT) could be useful in studying the critical behavior of many-body systems. Besides,
new protocols for quantum computation and communication could be formulated based on such systems.
Along these lines many theoretical tools (in particular entanglement quantifiers)
originally developed to tackle QIT problems
were employed to determine the CPs of QPTs at $T=0$ \cite{Nie99}.
Later L.-A. Wu et al.\cite{lidar} proved a general connection between non-analyticities in
bipartite entanglement measures and QPTs while in Ref.~\refcite{oliveira}
this connection was extended to multipartite entanglement.

Recently another quantity, namely, quantum discord (QD), has attracted the attention of the
quantum information community. QD goes beyond the concept of entanglement and captures
in a certain sense the ``quantumness'' of the correlations between two parts of a system.
It was built by noting the fact that two classically
equivalent versions of the mutual information are inequivalent in the quantum domain \cite{zurek,vedral}.
Using QD one can show that quantum entanglement does not describe all quantum correlations existing
in a correlated quantum state. In other words, it is possible to create quantum correlations
other than entanglement between two quantum systems via local operations and classical communication (LOCC).

The first study showing a possible connection between QD and QPT was done by
R. Dillenschneider\cite{Dil08}  in the context of spin chains at $T=0$,
where a CP associated to a QPT was well characterized by QD.
Further studies subsequently have confirmed the usefulness of QD in
describing other types of QPTs at zero temperature\cite{disT0}.
Moreover, it is important to note
that such quantum informational approaches to spotlight CPs of QPTs do not
require the knowledge of an order parameter (a macroscopic quantity
that changes abruptly during the QPT); only the extremal values or
the behavior of the derivatives of either the entanglement or QD is sufficient.

The previous theoretical studies, however, were restricted to the zero temperature
regime, which is
experimentally unattainable
; the third law of thermodynamics
dictates that it is impossible to drive a system to $T=0$ by a finite amount of
thermodynamic operations.
Due to this limitation it is therefore not straightforward to directly compare
those theoretical results with experimental data obtained at finite $T$.
In order to overcome this problem
one has to study the behavior of
quantum correlations in a system at thermal equilibrium,
which is described by the canonical ensemble
$\rho_T=e^{-H/kT}/Z$, with $H$ being the system's Hamiltonian, $k$ the Boltzmann's
constant and $Z=\mbox{Tr}\left(e^{-H/kT}\right)$ the partition function.
{\it Thermal entanglement}, i.e. the entanglement computed for states described
by $\rho_T$, was first studied by M. C. Arnesen {\it et al.}\cite{Ved01} for
a finite unidimensional Heisenberg chain. Other interesting works followed this one,
where thermal entanglement has been considered for other Hamiltonians, both
for finite\cite{Kam02,Rig03} and infinite chains\cite{ltte}. See Ref. \refcite{reviews}
for extensive reviews on entanglement and QPT.
However, the focus of the aforementioned works was not the study of the ability of
thermal entanglement to point out CPs when $T>0$.
In Ref.~\refcite{Wer10}, two of us introduced the analogous to thermal entanglement,
namely, {\it thermal quantum discord} (TQD), and we studied the behavior of this quantity
in a system consisting of two spins described by the $XYZ$ model in the presence of an
external magnetic field. In this work it was observed for the first time that
TQD could be able to signal a QPT at \textit{finite} $T$.

In order to fully explore the previous possibility, highlighted by solving
a simple two-body problem\cite{Wer10}, we tackled
the XXZ Hamiltonian in the thermodynamic
limit for several values of $T>0$\cite{werPRL}. Now, working with infinite chains,
we were able to show for the
first time that TQD keeps its ability to
detect the CPs associated to the QPTs for the XXZ model even if $T\neq 0$, while
entanglement and other thermodynamic quantities are not as good as TQD. Also,
we showed that these quantities when contrasted to TQD lose for increasing $T$ their
CP-detection property faster than TQD. Later\cite{werPRA}
we generalized those results considering $(i)$ the XXZ Hamiltonian and
$(ii)$ the XY Hamiltonian both in the presence
of an external magnetic field. For these two models it was shown that
among the usual quantities employed to detect CPs, TQD was the best suited to
properly estimate them when $T>0$.

Our goal in this paper is to present a short but self-contained review of
our aforementioned results
about TQD and its application as a CP detector of QPTs.
To this end we structure this paper as follows.
In Sec. 2 we present a brief review about quantum correlations
where QD and the entanglement of formation take on a prominent role.
In Sec. 3 we review the behavior of TQD in the context of simple two-qubit models.
We then move on to the analysis of the ability of quantum correlations,
in particular TQD, to spotlight the CPs of QPTs when the system is
at $T>0$ and in the thermodynamic limit. In this context we study the
XXZ and XY models with and without an external magnetic field.
Finally, we conclude and discuss future directions in Sec. 4.

\section{Quantum Correlations}

The superposition principle of quantum mechanics is directly related to
the existence of (quantum) correlations that are not seen
in classical objects. This principle together with the tensorial nature of
combining different quantum systems (Hilbert spaces) lead to entanglement,
which implies intriguing correlations among
the many constituents of a composite system that puzzle our classical minds.
It is worth mentioning that this tensorial nature from which a composite quantum
system is described in terms of its parts is not a truism. Indeed, the principle of
superposition is also present in classical physics, for example in the classical theory
of electromagnetism. However, in classical physics this tensorial nature for combining systems is missing,
which helps in understanding why ``weird'' quantum effects such as non-locality\cite{EPR} are not
seen in a classical world.

During many decades after the birth of quantum mechanics
quantum correlations were thought to be necessarily linked to non-locality,
or more quantitatively, to the violation of a Bell-like inequality\cite{Bel64}.
The non-violation of a Bell-like inequality implies that
the correlations among the parts of a composite system
can be described by a local realistic theory.
This fact led many to call a state
not violating any Bell-like inequality a classical state.

This situation changed by the seminal work of R. F. Werner\cite{Wer89}, who showed
that there are mixed entangled states that do not violate any Bell-like inequality.
Therefore, according to the Bell/non-locality paradigm these states should be considered
examples of classical states although possessing entanglement.
This state of affairs was unsatisfactory and the notion of classical states was
expanded. A classical (non-entangled) state was then defined as any state
that can be created only by local operations on the subsystems and classical communication
among its many parts (LOCC)\cite{nielsen,Hor09}. For a  bipartite system described by the density
operator $\rho_{AB}$, the states created via LOCC (separable states) can
be generally written as
\begin{eqnarray}\label{sstate}
\rho_{AB}=\sum_jp_j\rho_j^A\otimes\rho_j^B,
\end{eqnarray}
where $p_j\geq 0$, $\sum_jp_j=1$, and $\rho_j^{A,B}$ are legitimate density matrices.
If a quantum state cannot be written as (\ref{sstate})
then it is an entangled state.

At this point one may wonder if this is a definitive characterization of a classical state.
Or one may ask: Isn't there any ``quantumness'' in the correlations for some sort of
separable (non-entangled) quantum states? Can we go beyond the entanglement paradigm?
As observed in refs.~\refcite{zurek,vedral} there exist some states written as (\ref{sstate})
that possess non-classical features. This fact led the authors of
refs.~\refcite{zurek,vedral,ved10b} to push further our definition of classical states.
Now, instead of eq. (\ref{sstate}), we call a bipartite
state classical if it can be written as
\begin{eqnarray}
\rho_{AB}=\sum_{jk}p_{jk}|j \rangle_A \langle j|\otimes |k\rangle_B\langle k|,
\label{two}
\end{eqnarray}
where $|j\rangle_A$ and $|k\rangle_B$ span two sets of orthonormal states.
States described by (\ref{two}) are a subset of those described by (\ref{sstate})
and they are built via mixtures of locally distinguishable states\cite{ved10b}.
Intuitively, classical states are those where the superposition principle does
not manifest itself either on the level of different Hilbert spaces (zero entanglement)
or on the level of single Hilbert spaces (no Schr\"odinger cat states leading to a mixture
of non-orthogonal states). Such states have null QD.

Let us be more quantitative and define QD for a bipartite quantum
state \cite{zurek,vedral} divided into parts $A$ and $B$.
In the paradigm of classical information theory\cite{nielsen}
the total correlation between $A$ and $B$ is quantified by the mutual information (MI),
\begin{eqnarray}\label{mi1}
\mathcal{I}_1(A:B)= \mathcal{H}(A)+\mathcal{H}(B)-\mathcal{H}(A,B),
\end{eqnarray}
where $\mathcal{H}(X)=-\sum_xp_x\log_2p_x$ is the Shannon entropy with $p_x$
the probability distribution of the random variable $X$. The conditional
probability for classical variables $p_{a|b}$ is defined by the Baye's rule,
that is, $p_{a|b}=p_{a,b}/p_{b}$, with $p_{a,b}$ denoting the joint probability distribution
of variables $a$ and $b$. Using the conditional probability one  can show that
$\mathcal{H}(A,B) = \mathcal{H}(A|B) + H(B)$, where $\mathcal{H}(A|B)=-\sum_{a,b}p_{a,b}\log_2(p_{a|b})$.
This last result allows one to write MI,
eq.~(\ref{mi1}), as
\begin{eqnarray}\label{mi2}
\mathcal{I}_2(A:B)= \mathcal{H}(A)-\mathcal{H}(A|B).
\end{eqnarray}
Note that $\mathcal{H}(A|B)\geq0$
is the conditional entropy, which quantifies
how much uncertainty is left on average about $A$ when one knows $B$.
The quantum version of eq. (\ref{mi1}), denoted by $\mathcal{I}^q_1(A:B)$,
can be obtained replacing the Shannon entropy by the von-Neumann entropy
$\mathcal{S}(X)=\mathcal{S}(\rho_X)=-\mbox{Tr}\left(\rho_X\log_2\rho_X\right)$,
where $\rho_X$ is the density operator of the system $X=A,B$. On the other
hand, a quantum version of eq. (\ref{mi2}) is not so straightforward because
the Bayes' rule is not always valid for quantum systems\cite{brule}.
For instance, this rule is violated for a pure entangled state. Indeed,
one can show that if we naively extend the usual quantum conditional entropy
to quantum systems as $\mathcal{S}(A|B)\equiv\mathcal{S}(A,B)-\mathcal{S}(B)$,
it becomes negative for pure entangled states. QD is built in a way to
circumvent this limitation.

In order to build a meaningful quantum version of the conditional entropy
$\mathcal{H}(A|B)$ it is necessary to take into account the fact that knowledge about
system B is related to measurements on $B$.
And now, differently from the classical case, a measurement in the quantum domain
can be performed in many non-equivalent ways (different set of projectors, for instance).
If a general quantum measurement, i.e. a POVM (positive operator valued measure)\cite{nielsen},
$\{M_b\}$ is performed
in the quantum state $\rho_{AB}$, then after the measurement the state is
described by $\sum_bM_b\rho_{AB}M_b^\dagger$.
The probability of the outcome $b$ of $B$ is $p_b=\mbox{Tr}\left[M_b\rho_{AB}M_b^\dagger\right]$ and
the conditional state of $A$ in this case is
$\rho_{A|b}=\left(M_b\rho_{AB}M_b^\dagger\right)/p_b$. Thus, the conditional
entropy with respect to the POVM $\{M_b\}$ is $\mathcal{S}(A|\{M_b\})\equiv\sum_bp_b\mathcal{S}(\rho_{A|b})$.
Therefore, in order to quantify the uncertainty left on $A$ after a measurement on $B$
one has to minimize over all POVMs. This leads to the following definition
of the quantum conditional entropy\cite{zurek,vedral}:
\begin{eqnarray}\label{qce}
\mathcal{S}_q(A|B)\equiv\min_{\left\{M_b\right\}}\mathcal{S}(A|\{M_b\}).
\label{cond}
\end{eqnarray}

At this point, the quantum version of the mutual information (\ref{mi2}),
denoted by $\mathcal{I}^q_2(A:B)$, is obtained replacing the classical
conditional entropy $\mathcal{H}(A|B)$ by its quantum analog $\mathcal{S}_q(A|B)$,
which does not assume negative values.  The {\it quantum discord} is
defined as the difference between these two versions of the quantum mutual
information\cite{zurek,vedral}:
\begin{eqnarray}\label{qd}
D(A|B)&\equiv&\mathcal{I}^q_1(A:B)-\mathcal{I}^q_2(A:B)\nonumber\\
&=&\mathcal{S}_q(A|B)-\mathcal{S}(A|B).
\end{eqnarray}
Note that QD is not necessarily a symmetric quantity, because the
conditional entropy (\ref{qce}) depends on the system in which the
measurement is performed. However, for the density operators studied
in this paper QD is always symmetric. Furthermore, while for mixed
states there are states with null entanglement and QD $>0$, for pure
states QD is essentially equivalent to entanglement. In other words,
only for mixed
states there might be quantum correlations other than entanglement.
As demonstrated recently\cite{ved10}, a quantum state $\rho_{AB}$ has $D(A|B)=0$ if,
and only if, it can be written as
$\rho_{AB}=\sum_jp_j\rho_j^A\otimes\left|\psi_j^B\right\rangle\left\langle \psi_j^B\right|$,
with $\sum_jp_j=1$ and $\{\left|\psi_j^B\right\rangle\}$ a set of orthogonal states.
This result shows the importance of the superposition principle
to explain the origin of the quantum correlations. It is due to
this principle that one can generate a set $\{\left|\psi_j^B\right\rangle\}$
of non-orthogonal states leading to states with nonzero QD.

For arbitrary $N\times M$-dimensional bipartite states the computation of QD involves a complicated
minimization procedure whose origin can be traced back to the evaluation of
the conditional entropy $\mathcal{S}_q(A|B)$, eq. (\ref{cond}).
In general one must then rely on numerical procedures to get QD
and it is not even known whether a general efficient algorithm exists.
For two-qubit systems, however, the minimization over generalized measurements
can be replaced by the minimization over projective measurements
(von Neumann measurements)\cite{minDIS}. In this case the minimization
procedure can be efficiently implemented numerically\cite{Dnum} for arbitrary
two-qubit states and some analytical results can be achieved for a restricted
class of states\cite{Danalitico}. In this work we will be dealing with density
matrices $\rho$ in the X-form, that is, $\rho_{12}=\rho_{13}= \rho_{24}=\rho_{34}=0$.
Moreover, in our models $\rho_{22}=\rho_{33}$ and all matrix elements are real, making the numerical
evaluation of QD simple and fast.

To close this section, we introduce the measure of entanglement
used in this paper, the {\it Entanglement of Formation} (EoF)\cite{Woo98}.
EoF quantifies, at least for pure states,
how many singlets are needed per copy of $\rho_{AB}$
to prepare many copies of $\rho_{AB}$ using only LOCC.  For an $X$-form
density matrix we have
\begin{eqnarray}\label{eof}
EoF(\rho_{AB})&=&-g\log_2g-(1-g)\log_2(1-g),
\end{eqnarray}
with $g=(1+\sqrt{1-C^2})/2$ and the concurrence\cite{Woo98}
given by $C=2\max\left\{0,\Lambda_1,\Lambda_2 \right\}$,
where $\Lambda_1=|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}}$ and
$\Lambda_2=|\rho_{23}|-\sqrt{\rho_{11}\rho_{44}}$.

\section{Results and Discussions}

\subsection{Two interacting spins}

In this section we consider a two
spin system described by the XYZ model with an external magnetic
field acting on both spins.\cite{Wer10}
The Hamiltonian of this model is
\begin{eqnarray}\label{hxyz}
H_{XYZ}=\frac{J_x}{4}\sigma^x_1\sigma^x_{2}+\frac{J_y}{4}\sigma^y_1\sigma^y_{2}+\frac{J_z}{4}\sigma^z_1\sigma^z_{2}
+ \frac{B}{2}\left(\sigma^z_1+\sigma^z_2\right),
\end{eqnarray}
where $\sigma_j^\alpha$ ($\alpha=x,y,z$) are the
usual Pauli matrices acting on the $j$-th site and
we have assumed $\hbar=1$. As mentioned above, the
density matrix describing a system in equilibrium with
a thermal reservoir at temperature $T$ is $\rho_T=e^{-H/kT}/Z$,
where $Z=\mbox{Tr}\left(e^{-H/kT}\right)$ is the partition function.
Therefore, the thermal state for the Hamiltonian (\ref{hxyz}) assumes the following form
\begin{equation}
\rho  =  \frac{1}{Z} \left(
\begin{array}{cccc}
A_{11} & 0 & 0 & A_{12}\\
0 & B_{11}  & B_{12} & 0 \\
0 & B_{12} & B_{11} & 0 \\
A_{12} & 0 & 0 & A_{22} \\
\end{array}
\right), \label{rho2}
\end{equation}
with $A_{11}$  $=$ $\mathrm{e}^{-\alpha}$ $(\cosh(\beta)$ $-$ $4B$ $\sinh(\beta)/\eta)$,
$A_{12}$ $=$  $-$ $\Delta$ $\mathrm{e}^{-\alpha}$ $\sinh(\beta)/\eta$,
$A_{22}$ $=$ $\mathrm{e}^{-\alpha}$ $(\cosh(\beta)$ $+$ $4$ $B$ $\sinh(\beta)/\eta)$,
$B_{11}$ $=$ $\mathrm{e}^{\alpha}$
$\cosh(\gamma)$, $B_{12}$ $=$ $-$ $\mathrm{e}^{\alpha}$ $\sinh(\gamma)$,
and $Z = 2\left( \exp{(-\alpha)}\cosh(\beta) + \exp{(\alpha)}\cosh(\gamma) \right)$,
where $\Delta = J_{x} - J_{y}$, $\Sigma = J_{x} + J_{y}$,
$\eta = \sqrt{\Delta^{2} + 16B^2}$, $\alpha = J_{z}/(4kT)$,
$\beta = \eta/(4kT)$, and $\gamma = \Sigma/(4kT)$.

The first important result appears in the absence of an external
field. As shown in Ref.~\refcite{Rig03}, when $B=0$, the entanglement
does not increase with increasing temperature. On the other
hand, as can be seen in Fig.~\ref{fig1} (panels $a$ and $b$) for the
XXZ model $\left(J_x=J_y=J \quad\mbox{and}\quad J_z\neq0 \right)$,
TQD begins with a non-null value at $T=0$ and increases as $T$ increases
before decreasing with $T$,
while EoF is always zero\cite{Rig03}.
Note that such effect is observed
for different configurations of coupling constants.
\begin{figure
\centerline{\psfig{file=fig1b.eps,width=4.00in}}
\caption{TQD for a two spin system as a function of the absolute temperature $kT$ for
the $XXZ$ model with $B=0$. (a) Here $J_z=-0.5$ and $J$ $=$ $0.1$ (solid line),
$0.2$ (dashed line), $0.3$ (dotted line), $0.4$ (dash-dotted line);
(b) Now we fix $J=0.4$ and $J_z=-0.8$ (solid line), $-0.7$ (dashed line),
$-0.6$ (dotted line), $-0.5$ (dash-dotted line). Here and in the
following graphics all quantities are dimensionless.}
\label{fig1}
\end{figure}

The possibility of
TQD to point out a QPT even when the system is at finite temperatures emerged from
our study about the XXX model for two spins\cite{Wer10}. The XXX model
is obtained from Hamiltonian (\ref{hxyz}) making $J_x=J_y=J_z=J$.
When $J\rightarrow\infty$ the density operator (\ref{rho2}) will be
the Bell state $\rho=\left|\psi\right\rangle\left\langle\psi\right|$,
with $\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left(\left|01\right\rangle-\left|10\right\rangle\right)$,
for any $T$. For the opposite limit $J\rightarrow-\infty$ the density
operator is the mixed state $\rho=\frac{1}{3}\left(\left|00\right\rangle\left\langle
00\right|+\left|11\right\rangle\left\langle11\right|+\left|\phi\right\rangle\left\langle \phi\right|\right)$
with $\left|\phi\right\rangle=\frac{1}{\sqrt{2}}\left(\left|01\right\rangle +\left|10\right\rangle\right)$.
In this case EoF is zero, while TQD assumes the value $1/3$.
Furthermore, as shown in Fig.~\ref{fig2}, EoF is zero in the
ferromagnetic region $(J<0)$ and non-zero in the antiferromagnetic
region $(J>0)$ only when $T=0$.
\begin{figure
\centerline{\psfig{file=fig2b.eps,width=4.00in}}
\caption{TQD (a) and EoF (b) for a two spin system as functions of the coupling $J$
for $kT=0.05$ (solid line), $0.1$ (dashed line), $0.5$ (dotted line),
$1.0$ (dash-dotted line). Both plots for the XXX model with $B=0$.}
\label{fig2}
\end{figure}
For $T>0$ EoF becomes non-zero
only for $J>J_c(T)=kT\ln(3)$. On the other hand, TQD is equal
to zero only at the
trivial
point $J=0$, even at finite $T$.
Although we are considering here only two spins, such result
suggests that TQD may possibly signal a QPT for $T>0$.

Let us now analyze the case where $B\neq 0$ and focus on
the XY model in a transverse magnetic field ($J_x$, $J_y\neq 0$, and $J_z=0$).
As noted in Ref.~\refcite{Rig03}
EoF shows a sudden death and then a revival (see Fig.~\ref{fig3}, panel b).
\begin{figure
\centerline{\psfig{file=fig3b.eps,width=4.00in}}
\caption{TQD (a) and EoF (b) for a two spin system as functions of the absolute temperature
$kT$ for the XY model with transverse magnetic field $B$. Here $J_x=2.6$
and $J_y=1.4$. The values for $B$ are $1.1$ (solid line),
$2.0$ (dashed line), and $2.5$ (dotted line).}
\label{fig3}
\end{figure}
However, TQD does not suddenly disappear as Fig.~\ref{fig3}, panel a,
depicts. Actually, TQD decreases with $T$ to a non-null value
and after a critical temperature $T_c$
it starts increasing again. This effect is called {\it regrowth}\cite{Wer10}.
Although the regrowth of EoF with temperature is not observed for
two spin chains, we showed in Ref.~\refcite{werPRA} that such interesting
behavior is possible in the thermodynamic limit. Also, if we carefully look at
Fig.~\ref{fig3} the distinctive aspects of these two types of quantum correlations
become more evident. For example, comparing panels a and b we see
regions where TQD increases while EoF decreases.  Finally, in a very interesting
and recent work, X. Rong \textit{et al.}\cite{Ron12} experimentally verified some of the
predictions here revised and reported in refs.~\refcite{Wer10,werPRL,werPRA}, namely,
the sudden change of TQD at finite temperatures while
changing the anisotropy parameter of a two spin XXZ Hamiltonian.

\subsection{XXZ Model}

Turning our attention to infinite chains ($L\rightarrow \infty$), let us start
working with the one-dimensional anisotropic spin-$1/2$
XXZ model subjected to a magnetic field in the $z$-direction.
Its Hamiltonian is
\begin{eqnarray}\label{hxxz}
H_{xxz}&=&J\sum_{j=1}^L\left(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y
+\Delta\sigma_j^z\sigma_{j+1}^z\right) - \frac{h}{2}\sum_{j=1}^L\sigma_j^z,
\end{eqnarray}
where $\Delta$ is the anisotropy parameter, $h$ is the external
magnetic field, and $J$ is the exchange constant ($J=1$). We have
assumed periodic boundary conditions $\left(\sigma_{L+1}^\alpha=\sigma_1^\alpha\right)$.
The nearest neighbor two spin state is obtained by
tracing all but the first two spins, $\rho_{1,2}=\mbox{Tr}_{L-2}(\rho)$,
where $\rho=\exp{\left(-\beta H_{xxz}\right)}/Z$. The Hamiltonian (\ref{hxxz})
exhibits both translational invariance and $U(1)$ invariance $\left(\left[H_{xxz},
\sum_{j=1}^L\sigma_j^z\right]=0\right)$ leading to the following nearest neighbor
two spin state
\begin{eqnarray}\label{rho}
\rho_{1,2}  = \frac{1}{4}\left(
\begin{array}{cccc}
\rho_{11} & 0 & 0 & 0\\
0 & \rho_{22}  & \rho_{23} & 0 \\
0 & \rho_{23} & \rho_{22} & 0 \\
0 & 0 & 0 & \rho_{44}  \\
\end{array}
\right),
\end{eqnarray}
where
\begin{eqnarray}\label{rhoe}
\rho_{11} &=& 1+2\left\langle \sigma^z\right\rangle+\med{\sigma_1^z\sigma_2^z},\nonumber\\
\rho_{22} &=& 1-\med{\sigma_1^z\sigma_2^z},\\
\rho_{44} &=& 1-2\left\langle \sigma^z\right\rangle+\med{\sigma_1^z\sigma_2^z},\nonumber\\
\rho_{23} &=& 2\med{\sigma_1^x\sigma_2^x}\nonumber.
\end{eqnarray}

The magnetization and the two-point correlations above are obtained in terms
of the derivatives of the free-energy\cite{NLIE}, $f=-\frac{1}{\beta}
\lim_{L\rightarrow \infty} \frac{\ln{Z}}{L}$,
\begin{eqnarray*}
\med{\sigma^z}&=& -2\partial_h f/J, \\
\med{\sigma_j^{z}\sigma_{j+1}^{z}}&=&\partial_{\Delta}f/J,  \\
\med{\sigma_j^{x}\sigma_{j+1}^{x}}&=&\frac{u-\Delta \partial_{\Delta}f+
h \med{\sigma^z}}{2J}, \\
\med{\sigma_j^{z}\sigma_{j+1}^{z}}&=&\med{\sigma_j^{x}\sigma_{j+1}^{x}}=
\frac{u + h \med{\sigma^z}}{3J}, ~~ \Delta=1,
\end{eqnarray*}
where $u=\partial_{\beta} (\beta f)$ is the internal energy. We explained
in details the procedures to determine the free-energy $f$ in Ref.~\refcite{werPRA}.
It is not a simple task and it involves the application of complicated
analytical and numerical computations. The CPs of the XXZ
model depend on the value of the magnetic field $h$\cite{GAUDIN}.
One of them, called $\Delta_{inf}$, is an infinite-order QPT
determined by solving the following equation,
\begin{eqnarray}\label{cpinf}
h=4J\sinh(\eta)\sum_{n=-\infty}^\infty\frac{(-1)^n}{\cosh(n\eta)},
\end{eqnarray}
with $\eta = \cosh^{-1}(\Delta_{inf})$. The other CP, called $\Delta_1$, is
a first-order QPT given by
\begin{eqnarray}\label{cp1}
\Delta_1 = \frac{h}{4J}-1.
\end{eqnarray}

The behavior of TQD and EoF for the XXZ model at finite $T$
and $h=0$ was initially studied in ref.~\refcite{werPRL}. For $h=0$
the XXZ model has two CPs\cite{TAKAHASHI}. At $\Delta_{inf}=1$
the ground state changes from an XY-like phase ($-1<\Delta<1$)
to an Ising-like antiferromagnetic phase ($\Delta>1$).
At $\Delta_1=-1$ it changes from a ferromagnetic phase ($\Delta<-1$) to
the critical antiferromagnetic phase ($-1<\Delta<1$). In ref.~\refcite{werPRL}
we analyzed the behavior of TQD and EoF only for $\Delta>0$.
Here we extend those results by computing the correlation functions
for the remaining values of the anisotropy parameter, namely, $\Delta<0$.
In Fig.~\ref{fig4} we plot
TQD (panel a) and EoF (panel c) as a function of $\Delta$ for
different values of $kT$ and $h=0$.
\begin{figure
\centerline{\psfig{file=fig4.eps,width=4.00in}}
\caption{TQD (panels a and b) and EoF (panels c and d) as functions
of the tuning parameter $\Delta$ for $h=0$ (panels a and c) and $h=12$
(panels b and d). From top to bottom in the region where
$\Delta_1<\Delta<\Delta_{inf}$, $kT=0,0.1,0.5,1.0,2.0$.}
\label{fig4}
\end{figure}
For $T=0$ we can see that both
TQD and EoF are able to detect the CPs. TQD is discontinuous
at $\Delta_1$ while at $\Delta_{inf}$ the first-order derivative of
TQD presents a discontinuity. Furthermore, EoF is zero for
$\Delta<\Delta_1$ and non-zero for  $\Delta>\Delta_1$ reaching a
maximum value at $\Delta=\Delta_{inf}$. However, as the temperature
increases the maximum value of EoF is shifted to the right. Besides,
EoF becomes zero also for $\Delta>\Delta_1$ as we increase $T$.
On the other hand, the first-order derivative of TQD is still discontinuous at
$\Delta_{inf}=1$ for finite $T$. We can also observe that TQD
increases for $\Delta<-1$ as $T$ increases while
its first-order derivative diverges at the CP $\Delta=-1$. As mentioned
in Ref.~\refcite{werPRL} the cusp-like behavior at CPs $\Delta_1=1$
and $\Delta_{inf}=-1$ is due to an exchange in the set of projectors
that minimizes the quantum conditional entropy (\ref{qce}).

The study about the XXZ model was further explored in
Ref.~\refcite{werPRA}, with the addition of an external
field. The effects of the magnetic field $h$ on the
quantum correlations are exemplified in
Fig.~\ref{fig4} (panels b and d), where we set $h=12$.
The values of the CPs for $h=12$ are calculated employing
Eqs. (\ref{cpinf}) and (\ref{cp1}), resulting in
$\Delta_{inf}\approx4.88$ and $\Delta_1=2$. Again,
for $T=0$ the CPs are detected by both TQD and EoF
and, differently from the case $h=0$, the behavior of
these two quantities is quite similar. Both quantities
are zero for $\Delta<2$ and non-zero for $\Delta>2$, with
their first-order derivatives diverging at the CP $\Delta_1=2$.
However, the infinite-order QPT is no longer characterized by a
global maximum of TQD or EoF, but by a discontinuity in their first-order
derivatives. Note also that TQD presents a cusp-like behavior
between the CPs. This behavior is once again related to the minimization
procedure of the quantum conditional entropy and so far it is
not associated to any known QPT for this model.
It is important to mention that entanglement measures may
also have a discontinuity and/or a divergence in their
derivatives that are not related to a QPT\cite{yang}.

\begin{figure
\centerline{\psfig{file=fig5.eps,width=4.0in}}
\caption{First-order (panel a) and second-order (panel b) derivatives
of TQD as functions of $\Delta$ for the XXZ model with $h=12$ and
for $kT = 0.02$ (solid line), $kT=0.1$ (dashed line), and $kT = 0.5$ (dotted line).
The derivatives plotted here are normalized, that is, each curve was
divided by the maximum value of the respective derivative. The
maximum of the first-order and second-order derivatives of TQD
are very close to the CPs $\Delta_1=2$
and $\Delta_{inf}\approx4.88$, respectively.}
\label{fig5}
\end{figure}
Now we move on to the cases where $T>0$. When $T$ increases both curves
of TQD and EoF become smoother and broader, with well defined derivatives
in the CPs. Besides, the cusp-like behavior of TQD previously mentioned tends
to disappear while both maximums of TQD and EoF decrease\cite{werPRA}.
We noted in Ref.~\refcite{werPRA} that some features of the derivatives
of these quantities remain for a finite, but not too high temperature.
To illustrate this fact, we plotted in Fig.~\ref{fig5} the first-order
(panel a) and second-order (panel b) derivatives of TQD with respect
to the anisotropy parameter $\Delta$ for $h=12$ and $kT=0.02, 0.1, 0.5$.
To plot the curves for different temperatures in the same graph we
normalized the derivatives of TQD, that is, for each $T$ we plotted
the derivative of TQD divided by the maximum value of the respective
derivative. For $T=0$ the divergence in the first-order derivative of both
TQD and EoF spotlights the CP $\Delta_1$ while the CP $\Delta_{inf}$ is
characterized by a divergence in the second-order derivative.
As can be seen in Fig.~\ref{fig5}, although the divergence at the
CPs disappears as $T$ increases, the derivatives reach their
maximum values around the CPs. We used these maximum values to
estimate the CPs at finite temperatures. The same analysis was
applied to estimate the CPs using EoF instead of TQD.
In Ref.~\refcite{werPRA}
we compared the ability of TQD, EoF, and pairwise correlations
($\med{\sigma_1^z\sigma_2^z}$ and $\med{\sigma_1^x\sigma_2^x}$)
to point out the CPs for $T>0$ and we showed that TQD is the
best candidate to estimate the CPs.

\begin{figure
\centerline{\psfig{file=fig6.eps,width=3.4in}}
\caption{The difference between the correct CP and the
CP estimated by TQD (square), EoF (circle), $\med{\sigma_1^x\sigma_2^x}$ (up arrow),
and $\med{\sigma_1^z\sigma_2^z}$ (down arrow) as a function of $kT$.
In (a) and (b) we have $h=6$ with a first (a) and an infinite-order
(b) CP; in (c) and (d) we have $h=12$ with a first (c) and an infinite-order
(d) CP. $\Delta_c$ denotes the correct value of CP while $\Delta_e$
denotes the value of  CP estimated by the extremum values of the
derivatives of the quantities involved.}
\label{fig6}
\end{figure}
To illustrate such result
we compared in Fig.~\ref{fig6} the difference between the correct
CP $\Delta_c$ and the CP estimated by our method $\Delta_e$ for
$h=6$ and $h=12$. One can see in this figure that from zero to
$kT\approx 1$ the CPs estimated by TQD are closer to the correct ones
than the estimated CPs coming from other quantities.

\subsection{XY Model}

The Hamiltonian of the one-dimensional XY model in a transverse field is given by
\begin{eqnarray}
\label{HXY}
H_{xy}&=& - \frac{\lambda}{2} \sum_{j=1}^L
\left[(1+\gamma)\sigma^x_j\sigma^x_{j+1}
+(1-\gamma)\sigma^y_j\sigma^y_{j+1}\right]
-\sum_{j=1}^L\sigma^z_j,
\end{eqnarray}
where $\lambda$ is the strength of the inverse of
the external transverse magnetic field and $\gamma$
is the anisotropy parameter. The transverse Ising model
is obtained for $\gamma=\pm1$ while $\gamma=0$ corresponds
to the XX model in a transverse field\cite{LSM}.
At $\lambda_c=1$ the XY model undergoes a second-order
QPT (Ising transition\cite{isingQPT}) that separates a
ferromagnetic ordered phase from a quantum paramagnetic
phase. Another second order QPT is observed for $\lambda>1$
at the CP $\gamma_c=0$ (anisotropy transition\cite{LSM,anisQPT}).
This transition is driven by the
anisotropy parameter $\gamma$ and separates a ferromagnet
ordered along the $x$ direction and a ferromagnet ordered
along the $y$ direction. These two transitions are of the same
order but belong to different universality classes \cite{LSM,anisQPT}.

The XY Hamiltonian is $Z_2$-symmetric and can be exactly
diagonalized \cite{LSM} in the thermodynamic limit $L\rightarrow\infty$.
Due to translational invariance the two spin density operator $\rho_{i,j}$
for spins $i$ and $j$ at thermal equilibrium is \cite{osborne}
\begin{eqnarray}\label{doXY}
\rho_{0,k}&=&\frac{1}{4}\left[I_{0,k}
+\left\langle \sigma^z\right\rangle\left(\sigma^z_0+\sigma^z_k\right)\right]
+ \frac{1}{4} \sum_{\alpha=x,y,z}
\left\langle \sigma^\alpha_0\sigma^\alpha_k\right\rangle \sigma^\alpha_0\sigma^\alpha_k,
\end{eqnarray}
where $k=\left|j-i\right|$ and $I_{0,k}$ is the identity operator of dimension four.
The transverse magnetization $\left\langle \sigma^z_k\right\rangle$
$=$ $\left\langle \sigma^z\right\rangle$ is
\begin{eqnarray}\label{tmag}
\left\langle \sigma^z\right\rangle=-\int_0^\pi
(1+\lambda\cos{\phi})\tanh{(\beta\omega_\phi)}\frac{d\phi}{2\pi\omega_\phi},
\end{eqnarray}
with $\omega_\phi=\sqrt{(\gamma\lambda\sin{\phi})^2+(1+\lambda\cos{\phi})^2}/2$.
The two-point correlation functions are given by
\begin{eqnarray}\label{tpcf}
\left\langle \sigma^x_0\sigma^x_k\right\rangle &=& \left|
\begin{array}{cccc}
G_{-1} & G_{-2} & \cdots & G_{-k}\\
G_{0} & G_{-1}  & \cdots & G_{-k+1} \\
\vdots & \vdots & \ddots & \vdots \\
G_{k-2} & G_{k-3} & \cdots & G_{-1}  \\
\end{array}
\right|,\\
\left\langle \sigma^y_0\sigma^y_k\right\rangle &=& \left|
\begin{array}{cccc}
G_{1} & G_{0} & \cdots & G_{-k+2}\\
G_{2} & G_{1}  & \cdots & G_{-k+3} \\
\vdots & \vdots & \ddots & \vdots \\
G_{k} & G_{k-1} & \cdots & G_{1}  \\
\end{array}
\right|,\\
\left\langle \sigma^z_0\sigma^z_k\right\rangle &=&
\left\langle \sigma^z\right\rangle^2 - G_k G_{-k},
\end{eqnarray}
where
\begin{eqnarray*}
G_k&=&\int_0^\pi \tanh{(\beta\omega_\phi)}\cos{(k\phi)}(1+\lambda\cos{\phi})
\frac{d\phi}{2\pi\omega_\phi}\\
&-&\gamma\lambda\int_0^\pi \tanh{(\beta\omega_\phi)} \sin{(k\phi)\sin{\phi}}
\frac{d\phi}{2\pi\omega_\phi}.
\end{eqnarray*}

The relation between TQD and QPT for
the Ising model (XY model with $\gamma=1$) at $T=0$ was
investigated initially by Dillenschneider\cite{Dil08} for
first and second nearest-neighbors. More general results were
obtained in Ref.~\refcite{Sar10} where TQD and EoF from
first to fourth nearest-neighbors was computed for different
values of $\gamma$. This study at $T=0$ showed that while EoF between
far neighbors becomes zero, QD is not null and detects the QPT. The
effects of the symmetry breaking process in entanglement and QD for the XY
and the XXZ models were discussed in Refs.~\refcite{Ami10}, where the
low temperature regime was taken into account. In Ref.~\refcite{werPRA}
we compared the ability of TQD and EoF for first and second nearest-neighbors
to detect the CPs for the XY model at finite temperature.

The behavior of TQD and EoF for first nearest-neighbors as a
function of $\lambda$ for $kT=0.01,0.1,0.5$ and $\gamma=0,0.5,1.0$
can be seen in Fig.~\ref{fig7}. First, note that TQD is more robust
to temperature increase than EoF. For $kT=0.5$ TQD is
always non-zero while EoF is zero or close to zero for almost all
$\lambda$ (see the blue/solid curves in Fig.~\ref{fig7}).
\begin{figure
\centerline{\psfig{file=fig7.eps,width=4.0in}}
\caption{(a)-(c) TQD and (d)-(f) EoF as functions of
$\lambda$ for $kT=0.01$ (black/dashed line), $kT=0.1$ (red/dotted line) and
$kT=0.5$ (blue/solid line) for nearest-neighbors.
We use three values of $\gamma$ as shown in the graphs.}
\label{fig7}
\end{figure}
As showed in Ref.~\refcite{werPRA}, for second nearest-neighbors the
situation is more drastic since EoF is always zero for $kT=0.5$. Now,
to estimate the CPs at finite $T$ we used the same procedure adopted for
the XXZ model. If the first-order derivative of TQD or EoF is divergent at
$T=0$ then the CP is pointed out by a local maximum or minimum at $T>0$;
if the first-order derivative is discontinuous at $T=0$ then we look after
local maximum or minimum in the second derivative for $T>0$. These extreme
values act as indicators of QPTs. The CPs estimated with such
method are denoted by $\lambda_e$ while the correct CPs are denoted
by $\lambda_c$. The differences between $\lambda_c$ and $\lambda_e$ as a
function of kT for $\gamma=0,0.5,1.0$ are plotted in Fig.~\ref{fig8}. We can
see in this figure that TQD provides a better estimate of the CP $\lambda_c=1$
than EoF.
\begin{figure
\centerline{\psfig{file=fig8.eps,width=4.25in}}
\caption{The difference between the correct CP, $\lambda_c$, and, $\lambda_e$,
the CP estimated either by TQD (square) or EoF (circle) as a function
of $kT$ for (a) $\gamma=0$, (b) $\gamma=0.5$, and (c) $\gamma=1.0$.
Note that both curves coincide at panel (b).}
\label{fig8}
\end{figure}

For $\gamma=0.5$ EoF and TQD give almost the same estimation of the CP
and for $\gamma=1.0$ TQD is better than EoF with predictions differing at
the second decimal place. For $\gamma = 0$, TQD outperforms
EoF already in the first decimal place. Moreover, for $\gamma=0$ TQD
is able to correctly estimate the CP for higher temperatures than
for $\gamma=0.5$ and $\gamma=1.0$.

So far we have studied a QPT driven by the magnetic field. However,
for $\lambda>1$ the XY model undergoes a QPT driven by the
anisotropy parameter $\gamma$, whose critical point is $\gamma_c=0$.
To study such transition we fixed $\lambda=1.5$. In Fig.~\ref{fig9}
we plotted TQD and EoF for the first-neighbors as
functions of $\gamma$ and for $kT = 0.001, 0.1, 0.5, 1.0,$ and $2.0$.
Note that the maximum of TQD and EoF is reached at the CP $\gamma_c=0$.
However, only TQD has a cusp-like behavior at the CP.
\begin{figure
\centerline{\psfig{file=fig9b.eps,width=4.00in}}
\caption{(a) TQD and (b) EoF for the first nearest-neighbors as
functions of $\gamma$. From top to bottom $kT=0.001, 0.1, 0.5, 1.0$, and
$2.0$. Here we fixed $\lambda = 1.5$ and the CP is $\gamma_c=0$.}
\label{fig9}
\end{figure}
This
pattern of TQD (maximum with a cusp-like behavior) remains up to $kT=2.0$.
On the other hand, the maximum of EoF at the CP can only be
seen as far as $kT<1$ for above this temperature EoF becomes zero.
In Ref.~\refcite{werPRA} we computed TQD and EoF for second-neighbors.
In this case TQD is able to detect the CP even for values
near $kT=1.0$ while EoF is nonzero only for $kT\lesssim 0.1$.

\section{Conclusions}

In this article we presented a review of our studies about the
behavior of quantum correlations in the context of spin chains at
finite temperatures. The main goal of this paper was to analyze
the quantum correlation's ability to pinpoint the critical points
associated to quantum phase transitions assuming
the system's temperature is greater than the absolute zero. The two
measures of quantum correlations studied here were
quantum discord and entanglement, with the former producing the
best results.

We first reviewed a work of two of us about a simple but illustrative
two spin-1/2 system described by
the XYZ model with an external magnetic field\cite{Wer10}. We showed many surprising
results about the thermal quantum discord's behavior. For example, and
differently from entanglement, thermal quantum discord can increase with temperature in
the absence of an external field even for such a small system; and that
there are situations where thermal quantum discord increases while
entanglement decreases with temperature.
Furthermore, for the
XXX model we observed for the first time that quantum discord could
be a good candidate to signal a critical point at finite $T$.

To check whether quantum discord is indeed a good critical point detector
for temperatures higher than absolute zero, we analyzed its behavior
for an infinite chain described by the XXZ model
without\cite{werPRL} and with\cite{werPRA}
an external magnetic field and in equilibrium with a
thermal reservoir at temperature $T$. Here we also extended our previous results by
computing the quantum correlations between the first nearest-neighbor spins
for the whole range of the anisotropy parameter (positive and negative values).
In this way we were able to describe the two critical points of the XXZ model in the
thermodynamic limit and the behavior of quantum discord near them.
The results presented here and in refs.~\refcite{werPRL,werPRA}
showed that quantum discord is far better the best critical
point detector for $T>0$ with respect to all quantities tested (entanglement,
entropy, specific heat, magnetic susceptibility, and the two-point correlation
functions).

Another model considered in this work was the XY model in a transverse
magnetic field\cite{werPRA}. Again, we computed the quantum correlations between the first
nearest-neighbors assuming the system in equilibrium with a thermal
reservoir at temperature $T$. This model has two second-order quantum
phase transitions, namely, an Ising transition and an anisotropy transition.
For the Ising transition we observed that the critical point is better
estimated by quantum discord. For the anisotropy transition both quantities,
entanglement and discord, provide an excellent estimate for the critical
point at low temperatures.
However, since for increasing temperatures quantum discord is more robust
than entanglement, the former was able to
spotlight the quantum critical point for a wider range of temperatures,
even for values of temperature where entanglement was absent.

In conclusion, we showed that for the spin models studied here and in
Refs.~\refcite{werPRL,werPRA}
quantum discord was the best quantum critical point estimator among all
quantities tested when the system assumes a finite temperature.
It is also important to mention that the knowledge of the order
parameter was not needed to estimate the critical points. Therefore,
we strongly believe that our results suggest that
quantum correlations - mainly quantum discord -  are important
tools to study quantum phase transitions in realistic scenarios,
where the temperature is always above absolute zero.

\section*{Acknowledgements}

TW and GR thank the Brazilian agency CNPq (National Council for Scientific and Technological
Development) for funding and GAPR thanks CNPq and FAPESP (State of S\~ao Paulo Research
Foundation) for funding. GR also thanks CNPq/FAPESP
for financial support through the National Institute of Science and Technology
for Quantum Information.