geo - encyryption is an enhancement to traditional security techniques in which the decryption process can only occur at a specific geographic location .clearly such an outcome , if possible , would have important implications for a wide range of scenarios .many attempts have been made to deliver a viable geo - encryption process , with varying degrees of success .however , the security of all known attempts at geo - encryption is beholden to some assumed restriction on the ability of an adversary , and/or tamper - proof assumptions on a specific device . ultimately , the necessity of such assumptions detract from the geo - encryption paradigm and commercial implementations of ithave not occurred .  in this workwe remove all debilitating assumptions from the geo - encryption paradigm by introducing quantum information into the decryption process .more specifically , we show how the use of quantum information embedded on a communication device can be used to simultaneously invoke quantum direct communication ( qdc ) and quantum location verification ( qlv ) that collectively realize a geo - encryption solution that is secure against any known attack .we further show how this new form of geo - encryption can be extended to make the decryption of data viable not only at a specific location , but also a specific time .the notion of geo - encryption first appeared in the scientific literature in 1996 where geo - encryption based on gps signals was discussed .the main idea behind such classical geo - encryption is the addition of a so - called _ geo - lock _ to the standard ssl ( secure sockets layer ) protocol .the main idea can be summarized thus ._ encryption:- _ ( i ) a message is encrypted in the normal manner by the sender using a conventional cipher and associated session key , randomly generated by the sender .( ii ) the sender decides the location and time for which decryption is to be allowed .( iii ) based on this and , the sender creates a _ geo - lock _ , which is a binary string determined using an ( known ) algorithm to convert information related to the anticipated signal metrics a receiver at the decryption location and time should obtain ( e.g. specific frequencies at specific signal strengths ) .( iv ) the sender uses the geo - lock to xor the key , forming a new key .this new key is then encrypted using a standard rivest - shamir - adleman ( rsa ) private key to form another new key , . _decryption:- _ ( v ) the receiver captures the encoded message and the key using his own radio antenna(s ) .( vi ) the receiver proceeds to decrypt using the sender s rsa public key to obtain .( vii ) from the received signal metrics the receiver computes , and applies that to to compute the session key .( viii ) the original message is decrypted .a schematic of this process is illustrated in fig . 1 .pivotal to the success of this protocol is that the received signal information associated with the geo - lock can not be spoofed .indeed , a series of steps can be put in place in order to minimize such spoofing in civilian receivers , and in the military sphere anti - spoof gps receivers based on encoded gps signals are available . however , such techniques are not immune to device tampering and/or the leaking ( or determination ) of gps decoding keys .this is particularly so , when a quantum adversary is in place - namely an adversary whose resources and abilities are confined only by the laws of physics . in the presence of such an all - powerful adversary , the above classical geo - encryption protocol is readily attacked . in this workwe will assume the presence of a quantum adversary .the notion of qlv first appeared in the scientific literature in 2010 as a means of securing real - time communications to a _unique _ spatial position . in was pointed out how by monitoring the location of a device using classical radio communications and quantum information embedded in the device , real - time data could be delivered to a specific geographic location .data would be sent to the receiver only whilst the qlv was validated . through its use of classical communications ,qlv can be made operational under most channel conditions including those without a direct line - of - sight ( los ) component .pivotal to this paradigm of communication throughput conditioned on the location of the receiver is the anti - spoofing attributes of the specific qlv protocol ( spoofing is always possible in classical - only systems ) . in general terms, such anti - spoofing properties can be attributed to the quantum no - cloning theorem coupled with the no - signaling principle of special relativity . since the work of qlv has been further studied in numerous works with its deployment and security under a wide range of conditions and assumptions investigated .details of this history and associated references can be found in where the notion of qlv is discussed in the context of a quantum - enabled autonomous vehicle . in arguments for qlv s security against any known attack are also presented .  in this workwe extend the notion of location - enabled communications discussed in to the decryption process itself . using any pre - existing secret key ( or any key ,e.g. one generated dynamically with pre - existing shared entanglement ) will not suffice for our purposes as it could be distributed to other adversaries at other locations ( even though one adversary is at the qlv validated location ) .however , as we show here , by using qdc and by obfuscating the quantum information needed by both the qdc and the qlv protocols until the time of the decryption and verification processes , we shall see how it is possible make the _ decryption _ of data itself conditional upon the location of the receiver .although more complex , and requiring some quantum technological developments ( e.g. long - lived quantum memory ) , this form of geo - encryption is free from the vulnerabilities that plague classical geo - encryption protocols .in quantum geo - encryption we assume that most of the quantum information to be used is pre - stored in the device that will invoke the decryption ( the decryptor ) .this information could have been previously delivered to the decryptor via a quantum information transfer link such as fibre optic cable or los laser communications ( or in real - time via teleportation ) .we will also assume that the stored quantum states are forced to be non - orthogonal , and in the wider context can take any allowable form , e.g. continuous variable ( cv ) states , qubits , qudits , hybrids , etc .we also allow for some states being entangled - possibly with states at multiple reference stations ( rss ) .the rss are at known locations , assumed to be legitimate members of the wider communication system , and assumed to have secure quantum and classical ( e.g. via qkd ) communications between themselves .the basic idea in quantum geo - encryption is that encoded signals ( here we will take these signals to be classical ) sent by the rss and arriving simultaneously at the receiver , instruct the decryptor what specific quantum states should be used ( and for what process ) , any information ( e.g. random unitary matrices ) needed to remove any non - orthogonality introduced , any decoding instructions , and instructions on what output to produce ( e.g. a classical output to be broadcast or a state to be teleported ( or directly transferred ) back to a specific rs ) . by measuring the round - trip time associated with these procedures the decryptor s location can be verified ( see for more details on qlv ) .  due to the wide variety of forms that quantum informationcan take , and the many variations on how that information can be transferred , stored , manipulated , and/or retrieved from the decryptor ; unlimited versions of the quantum geo - encryption protocol are possible .for the purpose of focus , we will concentrate here on just one implementation of a geo - encryption protocol involving qubits , gaussian states , and hybrids formed from combinations of such states .we choose this version simply for its ease in demonstrating the most important concepts of a quantum geo - encryption application .  in terms of the annihilation and creation operators , the quadrature operators defined for photon states are ( assumed ) and , satisfying = 2i ] , where . from the symplectic spectrummany fundamental properties of gaussian states can be derived ( see ) .an important gaussian state is the two - mode squeezed vacuum ( tmsv ) state , described for two modes and as where , ] .the covariance matrix for the tmsv state can then be written where the quadrature variance , and  : = diag(1,-1 ) . in wider geo - encryption protocols ,tmsv states will be deployed partly for their use in holding entanglement between cv states at the rs and cv ( or hybrid ) states at the decryptor .in addition to gaussian cv states we will also utilize discrete two - dimensional states of the generic qubit form , , which can be entangled together to form higher multi - dimensional states of various levels of entanglement . we will also utilize combinations such as to form hybrid forms of such states such as for some arbitrary phase ( such states are not an absolute requirement for geo - encryption - but they will help illustrate its operation better ).  in qdc we negate the use of qkd , but instead encrypt the message we wish to send directly into a series of quantum states ( see for recent review ) . here, we will utilize a form of qdc based on entangled qubit pairs , namely the so - called ping - pong protocol . in the ping - pong protocol a maximally entangled qubit pair , say , is shared between the sender and receiver . depending onwhether a 0 or a 1 is to be encoded , the sender applies to its qubit an or an operation , respectively .this qubit ( the travel qubit ) is sent to the receiver , who then performs a bell measurement on both qubits . from the possible two results of this measurement , or ,the encoded message is inferred . during this processthe sender - receiver pair can do other tests to monitor for the presence of an eavesdropper . in our specific form of the geo - encryption protocolwe assume the ( travel ) qubit sent by the sender to the receiver , is teleported to the receiver using another pair of maximally entangled qubits that pre - exist between the sender and the receiver .the classical radio communication associated with this teleportation will form part of the geo - encryption protocol .    in our geo - encryption protocol we will obfuscate all quantum information to be used in several ways .first , all stored information used in the wider geo - encryption protocol , including all qubits , will have random unitary transformations applied to each state prior to being transferred to the decryptor .the classical information describing each of these unitaries will be held by the legitimate system in an encoded and distributed form - and will also form part of the radio communications of the geo - encryption protocol .some qubits will be part of the qlv protocol and some will be part of the qdc protocol .some qubits will be entangled with other qubits and some qubits will be entangled with cv states .some of the entangled states held by the decryptor will be entangled with states still held by the legitimate system .the qlv protocol will also partially use states from this same qubit grouping - the idea being that since the receiver does not know which states are to be used in each protocol - and since the qlv protocol demands the states it needs be at the correct location - then the qdc protocol is also tied to states that must be at the same location . any attempt to move quantum information that is required by the qdc protocol ( to another device at another location )will inevitably lead to a violation of the qlv protocol . by checking simultaneouslythat the qlv passes all verification tests , we can ensure the decryption process occurs only at the required location .since we can also dictate the time when the required classical information sent by the rss reach the decryptor , we can also ensure the decryption occurs at a specific time ( at least not before a specific time ) . avoiding delayed decryption of a complete message streamwould require the qdc and qlv outputs be intertwined ( e.g. embed some of the information needed by the qlv protocol within the qdc protocol ) .a schematic of a geo - encryption protocol is given in fig .2 . not explicitly shown in fig.2are other quantum states ( possibly entangled ) and other entanglement between quantum states stored in memory and states stored at the rss . also not shownare quantum transmitters ( receivers ) that may be added so as to transmit ( receive ) quantum information to ( from ) the decryptor if dictated to do so by the qlv protocol when conditions allow .  using other quantum states held by the decryptor designated for tasks beyond qdc and qlv , straightforward extensions of the geo - encryption protocolcan be used to produce any quantum task ( or output ) .for example , quantum computing steps could be forced to occur at a specific computing chip at a specific location .such quantum computation steps could be part of wider distributed quantum algorithm thereby adding a new layer of security to distributed computing not offered by classical machines .routing of information - classical and/or quantum - could also be enhanced in the sense that routing decisions could be enforced to be viable only at certain location - specific routers .this would allow for a chosen geographic path to be chosen as the sole route for information transfer .clearly , many other enhanced application scenarios exist for the geo - encryption paradigm .quantum geo - encryption is a rather more complex beast than its classical counterpart , and a series technological advances are required before its real - world implementation can occur .however , in the simplest forms of quantum geo - encryption the main technological advance needed is the development of reliable and long - lived quantum memory .rapid progress is being made in this endeavor ( e.g. see ) , and as such it is not inconceivable that quantum geo - encryption will be deployable in the not - so - distant future .thus far we have assumed the decryptor can be forced to invoke decryption at a unique location - a specific point in geographical space . in reality , due to inevitable noise ( and any processing delay ) in the location verification procedures this is not actually true in practice - devices that appear close to ( but not at ) the location can ( and should ) pass the verification test .what occurs in practice is we limit the decryption process to occur in an _ effective region _ or area of geographical space .the dimensions of this region can be thought of as being set according to a probability that the received timings are consistent with the decryptor being at the location it claims to be at ( the likelihood of the timing measurement is used by the qlv decision - making process ) .setting a required can therefore be visualized as setting an effective region in space ( e.g. an error ellipse ) within which a position - determined from the timings - must lie within with probability .the probability of any adversary spoofing the system approaches zero the further she actually is from this region .the size of the effective region is highly dependent on the nature of the radio communications channel used in the quantum geo - encryption protocol , most importantly on the weight of the non - los component in the channel relative to scattered components in the channel .note that non - los components can be of a real or virtual nature . in the latter case ,no visible los between the sender - receiver pair exists , but energy in the los component is still detected by the receiver .such an effect is due to the well - known penetration power ( dependent on the radio frequency and the composition of the obstructing object ) of radio waves .such penetration capabilities underpin so - called ` see - through - the wall ' radars and ultra - wide - band ( uwb ) location systems .recent advanced deployments have demonstrated virtual los detection through 20 cm thick concrete walls .value ) . in the rare circumstance where no set - up can allow for a virtual los component to be present at the decryptor ,the error introduced can be estimated and also folded into a new effective region .details of such issues and strategies around optimal nlos detection / mitigation are beyond the scope of the present work . ]  to understand in general terms the scale of the decrypting region let us consider a very general model for the wireless channel - the rician channel .such a channel between a sender with antennae ( linear array ) and a receiver with one antenna can be described by ( the channel vector ) , where where denotes the rician -factor of the channel , denotes the los component of the channel , and denotes the scattered component of the channel - the elements of which are assumed to be i.i.d complex gaussian random variables with zero mean and unit variance , i.e. , . in the above , can be expressed as where denotes the constant spacing ( in wavelengths ) between the antennas of the linear array , and is the angle defined by the direction of the linear array and the direction to the intended receiver . in the following to simplify the discussion we will ignore the directional information afforded to us by the linear array ( i.e. we ignore angle - of - arrival information ) - bearing in mind that its inclusion usually assists in the performance of any location - verification algorithm .the critical determination to be made in any channel is the time - of - arrival ( toa ) information obtained by the rss .the best approach to use for this problem is strongly dependent on the signal - to - noise ( snr ) ratio of the channel . in the high - noise snr regimeit is known that the cramer rao bound ( crb ) provides a tight bound on the timing error , whereas in the low - noise regime the ziv - zakai bound ( zzb ) is known to be tighter ( more pragmatically useful ) e.g. .  consider some signal pulse , for some signal template , time , channel coefficient , toa , and additive white gaussian noise with zero mean and spectral density .the crb on the toa is then given by , where , and where is the frequency and is the fourier transform of .as we can see , in principal for large enough snr the crb can be driven to zero .this crb result is modified in the presence of multipath components ( i.e. some finite ) . nonetheless ,as long as , in principal the toa can still be driven to zero .the zzb bound has the advantage that _ a priori _ channel information can be easily encapsulated to form a better estimator . in the high snr limit and for unbiased conditionsthe zbb provides the same bounds as the crb .the optimal detector for the zzb toa estimator has been derived in for uwb rician channels .we note the zbb is based on constructions of a likelihood ratio test ( lrt ) for optimum decision rules .once having determined the timing error associated with the relevant toa its role in setting the scale of the decrypting region can be determined via analysis .  for ease of exposition , here we will determine a crb estimate of the location error bounds given some gaussian distribution on the toa timing errors ( e.g. ) . in the followingwe denote the true location of the decryptor by  ] . to counteract the impact of unknown bias in the timings ( e.g. synchronization offsets or constant nlos bias ) it is normally useful to move to a time difference of arrival setting ( tdoa ) . to achieve thiswe denote the time difference relative to that measured by rs at the rs as , then the logarithm of the distribution of is obtained as where is the speed of light , is the variance of the timings , and the distance between the rs and the decryptor is expressed as the fisher matrix for the tdoa scheme can then be written ,\ ] ] where  and here , . ] , and the correlation coefficient by making the usual assumption that the likelihood function of the parameters to be determined is gaussian in the vicinity of the true values , the distribution of the estimated decryptor s location can be expressed as      using the above analysis , fig .3 displays the anticipated crb s for a set - up of four rss .here we have assumed m and the rss are of order 100 m away toward the corners of the simulation area ( moving the rss beyond about 1 km has the simple effect of flattening the crb plane to a constant error of about 7 m ) . as we can see very reasonable errors in the location verificationwill be possible for such settings .we have also shown the error ellipse for two specific points ( the ellipse center indicates the point ) using an eigenvalue analysis of the inverse fisher matrix to set the scale and orientation of the ellipse axes .multiplying these axes by a factor of three will lead to a scenario in which a location estimate of the decryptor ( which if truly at the ellipse center ) would be placed outside the expanded ellipse with probability .a more sophisticated analysis of the decrypting region can be given , as well as a more detailed probability analysis , that takes into account all _ a priori _ information and allows for decision - making even when a location can not be determined - for example when less than three rss with accurate toas are available ( e.g. see where a more direct decision - making process for qlv was adopted using optimal decision theory derived from a lrt ) .nonetheless , the analysis provided here already shows that for anticipated conditions a well - defined and pragmatically - useful decryption region can be readily determined . for a given , the size of the region can be very small ( of order of cm ) orvery large ( of order of km ) , dependent on many factors such as timing resolution , frequency bandwidth , transmitter power ( regulatory limits etc ) and channel conditions . as such , quantum geo - encryption can in principle be used in a wide range of real - world scenarios .in this work we have introduced quantum geo - encryption - a protocol that can allow data - decryption to occur at only one location and one specified time .we have also discussed how this same protocol can be extended to provide the same spatial and temporal characteristics to any quantum task .other useful security outcomes , such as enforcing a pre - specified route for data transfer , are also possible . in reality , issues arise when using an actual specific _ point _ for the allowed location . inreal - world deployments the allowed location in practice becomes an allowed effective region .this region accommodates noise issues that allow for timings that are not exactly consistent with a single geographical point .however , when assessment of the channel conditions is taken , the size of such an effective region can be readily determined and the probability of spoofing the system assessed .the specific protocol we have used in this work for the geo - encryption can be replaced by many different variants , each having their own trade - offs in terms of complexity versus performance .quantum geo - encryption opens up , for the first time , the possibility of delivering encrypted communications in the knowledge that the received signals can only be decrypted by a device at a unique geographical location ( free from the type of attacks that have hindered deployment of classical geo - 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Abstract: in this work we introduce the concept of _ quantum geo - encryption _ - a protocol that invokes direct quantum encryption of messages coupled to quantum location monitoring of the intended receiver . by obfuscating the quantum information required by both the decrypting process and the location verification process , a communication channel is created in which the encrypted data can only be _ decrypted _ at a specific geographic locale . classical wireless communications can be invoked to unlock the quantum encryption process thereby allowing for any deployment scenario regardless of the channel conditions . quantum geo - encryption can also be used to realize quantum - computing instructions that can only be implemented at a specific location , and allow for a specified geographical data - route through a distributed network . here we consider the operational aspects of quantum geo - encryption in generic rician channels , demonstrating that the likelihood of a successful spoofing attack approaches zero as the adversary moves away from the allowed decrypting location . the work introduced here resolves a long - standing quest to directly deliver information which can only be decrypted at a given location free of assumptions on the physical security of a receiver .


the cosmological principle which assumes that the universe is statistically homogeneous and isotropic on very large scales is one of the fundamental pillars of modern cosmology .this can not be proved in a mathematical sense and can be only verified from observations and various predictions of the physical theories based on it .the cosmic microwave background is by far the best conclusive evidence in favour of isotropy which also strongly supports large scale homogeneity in the early universe .various other observations like the isotropy in angular distributions of radio sources and the isotropy of the x - ray background support the assumption of cosmic homogeneity on large scales .the isotropy does not by itself guarantee homogeneity and it implies homogeneity only when there is isotropy around every points .the present universe is known to be highly inhomogeneous on small scales and there are important consequences if the inhomogeneities persist on large scales .the most important implication of inhomogeneities comes from the averaging problem in general relativity through their effect on the large scale dynamics known as backreaction mechanism .the backreaction mechanism can cause a global cosmic acceleration without any additional dark energy component although it seems unlikely that this can explain all of it .the implications of inhomogeneities and backreaction for cosmology are still considered to be important even if it does not provide an alternate explanation of dark energy .the principle of cosmic homogeneity demands that the statistical properties of the observed galaxy distribution in a given finite volume does not depend on the location of that volume in the universe .the statistical properties of galaxy distributions are characterized by the correlation functions .the two point correlation function on small scales , is well described by a power law of the form , with correlation length and slope . vanishes at scales is consistent with large scale homogeneity .however the problem with correlation function analysis is that it assumes a mean density on the scale of survey which is not a defined quantity below the scale of homogeneity .most of the statistical tools for homogeneity analysis are based on the simple number counts in spheres of radius which is expected to scale as for a homogeneous distribution . the conditional density measures the average density in these spheres which is expected to flatten out beyond the scale of homogeneity .the fractal analysis uses the scaling of different moments of to characterize the scale of homogeneity .some of the studies carried out with these methods on different galaxy surveys claim to have found a transition to homogeneity on sufficiently large scales  whereas some studies claim the absence of any such transition out to scale of the survey .the disagreements between various studies indicate the need for some alternative measures of homogeneity which could capture interesting information on different aspects of a homogeneous distribution and serve as complimentary and alternative tool to the existing methods in the literature . in the present workwe introduce a method to asses homogeneity which is based on the evenness , a more general and robust aspect of any homogeneous distribution .we employ the shannon entropy to measure the unevenness characterizing inhomogeneities in a distribution and explore the possibility of employing the proposed method for exploring the scale of homogeneity .  a brief outline of the paper follows .we describe our method and various effects in section 2 , describe the tests and the data in section 3 and presents the results and conclusions in section 4 .our method is based on the shannon entropy in information theory originally proposed by claude shannon to quantify the information content in strings of text .  in information theory entropyis a measure of the amount of information required to describe the random variable .the shannon entropy for a discrete random variable with outcomes is a measure of uncertainty denoted by defined as ,    where is the probability distribution of the random variable .increase in shannon entropy increases the uncertainty and decreases the information about the knowledge of the random variable .another interesting aspect of shannon entropy is its entropy - maximizing property for an uniform distribution .we propose a method to study inhomogeneities in a 3d distribution of points .given a set of points distributed in 3d we consider each of the points as center and determine the number of other points within a sphere of radius as , + where is the heaviside step function and and are the radius vector of and points respectively .to avoid any edge effects we discard all the points as centers which lie within a distance from the survey boundary .clearly the number of valid centers will decrease with increasing for any finite volume sample .we define a separate random variable for each radius which has possible outcomes each given by , with the constraint . here is the density at the center .note that for a given sample with a finite volume , is the maximum numbers of valid centers available at a radius i.e. there are no provision for projecting further another sphere of radius within the given volume .  the shannon entropy associated with the random variable can be written as , where the base of the logarithm is arbitrary and we choose it to be .note that in , and , is a just a label for a number and not an argument .in an ideal situation when all the spheres around the valid centers are equally populated then one gets an uniform value of for all the centers maximizing the uncertainty .then the shannon entropy ( equation 3 ) has its maximum value for radius .the relative shannon entropy at any quantifies the degree of uncertainty in the knowledge of the random variable . when the knowledge about the random variable becomes most uncertain .the distribution of also become completely uniform when is reached .equivalently one can use to quantify the information available in at any .the joint shannon entropy of a set of independent random variables is simply the sum of the individual entropies associated with each of the random variables .but if they are dependent then the total entropy is a sum of the conditional entropies .entropy measures uncertainty in the random variable whereas the information is a difference in uncertainty that is a difference in entropies .the mutual information characterizes the reduction in the uncertainty in the one random variable due to the knowledge of the other .  in our case the mutual information between the random variables are always positive as the random variables are not independent due to the fact that the density measurements around centers at each radius are taken from the same finite volume sample .given that the mutual information are always positive for correlated random variables , an increase in information in at one would lead to decrease in information in at other values . apart from the correlations from finite sample ,the random variables could have extra correlations if the points are clustered or if the points are distributed in a preferred way .these extra correlations increase the mutual information in the random variables .the decrease of information in with increasing is evident irrespective of a homogeneous / inhomogeneous distribution but it would diminish differently depending on the nature and the degree of inhomogeneity present in the distribution . maximum uncertainty or the complete loss of information in the knowledge of the random variable at a radius suggests that beyond the random variables would be independent and completely uninformative about each other . in an infinite perfectly homogeneous system if the set of spheres used for density measurements are completely independent then the random variables at each has no knowledge about each other making them maximally uncertain and devoid of any information. when no inhomogeneities are present due to other sources , the departure of from would be solely due to the correlations among caused by the finite volume of the sample .given the other sources are present one would expect a larger departure of this ratio from and a more uneven distribution of .the scale where this departure levels up with indicates absence of any correlations among beyond that scale .but this ideal situation may never occur in an exact sense in a finite sample as the correlations between introduced by the finite volume and confinement bias persists over the whole range of length scales .this would always give some residual information in even at the largest value of .but the mutual information content due to clustering or any other source of inhomogeneity are expected to diminish with increasing length scale provided a scale of homogeneity exist for the distribution . as the exact transition marked by may never happen in a finite volume sampleso one could set a very small limiting value for to identify the scale of homogeneity . in our analysiswe set this limit to .clustering of the points is the most important source of correlations between the random variables .clustering produces fluctuations in which is directly related to the fluctuations in . when the points are clustered we expect on average number of points in a volume where is given by , + here is the mean density of the distribution and is the volume of the sphere used .the first term has fluctuations from the poisson noise of the same order .the poisson noise rapidly decreases with increasing and is only important on small scales .if the distribution itself is intrinsically inhomogeneous then poisson noise will be modulated according to the spatially dependent intensity parameter .the second term takes into account of clustering of the points .the variance of is , +   going back to our definition of , the fluctuations in this quantity is closely related to the fluctuations in .the variance in is ,   ^ 2 } \label{eq : fvar}\ ] ]  which is times the normalized variance of .one can quantify the correlations of any two random variables at two different by estimating their covariance .the correlation coefficient is , where the indices and takes value between to . here is the total number of values used in the analysis . in this casewe have a positive correlation between and and the correlations would be higher when the differences between and are smaller .the random vector has equally likely outcomes and in principle one can estimate the full covariance matrix of all the random variables at different to compute their correlations .  even in the absence of any clustering and inhomogeneityone would expect a positive correlations between the random variables as their probability distribution is derived from the same finite volume sample .the set of centers at each is a subset of the centers at the preceding .more specifically all the set of centers at each is a subset of the set of the centers at the smallest implying the spheres at different share some common regions .the fractional amount of share is larger when the difference between is smaller thereby introducing larger correlations between the from neighboring values .so the finite volume introduces correlations between the random variables at different which increases the mutual information in the random variables .one should also keep in mind that the spheres used for density measurements are not independent and could share large overlapping regions . at each use a finite set of spheres for density measurements . in generalthese spheres overlap with each other .the probability that a random point drawn out of the distribution would lie in a particular sphere is where is the total number of points in the distribution .the probability that the randomly drawn point would appear somewhere in the sample is . if the spheres are disjoint then at any there are outcome of this experiment i.e the point would lie either in any one of the spheres or somewhere in the sample outside the spheres . butgiven the fact that the spheres overlap the point could also appear at the intersections of multiple spheres .given a finite sample the total probability can be written as , the first sum in above equation gives the sum of probabilities for individual spheres provided they are disjoint .as the spheres overlap one needs to take into account the probabilities for all the possibilities .the second sum is over all distinct pairs of spheres , third sum is over all distinct triples of spheres and so forth .the last term which gives the probability that the point would come up in a region outside the union of spheres is important only at small and becomes insignificant afterwards .it may be noted that in an overlapping scenario the different terms in equation [ eq : overlap ] can be much larger than but when summed together they will always give . in equation [ eq : overlap ] , contains all the terms from overlapping spheres and successive terms in has alternate sign . at smaller most dominant contribution to comes from all pairs of overlapping spheres .the contribution from all triples of overlapping spheres would be the next dominant term and so on . thus at smaller all the individual sums in are different and the magnitude of each sum is lesser than its preceding sum resulting into a positive value of .new sums appear in with increasing thereby increasing its value .but at the same time the differences between the successive sums starts decreasing due to larger overlap thereby decreasing the value of .there is a competition between this two effect and initially the first effect is more dominant than the second leading to an overall increase in with increasing .as we increase the radius the numbers of valid centers decreases and they preferentially get more confined near the center of the sample . due to the finite volume of the sample ultimately the second effect will dominate on some scale depending on the size of the finite sample and also depending on the nature of inhomogeneity to some extent .for example the confinement bias is expected to be even higher for an inhomogeneous distribution which preferentially has more particles residing near the center of the sample .when the differences between the successive sums become smaller then there would be a larger net cancellation leading to smaller values of .as the second effect starts dominating the first one , the value of would start decreasing and finally on the scale of the largest sphere that would fit inside the finite sample , all the individual terms in all the individual sums in would be of the same order i.e. .this would lead to a very large mutual cancellation of all the sums in decreasing its value to or .this finite volume effect eventually introduces an artificial evenness in the distribution of at some depending on the sample size .we see that the effect of overlapping starts dominating much before the scale of the largest possible sphere ( figure [ fig : ovp ] ) making an interpretation difficult on large scales .we note that all the other methods of testing homogeneity based on count in spheres are also affected by the same problem making the interpretations on large scales equally difficult .  in order to avoid these complicacy of correlations introduced by the overlapping of the spheres we also consider non - overlapping spheres of different radii .but the statistics becomes too noisy due to very small number of independent spheres at progressively larger radii which consequently prohibits us to address the issue of homogeneity on large scales .a point of caution in this context is that in our method at very large length scales does not necessarily indicate a real transition to homogeneity which is an obvious outcome forced by the confinement bias resulting from the finite volume of the sample . but with a large enough sample which ensures spheres upto a sufficiently large without significant confinement of the centers could detect the scale of homogeneity if the transition happens before the scale where the confinement bias completely dominates the statistics .in order to study the prospects and limitations of the proposed method we carry out some preliminary tests by applying it to some simple distributions . we consider ( i ) homogeneous distribution without any clustering , ( ii ) inhomogeneous distribution without any clustering and ( iii ) strongly clustered distribution.  in above three cases the unevenness in type ( i ) is only due to poisson noise .the unevenness in type ( ii ) distribution is controlled by poisson noise too but it has the additional complexity that the contribution to the unevenness is governed by the distribution of the spatially dependent intensity parameter . in type ( iii ) distribution both poisson noise and the clustering together contributes to the unevenness .for the first two types we generate a set of monte carlo realizations of some simple homogeneous and inhomogeneous point process . for the third type we use a set of n - body simulations where the points are strongly clustered .we generate a set of monte carlo realizations of different types of homogeneous and inhomogeneous distributions .  for the sake of simplicity we consider some simple radial density distributions where ( i ) , ( ii ) and ( iii ) . is a normalization constant .the distributions in ( i ) and ( ii ) are inhomogeneous poisson distributions which are isotropic about only one point and the inhomogeneities in these distributions persist on all scales . the distribution in ( iii ) is a homogeneous poisson point process which has a constant density everywhere .  enforcing the desired number of particles within radius one can turn the radial density function into a probability function within to which is normalized to one when integrated over that interval .so the probability of finding a particle at a given radius is which is proportional to the density at that radius implying more particles in high density regions .we generate the monte carlo realizations of these distributions using a monte carlo dartboard technique .the maxima of the function in is at in ( i ) and ( ii ) whereas in ( iii ) it is same and constant everywhere .we label the maximum value of as .we randomly choose a radius in the range and a probability value is randomly chosen in the range .the actual probability of finding a particle at the selected radius is then calculated using expression for and compared to the randomly selected probability value . if the random probability is less than the calculated value , the radius is accepted and assigned isotropically selected angular co - ordinates and , otherwise the radius is discarded . in this way ,radii at which particle is more likely to be found will be selected more often because the random probability will be more frequently less than the calculated actual probability .we choose in unit and .we generate realizations of each of the above density distributions and analyze them separately with the method described in section 2 .we simulate the dark matter distribution using a particle - mesh ( pm ) n - body code .the simulations use particles on a mesh and cover a comoving volume of ^ 3 $ ] .we use for the cosmological parameters along with a power spectrum with spectral index and normalization ( ) .a simple `` sharp cutoff '' biasing , scheme was used to extract particles from the simulations that are biased relative to the dark matter and are labelled as galaxies .the bias parameter of each simulated biased sample was estimated using the ratio where and are the galaxy and dark matter two - point correlation functions respectively .this ratio is found to be constant at length - scales and we use the average value over .we use this method to generate galaxy samples with bias values and .we run the simulation for three different realizations of the initial density fluctuations and extract biased samples from each of them .we extract randomly particles from three non overlapping spherical regions of radius from each of the simulation boxes giving us total nine samples for each of the bias values .the numbers and and the specific spherical geometry are chosen just to maintain the uniformity in all the analysis presented here .generally one can choose any number of particles and any geometry for the samples .in the figure [ fig : shratio1 ] we show the variations of with distance for some homogeneous and inhomogeneous distributions described in subsection 3.1 . for the poisson distributionthe value of shows a small departure from at smaller values of which decreases with increasing . the ratio reaches a value at .as the points are uncorrelated , the void probability function for a homogeneous poisson distribution is .it is expected that such a distribution would become homogeneous on a scale , which is a measure of the average size of voids in the distribution .since the correlations between the random variables in the homogeneous poisson distribution mainly comes from the finite volume effect one would expect a lesser amount of mutual information in the set of random variables and consequently a faster information leakage but the information in i.e. would never be exactly zero due to persistence of the finite volume correlations at all .the values at smaller values shows relatively larger departure from for both the inhomogeneous distributions considered here .the ratio has a larger departure for the distribution with than the distribution with on all scales .this shows that the ratio describes the degree of inhomogeneity present in two different anisotropic distributions in correct order .both these distributions are only isotropic about one point and have a radial density distribution which does not allow them to be homogeneous on any scales .the ratio at for both of them simply shows the importance of confinement bias which ultimately is expected to take over the situation at some depending on the sample size and type of inhomogeneity present .the confinement bias gets a larger boost in both of these distributions ( figure [ fig : ovp ] ) as there are already more particles preferentially located near the center of the samples .this effect produces larger overlap and hence larger mutual information are stored in the random variables from neighbouring .consequently this accelerates the leakage of information in with increasing finally forcing the ratio .it may be noted that for the homogeneous poisson distribution there are no such additional boost coming from preferential deposition and further there are no extra correlations due to clustering .the transition is reached there at a much before the scale where the confinement bias play a dominant role indicating a real transition to homogeneity .  in the figure [ fig : shratio2 ]the variations of with distance are shown for n - body simulations with different linear bias values .the assumption of linear bias holds reasonably well on large scales .different types of galaxies are biased differently with respect to the dark matter and the inhomogeneities in the distributions of different types of galaxies could be modulated by their bias values .we would also like to investigate here how well the ratio can track the variation in inhomogeneities in a distribution on different scales and give some useful information about the characteristics of the inhomogeneities present in them .we see that the ratio initially depart from at small for all the distributions with higher bias values showing systematically lesser inhomogeneity ( lesser information in ) than lower bias values . in the unbiased model the centers used for calculating are residing in all types of environments ( clusters , sheets , filaments and voids ) .the fact that the centers are distributed across different types of nonlinear structures increases the unevenness in the distribution of and the mutual information in . whereas with increasing bias values the particles in a biased distribution preferentially represent progressively higher density peaks in the density field homogenizing the distribution on the corresponding length scales .this is primarily due to the fact that in a biased distribution the centers are located in less diverse environments .a complete reversal of this behaviour is seen at after which with increasing bias values the distributions systematically show larger inhomogeneity and more information in at all scales until they eventually merge to .this behaviour is related to the fact that increasing the radius beyond the typical scale of the nonlinear structures in the biased distributions would lead to larger disparity in and values and hence the larger inhomogeneities .the scales corresponding to the reversal in the behaviour of ratio corresponds to the typical scale of the nonlinear structures present in the biased distributions .further on large scales the correlation functions in biased distributions are times larger than the in unbiased distributions introducing larger correlations among all the random variables at larger .eventually all the curves merge to in the range with the difference that the transition appears to happen at a relatively larger values for larger bias .although this can not be emphasized due to the size of the overlapping error bars at those length scales . it may be noted that the inhomogeneities in the model with different biases are much smaller as compared to the inhomogeneous radial density distributions given by .the information in for a poisson distribution mostly comes from the finite volume effect whereas the very large mutual information in the random variables in two inhomogeneous distributions is the combined outcome of finite volume correlations and large degree of confinement bias . in the unbiased and biased models the mutual information are generated by correlations from confinement bias and clustering together .it may be noted that the confinement biases are much smaller and very similar in the models and the homogeneous poisson distributions ( figure [ fig : ovp ] ) .the confinement biases become very large in the two anisotropic distributions considered here which eventually force the ratio to at some scales .if a scale of homogeneity exists then ideally one would expect a naturally emerging evenness in when the spheres of that radius around each centers statistically include similar numbers of different types of nonlinear structures .so provided that the correlations introduced by the finite volume of the sample and the confinement bias are much smaller compared to the correlations induced by clustering then the leakage of information in would ideally track the variation of inhomogeneity in the distribution with scale .but unfortunately in a real situation one has to also deal with the correlations due to finite volume and confinement bias modulating the information in or the evenness of .we find that the confinement bias could start to dominate even much before the scale defined by the largest sphere included in any finite sample ( figure [ fig : ovp ] ) .the degree of correlations introduced among would also depend on the type of inhomogeneity present in the distribution specially if it is overpopulated near the center of the sample with respect to the rest of the volume resulting into larger overlap between the spheres .when dealing with galaxy redshift surveys one has to also properly take into account the different selection effects involved , redshift space distortions and the specific geometry of the samples .we plan to carry out analysis in the publicly available catalogues from large galaxy redshift surveys ( e.g. 2dfgrs , ; sdss , ; 2mass , ) in future works and investigate these issues further .it may be noted here that a caveat of our method is we assume that we always have access to the data samples on a spatial hypersurface of constant time .though this assumption is approximately valid in case of low redshift galaxy samples in the nearby universe or data samples from a snapshot of n - body simulations , it is not strictly true as the entire observational samples do not consist of objects on a constant time hypersurface but rather on a light cone .taking this into account it is hard to distinguish radial inhomogeneity from time evolution without assuming a cosmological model . consequently this could make an inhomogeneous but isotropic model ( e.g. anti - copernican void models ) to look like a homogeneous one on large scales .fortunately these models can be constrained with other observations such as sne , cmb , bao and measurements of hubble parameter .finally we note that the method presented here has the desired ability to characterize inhomogeneities and their variations on different length scales in any 3d distribution of points .the method has the potential to successfully identify any existing scale of homogeneity given a sufficiently large volume which can ensure negligible confinement bias and less effective overlap between the spheres upto large length scales .given the large survey volumes that are currently available from the modern redshift surveys the method could provide an efficient tool for exploring the issue of cosmic homogeneity .the author thanks somnath bharadwaj for useful comments and discussions . the author would also like to thank an anonymous refree for insightful comments about the paper .the author acknowledges the alexander von humboldt foundation for financial support and the computing center of the max planck society in garching ( rzg ) for the computing facilities provided for the work .

Abstract: we propose a method for testing cosmic homogeneity based on the shannon entropy in information theory and test the potentials and limitations of the method on monte carlo simulations of some homogeneous and inhomogeneous 3d point process in a finite region of space . we analyze a set of n - body simulations to investigate the prospect of determining the scale of homogeneity with the proposed method and show that the method could serve as an efficient tool for the study of homogeneity .   methods : numerical - galaxies : statistics - cosmology : theory - large scale structure of the universe .