a crucial difference between the problem solving strategies used by experts in physics and beginning students lies in the interplay between how their knowledge is organized and how it is retrieved to solve problems  .categorizing or grouping together problems based upon similarity of solution can give a glimpse of the pattern " an individual sees in a problem while contemplating how to solve it  .in a classic study by chi et al . , a categorization task was used to assess introductory physics students level of expertise in physics . in chi s study , eight introductory physics students were asked to group together introductory mechanics problems into categories based upon similarity of solution .they found that , unlike experts ( physics graduate students in their study ) who categorized them based on the physical principles required to solve them , introductory students categorized problems involving inclined planes in one category and pulleys in a separate category  .previously , we conducted a categorization study in which 7 professors , 21 physics graduate students and more than a hundred introductory physics students in a classroom environment were asked to group together introductory physics problems based upon similarity of solution  .we found that the professors significantly outperformed both the graduate students and introductory physics students in grouping together problems based upon the physics principles involved rather than basing the grouping of the problems on the surface features of the problems and they created very similar categories  .the graduate students performed better than the introductory physics students in the categorization task .however , there is a large overlap in the performance of graduate students and introductory students in the calculus - based courses on the categorization of introductory physics problems into groups based upon the fundamental principles of physics required to solve the problems  .  while learning introductory physics is challenging , learning quantum mechanicsis perhaps even more so  . unlike classical mechanics, we do not have direct experience with the microscopic quantum world .also , quantum mechanics has an abstract theoretical framework in which the most fundamental equation , the time - dependent schroedinger equation ( tdse ) , describes the time evolution of the wave function or the state of a quantum system according to the hamiltonian of the system .this wave function is in general complex and does not directly represent a physical entity .however , the wave function at a given time can be used to calculate the probability of measuring a particular value for a given physical observable associated with the system .for example , the absolute square of the wave function in position - space gives the probability density .since the tdse does not describe the evolution or motion of a physical entity , unlike newton s second law , the modeling of the microscopic world in quantum mechanics is generally more abstract than the modeling of the macroscopic world in classical mechanics .the conceptual framework of quantum mechanics is often counter - intuitive to our everyday experiences . according to quantum theory ,the position , momentum , energy and other observables for a quantum mechanical entity are in general not well - defined .we can only predict the probability of measuring different values based upon the wave function when a measurement is performed .this probabilistic interpretation of quantum mechanics , which even einstein found disconcerting , is challenging for students .moreover , according to the copenhagen interpretation of quantum mechanics , which is widely taught to students , measurement of a physical observable collapses " the wave function into an eigenstate of the operator corresponding to the observable measured .thus , the usual time evolution of the system according to the tdse is treated differently from measurement processes .students often have difficulty with this notion of an instantaneous change or collapse " of the wave function during the measurement [ 13 ] . the proper way to interpret quantum mechanics is still the subject of debate , making the subject even more challenging for physics instructors .  here , we discuss a study in which 22 physics juniors and seniors in two undergraduate quantum mechanics courses and six physics faculty members ( professors ) were asked to categorize 20 quantum mechanics problems based upon similarity of solution .we also interviewed some faculty members concerning issues related to categorization of quantum mechanics problems .all but one faculty member had taught an upper - level undergraduate or graduate level quantum mechanics course .the faculty member who had not taught quantum mechanics regularly teaches other physics graduate core " courses including electricity and magnetism and statistical mechanics .all undergraduate students in the upper - level quantum mechanics classes ( 12 and 10 students in the two classes who were present on the day the categorization task was given as a quiz ) participated .the students were given 35 - 40 minutes to perform the categorization .the faculty members performed the categorization at a time convenient to them .except for the faculty member who had not taught quantum mechanics and took longer to categorize the problems , other professors noted that it took them less than 30 minutes to perform the categorizations .the 20 problems to be categorized ( given in the appendix ) were adapted from the problems found among the end of the chapter exercises in commonly used upper - level undergraduate quantum mechanics textbooks . all those who performed the categorizationwere provided the instructions given at the beginning of the appendix .the sheet on which individuals were asked to perform the categorization of problems had three columns . in the first column , they were asked to place their own category name " for each category ( in other words , they had to come up with their own category names ) , in the second column , they had to place a description of the category that explains why those problems can be grouped together ; in the third column , they had to list the problem numbers for the problems that should be placed in that category .we note that for solving a problem , more than one approach may be useful .the instruction for the categorization explicitly noted that a problem could be placed in more than one category .the goal was to investigate differences in categorization by faculty members and students and whether there are major differences in the ways in which individuals in each group categorize quantum mechanics problems .this study was partly inspired by the fact that a physics faculty member who was teaching advanced undergraduate quantum mechanics in a previous semester had given a take - home exam in which one problem asked students to find the wave function of a free particle after a time given the initial wave function ( which was a gaussian ) .two students approached the faculty member complaining that this material was not covered in the class .the faculty member pointed out to them that he had discussed in the class how to find the wave function after a time given an initial wave function in the context of a problem involving an infinite square well .but the students insisted that , while the time - development of the wave function may have been discussed in the context of an infinite square well , it was not discussed in the context of a free particle .it appears that the two students did not categorize the time - development issues for the infinite square well and the free particle in the same category .they did not realize that a solution procedure very similar to what they had learned in the context of the time - development of the wave function for an infinite square well should be applicable to the free particle case except they must use the energy eigenstates and eigenvalues corresponding to the free particle and replace the discrete sum over energy levels for an infinite square well by an integral since the energy levels for a free particle are continuous .this difficulty in discerning that the same concepts and procedures should be applicable in both contexts is similar to the difficulty introductory students have in discerning that the same principle is applicable in two problems that have different contexts .we note that each individual who categorized the problems had to come up with his / her own category names and justify why each problem should be placed in a particular category .the 20 questions for categorization were such that the context " in four of them was the hydrogen atom , the harmonic oscillator , the infinite square well , and the free particle ( see the appendix ) .three of the problems were related to the spin angular momentum and one was about the dirac delta function . within these different contexts ,there were questions about the time evolution of wave function , time dependence of expectation value , measurement of physical observables , expectation value including uncertainty , commutation relation between different components of the spin angular momentum etc .as noted earlier , we wanted to investigate if the questions were grouped together based upon the physics concepts and procedures required for solving them or the surface features " of the problems such as the contexts used . for problems related to the time - dependence of wave function or time - dependence of expectation value , we wanted to investigate if the faculty members and students categorized problems involving the stationary states differently from those involving the non - stationary states .we find that the categorizations of a problem performed by the students were diverse and they seldom placed a problem in more than one category although they were explicitly told they could do so if they wish .moreover , the faculty members often used a diverse set of categories unlike the highly uniform categorization by faculty members for introductory physics problems  .  to analyze the quality of categories created by the professors and the students quantitatively, we placed each category created by each individual into a matrix which consisted of problem numbers along the columns and categories along the rows .a 1 " was assigned if the problem appeared in the given category and a 0 " was assigned if the opposite was true .categories that were very similar were combined , e.g. , time - dependence of wave function " , time - development of wave function " or dynamics of wave function " were combined into a single category . in order to score the categorizations by students and faculty members , three faculty members ( a subset of those who had categorized the problems themselves ) were recruited .they were given the categorizations by students and faculty in the matrix form we had created ( without identifiers and with the categorizations by the faculty and students jumbled up ) . for example , all the different categories created by different individuals for problem ( 1 ) were placed one after the other to aid faculty members who were scoring the categorizations .for each question , the three faculty members doing the scoring were advised to read the question , think about how they would categorize it and then evaluate and score everybody s categorization .they were asked to evaluate whether each of the categories created by an individual should be considered ( assigned a score of 2 ) , moderate " ( assigned a score of 1 ) , or poor " ( assigned a score of zero ) .we note that if all three faculty members scored a particular problem for an individual as good " , the score of that individual on that problem will be 6 ( maximum possible ) .if one faculty member scored it as good " but the other two scored it as medium " , the score of that individual on that problem will be 4 .each of the 22 students and 6 faculty members categorized 20 problems .faculty members often placed a problem in more than one category .as noted earlier , some of the categories created for a problem by more than one individual were the same or similar .several categories that were similar were combined into a single category .all the three faculty members noted that evaluating and scoring other people s categorization was a very challenging task and required intense focus .one faculty member noted that it took him several hours to complete the scoring .moreover , two of the faculty members who evaluated everybody s categorization noted that they would prefer not to use the terms good " or poor " for judging the categories although some categories were better than others .the faculty members who scored others categorizations also noted that sometimes they liked the categorizations of a problem by others much more than their own .interestingly , in our earlier studies with introductory physics categorization , we had asked three faculty members to evaluate the categorizations of a subset of randomly selected individuals ( in that case we did not ask them to score all categorizations because the introductory physics classes had several hundred students )  . in scoring introductory physics categorizations ,faculty members were not hesitant in calling the categories good / poor and they did not say that the task was challenging  . they also never said that they preferred other s categorizations of a problem more than their own perhaps because there was a great conformity in faculty categorizations ( which were based upon physics principles such as the conservation of mechanical energy , conservation of momentum , conservation of angular momentum , newton s second law etc . )  .table 1 shows examples of category names for each question divided into three groups with a score of 5 or 6 " , 3 or 4 " or less than 3 " . with each category name, many faculty members and students provided an explanation justifying why certain problems should be placed in that category .inspection of table 1 shows that the categories that obtained a total score of less than 3 ( out of 6 ) included both concrete and abstract categories .for example , change in basis " for problem ( 1 ) , commutation relation " for problem ( 12 ) , matrix element " for problem ( 17 ) , rotation group " for problem ( 19 ) etc .are abstract categories that received a score of less than 3 . on the other hand , infinite square well " for problems ( 4 ) , ( 12 ) , ( 16 ) and ( 18 ) and free particle " for problems ( 3 ) , ( 7 ) and ( 10 ) etc .are examples of concrete categories that received a score of less than 3 .figure 1 shows a histogram of the percentage of people ( students or faculty ) vs. percentage of problems with a score of of better ( at least 3 out of 6 ) and figure 2 shows a histogram of the percentage of people vs. average score on the categorization task out of a maximum of 6 ( averaged over all problems ) .we note that what one faculty member scored as good " was often scored as medium " by another . while three of the six faculty members who categorized the problems were recruited to score all of the categories by all faculty members and students , the average score of the three faculty members who scored all problems was lower than those of the other three faculty members who did not score the categories .also , faculty members who scored the categorizations explicitly noted that they sometimes preferred others categorizations more than their own .thus , we do not believe that the faculty members who scored everybody s categorizations were partial to their own categories . it is interesting to note that the faculty member who had never taught quantum mechanics ( but had taught statistical mechanics and electricity and magnetism at the graduate level ) performed slightly better on average ( though not statistically significant ) than the faculty members who scored the categorizations .in fact , the faculty member who had never taught quantum mechanics but performed the categorization commented that he would like to teach quantum mechanics but was not assigned that course despite asking for it .he added that the main reason was that many other faculty members wanted to teach quantum mechanics but they did not want to teach the other graduate level courses that he was assigned .figure 1 shows that the categorizations by faculty members were rated higher overall than those by students , despite the diversity in faculty responses .we find that the faculty members were more likely to categorize the problems based upon the procedures and concepts required to solve the problems rather than the contexts involved .but figure 2 shows that none of the faculty members had an average score of 5 - 6 on the categorization task implying none of the faculty member placed all 20 problems in categories that were considered uniformly excellent ( although their categories were on average better than those of the students ) .faculty members sometimes categorized problems based upon the contexts used , e.g. , hydrogen atom , simple harmonic oscillator , angular momentum etc . however , most of the time when they did such categorizations , they also categorized the same problems in other categories which were based upon the procedures for solving the problems .they were also more likely than students to make use of the nuances in the questions to group problems , e.g. , whether the system was in a stationary state in order to categorize problems involving the time - dependence of wave function or the time - dependence of expectation value .the same category was sometimes assigned different scores for different questions depending upon whether the faculty members who scored them felt they were appropriate categories for those questions .for example , for question ( 15 ) , the category stationary state " obtained a score of at least 3 because the faculty members felt that it was relevant for determining the expectation value of momentum and for explaining whether it should depend on time . on the other hand , for question ( 20 ), stationary state " category obtained a score less than 3 ( see the appendix ) because it was not considered relevant for finding the possible values of energy after the measurement of the distance of the electron from the nucleus .faculty members who scored the categories were careful to distinguish between the categories uncertainty " and uncertainty principle " ( or uncertainty relation ) .for example , in questions ( 10 ) and ( 17 ) , expectation value and uncertainty " category obtained an average score of 5 or 6 whereas uncertainty principle " or uncertainty relation " obtained a score of less than 3 .individual discussions with the faculty who scored the categorization suggests that they saw a clear distinction between these categories . in particular , they asserted that calculating the standard deviation was about calculating the uncertainty in position but it was not about uncertainty principle " or uncertainty relation " .the question did not ask whether the product of the uncertainties in position and momentum is greater than or equal to .the overall scores ( by the three faculty members who evaluated all of the categorizations ) on concrete or context - based categories such as hydrogen atom " or harmonic oscillator " were higher than other concrete categories such as infinite square well " or free particle " , ( where four questions out of 20 given in the categorization task belong to each of these four systems as noted earlier ) .discussions with individual faculty suggest that they have a notion of a canonical quantum system that they use for thinking about concepts and to help clarify ideas about quantum mechanics . hydrogen atom " and harmonic oscillator " fit their notion of canonical quantum systems .one faculty member explicitly noted that the hydrogen atom and harmonic oscillator are quintessential in quantum mechanics .he added that the hydrogen atom embodies many essential features of other complex quantum systems but is exactly soluble and widely applicable .similarly , the harmonic oscillator is used as a model to understand diverse quantum systems such as molecular excitations and quantum optics . such explanations about why the average score for hydrogen atom " as a category was at least ( 3 out of 6 including the scores of all the three faculty members who evaluated the responses ) for three of the four questions that related to the hydrogen atom but was not for any of the four questions related to the infinite square well shed some light on why the faculty do not view all concrete " categories on the same footing .as noted earlier , most of the time when faculty members placed problems in a category involving context , such as hydrogen atom " or simple harmonic oscillator " , they also placed the same problem in another category based on the procedure involved in solving the problem .but sometimes they placed some of the problems only in concrete categories .for example , one faculty member grouped some problems about hydrogen atom in hydrogen atom " category or in a category based upon the procedure for solving the problems , e.g. , measurement " or time evolution of wave function " or in both these types of categories . during individual discussions , these faculty members were asked why their choices were more context - based in some of their groupings and more focused on the procedures and concepts to solve the problems for creating other categories . in response, some faculty members reasoned that they were perhaps using lenses with different zoom factors " for categorizing different problems .they noted that the categorization task was challenging and they sometimes zoomed in and out while categorizing different problems focusing on the contexts or the procedures for solving them .faculty members who scored the categorizations also noted that while scoring others categorizations they realized that there were many different ways to categorize the problems and sometimes others categorizations were better than their own .the faculty members were reminded during the individual discussions that while categorizing introductory physics problems , faculty always scored inclined plane category " , cliff category " or spring category " as poor categories explaining that they were based on the surface features " of the problems rather than the deep " features ( fundamental principles of physics required to solve them ) .they were asked to comment on whether making categories such as angular momentum " or hydrogen atom " was also based on the surface features " of the problems rather than the procedures relevant for solving the problems . in response to such questions , faculty members often noted that while these categories were less directly related to the procedure for solving the problems , they were hesitant to call them poor " categories .they noted , e.g. , that the knowledge about the hydrogen atom is relevant for solving the problems involving hydrogen atom even though that knowledge alone may not be the central component of how to set up the solution of the problem .for example , questions ( 11 ) and ( 14 ) in the appendix are about the hydrogen atom in a linear superposition of stationary states . in question ( 11 ) , the knowledge that the expectation value of an operator corresponding to a physical observable which does not commute with the hamiltonian depends on time in a non - stationary state is relevant to solve the problem .similarly , in question ( 14 ) , knowledge about the time - dependence of wave function in a non - stationary state is relevant for solving the problem .simply categorizing these problems in the hydrogen atom category " does not indicate whether the individual knows the procedure for solving the problem .while the faculty members agreed that some of these concrete categories may not be the best way to categorize the problems , they sometimes scored some of these context - based categories ( even if they did not give an indication of the procedures for solving the problems ) as 1 " instead of 0 " ( but rarely gave it a score of 2 " ) .as shown in table 1 , angular momentum " for questions ( 1 ) and ( 2 ) , harmonic oscillator for questions ( 5 ) and ( 6 ) , hydrogen atom for questions ( 8) , ( 14 ) and ( 20 ) are examples of such context - based categories that were judged favorably .  individual discussions with faculty members suggest that some felt that the structure of knowledge in quantum mechanics is more complex than that in introductory physics. moreover , the complexity of knowledge structure in quantum mechanics is due to both the requisite conceptual and mathematical knowledge .this complexity may make it difficult for everybody to focus on the same aspects of solution when asked to categorize ( although there are often underlying relations in faculty categorizations ) .one possible implication is that the way concepts are emphasized in a quantum mechanics course may differ based upon the patterns " that appear to be most central to the faculty member teaching the course .for example , one faculty member may emphasize the conceptual aspects while the other may emphasize the mathematical aspects .  during individual discussions ,faculty members were asked if they were surprised that the categories in which a problem was placed by different faculty members were not always similar and some faculty came up with categories that were more abstract than others . they were also asked to comment on the fact that the faculty members who scored the categorizations gave low scores not only to the concrete categories but also to some abstract categories .for example , as noted earlier , matrix element " for question ( 17 ) and rotation group " for question ( 19 ) received a cumulative score of less than 3 . in response to these questions, faculty members asserted that they were not very surprised about these because they felt that how one teaches quantum mechanics and how abstractly or concretely one presents the material depends strongly on the instructor . during discussions, several faculty members pointed out that if one takes a look at the quantum mechanics textbooks , he / she will realize that the textbooks are laid out very differently and emphasize different things . some faculty members mentioned that some undergraduate textbooks do not emphasize the postulates of quantum mechanics .also , the postulates in different textbooks are not identical ( e.g. , only some of the textbooks list the time - dependent schroedinger equation as a postulate ) .some textbooks are hesitant to mention the collapse " of the wave function during measurement while others discuss these issues in detail .they also mentioned that some textbooks start with the infinite dimensional vector space while others start with the quantum mechanics of a spin - half particle .the proponents of the spin - half first believe that it provides a simple two dimensional vector space to teach the foundations of quantum mechanics whereas those who discuss , e.g. , the infinite square well , first believe that spin is too abstract and continuity with the topics covered in the earlier courses is important . the extent to which symmetry ideas are emphasized and the conservation laws derived from them also varies in the undergraduate textbooks .discussions suggest that most faculty members believed that if there is no agreement on the basic issues about teaching undergraduate quantum mechanics , the differences in how the faculty members categorize problems , teach their courses and what they emphasize is perhaps expected .another common theme that emerged is that categorization of introductory physics problems involves identifying fundamental principles relevant for the problems , whereas in the upper - level undergraduate quantum mechanics problems , it mainly involves identifying concepts and procedures , because problem solving in such a course is tied to conceptual and procedural knowledge .some faculty members asserted that the fundamental principles of physics such a conservation of energy and conservation of momentum are important even for understanding quantum processes .however , the application of fundamental principles to quantum processes is not typically the focus of an upper - level undergraduate course .for example , one faculty member noted that for understanding the properties of a solid using neutron scattering , one will have to carefully account for the conservation of energy and momentum but questions involving these topics are typically not common in an undergraduate quantum mechanics course .he added that if such questions were given in the categorization task , there may be more uniformity in the faculty responses .the categorization of problems by students in a quantum mechanics course can be a useful tool for understanding the patterns students see in a problem when contemplating how to solve it . even in the context of quantum mechanics problems ,professors overall scored higher than students in grouping together problems based on similarity of solutions .however , unlike the categorization of introductory physics problems , in which professors categorizations are generally uniform , their categorizations were more varied in the context of quantum mechanics .the diversity of categories created for quantum mechanics may partly be due to the fact that the solution to a typical quantum mechanics problem in an upper - level quantum mechanics course typically requires the knowledge of requisite concepts and procedures .on the other hand , categorization in introductory physics is typically based on the fundamental principles of physics .faculty members noted that the fundamental principles , e.g. , conservation laws , are also important in understanding quantum processes but they are not the focus of an upper - level undergraduate quantum mechanics course . some faculty members created more abstract categories than others .it will be useful to investigate how different is the teaching emphasis of faculty members in a quantum mechanics course depending upon the types of categories they created .we thank all faculty and students who participated in this study for their help .this work is supported by the national science foundation ( phy-0653129 and 055434 ) . your task is to group the 20 problems below into various groups based upon similarity of solution on the sheet of paper provided .you can create as many categories as you wish .the grouping of problems should not be in terms of easy problems " , medium difficulty problems " and difficult problems " but rather it should be based upon the features and characteristics of the problems that make them similar . a problem can be placed in more than one group created by you . please provide a brief explanation for why you placed a set of questions in a particular group .you need not solve any problems .+ the first two questions refer to the following system : an electron is in an external magnetic field b which is pointing in the z direction .the hamiltonian for the electron spin is given by where is the gyromagnetic ratio and is the z component of the spin angular momentum operator .if the electron is initially in an eigenstate of , does the expectation value of depend on time ?justify your answer .2 .  if the electron is initially in an eigenstate of , does the expectation value of depend on time ?justify your answer .3 .  a free particle has the initial wave function where , , and are constants ( and are real and positive ) .find .4 .  a particle in an infinite square well ( ) has the initial wave function . find the uncertainty in position and momentum .5 .  in the ground state of the harmonic oscillator, what are the expectation values of position , momentum and energy ?do these expectation values depend on time ? 6 .a particle is in the first excited state of a harmonic oscillator potential . without any calculations , explain what the expectation value of momentum is and whether it should depend on time . 7 .  a free particle has the initial wave function where , and are constants ( is real and positive ) .find .an electron is in the ground state of a hydrogen atom .find the uncertainty in the energy and the component of angular momentum .make a qualitative sketch of a dirac delta function .then , make a qualitative sketch of the absolute value of the fourier transform of .label the axes appropriately for each plot .a free particle has the initial wave function where , , and are constants ( and are real and positive ) .find , , , , , .an electron in a hydrogen atom is in a linear superposition of the first and third excited states .does the expectation value of its kinetic energy depend on time ? 12 .suppose that the measurement of the position of a particle in an infinite square well ( ) yields the value at the center of the well .show that if energy is measured immediately after the position measurement , it is equally probable to find the particle in any odd - energy stationary state . 13 .an electron is in a linear combination of the ground and fourth excited states in a harmonic oscillator potential .a measurement of energy is performed and then followed by a measurement of position .what can you say about the possible results for the energy and position measurements ? 14 .an electron in a hydrogen atom is in a linear superposition of the first and third excited states .find the wave function after time .a particle is in the third excited state of a harmonic oscillator potential . without any calculations , explain what the expectation value of momentum is and whether it should depend on time . 16 .a particle in an infinite square well ( ) has the initial wave function . without normalizing the wave function ,find .a free particle has the initial wave function where , and are constants ( is real and positive ) .find , , , , , .a particle is initially in a linear combination of the ground state and the first excited state of an infinite square well . without any calculations ,explain whether the expectation value of position should depend on time .what is the commutation relation [ ? 20 .a hydrogen atom is in the first excited state .you measure the distance of the electron from the nucleus first and then measure energy .describe the possible values of energy you may measure ..examples of categories created for each question divided into three groups with a score of 5 or 6 " , 3 or 4 " or less than 3 " . ev " is an abbreviation for expectation value " and ft " is an abbreviation for fourier transform " .[ cols="^,^,^,^ " , ]

Abstract: we discuss the categorization of 20 quantum mechanics problems by physics professors and undergraduate students from two honors - level quantum mechanics courses . professors and students were asked to categorize the problems based upon similarity of solution . we also had individual discussions with professors who categorized the problems . faculty members categorizations were overall rated higher than those of students by three faculty members who evaluated all of the categorizations . the categories created by faculty members were more diverse compared to the categories they created for a set of introductory mechanics problems . some faculty members noted that the categorization of introductory physics problems often involves identifying fundamental principles relevant for the problem , whereas in upper - level undergraduate quantum mechanics problems , it mainly involves identifying concepts and procedures required to solve the problem . moreover , physics faculty members who evaluated others categorizations expressed that the task was very challenging and they sometimes found another person s categorization to be better than their own . they also rated some concrete categories such as hydrogen atom " or  simple harmonic oscillator " higher than other concrete categories such as infinite square well " or free particle " .   2em -0.25 in 6.9 in 9.62 in 1ex -.75 in


the human immunodeficiency virus encodes a protease ( hiv-1 pr ) cleaving the gag and the gag - pol viral polyproteins into enzymes and structural proteins .the discovery that inhibition of this protease ( a homodimer of 198 amino acids ) causes the formation of non - infectious virus particle has prompted an enormous effort to design efficient inhibitors against aids attack .currently , five antiviral agents are approved by fda : saquinavir ( sqv ) , ritonavir ( rtn ) , indinavir ( ind ) , nelfinavir ( nlf ) , amprenavir ( apr ) and several others are under clinical trials .therapeutic benefit is unfortunately short - lived as the virus strains - evolving under drugs selective pressure - encode hiv-1 pr multiple mutants with low drugs affinity : mutants resistant to protease inhibitors can emerge in vivo already after less than one year .the problem of drug - resistance persists also when a combination of pr and reverse transcriptase ( rt ) inhibitor are used .the occurrence of mutations withstanding antiviral drugs is not a mere consequence of drug action , rather it results from viral replication itself .indeed , mutations are found rather irrespective of drug structural diversity involving virtually every protein domain : this is the case for hiv-1 pr mutants which are resistant against fda approved drugs . among such mutations , a few involve the active site ( residue 25 - aspartic acid ) while the others belong to protein regions away from it .  by means of molecular dynamics ( md ) simulations within the framework introduced in ref . ( see next section ) , we show here that the positions where these mutations occur play a key role in the proximity of temperatures where the specific heat peaks occur .moreover , we shall argue that residues involved in a frequently observed covariant mutation , are statistically correlated .indeed , while the study of the native state structure can lead to a rational design of drugs binding the active site ( or otherwise disrupt the biological function of the agent by acting on its native structure ) the analysis of the folding pathways can provide fundamental information . in particular , it can reveal , as in the case of hiv-1 pr , the presence of kinetic bottlenecks associated to severe entropy reduction that inhibits the progress toward the native state .these bottlenecks represent the most delicate part of the folding process .they are followed by the sudden formation of specific native - like protein sub - regions and afterwards the folding process proceeds rapidly until another bottleneck is reached .the identification of sites involved in the bottlenecks and their correlation with the active site is crucial pharmaceutically because they are the ideal targets of effective drugs . from this point of view , due to the large amount of data available on drug resistance , hiv-1 pr is an excellent candidate to validate our automatic strategy to identify key folding sites . in the following , we shall present evidence showing that the crucial sites can be identified with good statistical confidence .the framework introduced here is general and , applied to other viral proteins , ought to be useful for suggesting which sites should be preferentially targeted by effective drugs .the strategy adopted here to identify the crucial sites for the folding and assembling of hiv-1 pr is based on a recent theoretical framework that allows to capture the main features of the folding process by a simplified description of both the protein structure and the folding dynamics . at the basis of the methodis the observation that the topology of the native state plays a crucial role in steering the folding process .this statement is supported by an increasing amount of experimental evidence .perhaps , the most notable examples are : ( a ) the close similarity of the transition - state conformations of proteins having structurally - related native states ( despite the very poor sequence similarity ) and ( b ) the strong influence that certain simple topological properties , such as the contact order , have on protein folding rates . such observations , and others summarised in the recent review of d. baker , complement the findings of anfinsen , who established that a protein s amino acid sequence uniquely encodes its native state . indeed ,since the topology of the native state influences the folding process , the amino acid sequence must also encode its possible folding pathways .we focus our attention on the topological rate - limiting steps along the pathways from unfolded states to the native one .such bottleneck stages , are usually found in correspondence of non - local amino acid interactions that require the overcoming of a large entropy barrier ( due to the flexibility of the peptide chain intervening between them ) ; the formation of such crucial contacts acts as a nucleus for the establishment of further native interactions and leads to a rapid progress along the folding reaction coordinate until another barrier is met .it is striking that the sites involved in the topological bottlenecks are those where the largest changes in the folding kinetics are observed in site - directed mutagenesis experiments , as first established for ci2 and barnase .this shows that nature has carefully optimised the protein sequence so to exploit the conformational entropy reduction accompanying the folding process through the careful choice of the amino acids forming the crucial contacts .  with the purpose of identifying the key sites we investigate the topological obstacles encountered during the formation of the native hiv-1 pr structure .such sites are , intuitively , the ideal candidate targets of effective drugs , as they take part to the most delicate steps of the folding process .this fact was first recognised by anfinsen in connection with the staphylococcal nuclease . the most effective strategy to preventthe protein formation is acting on residues involved in the key contacts and undermining the formation / overcoming of bottleneck stages .one of the distinctive features of the hiv virus is the extremely high rate of mutations .the capability of encoding several mutants provides a possibility for hiv-1 pr to elude the disruptive action of the drug by intervening on the key sites .this seems an unavoidable countermeasure since the viable mutants ( i.e. those with native - like enzymatic activity ) retain the original native structure and hence , arguably , encounter the same bottlenecks as the wild - type . within this framework , the lapse of time during which the drug therapy is temporarily effective , corresponds to the time taken by the virus to encode , through random mutations , a mutant form of hiv-1 pr where the crucial sites have been fine - tuned to overcome not only the kinetic bottlenecks ( as for the wild type ) but also the additional drug attack .the key sites identified through the method explained in the next sections , have been compared with the known key mutating positions of hiv-1 pr , finding a highly significant correlation between the two of them .in addition , previously unexplained co - variant mutations seen in hiv-1 pr are explained as arising due to the correlation between distinct topological bottlenecks .the model that we adopted encompasses an energy - scoring function of the go - type .this is one of the simplest energy functionals and provides a natural topological bias to the native state by rewarding the formation of native pairwise interactions . in the version used here , which is a generalization of ref . apt for molecular dynamics studies , the cooperativity of the folding process is enhanced by the introduction of repulsive non - native interactions . in our hamiltonian , each pair of non - consecutive amino acids interacts with the following strength :   +  v_1 ( 1-\varepsilon^n_{ij } ) \left(\frac{r_0}{r_{ij}}\right)^{12 } , \label{eq : inter}\ ] ]  where , denotes the distance of atoms of amino acids and in the native structure and is the native contact map , whose entries are 1 ( 0 ) if and are ( not ) in contact in the native conformation ( i.e. below or above 6.5 ) . and are constants controlling the strength of interactions ( , in our simulations ) . in addition , the peptide bond between two consecutive amino acids , , at distance is described by the unharmonic potential : with parameters , , and is the rest distance between consecutive atoms .it is important to notice that in eq .( [ eq : inter ] ) the formation of any native contact is rewarded in the same way , since does not depend on and .this choice is done deliberately , so that the only information entering eq .( [ eq : inter ] ) is the native contact map and not the types of interacting amino acids ( i.e. no sequence information ) . while by construction, the minimum of the energy scoring function is achieved in correspondence of the native state , there is no _ a priori _ guarantee that the folding process , under the influence the pair interaction ( 1 ) , occurs , on average , through the same stages encountered in nature , or even in a more sophisticated atomic md simulation with _ ab initio _ force fields .certainly , there are situations where the influence of the native - state topology on the folding process may be overridden by strong chemical propensities to form definite pairs of amino acids ( such as disulfide bridges ) . in addition , given the explicit bias towards the native state , one should not expect that it would be possible to observe intermediate states with low concentration of native contacts . aside from similar circumstances, it is appropriate to ask whether one can reproduce the key steps of the folding process by exploiting only the structural information of the native state . the basis and justification for the present study is the growing evidence that the above question has a positive answer .in fact , starting from the work of ref . and later of refs . , it has become clear that the characterization of the transition states can be confidently done within a go - model scheme for a variety of proteins .  in the present study , the starting structural model ( target )is the free enzyme which is a homodimer with each subunit composed by 99 residues , see fig [ fig : dimero ] .following  , the crystallographic c2 symmetry was enforced during md simulations to reduce the computational effort .the progress towards the fully folded ( native ) state was estimated in terms of the fraction of native contacts formed at any given time in the partially folded structure ,  .this quantity , also termed overlap , is defined as   where is the contact matrix of .constant temperature md simulations were carried out for several decreasing temperatures from the unfolded to folded state of the protein .the equations of motion for the atoms were integrated by a velocity - verlet algorithm with time step combined with the standard gaussian isokinetic scheme .we performed unfolding simulations within the same framework by starting from the native structure and taking it through a sequence of increasing temperatures ( heat denaturation ) .the temperature was measured in reduced units v/k ( being k the boltzmann constant , and v the energy of the native contacts in eq .( [ eq : inter ] ) ) . at each temperature , we let the system equilibrate from the last structure reached at the previous run at a nearby temperature . each equilibration involved md steps , a time much longer than the largest correlation times observed for the system .after equilibration , we sampled structures again at time intervals twice the estimated correlation time . at each temperature , we collected the energy histogram of such uncorrelated structures . using the multiple histogram techniques , the energy measurements for all temperatureshave been reweighted to provide optimal estimates of thermodynamic quantities such as the average energy and the specific heat ( by differentiation of the former ) for a continuous range of temperatures .the statistical significance of the data collected in our runs was checked by verifying that the reweighted thermodynamic quantities did not change by more than a few percent upon addition of energy histograms obtained from folding / unfolding simulations with different temperature schedules or initial conditions .  within the approximation where the reaction coordinate is the internal energy , from the high temperature side ,the slowest dynamics occurs at temperatures near the specific heat peak , with a relaxation time at least of order , where is a suitable coefficient dimension of seconds / joules .thus the contacts contributing more to the specific heat peak are identified as the key ones belonging to the folding bottleneck and sites sharing them as the most probable to be sensitive to mutations .furthermore , by following several individual folding trajectories ( by suddenly quenching unfolded conformations below the folding temperature , ) we ascertained that all such dynamical pathways encountered the kinetic bottlenecks described in the next section .  a reliable and convenient way to identify and characterize the kinetic bottlenecks is through the location of peaks and shoulders in the specific heat ( which denote the overcoming of free energy barriers ) .moreover , since the specific heat results from the contribution of each pair of native contacts , it is also useful to monitor the formation of each native interaction throughout the folding process .indeed , the probability of formation of a native contact is a decreasing function of and has a sigmoidal shape fitted by suitably shifted hyperbolic tangent ( see fig . [fig : fit ] ) .the smooth interpolation allows to identify a crossover temperature , , where the slope reaches its maximum , ( that is the inflection point of the curves in fig .[ fig : fit ] ) . defines a local `` transition '' temperature at which each contact is locked , whereas , which represents the `` rapidity '' of its formation , can also be regarded as a measure of the local contribution to the specific heat .at very low temperatures , the observed structures have nearly 100 % native - state similarity , measured as the fraction of established native contacts ( eq . [ eq : ovlp ] ) .further increase in temperature causes structural rearrangements into a configuration that can not be assembled into a dimer anymore ( fig .[ fig : diss ] ) : the number of subunit - subunit contacts vanished and the two subunits behaved independently . the dissociation mechanism is well described by fig .[ fig : ovlp ] where we report , for several temperatures , the fractional occupation of native contacts for the individual subunits and at the monomer - monomer interface .the dissociation is also signalled by an abrupt increase of the specific heat of the dimer , see inset of fig.[fig : specheat ] , ( this defines the the dissociation temperature ) . a typical structure at this temperatureis shown in fig .[ fig : diss]a . at even higher temperatures ( ) , a large increase of specific heat is observed , indicating the presence of a strong transition of the single subunits , see fig .[ fig : specheat ] .this temperature is identified with the folding temperature , .consistently with other studies on different proteins , the native overlap at was about 50 . a typical structure at this temperatureis shown in fig .[ fig : diss]b .  a further set of bottlenecks is encountered at , where the formation of the three -strands of hiv-1 pr is involved . upon increasing the temperatureone encounters -sheet , then , then .it is found that the kinetic bottleneck for the a general -sheet formation is not the establishment of the contact(s ) closest to the turn ( that involves amino acids near in sequence ) but it is located further away. a quantitative analysis of the amino acids most involved in the folding bottleneck is again obtained by monitoring during the folding / unfolding process each pair of amino acids which are in contact in the native state .examples of the probability with which individual contacts are formed is shown in fig .[ fig : fit ] .  at each temperature where the dynamical evolution of the hiv-1 pr is followed ,the formation probability of each native contact ( fractional occupation ) is calculated .such quantities are at very low temperatures ( all native contacts always present ) and decrease to zero at temperatures larger than the folding temperature .it may be anticipated that the rate of decrease as a function of temperature will not be the same for all contacting pairs .in particular , trivial local contacts between residues with a small sequence separation will have a large probability of being formed even at high temperatures .our interest focuses on those contacts which show a dramatic increase of the fractional occupation near the folding transition. those will be the key contacts responsible for the appearance of the specific heat peak .examples of the fractional occupation for three native contacts is shown in fig .3 . given the monotonic behaviour of the fractional occupation , one may synthetically characterize the formation of each contacting pair by the temperature at which the point of inflection of the curve is seen and also by the slope at that very same point .both data can be conveniently summarised in two scatter plots where the slope , , and the temperature of formation , , are reported for each residue taking part in native contacts .such graphs are reported in the scatter plot of fig .[ fig : flessi ] .notice that , for each site , there are as many entries as the number of contacts involving it ( a number that typically differs from site to site ) .figure [ fig : flessi ] clearly shows that there are clusters of contacts that are turned on at similar temperatures .the bottlenecks for the folding process are identified by isolating the contacts having both a formation temperature , matching the location of the peaks and shoulders of the specific heat , and a high rapidity of formation , .figure [ fig : flessi]a reveals the presence of four distinct clusters of contacts .the first three , labelled , , , are associated with the formation of the three antiparallel -sheets in hiv-1 pr .their temperature of formation is about , and corresponds to the shoulder visible in the larger plot of the specific heat of fig .[ fig : specheat ] .the sites sharing the most important contacts involved in such three bottlenecks are listed in table [ tab : tab ] and highlighted in fig .[ fig : diss]c , where a typical structure at is shown .it is interesting to see that is maximum for sites close to the -turn , in accord with the intuitive expectation that the formation is initiated at the turn . in fig .[ fig : pendenze ] a , b and c , we have reported the values of only for the pairs of contacting sites in the sheet .it is seen that the sites closest to the turn have a small formation rapidity .this can be understood since , being very close along the sequence , they can be easily formed / broken .the highest rapidity , , i.e. the highest difficulty of formation , is encountered typically 3 - 4 sites away from the turn .the corresponding contacts are then identified as the bottleneck for this particular folding stage . for the -sheets ,the bottlenecks involve amino acids that are typically 3 - 4 residues away from the turns .going back to fig . 5, one observes that there is a fourth group of contacts around residues 30 and 86 , labelled tb after `` transition bottleneck '' , that are formed cooperatively at the folding transition .the sites involved in the tb contacts are listed in table [ tab : tab ] . among those contactswe have recorded the largest values of , as shown more clearly in fig .again , we considered the sites with the highest values of as responsible for the main bottleneck of the folding process .the highest `` rapidity '' is measured in correspondence of contacts 29 - 86 and 30 - 84 ( see also fig . [fig : pendenze ] ) which are , consequently , identified as the most crucial for the folding / unfolding process .the sites involved in the main folding bottleneck ( tb ) are located at the active site of hiv-1 pr , which is targeted by anti aids drugs .hence , within the limitations of our simplified approach , we predict that changes in the detailed chemistry at the active site affect also a key step of the folding process . to counteract the drug action, the virus has to perform some very delicate mutations in correspondence of the key sites ; within a random mutation scheme this requires many trials ( occurring over several months ) .the time required for the biosynthesis of a mutant with native - like activity is even longer if the drug attack correlates with several bottlenecks simultaneously .this is certainly the case for several anti - aids drugs .indeed table [ tab : tab ] summarises the mutations emerged for the fda approved drugs .remarkably , among the first 23 most crucial sites predicted by our method and listed in table [ tab : tab ] ) , there are 7 sites in common with the 16 distinct mutating sites of table [ tab : tab ] .the probability that two sets of 16 and 23 sites randomly taken from a total population of 99 ( the length of the hiv-1 pr monomer ) share at least 7 sites is only 3 % . also note that , all the mutation sites of table [ tab : tab ] except 82,35,36 and 90 fall within a mismatch of at most one position from the sites of table [ tab : tab2 ] .these results highlight the highly statistical correlation between our prediction and the evidence accumulated from clinical trials .all mutations causing resistance involve crucial residues for the main folding bottleneck , ( particularly residue 84 ) in combination with key sites for one or more of the sheets .mutation in this sites are expected to modify the energetics and structure of partially folded states .in contrast the folded state appears to be weakly affected by specific mutations , such as m46i , l63p , v82 t , i84v , which lead to a rms distance of 0.5 from the wild - type . in the light of these results , it is possible to interpret the experimental evidence for the existence of correlations between mutations at residue 82 and residues 10 , 54 , 71 as correlations between the main kinetic barrier , tb and the others , , .the large separation of these associated sites , both along the sequence and in space , suggests that their correlations arise by virtue of the folding process itself .this kinetic effects is particularly clear in one of these cases , namely the co - mutation of sites in the tb and at residue 10 , which occurs under ind therapy .the mediator of the correlation is residue 23 which takes part to two bottlenecks : tb and through direct contact with residues 84 and 10 , respectively .co - varying mutations between the two sites are observed because changes in tb will affect the environment of the other key site 10 , which has to mutate accordingly .the strategy presented here allows both to identify the bottleneck of the folding process and to explain their highly significant match with known mutating residues . this approach should be readily applicable to identify the kinetic bottlenecks of other viral enzymes of pharmaceutical interest , thus aiding the development of novel inhibitor targetting the kinetic bottlenecks .this is expected to enhance dramatically the difficulty for the virus to express mutated proteins which still fold efficiently into the same native state with unaltered functionality .

Abstract: drug resistance to hiv-1 protease involves accumulation of multiple mutations in the protein . here we investigate the role of these mutations by using molecular dynamics simulations which exploit the influence of the native - state topology in the folding process . our calculations show that sites contributing to phenotypic resistance of fda - approved drugs are among the most sensitive positions for the stability of partially folded states and should play a relevant role in the folding process . furthermore , associations between amino acid sites mutating under drug treatment are shown to be statistically correlated . the striking correlation between clinical data and our calculations suggest a novel approach to the design of drugs tailored to bind regions crucial not only for protein function but also for folding .  2.0 cm


entanglement is a fragile feature of composite quantum systems that can easily diminish by uncontrollable interactions with the environment . at the same timehowever carefully crafted entangled states can protect quantum coherence from the deleterious effects of those random interactions .this idea underlies the principles of quantum error correcting codes that strengthen the optimism regarding the feasibility of implementing in practice complex quantum information processing tasks .  in this paperwe demonstrate how quantum entanglement can help in the task of classical communication . to this end, we develop a simple model of a noisy communication channel , where the noise affecting consecutive transmissions is correlated . within this model ,we derive bounds on the classical channel capacity assuming either separable or entangled input states , and we show that using collective entangled states of transmitted particles leads to an enhanced capacity of the channel .the motivation for our model comes from classical fiber optic communications . in practice ,light transmitted through a fiber optic link undergoes a random change of polarization induced by the birefringence of the fiber .the fiber birefringence usually fluctuates depending on the environmental conditions such as temperature and mechanical strain . at first sight , this makes the polarization degree of freedom unsuitable for encoding information , as the input polarization state gets scrambled on average to a completely mixed state .however , the birefringence fluctuations have a certain time constant which means that the transformation of the polarization state , though random , remains nearly the same on short time scales .consider now sending a pair of photons whose temporal separation lies well within this time scale .although the polarization state of each one of the photons when looked at separately becomes randomized , certain properties of the joint state remain preserved .for example , this is the case of the relative polarization of the second photon with respect to the first one .we can therefore try to decode from the output whether the input polarizations were mutually parallel or orthogonal .this property can not be determined perfectly , as in general we can not tell whether two general quantum states are identical or orthogonal if we do not know anything else about them , but even the ability of providing a partial answer establishes correlations between the channel input and output that can be used to encode information into the polarization degree of freedom . the situation becomes even more interesting when we allow for entangled quantum states .then the singlet polarization state of the two photons , when sent as the input , remains invariant under such perfectly correlated depolarization , and it can be discriminated unambiguously against the triplet subspace .therefore we can encode one bit of information into the polarization state of two photons by sending either a singlet state or any of the triplet states .we shall see that these simple observations will also emerge from our general analysis of the channel capacity .the first example of entanglement - enhanced information transmission over a quantum channel with correlated noise has been recently analyzed by macchiavello and palma .our model assumes a different form of correlations , and its high degree of symmetries has allowed us to perform optimization of the channel capacity over arbitrary input ensembles .although we analyze only zero- and one - photon signals , we define the action of the channel in terms of the transformations of the bosonic annihilation operators , which sets up a framework for possible generalizations , such as use of multiphoton signals .this application of entanglement in classical communication is a distinct problem from entanglement - assisted classical capacity of noisy quantum channels studied by bennett __ in ref . , where it has been shown that prior entanglement shared between sender and receiver can increase the classical capacity .we also note that the non - zero time constant of phase and polarization fluctuations can be used in robust protocols for long - haul quantum key distribution .  before passing on to a detailed discussion of the problem in the subsequent sections ,let us introduce some basic notation .the action of a channel is described by a completely positive map that we will denote by .the sender selects messages from an input ensemble , where is the probability of sending the state through the channel .the capacity of the channel is a function of the mutual information between the input ensemble and measurement outcomes at the receiving stations : it characterizes the strength of correlations between these two that are preserved by the channel .the mutual information itself involves a specific measurement scheme ; however , it has a very useful upper bound in the form of the holevo quantity that depends only on the output ensemble of states emerging from the channel : where is the von neumann entropy .as we will see , in our model the holevo quantity will provide a tight bound on the mutual information that could be achieved in practice using a simple measurement scheme .the classical channel capacity is obtained by assuming arbitrarily long sequences of possibly entangled input systems , and calculating the average capacity per single use of the channel . in our analysis, we will perform a restricted optimization by considering only two consecutive uses of the channel .we will start our discussion by proving a rather general lemma about channels that can be decomposed into a direct sum of maps acting on subspaces of the hilbert space of the input systems . in physical terms ,such channels remove quantum coherence between the components of the input state that belong to different subspaces , by zeroing the respective off - diagonal blocks of the density matrix characterizing the input state .this lemma will greatly simplify our further calculations ._ lemma 1 : _ suppose that we can decompose the hilbert space of the system into a direct sum of subspaces such that for an arbitrary input state the state emerging from the channel can be represented as where is the input state truncated to the subspace , and each is a certain trace - preserving completely positive map acting in the corresponding subspace .then the optimal channel capacity can be attained with an ensemble in which each state belongs to one of the subspaces .  _proof : _ indeed , suppose that there is a state that does not satisfy the above condition , i.e. it is defined on more that one subspace . we can replace it by a sub - ensemble , obtained by truncating the state to the subspaces and normalizing the resulting density matrices .in other words , whenever the sender is supposed to transmit , she replaces it by one of the normalized truncated states with the corresponding probability .it is straightforward to verify that the average state obtained from such a subensemble is identical with .the above observation has a useful consequence when optimizing the holevo bound on channel capacity .if the input ensemble is of the form discussed above , then it can be split into subensembles of states that belong to separate subspaces , with the probability distributions normalized to one within each subensemble , and denoting the probability of sending a state from the subensemble .it is then easy to check that the holevo quantity is given by the following expression : where is the holevo quantity for the subensemble .therefore , the maximization of the holevo quantity can be performed in two steps .the first one is the optimization of each of separately , assuming an input ensemble restricted to the subspace .the second step consists of optimizing the probability distribution with the normalization constraint , and it can be performed explicitly using the method of lagrange multipliers .indeed , if we denote the lagrange multiplier as , then differentiation over yields : this formula allows us to express the probabilities in terms of the lagrange multiplier as : and furthermore summation over and using the fact that gives the value of the lagrange multiplier as : finally , inserting eqs .( [ eq : pl ] ) and ( [ eq : mu ] ) into eq .( [ eq : chi = sumpkchik ] ) yields the maximum value of the holevo quantity equal to : we will later find this expression useful in calculating the channel capacity in our model .the physical reason for this is that we will be able to decompose the set of states used for communication into subensembles with a fixed number of photons , and then optimize the holevo quantity separately in each subspace .let us now introduce a mathematical model for the random transformation of polarization during transmission through the channel .a general linear transformation between two annihilation operators corresponding to a pair of orthogonal modes is given by unitary matrices that form the lie group u(2 ) . in situationswhen only the relative phase between the two polarization modes is relevant , the overall phase of the transformation can be assumed to be fixed , which reduces the group of transformations to su(2 ) .however , in our case the overall phase shift can vary between the consecutive temporal slots , and therefore we need to keep it as an independent parameter .we note that any u(2 ) matrix can be mapped onto a rotation in the three dimensional physical space .such a rotation describes the corresponding transformation of the poincar sphere used to represent the polarization state of light in classical optics .we will label elements of u(2 ) as and use a dot to denote the multiplication within the group .the u(2 ) group has a natural invariant integration measure which we assume is normalized to one .this measure defines a uniformly randomized distribution of polarization transformations that scrambles an arbitrary input polarization to a completely mixed one .suppose now that two consecutive temporal slots labelled by and , each comprising two orthogonal polarizations , are occupied by a joint state of radiation , as shown schematically in fig .[ fig : tempslots ] .we will assume that the polarization transformation affecting the slot is completely random , but that the transformation is correlated with the first one through a conditional probability distribution .the resulting transformation of the joint two - slot state is therefore given by the following completely positive map : here is a unitary matrix acting in the hilbert space of one of the slots that represents the polarization transformation .we will now assume that the conditional probability depends only on the relative transformation between the slots and and that it can consequently be represented as .in such a case , we can substitute the integration variables in the second integral according to , and make use of the invariance of the integration measure .this procedure shows that the map can be represented as a composition of two maps : . the first one of them , , acts on both the temporal slots and it depolarizes them in exactly the same way : the second map , , acts only on the slot , and it introduces additional depolarization relative to the slot according to the probability distribution : we will assume later that the distribution has sufficient symmetry to describe the action of the map in the relevant hilbert space with the help of two simple parameters .we now introduce a further simplification by imposing a condition that each temporal slot may contain at most one photon .therefore the relevant hilbert space for each slot is spanned by three states : the zero - photon state , and horizontally and vertically polarized one - photon states and .we can conveniently write the explicit form of the unitary transformation using the irreducible unitary representations of the group su(2 ) .we will denote by a matrix that is a -dimensional representation of an su(2 ) element obtained from by fixing the overall phase factor to one .these matrices are well known in the quantum theory of angular momentum as describing transformations of a spin- particle under the rotation group .we will also denote by the overall phase of the element .then the unitary transformation of the input state corresponding to the polarization rotation is given by the matrix : in this formula , the one - dimensional representation is identically equal to one , and is a unitary matrix itself ; however , we will keep this more general notation in order to be able to use results from the theory of group representations .in particular , the following property of the rotation matrix elements will allow us to evaluate directly a number of expressions : ^{\ast } { \cal d}^{j'}_{m'n'}({\bf\omega } ) = \frac{1}{2j+1 } \delta_{jj ' } \delta_{mm ' } \delta_{nn'}.\ ] ]  the action of the map on a joint two - slot state can be analyzed most easily if we decompose the complete hilbert space into a direct sum of subspaces with a fixed number of photons : , where the upper index labels the number of photons .the zero - photon subspace is spanned by a single state .the one - photon space has a basis formed by four vectors : , , , and .finally , in the two - photon subspace we will introduce a basis that consists of the singlet state and the three triplet states , , and .the reason for this choice is that then the action of the tensor product on a two - photon state can be decomposed into the sum : where acts on the singlet state , and is a three - dimensional matrix acting in the triplet subspace .using our decomposition of the complete hilbert space , the action of the tensor product on a general two - slot state in the basis specified above is given by :    if we now insert this fomula into eq .( [ eq : lambdaperf ] ) , it can be easily seen that the invariant integration over the overall phase factor kills all the off - block diagonal elements of the density matrix that link different subspaces . in other words ,all the coherence between states with different photon numbers is completely removed by the phase fluctuations .furthermore , the operation , acting only on the second slot , does not mix subspaces with different photon numbers . therefore the conditions of our lemma are satisfied and we can consider only states with a definite number of photons as elements of the input ensemble .thus we need to calculate are three corresponding holevo quantities , , and that can be combined into a holevo bound for the overall channel capacity according to eq .( [ eq : chisum ] ) .this calculation forms the contents of the next section .the communication capacity of the zero - photon subspace itself is naturally zero , as we have only a single state at our disposal .this state can of course be used as an element of a larger ensemble thus contributing to the overall capacity .this fact is reflected in the form of eq .( [ eq : chisum ] ) , where indeed does increase the total value of .a less trivial problem to calculate is the capacity of the one - photon subspace . if we assume a normalized input state from the subspace , then the action of the channel restricted to this subspace is given by : where the parameters and are defined in terms if the input density matrix as : for the form of density matrix given in eq .( [ eq : lambda(1)perfrhoin ] ) , the depolarizing channel affects only the off - diagonal elements and .we will assume that the symmetry of the distribution is such that the effect of is a rescaling of these elements by a real parameter bounded between and .it is now easy to check that the entropy of the one - photon state emerging from the channel can be written as where the matrix appearing in the second term can be interpreted as a state of a qubit .therefore , the second term is bounded by and , and consequently .it is a straightforward observation that the holevo quantity is bound from above by the difference between the maximum and the minimum possible entropies of states emerging from the channel .therefore we obtain that .this inequality can be saturated simply by taking a one - photon state confined either to the first or to the second temporal slot , with an arbitrary polarization .thus , the channel capacity is not enhanced in the one - photon sector .the most interesting regime is when both the temporal slots are occupied by photons . as we will see below , in this case quantum correlationscan then enhance the capacity of the channel .if we take a normalized input state from the two - photon subspace , then the map produces a werner state : where we have introduced the following notation : and we will use for the name of the werner parameter of the input state , defined as : ) .the second operation affecting the input state is the partially depolarizing channel .we will assume that the action of the map acting on the photon in the second temporal slot is simply isotropic depolarization shrinking the length of the bloch vector by a factor satisfying .such an operation preserves the werner form of the transmitted state , and its only effect is the multiplication of the parameter by the factor .thus , the state emerging from the channel is given by : with the parameter defined by the input state according to eq .( [ eq : c ] ) .  at this pointthe possibility of enhanced communication capacity by exploiting entanglement manifests itself .the difference between the separable and entangled alphabets can be seen by comparing the allowed ranges of the parameter .the positivity of the input density matrix requires that and this is the only condition if we consider the most general , possibly entangled input states .however , if the input states are restricted to _ separable _ones , then as shown by horodeccy , the allowed range for the parameter is reduced to this limitation will underlie the reduced channel capacity in the case of separable states .  as the two - photon states emerging from the channel are fully characterized by the werner parameters of the respective input states , optimization of the holevo quantity can be carried out over the ensemble of the probabilities of sending the state with the werner parameter equal to .the output states emerging from the channel is therefore given by an ensemble of werner states . because a statistical mixture of werner states is also a werner state with the average parameter: holevo quantity can be expressed with the help of a single real - valued function : where the explicit form of the function is given by : the optimization of the holevo quantity , which in principle needs to be performed over an arbitrarily large input ensemble of permitted quantum states , can be greatly simplified using the following observation ._ lemma 2 _ : let be a concave function defined on a closed interval  ] .as we have seen , the permitted range of the parameters characterizing the states belonging to the input ensemble depends on whether we allow most general , possibly entangled states , or rather restrict the input to separable states only .if we assume that this range spans from to : then we can easily apply lemma 2 to the expression of the holevo quantity in terms of the function that has been given in the second line of eq .( [ eq : chi2 ] ) . taking and and using the explicit value of the turning point derived in eq .( [ eq : turningpoint ] ) yields the following bound : where is given in terms of the input ensemble characteristics as : we will analyze in detail numerical values of the channel capacity in the next section . before doing so , we will close this section by describing a simple intuitive picture of lemma 2 that gives an additional insight into the form of the input ensemble .the result of lemma 2 can be visualized using the following geometrical reasoning depicted in fig .[ fig : graphill ] . consider a graph of the function versus its argument .the numbers and the corresponding values of the function are given by a set of points in the plane of the graph .the probability distribution for the arguments defines an average that can be interpreted as a center of gravity for the system of points that have been assigned respective masses . obviously , if the probability distribution is arbitrary , then this average can lie anywhere within the convex polygon spanned by the points .since the function is strictly concave over the range considered , the whole polygon lies within the area bounded by the graph of the function on one side , and a straight line connecting the points and on the other side .this straight line is given by a function defined as :   the left hand side of eq .( [ eq : fsumqgamma ] ) is now given by the length of a vertical line connecting with the point on the graph of the function , where .clearly , the line will be always equal in length or shorter than the line where the point lies on the graph of the function .furthermore , in order to find the maximum possible length of the line , it is clear from this geometric construction that we need to maximize the difference over belonging to the interval  ] can only increase the value of the upper bound given in eq .( [ eq : fsumqgamma ] ) .this implies two rather straightforward observations .first , the use of entangled states should give a larger capacity compared to separable states .secondly , a lower value of the parameter meaning weaker correlations between consecutive polarization rotations results in a decreased channel capacity .the graphical construction presented above also gives a simple recipe for constructing an output ensemble that saturates the bound on the holevo quantity .it is sufficient to take a two - element ensemble with the extreme points of the allowed interval as the parameters of the werner states emerging from the channel : and .the optimal probabilities of using the two states need to be selected in such a way that the weighted sum of the points corresponding to these states gives the point maximizing the difference .explicitly , these probabilities are respectively given by and .the actual graph of the function with the permitted ranges of the werner parameter for perfectly correlated noise and entangled and separable inputs is shown in fig . [fig : ens ] .the holevo quantity is only an upper bound on the channel capacity and therefore is not necessarily attainable .users of a communication channel need two relevant pieces of information .the first one is the optimal form of the input ensemble that should be used by the sender .the second one is a measurement scheme that should be employed at the output of the channel in order to optimize the capacity .let us start by summarizing the results of the preceding section and specifying the input ensemble implied by these considerations .we have seen that in the zero- and one - photon subspaces the channel capacity can not be enhanced by exploiting the polarization degree of freedom .therefore as the elements of the input ensemble we can take for example states , , and , where for concreteness we have fixed the polarization of single - photon states to vertical .the polarization degree of freedom starts to play a nontrivial role when both the temporal slots are occupied by photons . in this subspace, we need to select two input states characterized by the werner parameters that are as distant as it is allowed by the constraints on the input ensemble .if we restrict ourselves to separable states , then according to eq .( [ eq : separableconstraint ] ) we need to take one separable state with and another one with .it is easy to verify using eq .( [ eq : c ] ) that the pair of separable states satisfying this condition can be taken as and .we thus see that in agreement with the simple picture developed in the introduction to this paper , the relevant quantity is the relative polarization of the photons occupying consecutive slots .if we allow for entangled input , then the lower limit for the werner parameters of the input states shifts down to .this value can be of course attained by taking the singlet state itself as one element of the input ensemble , and any state with , for example again as the second one .  in order to complete the description of the communication protocol , we need to specify the measurement applied to the states emerging from the channel .this task can be decomposed into two steps .the first one is the determination of the total number of photons contained in the two slots and it can in principle be accomplished by a collective quantum non - demolition measurement on all the modes involved that would determine the total photon number without destroying coherence between the modes .depending on the outcome , the second step needs to be either finding the temporal slot occupied by a photon in the one - photon subspace which can be realized by direct temporally resolved detection , or discriminating between the states used to encode information in the two - photon subspace .it is easy to see that this discrimination takes a simple form in the case of perfectly correlated noise and entangled input states : we need to determine whether the received states belong to the singlet or the triplet subspace , which corresponds to a two - element projective measurement : it turns out that the same measurement saturates the holevo bound also in the general case of any value of the parameter with either entangled or separable input states . in fig .[ fig : arrows](a ) we depict conditional probabilities of obtaining the singlet or the triplet outcomes for a two - element input ensemble characterized by werner parameters and .a lengthy but straightforward calculation shows that if we take as the input probabilities the values discussed in the preceding section , the mutual information is given exactly by the right hand side of eq .( [ eq : chi(2)le ] ) .thus the described procedure indeed maximizes the channel capacity in the two - photon subspace .( 340,480 )  ( 200,470)(a ) ( 40,450) ( 40,320) ( 40,440)(4,0)300 ( 40,340)(4,0)300 ( 40,340)(3,1)300 ( 40,440)(3,-1)300 ( 340,435 ) ( 340,335 ) ( 220,320) ( 220,450) ( 270 , 370) ( 270,400)  ( 200,280)  ( 40,270 ) ( 40,180 ) ( 40,260)(4,0)300 ( 40,200)(4,0)300 ( 340,255 ) ( 340,195 ) ( 250,180)1 ( 250,265)1  ( 200,140)(c ) ( 40,130 ) ( 40,0 ) ( 40,120)(4,0)300 ( 40,20)(4,0)300 ( 40,120)(3,-1)300 ( 340,115 ) ( 340,15 ) ( 250,0)1 ( 250,55)1/2 ( 250,105)1/2  it is instructive to compare the above diagram for optimal entangled and separable input ensembles in the case of perfect correlations . for the optimal entangled ensemble , shown in fig .[ fig : arrows](b ) we can distinguish perfectly between the two inputs as they belong to orthogonal subspaces even after the transmission .for the separable ensemble , the emerging states can no longer be perfectly discriminated as seen in fig .[ fig : arrows](c ) .  the complete channel capacity obtained by combining eq .( [ eq : chisum ] ) with the results of sec .[ sec : channelcapacity ] is shown as a function of in fig .[ fig : holevoplot ] .it is seen that using an entangled input ensemble gives a clear advantage over the separable states over the complete range of the correlation parameter .we note that the measurement discriminating between the singlet and the triplet subspaces can be implemented using the braunstein - mann scheme based on linear optics , as we do not have to distinguish between all four bell states . after overlapping temporally the received photons and interfering them on a 50:50 non - polarizing beam splitter, their detection in the same output port corresponds to a projection onto the triplet subspace , whereas measuring them in the separate output ports of the beam splitter identifies the singlet state .we have introduced a model of a communication channel with correlated noise motivated by random birefringence fluctuations in a fiber optic link . within this model , we have demonstrated that introducing quantum correlations between consecutive uses of the channel increases its capacity .this demonstrates how specifically quantum phenomena such as entanglement can be helpful in the task of transferring classical information .making use of entanglement requires more complex preparation procedures that provide joint input states extending over a number of temporal slots .a related question is the role of collective quantum measurements on the output of the channel rather than detecting radiation in each of the slots individually and combining classical outcomes of separate measurements .the action of the channel has been defined in terms of transformations of the bosonic field operators .this opens up a route towards interesting generalizations of the present work , for example including arbitrary multiphoton states .another direction would be extending the model to an arbitrary number of temporal slots rather than just allowing for correlations between pairs of consecutive slots as in our example .it is easy to give a simple protocol showing that in this case the channel capacity can be enhanced even further .suppose that the sender generates a train of zero- and one - photon states with the same probabilities equal to one half .the first time she is to transmit a photon , she sends half of maximally entangled pair . in the second instance when a one photon should be transmitted, she sends the remaining member of the pair transforming it in such a way that the joint two - photon polarization state belongs either to the singlet or the triplet subspace .the receiver implements a polarization - independent quantum non - demolition measurement on each temporal slot .when a photon is detected , it needs to be stored until the arrival of the second member of a pair , when the discrimination between the singlet and the triplet subspaces can be performed with the help of a joint measurement .if the fluctuations in random birefringence can be neglected over the temporal separation between the photons in a pair , this procedure allows one to encode one extra bit of information into each pair of transmitted photons .this gives the average channel capacity equal to per a pair of temporal slots , enhancing further the optimal value shown in fig .[ fig : holevoplot ] .this research was supported by epsrc and polish kbn .

Abstract: we develop a model for a noisy communication channel in which the noise affecting consecutive transmissions is correlated . this model is motivated by fluctuating birefringence of fiber optic links . we analyze the role of entanglement of the input states in optimizing the classical capacity of such a channel . assuming a general form of an ensemble for two consecutive transmissions , we derive tight bounds on the classical channel capacity depending on whether the input states used for communication are separable or entangled across different temporal slots . this result demonstrates that by an appropriate choice , the channel capacity may be notably enhanced by exploiting entanglement .