due to the importance of entanglement as a resource for quantum information processing , the task of determining whether a quantum state is entangled or not plays a crucial role for the theoretical and practical developments of the field .while the usefulness of bipartite entanglement is well established , the development of applications using entanglement in multipartite systems is still in its early stages , however , it has already been shown to play a fundamental role for universal quantum computation in the measurement - based quantum computation paradigm and for cryptographic tasks such as secret sharing . concerning the characterisation of entanglement ,one of the biggest challenges to start with is the fact that it is np - hard in the hilbert space dimension to decide whether a given quantum state is entangled .since the dimension grows exponentially in the number of involved parties , the exact answer to that question will probably remain elusive for many - body systems .nonetheless there exists an abundance of necessary separability criteria capable of certifying entanglement in a practically satisfying way . for bipartite entanglement , the two most paradigmatic techniques are positive maps which are not completely positive and entanglement witnesses .these two concepts are intimately related , as every positive map directly leads to a multitude of entanglement witnesses and every witness can be associated to a positive map .both concepts are sufficient for the characterisation of entanglement , in the sense that , for every entangled state , there exists both a positive map and an entanglement witness certifying its entanglement .  turning to multipartite systems, the potentially underlying separability structure adds a layer of complexity to the detection of entanglement .as opposed to just entanglement in multipartite systems , mixed multipartite states pose another fundamental challenge .counterintuitively , multipartite systems can exhibit entanglement across every partition , yet still not feature any genuine multipartite entanglement ( gme ) . to certify gme ,an abundance of entanglement witnesses were derived ( see e.g. reviews ) and a characterisation in terms of semi - definite programs ( sdp ) was developed .there is however no direct correspondence to the concept of positive map based criteria , not even for the undoubtedly most used criterion of positivity under partial transposition ( ppt ) .indeed , the naive application of a positive map to a subsystem in correspondence to the bipartite case must inevitably fail , as all it can reveal is entanglement across that bipartition , which , as we just mentioned , is never enough to infer gme .  in this manuscriptwe expand the notion of positive map based criteria from the bipartite to the multipartite case , thus filling a gap in the set of available tools .our main idea is to derive maps which are positive on all biseparable states , whilst not positive on the set of all states .we first give a general description of the approach and then showcase some exemplary maps .the maps we derive are based on convex combinations of positive maps that are used in the bipartite case , thus generalising the ppt criterion and others to the multipartite case .the manuscript is organised as follows : in section [ sec.preliminaries ] we set the stage by recalling some important bipartite and multipartite entanglement definitions and giving the motivation of this work . in section [ sec.results ]we introduce our framework and general construction method based on positive maps .we also make some general existence considerations . in section [ sec.examples ]we give some explicit examples of our method .in particular , a remarkably simple partial transposed based criteria is worked out and a choi map construction is shown to robustly detect noisy -qudit ghz - like states as well as ppt states . finally in section [ sec.discussion ]we discuss possible extensions and applications of our method .to set the stage and notation we first remind the reader of the concepts of separability , entanglement witnesses and positive maps in the bipartite case .a finite dimensional state is called separable iff it can be decomposed into a convex combinations of pure product states , i.e. where is a probability distribution .  the states which can be written in the form of eq .form a closed convex set and therefore , as a consequence of the hahn - banach theorem ( see ) , the set of separable states can be separated from any point in its complement by a hyperplane that can be written as for a self - adjoint operator ( observable ) .now if ( by convention ) and there exits at least one ( entangled ) state such that , the operator is referred to as an _ entanglement witness_. the above thus states that any entangled state can be detected by an entanglement witness .  from another perspective a linear positive map , =\alpha \lambda[\rho]+\beta\lambda[\sigma] ] , can be used to detect entanglement if it is not completely positive , since the application of a non - completely positive map to an entangled state can result in a non - positive operator , \ngeq 0 ] , it implies that there exists a pure state such that )<0 ] , it can immediately be seen that  ] is a gme - witness .this looks all good but upon inspecting the gme - map defined by eq ., one remarks that it is only gained from positive maps in an indirect way .indeed from the family of maps ( bottom left of fig .[ diagram ] ) , a family of bipartite witnesses was first associated ( bottom right of fig .[ diagram ] ) , from which a gme - witness was extracted ( top right of fig .[ diagram ] ) .this gme - witness finally defined the non - positive map of eq .( top left of fig .[ diagram ] ) .now , our goal is to explore a direct lifting method , illustrated by the red arrow from to in fig .[ diagram ] , from positive maps to gme maps , without passing through the witnesses .the reason for pursuing this is that gme - maps of the form of eq .are quite trivial in the sense that the boundary of the set is a hyperplane rather than the boundary of a more complex convex set , which could possibly explore more subtleties of genuine multipartite entanglement .inspiring ourselves from the structure of in eq ., our goal is to seek for maps of the form where is a positive map , :=\sum_i p_i^{(a ) } \ , u_i^{(a)}\rho  \left ( u_i^{(a)}\right)^\dagger ] . in appendix [ app.generalphi ]we derive some general conditions as to when one can ensure that for a given gme - state a gme - map of the form of eq . detecting this gme - state exists .unfortunately , it is not clear from those conditions whether the construction of via lifting of positive maps is general enough , that is , whether for every gme - state , there exists a gme - map of the form of eq . detecting .after those general considerations , we want to give in the remaining of the manuscript concrete examples of our method by exhibiting gme - maps constructed via the lift of positive but non - completely positive maps .our first example is possibly the most naive attempt , where we set , and , the transposition , in a tripartite setting , and find the value of for which defined by eq . is a gme - map .that is , we consider the map =(t_a \otimes \mathbb{i}_b \otimes \mathbb{i}_c + \mathbb{i}_a \otimes t_b \otimes \mathbb{i}_c  + \mathbb{i}_a \otimes \mathbb{i}_b \otimes t_c  + c \,\mathbb{i}\cdot \text{tr})[\cdot].\end{aligned}\ ] ]  [ thmphit ] for it holds that for all tripartite biseparable states  \geq 0 , \end{aligned}\ ] ] and this value of is optimal , i.e. it is the least compatible with the above constraint .the proof of theorem [ thmphit ] consists in determining the minimal eigenvalue of the operators , which turns out to be for any partition ( see ) .the detailed proof of theorem [ thmphit ] is presented in appendix [ aproofphit ] .the intuition behind this construction rests upon the prerequisite that the negative eigenstates under application of partial transposition for every cut have at least some nonzero overlap . as we will see in the following examples, some states will exhibit this property and the map works in a straightforward way . the additional unitary transformation in eq.([eq : phidef ] )is supposed to systematically map the corresponding negative eigenstates into the same space , opening up the possibility to detect many more states .we will see that such unitary corrections can be readily constructed , but their success depends also on the symmetries exhibited by the state .  to start ,let us consider an example where the map directly works .if we consider the state we find \ngeq 0 ] .let us consider =(\tilde{\sigma}_x\circ t_a \otimes \mathbb{i}_b \otimes \mathbb{i}_c + \mathbb{i}_a \otimes \tilde{\sigma}_x\circ t_b \otimes \mathbb{i}_c  + \mathbb{i}_a \otimes \mathbb{i}_b \otimes \tilde{\sigma}_x \circ t_c  + \mathbb{i}\cdot \text{tr})[\cdot]\end{aligned}\ ] ] where the positive map denotes the transposition followed by the application of the unitary pauli operator .since we only added a local unitary operation , the map remains positive on all biseparable states .one calculates \ngeq 0 ] for all , with negative eigenvalue which implies where is the -partite ghz state .the map can also be similarly generalised for parties , however , already for , the -partite state is no more detected by it .we therefore see that this construction although generating gme - maps , does not provide a noise resistant detection .this might have two causes , namely the choice of the positive map or the naive choice of m. see sections [ sec : diagm ] or [ sec : choi ] for less naive constructions .   in this sectionwe will modify the map of section [ sec : trans ] by looking at a slightly better choice for the correction map .this will enable us to detect the ghz state with an optimal noise resistance .we will choose , where here and  ] , for all n - qubit biseparable states .the proof relies on the following fact :  [ lemmaodt ] = ( 2^{n-1}-1 ) \text{od}[\tilde{\sigma}_x^a \circ t_a \otimes\mathbb{i}_{\bar{a } } [ \mathcal{x}_n[\rho ] ] ] , \quad \forall \rho , \forall a,\ ] ] where := x- \text{diag } [ x] ] projects any -partite -dimensional state into the subspace .the proof of proposition [ prop - main ] can be found in appendix [ aprojection ] .now that we have a way to project any -qudit state into the subspace spanned by _ ghz - like cyclically permuted _ states , we can continue to discuss the positive maps employed .the choi - map of dimension is defined as follows =2 \text{diag}[\rho]+\sum_{j=1}^{d-2}x_d^j\text{diag}[\rho]{x_d^j}^{\dagger}-\rho ,  \end{aligned}\ ] ] where is the shift operator defined in eq . .we next want to look at where and denote the choi - map applied to the subsystems contained in . with the above we can consider the map   & = \phi[\rho ] + ( 2^{n-1}-2)\left[\text{diag}[\phi[\rho]]-\sum_{a } \text{diag}[\rho ] \right].\end{aligned}\ ] ]  now we are ready to state the main result of this section .[ thm_choigme ] \geq 0 ] .theorem [ thm_choigme ] gives us a sufficient condition for a state to be gme : if upon projecting an -partite state into the ghz - like cyclically permuted subspace and applying the map results into a negative eigenvalue , one can assure that the state is genuinely -partite entangled .the proof is analogous to the one of theorem [ thm_eta ] of section [ sec : diagm ] .indeed , we make use of the following lemma that will be proven in appendix [ aprooflemma ] .[ lemmaod ] = ( 2^{n-1}-1)\text{od}[\lambda_a\otimes i_{\bar{a}}[\mathcal{x}_{n}^d[\rho]]]\;\ ; \forall \ , \rho , \forall \ , a ,  \end{aligned}\ ] ] where =x-\text{diag}[x] ] , where for each a , is a positive map detecting and is chosen such that . then by defining + - t_a ] \geq 0;\ ] ] we have that + m_{\{\lambda_a , { \left| \psi_a \right\rangle}\}_a}\\ & = \sum_a ( q+t_a)+ \sum_a ( ( 2^{n-1}-1 ) [ t_a]_+ - t_a)\\ & = ( 2^{n-1}-1 ) w;\end{aligned}\ ] ] proving that there exists a gme - witness detecting of the desired form , namely .in this section we want to derive some general conditions about the existence of a gme - map of the form detecting a given gme - state . quite trivially , we first note that  [ propidea ] for every family of positive but non - completely positive maps , the map for a family of convex combination of local unitaries and positive satisfying \geq -\sum_{a } \lambda_a \otimes \mathbb{i}_{\bar{a}}\circ \mathcal{u}^{(a)}[\rho_{2-sep}]\;\ ; \forall \rho_{2-sep},\end{aligned}\ ] ] is such that \geq 0 ] it follows that is a gme - map constructed from the lifting of positive but non - completely positive operatorsof course , given a gme - state , it is not clear from proposition [ propidea ] if a family and a map m exist such that as in eq . is a gme - map .conditions for the existence of such a map are given by the following  [ prop : condphi ] given a gme - state and a gme - witness detecting of the form ,\ ] ] there exists a gme - map detecting of the form with and a family of local unitaries if  * there exists a family of local unitaries such that are equal for all a , * is independent of for any chosen family .first of all recall that the states were chosen such that { \left| \psi_a \right\rangle } < 0 ] and . since by ii ) is independent , we can define := \text{tr}(\rho m_{\{\lambda_a\}_a,{\left| \phi \right\rangle } } ) { \left| \phi \right\rangle } { \left\langle \phi \right|},\ ] ] where we did not explicitly write the dependence of on the choice of .note that \geq 0 , \ ; \forall \rho ] .we then have + m [ \rho ] ) { \left| \phi \right\rangle}.\ ] ]  since by construction detects , equation [ equ : opdef ] tells us that   furthermore we have from the definition of that for all biseparable state , .in fact , one can plug any family of states in the defining formula of instead of and still holds for any biseparable ; denoting the newly obtained operator . in particular , the states can be of the form , for an arbitrarily chosen .as before , we obtain + m[\rho ] { \left| \tilde{\phi } \right\rangle } \geq 0 , \quad \forall \text { biseparable } \rho.\ ] ]  since is arbitrary , this means that + m[\rho ] \geq 0 ] .furthermore , since the latter becomes diagonal upon applying , it follows ] = \text{od } [ \lambda_a \otimes i_{\bar{a } } [ \mathcal{x}_n^d [ \rho ] ] ] ,  \label{equ : odcommute}\ ] ] such that ] & = \text{od}\left [ \sum_{b \mid \bar{b } } \lambda_b \otimes i_{\bar{b}}\right]\\  & = \sum_{b \mid \bar{b } } \text{od } [ \lambda_b \otimes i_{\bar{b } } [ \mathcal{x}_n^d [ \rho]]]\\  & { \mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily \ref{equ : odcommute}}}}{= } } } \sum_{b \mid \bar{b } } \lambda_b \otimes i_{\bar{b } } [ \text{od } [ \mathcal{x}_n^d[\rho]]]\\  & { \mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily \ref{equ : odpartition}}}}{= } } } \sum_{b \mid \bar{b } } ( - \text{od } [ \mathcal{x}_n^d[\rho]])\\  & = ( 2^{n-1}-1 ) ( - \text{od } [ \mathcal{x}_n^d[\rho]])\\  & { \mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily \ref{equ : odpartition}}}}{=}}}(2^{n-1}-1 ) \lambda_a \otimes i_{\bar{a } } [ \text{od } [ \mathcal{x}_n^d[\rho ] ] ] , \quad \forall a.  \end{aligned}\ ] ]  lemma [ lemmaodt ] is the same assertion as lemma [ lemmaod ] but for instead of .the proof is also identical and can be carried through by systematically replacing by , setting and noting that in the above proof .only equation [ equ : odpartition ] has to be adapted to ] = \text{od}[\mathcal{x}_n[\rho]]\ ] ] since acts as the identity on $ ] , not as minus the identity .

Abstract: positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement . in the multipartite case , however , the direct application of a positive but not completely positive map can not distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition . we thus generalise this bipartite concept to the multipartite setting by introducing non - positive maps that are positive on the subset of bi - separable states , but can map to a non - positive element if applied to a genuine multipartite entangled state . we explicitly construct examples of multipartite non - positive maps , obtained from positive maps via a lifting procedure , that in this fashion can reveal genuine multipartite entanglement in a robust way .


hash table is a common data structure used in large set of data storage and retrieval .it has an o(1 ) lookup time on average , but the worst case lookup time can be as bad as o()( is the size of the hash table ). such a time variation is essentially caused by possibly many collisions during keys hashing . in this paper, we present a set of hash table schemes called cascade hash tables which consist of several levels( ) of hash tables with different size .after constant probes , if an item ca nt find a free cell"(slot ) in the first level table , it will try to find a cell in the second level , or subsequent lower levels . with this simple strategy, these hash tables will have descendent load factors , therefore descendent collision probabilities .so finally the probability that the item can not find an empty cell in any hash table is slight .this enables the whole hash table to reach a high load factor with constant probes in random generated test set , before a _crisis_(the situation that when an item comes , we can not find a free slot in limited probes in any hash table ) happens .a hash table is a data structure that associates keys with values .the primary operation it supports efficiently is a _lookup _ : given a key ( e.g. a person s name ) , find the corresponding value ( e.g. that person s telephone number ) .it works by transforming the key using a _ hash function _ into a hash a number that the hash table uses to locate the desired value .a hash function is a many - to - one mapping , which maps keys in a large domain to hashes in a relative small range . so collisions among keys which are mapped to the same hash are inevitable .differences among hash table schemes lie in hash function and collision resolution strategy .generally a string type key is hashed into an integer by a hash function , then mapped into an index not bigger than the table size(a common method is to compute the hash value modulo the table size .there are various hash functions on strings , such as crc , _lookup2 _ and md5 . as to integer type keys ,they are directly mapped into indices .   if two keys hash to the same index , the corresponding records can not be stored in the same location .so , if it s already occupied , we must find another location to store the new record , and do it so that we can find it when we look it up later on .the most popular collision resolution techniques are _ chaining _ and _ open addressing_.  in the chained hash table technique , each slot in the array references a linked list of inserted records that collide to the same slot .insertion requires finding the correct slot , and appending to either end of the list in that slot ; deletion requires searching the list and removal .this technique is intuitive and the performance degrades gracefully when the load factor increases .but if the record size is small , the overhead of the linked list is significant .additionally , traversing a linked list has poor cache performance .some popular hash table implementations , such as stl , use this technique .open addressing hash tables store the colliding records directly within the array .a hash collision is resolved by probing through alternate locations in the array(the probe sequence ) until either the target record is found , or an unused array slot is found , which indicates that there is no such key in the table .well known probe sequences include : linear probing , in which the interval between probes is fixed often at 1 ; quadratic probing , in which the interval between probes increases linearly ( hence , the indices are described by a quadratic function ) ; double hashing , in which the interval between probes is fixed for each record but is computed by another hash function .any of these methods may probe indefinite number of locations , even as many as in the worst case !we call these methods _ unlimited_. a method is _ limited _ , if the number of probes can not exceed some limit .in the implementation of -level cascade hash table , we have hash tables , and use limited double hashing in every level of table . in turn , the hash table size is half of its preceding hash table(the proportion is chosen empirically ) .we limit the total number of probes to .thus in every level , the probe number is . here is a factor of , so . if an item ca nt find a free cell in level in probes , it will probe in level 2 , and if still with bad luck , it turns to search lower levels .if a crisis happens , the hash table will be enlarged and rehashed .the lookup procedure is similar to the insertion procedure .it also takes not more than steps .  clearly , insertion andlookup both take at most probes , so the time complexity of cascade hash table is o(1 ) .  specially , when , it s the ordinary ( limited ) double - hashing scheme . when (one probe every level ), it s the multilevel adaptive hashing " scheme presented by paper .experiments show that it is not the best configuration .are all occupied ) ]by experiments on random data sets , we get the result as in table [ result1 ] , [ result2 ] and [ result3 ] .  .experiment result 1 [ cols="<,^,^,^,^,^,^ " , ]   from these tables , we can see that in a certain level of a multiple level hash table , the load factor decreases drastically .take three - level hash table for example , when hash tables are full"(not really full , but we can not insert the coming new item into it in limited probes ) , is around , is , but is only around .so if a new item comes , the _ crisis rate_(the probability that the item ca nt find an empty room in any hash table ) is not bigger than .given a one level hash table , assume it has a load factor of , then it will take at least probes on average to obtain the same small crisis rate(which ensures a high load factor ) as 3-level cascade hash table .but cascade hash table just makes probes meanwhile .  to our surprise, the space efficiency of -level hash table does nt increase monotonically with . at ,the space efficiency hits the climax , then falls down at .in this paper , we introduce a series of hash table schemes cascade hash tables .it uses levels of hash tables ; in every level , we use limited double hashing to make probes .smaller hash tables work as fail - safes of bigger hash tables . roughly speaking ,different tables are similar with sieves with holes in different shapes .we hope no object escape through these sieves . with more sieves ,the chance that an object is screened in some level is bigger .the idea is simple , but its performance exceeds the ordinary one level hash table dramatically when there are more than levels . by choosing , it s much better than the hash scheme proposed in paper .obviously , if we permit a larger total probe count , we can achieve higher load factor .but the average speed will be slower .so a user can choose an appropriate configuration which balances best between speed and space efficiency to him .12 andrei z. broder , anna r. karlin .multilevel adaptive hashing . in proceedings of the 1st acm - siam symposium on discrete algorithms ( soda ) , pp .43 - 53 , 1990 .bill mcquain .hashing , slides for class cs 2604 .http://courses.cs.vt.edu/~cs2604/summerii_2005/notes/c10.hashing.pdf thomas h. cormen , charles e. leiserson , ronald l. rivest and clifford stein .introduction to algorithms , second edition . the mit press hash table " entry in wikipedia .

Abstract: in this paper , the author proposes a series of multilevel double hashing schemes called cascade hash tables . they use several levels of hash tables . in each table , we use the common double hashing scheme . higher level hash tables work as fail - safes of lower level hash tables . by this strategy , it could effectively reduce collisions in hash insertion . thus it gains a constant worst case lookup time with a relatively high load factor( ) in random experiments . different parameters of cascade hash tables are tested .


modern quantum technologies harness characteristic features of quantum systems to gain performance that is otherwise unattainable through classical means .this progress ultimately relies on the ability to create , manipulate , and measure quantum states .all of these tasks require a step - by - step verification in the experimental procedures ; this is essentially the scope of quantum tomography  .  typically , a tomographic protocol attempts to infer the unknown quantum state from the distinct outcomes of a collection of measurements performed on a finite set of identical copies of the system . with these limited resources ,the choice of optimal measurements and the design of efficient reconstruction algorithms are crucial .when the measurement outcomes form an informationally complete set  , the data obtained contain maximal information and a unique state estimator can be inferred .unfortunately , as we probe more intricate quantum systems  , such an informationally complete set of measurements becomes extremely difficult to implement .in addition , the complete knowledge of the quantum state of a system with many degrees of freedom is usually not needed , as one could very well be interested in a few parameters only , such as the fidelity with respect a target state or a measure of entanglement .we are thus inevitably led to the consideration of alternative techniques . a promising class of new protocols are explicitly optimized for particular kinds of states .this includes states with low rank  , such as matrix product states , or multi - scale entanglement renormalization ansatz states  .the specific but pertinent example of permutation invariance was also examined  .  in the same spirit , several algorithms for estimating quantum states from incomplete measurement data have been reported recently  . here , we revisit the problem in the context of mutually unbiased ( mu ) measurements , which are known to be optimal for state reconstruction  if a complete set of bases is to be measured . at the fundamental level , mutually unbiased bases ( mub )are part of the mathematical framework for an explicit formalism of the complementarity principle .the idea emerged in the pioneering work of schwinger  , and has since been integrated into the foundation of quantum theory : apart from their role in tomography , mub are instrumental in addressing a number of enthralling questions  .  a first , self - suggesting , if naive , approach could be to assume a uniform distribution for the outcomes of the unmeasured bases from a set of mub , as this seems to be consistent with the very physical nature of mub , which minimize the statistical uncertainty hypervolumes .however , as we show here , this is often incompatible with the positivity required by quantum mechanics , and even when the resulting estimator is physical , it is usually not the estimator with the largest von neumann entropy  .the bases of eigenstates of complementary observables are called _ unbiased _ because we can not predict at all the outcome of a projective measurement in one basis if the system is prepared in a state from another basis all outcomes are equally probable , there is no preference in our prediction .this notion of being unbiased must not be confused with other uses of the adjective .in fact , there are many different meanings and connotations associated with `` bias '' : statistical bias , bias in a sample , cultural bias , and media bias are but a few uses of the word .  in the context of state estimation ,_ statistical bias _ is of some importance ; an estimator has a statistical bias if its average over all thinkable compositions of a finite sample of measurement results deviates from the true value .this will be of no concern here because we shall take for granted that the measured sample is so large that statistical noise in the data can be safely ignored . also , we do not have to worry about a _ biased sample _ , a common problem when polls are taken .the simple `` estimators of unbiased linear inversion , '' which we shall introduce in sec .[ sec:2c ] , are unbiased in the sense that they assume equal probability for all outcomes of the unmeasured bases from a set of mub ; they are , however , estimators with a statistical bias ( for data from a finite sample ) , a property that can not be avoided  if one insists , as we do , that all permissible estimators are physical they must be _ bona fide _ density operators . as noted in the preceding paragraph ,the statistical bias is irrelevant in the current context , and it is worth recalling that , despite the negative connotation of `` bias , '' a statistical bias is not only harmless , but can be rather beneficial ( see jaynes s discussion in sections 17.2 and 17.3 of ref . ) .  regarding _ cultural bias _ and _ media bias _ , we are confident that we do not suffer from the former and hope that we will not be a victim of the latter . may bias , melampus s brother and pero s and iphianassa s husband ( consecutively ) , guard this work against biased judgment .the plan of this paper is as follows . after introducing background material and notational conventions in sec .[ sec : background ] , we illustrate various aspects of the said self - suggesting approach in sec .[ sec : ulinexamples ] , with particular emphasis on unphysical properties of the resulting estimators .we conclude that the `` estimators of unbiased linear inversion '' are often unphysical they are _ not _ estimators .then , in sec . [ sec : least - bias ] , we introduce a natural alternative that keeps these estimators whenever they are physical and , when they are not , replaces them with physical estimators . rather than being unbiased about the probabilities of unmeasured bases , the physical estimators are minimally biased .in this alternative approach , we maximize the shannon entropy with due attention to the physical constraints and so minimize the bias .this entails a simple optimization algorithm over the state space , in which a single equation is iterated . in sec .[ sec : lbcomp ] , we consider alternatives to maximizing the shannon entropy . some of them minimize the bias with respect to another criterion , others are characterized in a different way .finally , we offer a summary and conclusions in sec .[ sec : con ] .the state of a -dimensional quantum system is specified by a positive semidefinite , unit - trace density operator .carefully note that a different symbol is reserved for its matrix representation , which requires independent real parameters for its complete characterization .if a von neumann maximal test is chosen to fix of these parameters , then a total of tests is necessary to reconstruct the state .this strategy is optimal when the bases in which the measurements are carried out are `` as different as possible ; '' that is , when these bases are mu  .  throughout this paper , we take the dimension to be a prime or a prime - power integer .then , a maximal number of mub  exist and can be explicitly constructed  .we denote by the ket in the basis of the set of mub ; here and below , greek indices ( , , ) label the distinct mub , whereas latin indices ( , , ) label the outcomes in each basis .we define mu projectors as .any two mu projectors satisfy the trace relation that states their orthonormality for and their mutual unbiasedness for .besides , these projectors constitute a complete set of projective measurements for each ,   to facilitate the discussion of the physics behind incomplete mub tomography , the number of copies of the system probed by the measurement apparatus is taken sufficiently large , so statistical fluctuations in the measurement data are negligible .  the problem we are studying here is , therefore , not one of state estimation _ sensu stricto _ , where the estimation of the probabilities from observed relative frequencies is the central theme .rather , we are dealing with the problem of converting the probabilities into a statistical operator , which requires a deliberate choice when the tomography is incomplete .the measured probabilities which obey the constraints , fully characterize the density operator of the system .first , if only one basis is measured the basis , say our state estimator is it is the most natural estimator , inasmuch as we estimate the probabilities of the unmeasured bases in the unbiased manner of for ; this is most adequate for mub if we are mindful of laplace s advice to assign equal probabilities to alternatives about which we have no information .second , after measuring all bases , we know the density operator completely , alternatively , we write while the mappings and are linear , the mappings in eqs .  or appear to be affine .in fact , these are linear mappings too , since we can make use of identities such as   { w_{\alpha k } } & = & { \sum_{j=0}^{d-1}p_{\alpha j }  \left  ( \delta_{jk}-\frac{1}{d } \right ) } \ , , \end{aligned}\ ] ] if we wish .here is a caveat .the linear map in for a single basis continues to yield physical estimators if one replaces the probabilities by the corresponding relative frequencies from an experiment with a finite sample .the resulting single - basis estimators have no statistical bias .for the full tomography , however , one can not replace the probabilities in by relative frequencies .if one does , one gets mock estimators that have no statistical bias but are unphysical , whereas physical estimators are statistically biased since they require a suitable nonlinear mapping from the relative frequencies to estimated probabilities. see refs . for various aspects of these matters .the second version of eq .  exploits a well - known important geometrical property of density operators : the difference between any and the completely mixed state is a traceless hermitian operator , and these operators are elements of a real vector space .the sum over in eq .  or is a sum over hyperplanes in this vector space ; there are of them , each -dimensional since .the hyperplanes are orthogonal in the sense of and in & = & \sum_{\alpha=1}^{d+1}\sum_{k=0}^{d-1 } ( w_{\alpha k}-w_{\alpha k}^{\prime } ) ^2\end{aligned}\ ] ] we recognize the euclidean metric of the vector space .the single - basis estimator in eq .  is obtained from the right - hand side of eq .  by setting for the basis indices that are different from the privileged index , if an intermediate number of bases from the set of mub has been measured say , the first bases with unbiasedness of the bases and the linearity just noted suggest to set for the unmeasured bases , which yields the estimator of _ unbiased linear inversion _ ( ulin ) , we can also regard the truncation of the summation as effected by a replacement of the coefficients in eq .  in accordance with   yet , irrespective of how we regard the map of the measured probabilities with onto the ulin estimator , the map simply amounts to estimating the probabilities of the unmeasured bases in the unbiased manner faithful to laplace s advice , so to say : all other estimators are characterized by the non - zero operator they assign to the unmeasured part in where is the estimator for the contribution of the unmeasured bases to the sums in eqs .  and .before proceeding , six comments are in order .first , the map projects onto the span of the first hyperplanes in the vector space discussed above ; applying the map a second time has no effect .this projection property implies for any two density operators and .second , we note that , since eq .  involves only the measured bases , it is also well defined for systems where mub exist , but this set can not be extended to bases .this is the case , e.g. , for and where only sets of at most three mub have been found thus far  .there are also non - extendable sets for prime - power dimensions  , although ulin estimators for these sets suffer from the same problems as the ulin estimators for subsets of full sets ( see sec .[ sec : ulinexamples ] ) : they are often unphysical .  also , whether the unmeasured bases are pairwise unbiased among themselves , or unbiased with some of the measured bases , is not important for the maximum property of the following third comment , nor for the physical least - bias estimators of secs .[ sec : lbphilo ] and [ sec : lbcomp-1 ] .if one wishes , one can choose the unmeasured bases such that they have largest average distance ( see ref . ) , so that they resemble a set of mub as best as they can .third , it is also worth noting that , of all estimators consistent with the measured probabilities , the ulin estimator maximizes the shannon entropy associated with the unmeasured probabilities , which must be calculated according to eq . .the ulin estimator has this property by construction , since it gives the uniform estimates of eq .  for the .  for simplicity, we are taking some liberties with the shannon entropy in : the probabilities are normalized to unit sum for each , rather than to total unit sum and we are using the natural rather than the binary logarithm .this betrayal of the pure doctrine is of no consequence , however .fourth , irrespective of the unbiased way of estimating the , the ulin estimator has a statistical bias ; recall the pertinent remarks in the introduction .fifth , as we shall see , after measuring bases , we actually have some information about the unmeasured bases .we are not really faithful to laplace s advice as long as we do not account for this information properly .for we recover the estimator ( [ eq : bm-1 ] ) with , whereas is the full - tomography case of in eq . .the reliability of for these limiting values of is , however , misleading , and so is the self - suggesting probability estimation ( [ eq : bm-5 ] ) .as we shall see and this is one main objective of this work the ulin estimator is not assuredly physical for intermediate values , except in the qubit situation of .that is , is possible .the correct implementation of laplace s advice is not achieved by the simple - minded estimates in .sixth , there is no other linear map that could be used instead of if we require for consistency , as we must , that for the bases that have been measured .there are , of course , legitimate state estimators that are consistent with the known probabilities for bases the physical `` other '' estimators of eq .and these estimators make up a convex set ; correspondingly , there is a convex set of estimated probabilities for .the ulin estimator of eq .  and the unbiasedly estimated probabilities may or may not be in the respective sets . when they are outside , we have to choose the best state estimator in accordance with a suitable optimality criterion .more about this in sec .[ sec : least - bias ] .as a first illustration , let us take the elementary example of a single qubit ( ) .the three mub here are just the eigenstates of the standard pauli operators , , and , so that , , and where is the three - dimensional bloch vector with , e.g. ,the eigenvalues of are , and the von neumann entropy grows monotonously with shrinking bloch - vector length  .when a single basis is measured ( ) , say that of , only is known , and we have clearly , this physical estimator maximizes the von neumann entropy , since give the shortest bloch vector consistent with the known probabilities .likewise , the ulin estimator for two measured bases , which gives the estimated probabilities for the unmeasured , maximizes the von neumann entropy regardless of the values for the observed probabilities and  .these properties of the ulin estimators for a qubit are consequences of the simple spherical geometry of the bloch ball and the orthogonality of the planes corresponding to the distinct single - qubit mu pauli observables .such lines of argument were employed for justifying the optimality of mub tomography in higher dimensions  : when picturing the results obtained with a finite number of copies using `` fuzzy '' hyperplanes , their mutual orthogonality makes the uncertainty hypervolume particularly small .so , for a qubit , the unbiased unmeasured probabilities of the ulin estimator , as quantified by the shannon entropy , optimize ( maximize ) the von neumann entropy , much like the unbiased mu observables optimize ( minimize ) statistical uncertainty .these two properties complement each other .in fact , for any and , if is equal to a projector of one of the measured bases , then the ulin estimator is the state itself according to the defining relation . in view of all this evidence , one might expect this mutual compatibility to extend to higher dimensions and so arrive at the ulin estimator of eq . .however , we shall see that this somewhat naive approach fails for .   for the case of a qutrit, we use this set of four mub : where the columns are the probability amplitudes of the basis kets with reference to the fourth basis , and is the basic cubic root of unity .the matrices that represent the respective single - basis estimators of eq .are where we have for the probabilities in terms of the , and the unitary matrices for the pairwise complementary observables are we note that , and the replacement for an unmeasured basis is here simply implemented by .consider now a qutrit state for which the density operator is an incoherent mixture of two projectors , one each from the first and the second bases , with . as an immediate consequence of , we have here , , and .it follows that , so that all ulin estimators are genuine density operators for this particular  .matters are quite different for for which and ; all have eigenvalues .while and are density operators , as they are always , here and are not acceptable estimators ; they have negative determinants of and , respectively .the last example illustrates the statement in sec .[ sec:2c ] that `` the ulin estimator is not assuredly physical for intermediate values , '' here for .analogous examples can be constructed for higher dimensions ; see sec .[ sec : qudit ] . regarding the other assertion in sec .[ sec:2c] that there is no consistent `` linear map that could be used instead of ''we shall now argue that one gets a contradiction when assuming that there is such a map .for this purpose , we look specifically at the case of , but the argument can clearly be modified for other dimensions and other intermediate values .suppose we have measured the probabilities of bases , , and but lack data for basis . then we know the values of , , and , which determine all off - diagonal elements in , and we need a linear ( or affine ) map for the estimation of the diagonal matrix elements . as an immediate consequence of the linearity, we note that convex sums of values yield respective convex sums of values , for .further , for with , we have with and , and the dots stand in for the yet - unknown diagonal entries . in the convex sum, both matrices must have and on the diagonal , which implies the same values for this .we conclude that which , by convexity , also holds for .analogous arguments for and establish obviously , there is a contradiction for , as we can not have and and also for . case closed .the example of eq .is easily generalized to any dimension .the ulin estimator for , is unphysical for all because it has the negative eigenvalue .yet another example of unphysical ulin estimators for is provided by a projector to a superposition of two states from the first basis [ the of eqs .  andproject on superpositions of two states from the basis ] ; that is , where is the row of kets from the first basis , as in eq . ,and is the adjoint column of bras .note that we are using square parentheses to denote matrices expressed with the first basis , as opposed to the usual computational basis .  here , accounts for all the diagonal elements whereas , for , the nonzero matrix elements of here , the matrix to the basis is diagonal with the entry equal to , whereas the matrix to the first basis has null entries on the diagonal and some , if not all , off - diagonal entries are nonzero ( indicated by the symbol ) .then , for , the matrix in has hermitian submatrices \not\geq0 12 & 12#1212_12%12[1][0] and , eds . , _ _ , , vol .( , , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ , edited by ( , ) * * ,  ( ) link:\doibase  10.1007/s00453 - 002 - 0980 - 7 [ * * ,  ( ) ] in  _ _ , , vol . , ( , , ) pp .  * * ,  ( ) link:\doibase  10.1016/j.laa.2004.04.025 [ * * ,  ( ) ]\doibase  http://dx.doi.org/10.1142/s0219025704001797 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) in _ _ , , vol ., ( , ) pp . _ _ , ( , ) * * ,  ( ) _ _ , mathematics and its applications ( , , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) ( ) `` , '' ( , , ) chap . ,

Abstract: measuring incomplete sets of mutually unbiased bases constitutes a sensible approach to the tomography of high - dimensional quantum systems . the unbiased nature of these bases optimizes the uncertainty hypervolume . however , imposing unbiasedness on the probabilities for the unmeasured bases does not generally yield the estimator with the largest von neumann entropy , a popular figure of merit in this context . furthermore , this imposition typically leads to mock density matrices that are not even positive definite . this provides a strong argument against perfunctory applications of linear estimation strategies . we propose to use instead the physical state estimators that maximize the shannon entropy of the unmeasured outcomes , which quantifies our lack of knowledge fittingly and gives physically meaningful statistical predictions .


background reduction is becoming more and more important for all the experiments looking for rare events , such as dark matter interactions , neutrino - less double beta decay ( 0 ) , or rare decays . among the several technological approaches proposed for these searches , cryogenic calorimeters ( or bolometers )stand out for their excellent energy resolution ( of the order of 0.1 ) and for the possibility of probing different compounds ; the crystals operated as bolometers can be grown starting from most of the possible 0 candidates or rare- emitters , and from several interesting targets for dark matter search .bolometers can be equally sensitive to s , electrons and , if the threshold is low enough , also to nuclear recoils. this could be an advantage , as experiments designed for a specific purpose could be competitive also in other physics sectors . on the other hand , the lack of tools to identify the nature of the interacting particles prevents an efficient background suppression . a possible solution consists in equipping the bolometer with a light detector , that enables particle identification exploiting the different light emission of s , electrons and nuclear recoils .this technique has been successfully exploited for bolometers emitting scintillation light , like znse  , znmoo  , limoo  and many others .the scintillation crystals were coupled to light detectors made by thin germanium disks equipped with neutron transmutation doped germanium sensors , that show a typical intrinsic energy resolution of about 80ev rms  .this sensitivity is high enough to enable an efficient particle identification with scintillation bolometers , but may be not sufficient for next generation experiments. a possible application of new light detectors with these characteristics could be the upgrade of the cuore experiment .the teo bolometers used by this experiment do not scintillate . on the other hand , coupling a sensitive light detector to a teo would allow to measure the tiny amount of cherenkov light emitted by electrons ( about 100ev at 0 energy ) and not by s , enabling the rejection of the dominant background  .the cupid interest group  , that is defining the road - map for a 0 next - generation project based on bolometers , identified the features of the desired light detector : a noise rms resolution lower than 20ev , high radio - purity , large active area ( 5 ) and ease in fabricating / operating up to 1000 channels .the growing interest in developing sensitive light detectors ( that would be important also for other applications  ) gave birth to several r activities exploiting different technologies  .none of these technological approaches is ( yet ) able to fulfill all the requirements for next generation projects .for this reason , the calder ( cryogenic wide - area light detectors with excellent resolution ) project proposes a new technique , based on kinetic inductance detectors .kinetic inductance detectors ( kids ) are superconductors biased with an ac current  .the energy of the electric field applied to the superconductor is stored in the kinetic energy of cooper pairs .the oscillation of the field and , as a consequence , of the cooper pairs , produces an additional inductance in the superconductor , called kinetic inductance l . by coupling the superconductor with a capacitor, we can realize a rlc circuit , that acts like a resonator with resonant frequency .the properties of superconductors allow to fabricate resonators with typical quality factor ( q ) of the order of 10 - 10 .a photon interacting in the superconductor breaks cooper pairs , increasing the average momentum per cooper pair and , as a consequence , increasing l .moreover , the breaking of cooper pairs into quasiparticles increases the dissipation of the resonator , thus reducing q. monitoring the changes in the resonance parameters ( frequency and phase ) we can infer the energy that was deposited by the photon interaction .this is the working principle at the basis of kids that , besides an excellent energy resolution ( ) bring the advantage of being naturally multiplexed in the frequency domain ; this means that hundreds of kids can be coupled to a single read - out wire , decreasing the number of electronics channels , as well as the heat - load on the cryogenic system .the main limit of this technology is that kids can feature a maximum active area of a few mm , while next generation experiments demand for light detectors with surface of tens of . in principle, we could realize a large area light detector with an array of hundreds of kids .this is not a realistic option , because photons do not interact efficiently in the superconductor and , in addition , the operation of about 1000 light detectors would require a very large number of kids ( hundreds of thousands ) .  for this reason , we deposit kids on an insulating substrate that is used to absorb the interacting photon .the absorption of a photon produces phonons , that travel in the substrate until they are absorbed by the kids or lost in the substrate supports or defects .the goal of the calder project is proving that this approach , proposed by swenson et al . and moore et al . for other applications , allows to realize a light detector with all the characteristics required by next - generation experiments .the project foresees three main phases .the fist stage is devoted to the development of all the necessary acquisition and analysis tools and to the optimization of the detector geometry . for this phase we decided to work with a well known material for kids applications , aluminum , that will allow to reach an rms energy resolution of about 80ev . in the second stagewe will move to more sensitive superconductors , such as tin , ti+tin , or tial , in order to lower the energy resolution below 20ev .finally , we will couple the optimized light detectors to an array of teo bolometer to prove the potential of this technology .the aluminum prototypes , produced at consiglio nazionale delle ricerche istituto di fotonica e nanotecnologie ( cnr - ifn , rome , italy ) , are fabricated using direct - write electron beam lithography ( vistec ebpg 5000 ) .the resonators are deposited on high resistivity ( cm ) si(100 ) substrates . for the first detectors we used 275 m thick substrates , but we will investigate soon other thicknesses to better understand the efficiency in photons absorption and phonons propagation .the aluminum layer is evaporated in an electron - gun evaporator that allows to deposit the desired superconductor thickness ( 25 or 40 nm for the detectors described later ) .after the lift - off , we cut the si wafer into a 2 chip , that is assembled in a copper structure using insulating ptfe elements , as shown in figure [ fig : photo ] .  a single al kid deposited on a 2 si substrate .the chip was assembled in a copper structure using four ptfe elements . ]the detectors are cooled below the critical temperature using a / dilution refrigerator with base temperature of about 10mk .the output signal is fed into a citlf4 sige low noise amplifier  , which is thermally anchored to the 4k plate of the cryostat .the rest of the electronics is located at room temperature and its features , together with the description of the cryogenic facility and the acquisition software , can be found in references  .we developed several analysis tools to fit the complex transmission function s in order to extract useful parameters , like the resonance frequency , the quality factor q , the internal and coupling quality factors q and q ( q=q+q ) . taking advantage from the analysis done by swenson et al . and khalil et al . , we developed a model of the resonator that allows to extract the parameters of interest even in presence of large distortions produced , for example , by impedance mismatches between the feed - line and the chip  .the resonance fit allows to determine also the center and radius of the resonance loop .when an interactions occurs , the phase and amplitude of the resonance vary with respect to this reference frame , producing two readable signals .since the phase signal has a much better signal to noise ratio , we use this estimator to reconstruct the energy spectrum of the detector . according to our signal model ,an energy release e produces a phase variation : the first term in this formula is mainly material dependent : is the fraction of kinetic inductance ( of the order of 415 for al ) , is a slow function of the temperature ( about 2.32.6 for our detectors ) , is the single spin density of states ( for al 1.72 ) , and 2 is the binding energy of cooper pairs ( about 200 for thin al films ) .the second term , containing the resonator and active volume , depends mostly on the geometry of the device and can be easily varied up to an order of magnitude . finally , is the detection efficiency .   to measure the efficiency of our device, we performed several tests using a 2 , 275 m thick silicon substrate .the substrate was illuminated with an optical source ( 400 nm room - temperature led coupled to an optical fiber ) producing pulses up to , and with / x - rays sources with peaks between 5.9 and 14.4kev , in order to cross - check the results .the calibration of the optical apparatus , indeed , is made with a photomultiplier at room temperature and is corrected using a monte carlo that accounts for the geometry of the set - up and for the optical properties of si . on the contrary , the nominal energy of x - rays is known with high precision , allowing to investigate possible systematic errors in the calibration of optical pulses .all the sources were always placed on the back of the substrate to prevent direct illumination of the resonators .since we expect the efficiency to scale with the detector volume , we tested different kids geometries : ( 1 ) a 25 nm thick al kid with active area of 2.4mm , ( 2 ) a 40 nm thick kid with the same active area , and ( 3 ) a 40 nm thick kid with an increased active area of 4.0mm .the results , reported in table [ tab : efficiency ] , were obtained by illuminating the substrate in the proximity of the resonator , or as far as possible from it ..[tab : efficiency]number of kids deposited on the 2 si substrate ; thickness ( t ) and area ( a ) of the inductor ; total detection efficiency measured with optical pulses illuminating the substrate region close to the kid ( ) and far from the kid ( ) .[ cols="<,^,^,^ " , ]   we have already realized a single pixel detector with geometry ( 3 ) by depositing ti+al ( 10 nm of ti and 25 nm of al ) on a 2 si substrate . the first prototype , that was produced in collaboration with csnsm ( orsay , france ) and institut nel , cnrs ( grenoble , france ) , is currently being tested and shows very encouraging results .in this paper we presented a novel technology for the development of sensitive light detectors , based on phonon - mediated kids .we presented the results obtained depositing prototypes of al kids with different geometries on 2 si substrates and we derived the detection efficiency .we evaluated the noise of the detectors and showed that the optimization of the design allowed to reach a baseline noise of about 90ev rms .finally , we discussed the perspectives of this technology , explaining how more sensitive superconductors will allow to reach the target sensitivity .this work was supported by the european research council ( fp7/2007 - 2013 ) under contract calder no .335359 and by the italian ministry of research under the firb contract no .rbfr1269sl .j. w. beeman , et al ., performances of a large mass znmoo(4 ) scintillating bolometer for a next generation 0vdbd experiment , eur . phys .j. c 72 ( 2012 ) 2142 . http://arxiv.org/abs/1207.0433 [ ] , http://dx.doi.org/10.1140/epjc/s10052-012-2142-7 [ ] .l. cardani , et al . , development of a li2moo4 scintillating bolometer for low background physics , jinst 8 ( 2013 ) p10002 .http://arxiv.org/abs/1307.0134 [ ] , http://dx.doi.org/10.1088/1748-0221/8/10/p10002 [ ] .j. w. beeman , et al ., characterization of bolometric light detectors for rare event searches , jinst 8 ( 2013 ) p07021 .http://arxiv.org/abs/1304.6289 [ ] , http://dx.doi.org/10.1088/1748-0221/8/07/p07021 [ ] .f. bellini , et al . ,measurements and optimization of the light yield of a teo crystal , jinst 9 ( 10 ) ( 2014 ) p10014 .http://arxiv.org/abs/1406.0713 [ ] , http://dx.doi.org/10.1088/1748-0221/9/10/p10014 [ ] .n. casali , et al ., teo bolometers with cherenkov signal tagging : towards next - generation neutrinoless double beta decay experiments , eur .j. c 75 ( 1 ) ( 2015 ) 12 .http://arxiv.org/abs/1403.5528 [ ] , http://dx.doi.org/10.1140/epjc/s10052-014-3225-4 [ ] .k. schaeffner , et al . ,particle discrimination in bolometers using light detectors read out by transition edge sensors , astropart .69 ( 2015 ) 3036 .[ ] , http://dx.doi.org/10.1016/j.astropartphys.2015.03.008 [ ] .m. willers , et al ., neganov - luke amplified cryogenic light detectors for the background discrimination in neutrinoless double beta decay search with teo bolometers , jinst 10 ( 03 ) ( 2015 ) p03003 .[ ] , http://dx.doi.org/10.1088/1748-0221/10/03/p03003 [ ] .m. biassoni , et al . ,large area si low - temperature light detectors with neganov luke effect , eur .j. c 75 ( 10 ) ( 2015 ) 480 .http://arxiv.org/abs/1507.08787 [ ] , http://dx.doi.org/10.1140/epjc/s10052-015-3712-2 [ ] .e. battistelli , f. bellini , c. bucci , m. calvo , l. cardani , n. casali , m. g. castellano , i. colantoni , a. coppolecchia , c. cosmelli , et al ., calder : neutrinoless double - beta decay identification in teo bolometers with kinetic inductance detectors , eur .j. c 75 ( 8) . http://arxiv.org/abs/1505.01318 [ ] .p. k. day , h. g. leduc , b. a. mazin , a. vayonakis , j. zmuidzinas , a broadband superconducting detector suitable for use in large arrays , nature 425 ( 6960 ) ( 2003 ) 817821 .l. j. swenson , a. cruciani , a. benoit , m. roesch , c. s. yung , a. bideaud , a. monfardini , high - speed phonon imaging using frequency - multiplexed kinetic inductance detectors , appl.phys.lett .96 ( 2010 ) 263511 . http://arxiv.org/abs/1004.5066 [ ] , http://dx.doi.org/10.1063/1.3459142 [ ] .d. c. moore , s. r. golwala , b. bumble , b. cornell , p. k. day , h. g. leduc , j. zmuidzinas , position and energy - resolved particle detection using phonon - mediated microwave kinetic inductance detectors , appl.phys.lett .100 ( 2012 ) 232601. http://arxiv.org/abs/1203.4549 [ ] , http://dx.doi.org/10.1063/1.4726279 [ ] .o. bourrion , a. bideaud , a. benoit , a. cruciani , j. f. macias - 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Abstract: in this paper we describe the current status of the calder project , which is developing ultra - sensitive light detectors based on superconductors for cryogenic applications . when we apply an ac current to a superconductor , the cooper pairs oscillate and acquire kinetic inductance , that can be measured by inserting the superconductor in a lc circuit with high merit factor . interactions in the superconductor can break the cooper pairs , causing sizable variations in the kinetic inductance and , thus , in the response of the lc circuit . the continuous monitoring of the amplitude and frequency modulation allows to reconstruct the incident energy with excellent sensitivity . this concept is at the basis of kinetic inductance detectors ( kids ) , that are characterized by natural aptitude to multiplexed read - out ( several sensors can be tuned to different resonant frequencies and coupled to the same line ) , resolution of few ev , stable behavior over a wide temperature range , and ease in fabrication . we present the results obtained by the calder collaboration with 2 substrates sampled by 1 or 4 aluminum kids . we show that the performances of the first prototypes are already competitive with those of other commonly used light detectors , and we discuss the strategies for a further improvement .   kinetic inductance detectors , light detector , background reduction , dark matter , neutrinoless double beta decay


we shall describe a series of numerical studies of the nature of singularities in cosmological models . since the interiors of black holes can be described locally as cosmological models , it is possible that our methods and results may be useful to the participants in this conference .the generic singularity in spatially homogeneous cosmologies is reasonably well understood .the approach to it asymptotically falls into two classes .the first , called asymptotically velocity term dominated ( avtd ) , refers to a cosmology that approaches the kasner ( vacuum , bianchi i ) solution as .( spatially homogeneous universes can be described as a sequence of homogeneous spaces labeled by .here we shall choose so that coincides with the singularity . )an example of such a solution is the vacuum bianchi ii model which begins with a fixed set of kasner - like anisotropic expansion rates , and , possibly , makes one change of the rates in a prescribed way ( mixmaster - like bounce ) and then continues to as a fixed kasner solution .in contrast are the homogeneous cosmologies which display mixmaster dynamics such as vacuum bianchi viii and ix and bianchi vi and bianchi i with a magnetic field .jantzen has discussed other examples .mixmaster dynamics describes an approach to the singularity which is a sequence of kasner epochs with a prescription , originally due to belinskii , khalatnikov , and lifshitz ( bkl ) , for relating one kasner epoch to the next .some of the mixmaster bounces ( era changes ) display sensitivity to initial conditions one usually associates with chaos and in fact mixmaster dynamics is chaotic .the vacuum bianchi i ( kasner ) solution is distinguished from the other bianchi types in that the spatial scalar curvature , ( proporional to ) the minisuperspace ( mss ) potential , vanishes identically . but arises in other bianchi types due to spatial dependence of the metric in a coordinate basis .thus an avtd singularity is also characterized as a regime in which terms containing or arising from spatial derivatives no longer influence the dynamics .this means that the mixmaster models do not have an avtd singularity since the influence of the spatial derivatives ( through the mss potential ) never disappears there is no last bounce .  in the late 1960 s ,bkl claimed to show that singularities in generic solutions to einstein s equations are locally of the mixmaster type .this means that each point of a spatially inhomogeneous universe could collapse to the singularity as the mixmaster sequence of kasner models .( it has been argued that this could generate a fractal spatial structure . ) in contrast , each point of a cosmology with an avtd singularity evolves asymptotically as a fixed kasner model .although the bkl result is controversial , it provides a hypothesis for testing .our ultimate objective is to test the bkl conjecture numerically .the work reported here was performed by using symplectic ode and pde solvers .while other numerical methods may be used to solve einstein s equations for the models discussed here , symplectic methods have proved extremely advantageous for mixmaster models and have also worked quite well in the gowdy plane symmetric and polarized symmetric cosmologies .consider a system with one degree of freedom described by and its canonically conjugate momentum with a hamiltonian note that the subhamiltonians and separately yield equations of motion which are exactly solvable no matter the form of .variation of yields , with solution variation of yields , with solution note that the absence of momenta in makes ( [ ber - hvsoln ] ) exact for any .one can then demonstrate that to evolve from to an evolution operator can be constructed from the evolution sub - operators and obtained from ( [ ber - hksoln ] ) and ( [ ber - hvsoln ] ) .one can show that  reproduces the true evolution operator through order .suzuki has developed a prescription to represent the full evolution operator to arbitrary order .for example \,{\cal u}_{(2)}(s\delta t)\ ] ] where . the advantage of suzuki s approach is that one only needs to construct explicitly . is then constructed from appropriate combinations of .the generalization of this method to degrees of freedom and to fields is straightforward . in the latter case , \to v[\vec q(\vec x , t)] ] in the gowdy models fail in generic models .this suggests that will always grow exponentially if the system tries to be avtd producing a mixmaster - like bounce .a bounce in the opposite direction will come , as in gowdy models , from terms in .this bouncing could continue indefinitely .unfortunately , the bounces off probably cause numerical instabilities that limit the duration of the simulations .spatial averaging has been used to improve stability but it is known to produce numerical artifacts .typical evolutions of at single spatial grid points are shown in fig .[ ber - fig8 ] and compared to similar quantities in mixmaster and gowdy models and , of course , to fig .[ ber - fig6 ] .while generic models are clearly different from polarized ones , it is not clear yet whether the observed bounces are mixmaster - like or will eventually die out as in gowdy models . recall that mixmaster universes are very close to the avtd kasner solution between bounces .one is also concerned about numerical artifacts although it is not clear that standard tests are helpful .[ ber - fig9 ] shows two frames from a movie of vs for two different spatial resolutions .features are narrower on the finer grid .while this usually indicates artifacts , we recall that this is precisely what happens in gowdy models where the resolution dependence is well understood .in the search for the nature of the generic cosmological singularity , we have obtained convincing evidence that both the gowdy universes and the polarized symmetric cosmologies have avtd singularities .these results are found in simulations which present no numerical difficulties .in contrast , we can not draw definite conclusions for generic models except to say that we have not found evidence that the singularities are avtd everywhere .  presumably , numerical difficulties in generic models are due to gowdy - like spiky features ( which are less easy to represent accurately in two spatial dimensions ) .it is possible that the new mixmaster algorithm which can be adapted to the gowdy model will help in the generic case where it can also be implemented .the hope is that better treatment of the bounces would give better local treatment of spiky features that arise from them .while we solve the constraints initially in models , we do nothing to preserve them thereafter . in the polarized case, they remain acceptably small ( and in fact converge to zero with increasing spatial resolution ) .we have learned from the mixmaster case that one must preserve the constraints .in fact , it is the kinetic part of the hamiltonian constraint which restricts the exponential factor in .an error in the constraints could give the argument of the exponential the wrong sign leading to the observation of qualitatively wrong behavior . by starting closer to the singularity , we can supress some of the numerical instabilities and study this controlling exponential .studies of this type have provided some evidence that it is essential to solve the constraints .work on implementing a constraint solver is in progress .b.k.b . and v.m .would like to thank the albert einstein institute at potsdam for hospitality .b.k.b . would also like to thank the institute of geophysics and planetary physics of lawrence livermore national laboratory for hospitality .this work was supported in part by national science foundation grants phy9507313 and phy9722039 to oakland university and phy9503133 to yale university .computations were performed at the national center for supercomputing applications ( university of illinois ) .berger , d. garfinkle , b. grubisic , v. moncrief , v. swamy , in _ proceedings of the cornelius lanczos symposium _ , edited by j.d .brown , m.t .chu , d.c .ellison , r.j .plemmons ( siam , philadelphia , 1994 ) .  b.k .berger , d. garfinkle , v. moncrief , c.m .swift , in _ proceedings of the ams - cms special session on geometric methods in mathematical physics _, ed . by j. beem and k. duggal ( americal mathematical society , 1994 ) .

Abstract: the spatially homogeneous , isotropic standard cosmological model appears to describe our universe reasonably well . however , einstein s equations allow a much larger class of cosmological solutions . theorems originally due to penrose and hawking predict that all such models ( assuming reasonable matter properties ) will have an initial singularity . the nature of this singularity in generic cosmologies remains a major open question in general relativity . spatially homogeneous but possibly anisotropic cosmologies have two types of singularities : ( 1 ) velocity dominated(reversing the time direction ) the universe evolves to the singularity with fixed anisotropic collapse rates ; ( 2 ) mixmaster the anisotropic collapse rates change in a deterministically chaotic way . much less is known about spatially inhomogeneous universes . belinskii , khalatnikov , and lifshitz ( bkl ) claimed long ago that a generic universe would evolve toward the singularity as a different mixmaster universe at each spatial point . we shall report on the results of a program to test the bkl conjecture numerically . results include a new algorithm to evolve homogeneous mixmaster models , demonstration of velocity dominance and understanding of evolution toward velocity dominance in the plane symmetric gowdy universes ( spatial dependence in one direction ) , demonstration of velocity dominance in polarized u(1 ) symmetric cosmologies ( spatial dependence in two directions ) , and exploration of departures from velocity dominance in generic u(1 ) universes .


suppose we are given a set of points , and some non - negative weights to these points .consider the problem where is the euclidean norm and is closed and convex .this is a convex optimization problem .the ( global ) minimizer of when is the entire space is called the _euclidean median _( also referred to as the geometric median ) .the euclidean median has classically been used as a robust estimator of centrality for multivariate data .this multivariate generalization of the scalar median also comes up in transport engineering , where is used to model the problem of locating a facility that minimizes the cost of transportation .more recently , it was demonstrated in that the euclidean median and its non - convex extensions can be used to perform robust patch - based regression for image denoising by generalizing the poplar non - local means algorithm .note that the minimizer of is simply the projection of the euclidean median onto .unfortunately , there is no simple closed - form expression for the euclidean median , even when it is unique .nevertheless , the euclidean median can be computed using numerical methods such as the iteratively reweighted least squares ( irls ) .one such iterative method is the so - called weiszfeld algorithm .this particular algorithm , however , is known to be prone to convergence problems .geometric optimization - based algorithms have also be proposed for approximating the euclidean median .more recently , motivated by the use of irls for sparse signal processing and minimization , an irls - based algorithm for approximating the euclidean median was proposed in .more precisely , here the authors consider a smooth convex surrogate of , namely , and describe an iterative method for optimizing .the convergence properties of this iterative algorithm was later studied in . in the last few years , there has been a lot of renewed interest in the use of variable splitting , the method of multipliers , and the alternating direction method of multipliers ( admm ) for various non - smooth optimization problems in signal processing .motivated by this line of work , we introduce a novel algorithm for approximating the solution of using variable splitting and the augmented lagrangian in section [ sec : admm ] .an important technical distinction of the present approach is that , instead of working with the smooth surrogate , we directly address the original non - smooth objective in .the attractive feature here is that the subproblems involved in the admm - based optimization of the augmented lagrangian can be resolved using simple closed - form projections . in section[ sec : results ] , the proposed algorithm is used for robust patch - based denoising of images following the proposal in . in particular , we incorporate the information that the pixels of the original ( clean ) patches take values in a given dynamic range ( for example , in [ 0,255 ] for grayscale images ) using an appropriately defined .numerical experiments show that the iterates of the proposed algorithm converge much more rapidly than the irls iterates for the denoising method in question. moreover , the reconstruction quality of the denoised image obtained after few admm iterations is often substantially better than the irls counterpart .since we will primarily work with in this paper , we will overload terminology and refer to the minimizer of as the euclidean median . the main idea behindthe admm algorithm is to decompose the objective in into a sum of independent objectives , which can then be optimized separately .this is precisely achieved using variable splitting . in particular , note that we can equivalently formulate as where and is the indicator function of  . in other words ,we artificially introduce one local variable for each term of the objective , and a common global variable that forces the local variables to be equal .the advantage of this decomposition will be evident shortly .note that in the process of decomposing the objective , we have introduced additional constraints . while one could potentially use any of the existing constrained optimization methods to address, we will adopt the method of augmented lagrangian which ( among other things ) is particularly tailored to the separable structure of the objective .the augmented lagrangian for can be written as where are the lagrange multipliers corresponding to the equality constraints in , and is a penalty parameter .we next use the machinery of generalized admm to optimize jointly with respect to the variables and the lagrange multipliers .the generalized admm algorithm proceeds as follows : in each iteration , we sequentially minimize with respect to the variables in a gauss - seidel fashion ( keeping the remaining variables fixed ) , and then we update the lagrange multipliers using a fixed dual - ascent rule . in particular , note that the minimization over at the -th iteration is given by where denotes the variable at the end of the -th iteration .a similar notation is used to denote the lagrange multipliers at the end of the -th iteration .next , by combining the quadratic and linear terms and discarding terms that do not depend on , we can express the partial minimization over as where therefore , where denotes the projection of onto the convex set .note that we have used the most recent variables in .finally , we update the lagrange multipliers using the following standard rule :   we loop over , and until some convergence criteria is met , or up to a fixed number of iterations . for a detailed exposition on the augmented lagrangian , the method of multipliers , and the admm , we refer the interested reader to .we now focus on the update in . by combining the linear and quadratic terms ( and discarding terms that do not depend on ), we can express in the following general form : where note that the objective in is closed , proper , and strongly convex , and hence exists and is unique .in fact , we immediately recognize to be the _proximal map _ of evaluated at  .for any  ] be closed , proper , and convex . then we can decompose any as where  ] ; e.g. , and for a grayscale image .the projection onto can be computed coordinate - wise at negligible cost . in particular , for , = \begin{cases } \x[i ] & \text{if } \l \leq \x[i ]  \leq u , \\l    & \text{if } \\x[i ]  < l , \text { and } \\ u    & \text{if } \\x[i ]  > u. \end{cases}\ ] ]  notice that is exactly the optimization in .in other words , we denoise each patch by taking the euclidean median of its neighbouring noisy patches . the center pixel of the denoised patch is taken to be the denoised pixel .the overall denoising scheme is called the non - local euclidean medians ( in short , nlem ) .l c rrrrr  image & method & +  & ( a ) & * 34.22 & 29.78 & 25.20 & 23.37 & 22.35 + _ house _ & ( b ) & 32.66 & 29.01 & 26.68 & 24.78 & 23.46 + & ( c ) & 34.13 & * 30.77 & * 27.04 & * 24.87 & * 23.47 + * * * * *  & ( a ) & 33.24 & 29.31 & 26.17 & 24.54 & 23.64 + _ lena _ & ( b ) & 32.54 & 29.48 & 27.31 & 25.38 & 24.37 + & ( c ) & * 33.57 & * 30.38 & * 27.59 & * 25.42 & * 24.38 + * * * * *  & ( a ) & * 32.32 & 27.66 & 23.11 & 21.03 & 19.99 + _ peppers _ & ( b ) & 30.95 & 26.95 & 24.31 & 22.84 & 21.24 + & ( c ) & 32.14 & * 28.54 & * 25.06 & * 22.98 & * 21.26 + * * * * *  & ( a ) & 32.37 & 27.39 & 23.53 & 22.05 & 21.34+ _ barbara _ & ( b ) & 30.77 & 27.28 & 25.55 & 23.77 & 22.61 + & ( c ) & * 32.43 & * 28.84 & * 25.67 & * 23.77 & * 22.62 + * * * * *  [ tablepsnr ]  the denoising experiments were performed on several grayscale test images .we used the standard nlm parameters for all the experiments : , and .the proposed ` em - admm ` algorithm is used for computing the solution of . notice that the main computation here is that of determining the distances in .since we require a total of such distance evaluations in dimensions , the complexity is per iteration per pixel .incidentally , this is also the complexity of the irls algorithm in .for all the denoising experiments , we initialized the lagrange multipliers in ` em - admm ` to zero vectors .we considered two possible initializations for , namely , the noisy patch and the patch obtained using nlm .exhaustive experiments showed that the best convergence results are obtained for .for example , figure [ psnrevolution ] shows the evolution of the peak - signal - to - noise ratio ( psnr ) over the first ten iterations obtained using the admm algorithm . also show in this figure is the evolution of the psnr for the irls algorithm from . in either case, we used the noisy patch as the initialization .notice that the increase in psnr is much more rapid with admm compared to irls .in fact , in about iterations , the optimal psnr is attained with admm .notice that at , the psnr from irls grows much more slowly compared to admm .  .the inset shows the position of the particular pixel where the patch regression is performed .the objective functions shown in the plot correspond to and that are respectively optimized by admm and irls ( note that the objectives are very close since we use in ) .the noisy image patch was used to initialize both algorithms . for ` em - admm ` , the objective function converges to the optimal value ( up to to three decimal places ) in just iterations .the convergence is relatively slow for irls ( accuracy up to three decimals obtained after iterations ) . ]the above observation is also supported by the convergence result shown in figure [ comparisonobj ] for the _ house _ image .notice that admm converges in just iterations , while irls takes about iterations to attain the same accuracy ( we recall that the cost per iteration is comparable ) .in general , we observed that the within iterations the admm result converges to an accuracy that is sufficient for the present denoising application . at higher noise levels ( ) , about iterations are required .this applies to all the test images that we have experimented with .the denoising results for some test images obtained using nlm and nlem are provided in table [ tablepsnr ] . for nlem, we used both the admm and irls solvers . following the previous observations , we used iterations for both solvers . for , we used the noisy patch to initialize the iterations , and for we used the nlm patch .this initialization scheme was found to give the best psnr results . at large noise levels, there is not much difference after iterations .however , at low noise levels , the psnr obtained using admm is often substantially larger than that obtained using irls after iterations .the reason for this is that irls converges really slow in such situations .this is evident from the psnr plots in figure [ psnrevolution ] at .for a visual comparison , a particular denoising result for the _ barbara _ image obtained using nlm and nlem ( using the admm solver ) is shown in figure [ comparison1 ] .roughly speaking , at the cost of just four nlms , we are able to improve the psnr by more than db . notice that the nlem result is much more sharp compared to the nlm counterpart .we proposed a new algorithm for computing the ( constrained ) euclidean median using variable splitting and the augmented lagrangian .in particular , we demonstrated how the admm - based optimization of the augmented lagrangian can be resolved using simple closed - form projections .the proposed algorithm was used for image denoising using the non - local euclidean medians , and was generally found to exhibit faster convergence compared to the irls algorithm .one interesting direction that we did not pursue is to adapt the penalty at each iteration , which is known to speed up the convergence .yet another interesting direction is the use of accelerated admm to further speed up the convergence .k. n. chaudhury and a. singer , `` non - local patch regression : robust image denoising in patch space , '' _ proc .ieee international conference on acoustics , speech and signal processing _ ,pp.1345 - 1349 , 2013 .i. daubechies , r. devore , m. fornasier , and c. s. gunturk `` iteratively reweighted least squares minimization for sparse recovery , '' _ communications on pure and applied mathematics _ ,63 , pp . 1 - 38 , 2009 .j. eckstein and d. p. bertsekas , `` on the douglas - rachford splitting method and the proximal point algorithm for maximal monotone operators , '' _ mathematical programming _ ,55 , no . 1 - 3 , pp .293 - 318 , 1992 .m. v. afonso , j. m. bioucas - dias , and m. a. figueiredo , `` fast image recovery using variable splitting and constrained optimization , '' _ ieee transactions on image processing _ , vol .9 , pp . 2345 - 2356 , 2010 .m. v. afonso , j. m. bioucas - dias , and m. a. figueiredo , `` an augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems , '' _ ieee transactions on image processing _ , vol .681 - 695 , 2011 .s. boyd , n. parikh , e. chu , b. peleato , and j. eckstein , `` distributed optimization and statistical learning via the alternating direction method of multipliers , '' _ foundations and trends in machine learning _ ,1 , pp . 1 - 122 , 2011 .m. aharon , m. elad , a. bruckstein , `` k - svd : an algorithm for designing overcomplete dictionaries for sparse representation , '' _ ieee transactions on image processing _ , vol. 54 , no .4311 - 4322 , 2006 .

Abstract: the euclidean median ( em ) of a set of points in an euclidean space is the point minimizing the ( weighted ) sum of the euclidean distances of to the points in . while there exits no closed - form expression for the em , it can nevertheless be computed using iterative methods such as the weiszfeld algorithm . the em has classically been used as a robust estimator of centrality for multivariate data . it was recently demonstrated that the em can be used to perform robust patch - based denoising of images by generalizing the popular non - local means algorithm . in this paper , we propose a novel algorithm for computing the em ( and its box - constrained counterpart ) using variable splitting and the method of augmented lagrangian . the attractive feature of this approach is that the subproblems involved in the admm - based optimization of the augmented lagrangian can be resolved using simple closed - form projections . the proposed admm solver is used for robust patch - based image denoising and is shown to exhibit faster convergence compared to an existing solver .   image denoising , patch - based algorithm , robustness , euclidean median , variable splitting , augmented lagrangian , alternating direction method of multipliers ( admm ) , convergence .