quantum coherence , as a prominent resource for quantum information processing , has found its various diversified applications in quantum cryptography , quantum computation and quantum metrology .nevertheless , until very recently , there still lacks a rigorous information - theoretic framework for characterizing coherence and a great deal of effort has been devoted to this significant and long - standing topic . exploiting the quantum resource theory , baumgratz _ et al . _proposed a framework for quantifying coherence based on distance or pseudo - distance measures , by noting that a similar line of thought has been successfully applied in the theory of quantum entanglement . within this framework ,several reasonable postulates has been proposed which should be satisfied by all _bona fide _ measures of quantum coherence .moreover , as a prerequisite , a _fiducial basis _ should be pre - selected for determining the exact value of quantum coherence , according to the specific theoretic considerations or physical implementations . in other words , the coherence measures defined in ref . are all _ basis - dependent _ quantities . for a simple example , the eigenvectors of pauli matrices and constitute two mutually unbiased bases , that is , each incoherent basis pure state is a maximally coherent state with respect to the other basis .therefore , a natural question is raised : what are the maximum and minimum coherences contained in a given quantum state with respect to the random bases ?notably , this problem is not only theoretically motivated but also experimentally relevant .since quantum coherence has been identified as the essential resources for certain quantum information tasks , it will be preferred to extract the coherence content of a given state as much as possible . obviously , the key issue is to find out a reference basis with respect to which the coherence value is maximal .it is noteworthy that a similar but distinct problem has been discussed by singh _ , where a notion of maximally coherent mixed states ( mcmss ) was considered and in fact they obtained the maximally achievable quantum coherence for a fixed mixedness in the computational basis .however , while in ref . the purity is the only independent variable , here we expect that the maximally achievable value of quantum coherence may depend on the entropy or eigenvalues of the given state .  in this work, we mainly focus on the coherence measures of relative entropy and norm , which are the only monotones that are found to satisfy all criterions proposed in until now .while any pure state can always be transformed to a maximally coherent state by a change of basis , we realize that for general mixed states the situation becomes much more complicated and subtle .although it seems counter - intuitive , we found that the condition for achieving maximum values of coherence is not universal and measure - dependent .the paper is organized as follows . in sec .[ sec2 ] , we briefly review the resource framework of quantum coherence and define the problem in standard notations . in sec .[ sec3 ] , we identify the basis in which the relative entropy of coherence of a given state is maximal and the significance of the fourier matrix ( complex hadamard matrices ) is illustrated . in sec .[ sec4 ] , we prove that this basis is ( at least ) suboptimal for optimizing the norm of coherence and the properties of the circulant matrix is emphasized . in sec .[ sec5 ] , we perform a detailed numerical simulation and demonstrate that the basis associated with the fourier matrix is not global optimal . the sec .[ sec6 ] is devoted to the conclusion and discussion of the main results and several open questions are presented for future investigation .throughout the paper , we adopt the resource theory of quantum coherence proposed by baumgratz __ in .this theoretical framework has been successfully applied for quantifying quantum entanglement . except for some basic notions , such as the free states and free operations defined in their own context, the quantum resource theories mainly rely on two ingredients : ( i ) a series of reasonable postulates that should be fulfilled by each measure or indicator of genuine resource ; ( ii ) a set of contractive geometric metrics . in the corresponding resource theory of coherence , the free ( incoherent ) states are those diagonal in a pre - selected basis in the -dimensional hilbert space , denoted by the set .accordingly , the the free ( incoherent ) operations are completely positive and trace preserving quantum maps admitting an operator - sum representation where every kraus operator will transform the set of incoherent states into itself , that is , .  except for the nullity condition and convexity requirement , each _ bona fide _ measures of quantum coherence are assumed to be a monotone function under the non - selective and sub - selective incoherent measurements , namely  where and . note that the latter constraint combined with convexity condition will lead to the former , which implies the latter is a stronger monotonicity requirement .based on these criteria , several potential candidates were put forward for the quantification of coherence .however , so far only two measures have been identified to fulfill all the requirements , that is , the relative entropy of coherence and the norm of coherence  where is the diagonal part of .it is worth pointing out that the coherence measures induced by norm and the fidelity do not constitute valid coherence monotones , since for both quantities the strong monotonicity ( [ sm ] ) do not hold in general .moreover , though the trace - norm measure of coherence was proved to be a strong monotone for all qubit and states , a recent work showed that the trace norm of coherence can not be regarded as a legitimate coherence measure for general states .in such a framework , a pre - determined _ fiducial basis _ is prior to any evaluation of the value of coherence .recall that a specific orthonormal basis corresponds to a particular matrix representation of a given density matrix ( the maximally mixed state is an exception since it is always diagonal in any bases ) . from the definitions of and , it is easy to see that any density matrix with off - diagonal entries in such a representation will be identified as a resource state ( having non - zero coherence value ) with respect to this fiducial basis . furthermore , it is noteworthy that any two distinct orthonormal bases are connected with a unitary operator and this remarkable fact indicates that with reference to the computational basis , any random basis can be fully characterized by a unitary operator drawn randomly from the set of unitary matrices , with a _one - to - one correspondence _ .therefore , for a given density matrix , the evaluation of quantum coherence in a random basis is equivalent to considering the coherence of in the the computational basis , that is   therefore , the problem of determining in which basis the matrix representation will display the maximum or minimum coherence is tantamount to finding out the corresponding unitary transformations .in fact , according to the spectral theorem , any normal operator is diagonalizable and thus , for a given density matrix , the minimum coherence is always zero in which case the eigenvectors of form the columns of this particular .however , on the other hand , it is not immediately intuitive that in which basis we can acquire the maximum coherence .now let us consider a fixed density matrix in the -dimensional hilbert space . in this section we will mainly concentrate on , since the entropy function solely depends on the eigenvalues of its argument and is usually easier to handle. to begin with , two observations caught our attention .first , regardless of the fiducial basis , is universally upper bounded by in fact , this inequality stems from the majorization relation , also known as the schur - horn theorem .note that though itself always remain unchanged with respect to the basis change , the diagonal part will definitely depend on different matrix representations .the tightness of this upper bound is equivalent to whether there exists a specific basis in which the matrix representation of has equal main diagonal elements .the second observation is recapitulated in the following lemma , which was proved by horn and johnson using two different approaches .denote the set of -dimensional square matrices by .then for each , there exists a unitary matrix such that all the diagonal entries of have the same value .this lemma is a rather general result and implies that can always achieve the upper bound of eq .( [ ub ] ) through the change of basis. however , the proves in ref . do not indicate the explicit form of this particular .the following theorem spells out the exact form of this type of and thus the corresponding basis .there exists a set of unitary operators such that achieves the maximum value for a given density matrix , where the reference basis is defined by .the unitary transformation has the form , where consists of the eigenvectors of as its column vectors and belongs to the set of ( rescaled ) complex hadamard matrices .[ t1 ]  _ proof ._ from eq .( [ er ] ) , the evaluation of coherence value of in the transformed basis is equivalent to that of in the computational basis . due to the spectral decomposition of , without any loss of generality , we assume by proper arrangement of the order of the eigenvectors in . then we have let us denote the elements of the matrix as {ij}=a_{ij} ] .note that the value of is upper bounded by  .thus , this result implies that every pure state in finite dimensions can be represented as a maximally coherent state through the change of basis and this optimal basis is the one defined by the fourier matrix ( in fact by all complex hadamard matrices ) .however , the optimization problem starts to become complicated even for general mixed qutrit states .in fact , a complete parameterization of the space of unitary matrices includes independent real parameters ( e.g. , euler angles ) .therefore , it seems clueless if we perform a brute - force optimization process over nine independent real variables .to gain more insight , we first observe the transformed state concerning this _ suspected _ optimal basis : where and .thus , for general qutrit states , is given by where we have used the relation .in addition to the expected symmetry of this expression , when and , coincides with the above pure state case .moreover , if ( that is , is the maximally mixed state ) , since the maximally mixed state is the only state that is diagonal in any basis .so far , we may be convinced that the basis associated with the fourier matrix is optimal for the norm of coherence .in fact , the lagrange multiplier method will give us further evidence .in essence , the issue discussed in this work is an optimization problem .more precisely , we pursue the maximum value of eq .( [ l1 ] ) subject to the unitarity property .therefore , we introduce the lagrange function where are the lagrange multipliers and to simplify the notation , we define note that is the matrix element of the given density matrix with respect to the basis and thus the symmetry holds .the ( local ) extreme value of corresponds to a stationary point for the lagrange function .the first - order partial derivatives are given by multiplying eq .( [ condition1 ] ) by and summing over , we obtain similarly , from eq .( [ condition2 ] ) we have therefore , is the very condition that should be satisfied by the optimal unitary matrix .now we can demonstrate that the condition is indeed fulfilled by the fourier matrix . for , the matrix elements reduce to therefore , apart from , there are two additional properties possessed by : ( i ) _ periodic property _ ; ( ii ) _ circulant property _ ] and especially the entries on the main diagonal are all equal to .another important property of circulant matrices is that they can always be diagonalized by the fourier matrix . in this work ,the diagonalization is of the form where with the eigenvalues of ( and also of ) .in fact , for general circulant matrices , the eigenvalues are given by it is easy to check that in our case by use of the identity eq .( [ orthogonality ] ) .as shown in appendix [ a1 ] , for the unitary matrix the transformed state is actually a circulant matrix . in this specific basis, the norm of coherence can be expressed as where we define .furthermore , we have },\end{aligned}\ ] ] which is the desired formula .99 n. gisin , g. ribordy , w. tittel , and h. zbinden , rev .modern phys . * 74 * , 145 ( 2002 ) .r. cleve , a. ekert , c. macchiavello , and m. mosca , proc .a * 454 * , 339 ( 1998 ) .m. a. nielsen and i. l. chuang , _ quantum computation and quantum communication _( cambridge university press , cambridge , 2000 ) . v. giovannetti , s. lloyd , and l. maccone , science * 306 * , 1330 ( 2004 ) . v. giovannetti , s. lloyd , and l. maccone , phys . rev .lett . * 96 * , 010401 ( 2006 ) .a. streltsov , g. adesso , and m. b. plenio , arxiv:1609.02439 .m. horodecki and j. oppenheim , int .j. mod . phys .b * 27 * , 1345019 ( 2013 ) .f. g. s. l. brando and g. gour , phys .115 * , 070503 ( 2015 ) .t. baumgratz , m. cramer , and m. b. plenio , phys .113 * , 140401 ( 2014 ) .v. vedral , m. b. plenio , m. a. rippin , and p. l. knight , phys .. lett . * 78 * , 2275 ( 1997 ) . v. vedral and m. b. plenio , phys .a * 57 * , 1619 ( 1998 ) .t. durt , b .-englert , i. bengtsson , and k. yczkowski , int . j. quantum inf . * 8 * , 535 ( 2010 ). u. singh , m. n. bera , h. s. dhar , and a. k. pati , phys .rev . a * 91 * , 052115 ( 2015 ) .shao , z. xi , h. fan and y. li , phys .rev . a * 91 * , 042120 ( 2015 ) .zhang , g. f. xu , and d. m. tong , arxiv:1606.03181 .j. watrous , _ theory of quantum information _( university of waterloo , waterloo , 2011 ) .i. bengtsson and k. yczkowski , _ geometry of quantum states : an introduction to quantum entanglement _( cambridge university , cambridge , england , 2007 ) .r. a. horn and c. r. johnson , _ matrix analysis _ ( cambridge university press , new york , 1985 ) .r. a. horn and c. r. johnson , _ topics in matrix analysis _( cambridge university , cambridge , england , 1991 ) .w. tadej and k. yczkowski , open syst .* 13 * , 133 ( 2006 ) .i. bengtsson , w. bruzda , .ericsson , j .- .larsson , w. tadej , and k. yczkowski , j. math .* 48 * , 052106 ( 2007 ) .f. szllsi , _ construction , classification and parametrization of complex hadamard matrices _( phd thesis , central european university , 2011 ) . m. van den nest , new j. phys . * 13 * , 123004 ( 2011 ) .y. yao , x. xiao , l. ge , and c. p. sun , phys .a * 92 * , 022112 ( 2015 ) .y. peng , y. jing , and h. fan , phys .a * 93 * , 032326 ( 2016 ) .d. gottesman , a. kitaev , and j. preskill , phys .a * 64 * , 012310 ( 2001 ) .s. bandyopadhyay , p. o. boykin , v. roychowdhury , and f. vatan , algorithmica * 34 * , 512 ( 2002 ) .r. bhatia , _ matrix analysis _( springer - 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Abstract: since quantum coherence is an undoubted characteristic trait of quantum physics , the quantification and application of quantum coherence has been one of the long - standing central topics in quantum information science . within the framework of a resource theory of quantum coherence proposed recently , a _ fiducial basis _ should be pre - selected for characterizing the quantum coherence in specific circumstances , namely , the quantum coherence is a _ basis - dependent _ quantity . therefore , a natural question is raised : what are the maximum and minimum coherences contained in a certain quantum state with respect to the random bases ? while the minimum case is trivial , it is not so intuitive to verify in which basis the quantum coherence is maximal . based on the coherence measure of relative entropy , we indicate the particular basis in which the quantum coherence is maximal for a given state , where the fourier matrix ( or more generally , _ complex hadamard matrices _ ) plays a critical role in determining the basis . intriguingly , though we can prove that the basis associated with the fourier matrix is suboptimal ( e.g. , a local extremum ) for the norm of coherence , numerical simulation shows that it is not a global optimal choice .


impedance surfaces for microwave frequency antennas and other devices have been recently introduced and actively investigated .these surfaces are mostly thought to be applied for reducing surface waves , for controlling the plane - wave reflection phase , and as artificial magnetic conductors .high - impedance surfaces are thin composite layers ( as a rule backed by a metal plane ) , whose surface impedance has a parallel resonance at a certain frequency or at several frequencies . within the resonant band the surface of the layer behaves as a magnetic wall for normally incident plane waves .that is why the his are often called _ artificial magnetic conductors ( amc)_. in practical cases , the thickness of an amc layer at resonance is much smaller than a quarter of the wavelength .it means that the reflection coefficient of a plane wave from the structure can be close to + 1 at a very small distance from the ground plane .this means that the electromagnetic interaction of a horizontal electric current with an amc can be constructive even for very small distances from the current to the ground plane .this is why amc are prospective for designing low profile antennas .infinite ( practically very high ) surface impedance means that the tangential magnetic field component is very weak at the surface of the wall .tangential electric field can be strong at this surface .the reflection properties of a his depend on the frequency , and the his resonant band can be defined as a frequency range at which the phase change of the reflected electric field is within certain limits ( usually from to ) .a his conventionally possesses a rather narrow band of amc operation .in general , these structures support surface waves , however there is an important difference with the case of a simple metal - backed dielectric layer : there are stop bands for surface waves .therefore , the same structures interacting with plane waves as his operate as _ ebg ( electromagnetic band gap ) structures _ with respect to the surface waves .sometimes they are called as _ 2d pbg ( photonic band gap ) structures _ which is not a proper term at microwaves . besides the high - frequency band gaps whose band structure is related with the spatial periodicity of the amc layer, there can be one or two low - frequency ones , related with the resonance of the surface impedance which can also hold in the regime of the surface wave .  while the amc are useful to design low - profile antennas , artificial material fillings are useful in the design of small antennas , especially magnetic materials can be utilized as a mean to reduce the antenna size . according to a simple theoretical model of a small patch antenna as a resonator , the resonant frequency and impedance bandwidth of a small antenna depend on the relative permittivity and permeability of the loading material as f_r ~ , bw ~.suphere means the proportionality . in practice , these expressions are not very accurate , and the result of loading depends on the antenna type , but still the tendency predicted by holds.  there has been some work on the effects of magneto - dielectric substrates on the bandwidth of microstrip antennas , but these results are contradictory .papers and conclude that the best filling is a magnetic material with large , although the authors of conclude that the best filling is a material with both and .nevertheless , the usefulness of magnetic filling is evident from the known literature .  in paper a material with a finite isotropic permeability was considered as a loading material .natural magnetic materials at frequencies higher than mhz are ferrimagnetic crystals ( for example , hexaferrites ) .these ferrites are not very attractive for applications in small antennas since they are lossy , heavy , and expensive . probably because of this ,antennas with ferrite or ferromagnetic filling have been analyzed mainly from the point of view of electrical control of their resonant frequency and radiation pattern , see , e.g. , . possible realizations of artificial magnetics at ghz can be based on the resonant principle : artificial magnetic materials can be formed by small particles with a resonant magnetic susceptibility .these should resonate at a frequency which is close to the resonant frequency of an unloaded antenna .such artificial magnetic materials are practically known in the frequency range ghz ( the structures formed by the so - called _ split - ring resonators _ ) .such loadings have been successfully tested with patch antennas in : a size patch antenna with the bandwidth of has been reported in that paper .  in certain frequency regions ,high - impedance surfaces behave as layers of effective magnetic materials , and can be used as magnetic fillings in the antenna design .amc are not similar to natural isotropic magnetics .these can not be properly described in terms of magnetic permeability .formally , one can introduce an effective permeability , but it will strongly depend on the incident plane wave polarization and the angle of incidence .this dependence is a spatial dispersion effect .however , this effect does not forbid one to use such composite layers as a magnetic filling material .recently , these artificial layers have been used to reduce the size of antennas and filters .the main idea of the present paper is to use an artificial material which would possess the properties of both amc and magneto - dielectric composite within a rather wide resonance band . in this paper , the newhis playing also the role of an artificial magnetic material ( amm ) is tested under a patch antenna .the choice of this material can be explained in terms of the angular stability of the resonant frequency .the known amc introduced in papers have the resonance depending on the angle of incidence and wave polarization ( te or tm ) .therefore the interaction of such amc with currents on a real antenna can not be completely constructive .for example , let the working frequency of the patch antenna correspond to the resonance of the conventional amc illuminated by a normally incident plane wave .then the narrow part of the angular spectrum of radiation centered at will interact with the amc as if it were a magnetic wall . for the other part of the angular spectrumthe conventional amc is not a magnetic wall , since its resonant frequency will be different . in the amcused here this shortcoming is absent as it was shown in .notice , that in paper another full - angle amc has been suggested and studied .this is a self - resonant grid on a simple metal - backed dielectric layer .however , our choice of the artificial magnetic material suggested in has additional advantages .first , in this amm there are vertical conductors ( vias ) .we expect that the tm - polarized surface waves are suppressed in this structure at low frequencies due to the presence of vias in the same way as in mushroom structures .second , the resonance of the structure takes place at lower frequencies ( compared to the structure period ) than the resonance of the amc described in .both these advantages are important for patch antennas .the second one is important for the antenna miniaturization .the first one is crucial when one uses an array of patches and the problem of mutual interaction appears .the behavior of the amm is investigated at different distances from the patch antenna with different configurations .also a simple comparison method is introduced in order to find the effective ( averaged ) permeability value of the layer .results from ie3d / fdtd simulations are shown and compared with the measurement results .structure introduced in is shown in fig .[ cellnew ] .this amm can be considered as a 2d grid of bulk unit cells with the horizontal period .every cell contains two orthogonal loops of length , so that the upper interface represents an array of metal crosses on the surface of a dielectric layer .every cross ( whose ends are connected to metal vias ) together with vias and patches form two orthogonal loops ( the loop length is the same as the length of a cross side ) .so , the effective vertical loop is formed by two vias perforating the dielectric layer and a horizontal strip lying on the dielectric interface .the loading capacitors are formed by metal patches and the ground plane .the patch array is separated from the ground plane by a thin dielectric layer . the standard printed - board circuit ( pbc )can be used to prepare both patch array ( which is located on one side of a pbc ) and the array of crosses ( located on the other one ) if the thin layer has no metallization ( teflon film ) .the analysis of the structure impinged by a plane wave becomes easier with the help of the image theory .every real loop complemented by its mirror image is a symmetrically loaded rectangular loop with sizes , where ( see fig .[ scheme ] ) .the electric field is zero at the loop center ( at the ground plane ) .therefore , the electric polarization of the loop is negligible and it can be considered as a horizontal magnetic dipole excited by an external magnetic field . following the image method , consider an array of loaded rectangular loops illuminated by two plane waves from both sides of the array. let the plane wave be polarized so that and the magnetic field is directed along the -axis .then the loops in the plane are not excited ( see fig .[ cellnew ] ) .the whole structure behaves as an array of parallel loops lying in the planes within the dielectric layer excited by two plane waves coming from and .since there is no electric dipole polarization of loops , the electric polarization of the whole structure is practically that of the dielectric layer .the reflected field is then the sum of the field produced by the single dielectric layer of thickness ( excited by two waves impinging the layer from the top and the bottom ) and the field produced by the magnetic moments of loops .the magnetization of loops is resonant due to the presence of capacitive loads , and the magnetic response at the resonance is comparatively strong due to high inductance of the loops .loops are made from thin electric conductors and their inductance is much higher than the effective inductance of the conventional amc ( which is practically determined by the thickness of the dielectric layer ) .this factor is responsible for rather low frequency of the resonance and for a wider bandwidth ( in parallel -circuits the higher is the inductance the larger is the resonant band ) .an analytical model of this structure and a comparison with the results of numerical simulations ( obtained with the hfss package ) are presented in a recent paper by c. simovski and a. sochava .the geometry of the structure that we use in this paper is different from the structure proposed in , although the operational principle is the same .this new structure is shown in fig .[ fig1 ] , where square horizontal loops replace the conducting crosses of the structure .for simplicity , let us consider the normal incidence of a wave whose electric field is polarized along one side of the square horizontal loop .then the two vertical -loaded loops ( formed by two sides of the horizontal loop , four vias connected to them and four capacitances between patches and the ground plane ) will be excited in every unit cell of the structure .a horizontal loop as such is not excited by the external magnetic field since this magnetic field is tangential .the electric connection of the two vertical loops does not change the operation and the theory developed in basically remains valid .  to realize the material a two sided tly-5 film with the thickness of mm and the dielectric constant has been used .this film separates small patches from the ground plane .horizontal parts of the loops and the patches are printed on the opposite sides of a printed circuit board layer with the dielectric constant .horizontal parts of the loops and the patches are connected by via wires ( round metal cylinders ) as shown in fig .the whole manufactured structure contains unit cells and has the dimensions of mm .this structure is a brick " from which larger samples of an amm can be built .test the performance of the new artificial magnetic layers with patch antennas we design and study square - patch antennas with different sizes . for measurements ,a large ground plane is used and for simulations the ground plane is infinite .this is done in order to exclude possible resonant effects of a finite ground plane .ie3d software as well as an in - house developed fdtd code are used to simulate antennas with and without the material filling .also , for the sake of comparison , we simulate the same antennas with an infinite dielectric material layer inserted between the ground plane and the patch .the dielectric has the relative permittivity and the slab thickness is mm ( the same as for the substrate used to manufacture the artificial magnetic layer ) . the antenna is fed from a side using a microstrip .the new amm is introduced under the patch gradually in order to save simulation time . in fig .[ fig2 ] , the configuration with three columns of the material under the patch antenna are shown .amm blocks are located at the two sides of the antenna symmetrically .antennas with the patch dimensions of , , , and mm have been considered .since patches of different sizes resonate at different frequencies from ghz to ghz , we could test the effectiveness of the material at different frequencies .the height of the antenna changes the field applied on the material which also changes the response of the material layer .the height of the patch antenna has been varied in order to find the best reduction for the resonance frequency . in these simulations , the mm antenna is chosen and only one column of the material is placed under each side of the patch . the return loss is compared for , , and mm heights of the patch antenna ( see fig . [ fig3 ] ) .the results show that the effect of the material is enhanced when the patch is closer to the material layer . herewe of course see also the effect of the relative permittivity of the loading layer .if there would be no magnetic behavior of the material , antenna would be thought as loaded with a dielectric layer with and the thickness 3 mm . in that casethe resonant frequency drops from to ghz for a patch antenna with the patch at mm above the ground plane . but with the magnetic material the resonant frequency drops to ghz . here , with only one column of the artificial materialwe have a reduction in the resonant frequency or , in other words , we have a patch antenna with a size of .when an antenna is totally filled with such complex material , it becomes very difficult to simulate it with ie3d , using the computer power we have .this is because the amc is made of a lot of small metal strips , patches , and vias .therefore we gradually fill the volume below the antenna .we start to fill from the sides of the antenna where the currents and magnetic fields are strong . herewe investigate the effect when the antenna is partially filled .the number of columns is increased and the effect in return loss is shown in fig .[ fig4 ] for the mm patch . in this examplethe antenna patch is at mm above the ground plane .the same exercise has been repeated with the patch at mm above the ground plane .the results are listed in table [ table1 ] .it is seen from the simulations that even with only one column we achieve a 16% reduction in the antenna resonant frequency for the patch at mm above the ground and for the mm height . with three columns we achieve about 25% and 33% reduction for heights 6 and 4 mm , respectively ..resonant frequencies of the mm antenna with and without amm for the patch heights and mm .the antenna resonant frequencies are also shown when there is an infinite dielectric material slab under the antenna patch with the mm thickness and . [ cols="^,^,^",options="header " , ]an antenna with the patch size mm has been built .the patch is first positioned at 6 mm above the ground plane , then at 4 mm above the ground plane .it is filled with the artificial material layer of the dimensions mm ( fig .[ fig10 ] ) . the antenna in fig .[ fig10 ] has also been simulated with a 3d fdtd code . in fdtd ,the particles are constructed from joint thin wires and small plates of approximately same size as in measurements ( within limits of finite cell sizes ) .  as can be seen from fig .[ fig12 ] , the results are similar even though there are differences in the resonant frequencies .of course in simulations we have an infinite ground plane , and only three columns of the artificial material partially fill the antenna volume in the case of ie3d simulations . on the other hand , in measurementswe have had a large but finite ground plane , and the material sample is of the same transverse dimensions as the patch .a new artificial magnetic material layer has been tested in order to shrink patch antenna dimensions . in simulations a antenna and in practice a antennahave been realized , with the 6-db bandwidth of 4.35% .the antenna bandwidth of these reduced - size antennas is practically of the same order as for usual air - filled patch antennas with the patch size , that resonate at the same frequency .thus , we have demonstrated in practice a technique to miniaturize patch antennas with the use of a certain high - impedance surface ( working as an artificial magnetic material layer ) without worsening the bandwidth .h. mossallaei and k. sarabandi , periodic meta - material structures in electromagnetics : concept , analysis , and applications , _ 2002 ieee aps international symposium digest , _ vol .380 - 383 , san antonio , tx , 2002 .f. martn , f. falcone , j. bonache , r. marqus , and m. sorolla , miniaturized coplanar waveguide stop band filters based on multiple tuned split ring resonators , _ ieee , microwave and wireless components letters , _vol.13 , no .12 december 2003 .simovski , a.a .sochava , s.a .tretyakov , new compact and wide - band high impedance surface , _ 2004 ieee antennas and propagation society international symposium , _ monterey , ca , usa , vol . 1 , pp .297 - 300 , june 20 - 25 , 2004 .yang , k. p. ma , y. qian , t. itoh , a novel tem waveguide using uniplanar compact photonic band - gap ( uc - pbg ) structure , _ ieee trans .theory techniques _ , vol .47 , no 11 , pp .2092 - 2098 , nov .1999 .simovski , p. de maagt , s.a .tretyakov , m. paquay , and a.a .sochava , angular stabilization of resonant frequency of artificial magnetic conductors for te - incidence , _ electronics letters _1 , pp . 1 - 3 , 2004 .  c.r .simovski , a.a .sochava , high impedance surfaces based on self - resonant grids . analytical modeling and numerical simulations ,_ progress in electromagnetics research _ , vol .pier 43 , pp .239 - 256 , 2003 .  c. simovski , p. de maagt , i. melchakova , high - impedance surfaces with angular and polarization stability , _ proc .27-th european space agency antenna workshop on innovative periodic antennas _, santiago de compostela , spain , march 9 - 11 , 2003 , pp .178 - 185 .

Abstract: a new type of high - impedance surfaces ( his ) has been introduced by c.r . simovski et al . recently . in this paper , we propose to use such layers as artificial magnetic materials in the design of patch antennas . the new his is simulated and patch antennas partially filled by these composite layers are measured in order to test how much the antenna dimensions can be reduced . in order to experimentally investigate the frequency behavior of the material , different sizes of the patches are designed and tested with the same material layer . also the height of the patch is changed in order to find the best possible position for minimizing the antenna size . this composite layer of an artificial magnetic material has made the antenna smaller while keeping the bandwidth characteristics of the antenna about the same . about of size reduction has been achieved .  artificial magnetic material , patch antenna , antenna miniaturization , high - impedance surface .


edison s invention of direct current ( dc ) preceeded tesla s invention of alternating current ( ac ) . however , once there were two possible modes of power available , an inevitable debate on the merits and demerits of the two started that led to what is called the `` war of currents '' .the inventor s became adversaries with edison promoting direct current ( dc ) for electric power distribution over the alternating current ( ac ) .the `` war of currents '' got so bitter that both inventors lost a lot of money and rumors have it , their nobel prize .  todaythe debate is more or less resolved with ac being accepted as the best method for electric power distribution , especially where the power requirement is large .it can be appreciated that since direct current can not be trivially stepped up or stepped down , the same voltage level is transmitted as required by the load .this resulted in large transmission losses .transmission loss takes place due to heat dissipation along the current carrying wires used for delivering power from generation point to consumer .the initial transmission networks laid were of copper , which is one of the best conductors with low resistivity . even with low resitivity , since the length of the transmission wires involved are large , they offered finite and non - negligiable resitance . thus introducing power loss during transmission .mathematically power dissipated is gien as where i , , l and a are the rms ( root mean square ) current , wire s resistivity , it s length and cross - sectional area respectively .thus , for transmitting a given power with minimum power loss , one would have to reduce the current while increasing the voltage .this is exactly what a transformer does for ac .thus for dc , transmission loss can only be minimised by using thicker copper wires . in turn, only fairly low dc power can be transmitted . since ,ac power can be stepped - up and down easily using transformers , the issue of transmission loss over thinner wires can be economically addressed .  with industrialization and increasing demands for higher levels of power ,even a ac distribution is not enough .hence , today distribution has moved to polyphase ( m ) distribution .polyphase voltages are also ac voltages but made up of multiple sinosudial varying voltages .of the possible polyphases , the three - phase supply is the most popular .it refers to three voltages that differ in phase by degrees from each other .the voltages go through their maxima in a regular order , after every .the phase sequence are named ` a ' , ` b ' and ` c ' .  0.5 cm 0.5 cmthe three phases are generally distributed using three wires .the phases are separated or collected at the load side using special transformers , called `` star '' or `` delta '' transformers ( fig 1a ) .each wire has the same current carrying capability as in case of single phase supply .however , the power delivered at the load can be far greater depending on how the two potential levels are selected .consider , in the star transformer the load is connected between point ` a ' and neutral ( fig 1b ) the output waveform would be that of phase a , with the neutral point acting as ` zero potential ' point .the power delivered to the load is .however , if the load is connected between point ` a ' and ` b ' , the net potential difference can be found ( first in general ) from    where is the peak voltage and the phase difference between the two phases . for a 2-phase supply ,where the phase difference between the two waveforms is , the potential difference would be this mathematics is graphically represented in fig 2 .it can be appreciated that the power delivered to the load now would be .this example shows how polyphase transmissions delivers more power .  while these are important issues to be appreciated by a physicist setting up a laboratory with instruments that require high power , schools and colleges do not help develop this concept with hand - on experimentation for fear of possible accidents that could even prove fatal .while kits are present in engineering colleges for electrical engineering , not much is found in the literature in way of simple experiemnts for physics and electronics students . in this direction , we have designed and tested a simple circuit , where two phases are generated from a sinusodial wave taken from a function generator .the proposed circuit requires three opamps , of which the first opamp is assembled in an inverting configuration . the sinusodial input from a function generator ( shown in fig 3 as ` 1 ' ) is given as an input to this inverting amplifier .this signal also acts as one of the 3 phases , namely phase a. the gain of this amplifier is kept as unity with .this circuit acts as a buffer preventing any loading by successive circuits and introduces a phase change of between output and input waveforms ( as indicative of the name `` inverting '' amplifier ) .the rc circuit is designed to introduce a phase difference of for the selected frequency . however ,a single combination of rc can at the most introduce a phase difference equal to or less than .hence , two sections are used , with each rc section introducing a phase shift of . the values of r and c are selected using the formula   _ in our study , for an 5khz wave provided from the function generator , we selected and ._ hence points ` 3 ' and ` 4 ' of the circuit would be and respectively out of phase with respect to the input .the impedence networks act as potential dividers and hence the voltage levels at ` 3 ' and ` 4 ' would be lower than that given as input . since the three phases would have the same amplitude , the signals at ` 3 ' and `4 ' would have to be amplified .we use as inverting amplifier again for point ` 4 ' .the voltage gain of this amplifier is give by .hence , has to be varied till the voltage signals amplitude is identical to the input signal .also , this inverting amplifier introduces a phase change of . the net phase differencehence would be which is nothing but a phase difference of .the output of this op amp circuit would be phase ` b ' .similar amplitude correction is required for the wave collected at point ` 3 ' . however , here we select a non - inverting amplifier circuit whose gain is given as and as the name suggests does not introduce any further phase shifts _ ( and used were pots and typically had values greater than )_. hence , the net shift at the output remains the same as that at point ` 3 ' at or .this is phase c of the simulated 3-phase supply .fig 4 shows the waveforms ( phase a and c ) as captured by picoscope ( model 2202 ) .since the model only gives dual trace , only two waves can be shown simultaneously . however , the data captured by picoscope shows all three phases simultaneously .curve fitting these data points give the phase difference between the various phases to be 1.95radians or . also , the peak voltage ( ) also works out to be 1.58v .  while the circuits achieves the purpose of mimicing a 3-phase supply ,experiments can be done to further understand applications of 3-phase supply .for example , one can study the rectification and advantage of generating dc from 3-phase supply .for this , all one needs is 3 diodes and a load resistance ( see fig 5 ) .the output waveform is dc and also notice even without any filtering circuit , the output is continuous with little ripple .this is because , as seen by the load , the input frequency is three times that of a single phase supply .the ripples in a rectified output is inversely proportional to the frequency , hence the low ripples here can be understood .also , this output gives visual idea and helps in easier understanding that each phase in a 3-phase supply is separated by . in this rectifier circuit , only that diode conducts , for which the phase connected to it has the highest instantaneous potential with respect to the neutral .fig 4 shows one phase to have highest potential with repect to the other phases for .thus , in one cycle ( ) there would be three peaks of the output dc wave .this is visiable in fig 5 .a simple circuit has been proposed to demonstrate the behaviour of 3-phase power distribution .the simple circuit discussed in this article cost rs ( indian rupee ) 20/- to assemble ( less than a dollar ) and can be a very usual experiment in schools and under - graduate laboratorys to learn more about 3-phase supply and it s application .the financial support of u.g.c ( india ) in the form of minor research project no.f.6 - 1(25)/2007(mrp / sc / nrcb ) is gratefully acknowledged .the authors would like to express their gratitude to the lab technicians of the department of physics and electronics , s.g.t.b .khalsa college , for the help rendered in carrying out the experiment .

Abstract: the power distribution of nearly all major countries have accepted 3-phase distribution as a standard . with increasing power requirements of instrumentation today even a small physics laboratory requires 3-phase supply . while physics students are given an introduction of this in passing , no experiment work is done with 3-phase supply due to the sheer possibility of accidents while working with such large powers . we believe a conceptual understanding of 3-phase supply would be useful for physics students with hands on experience using a simple circuit that can be assembled even in a high school laboratorys .


the phenomenal argument by einstein , podolsky and rosen ( epr ) in 1935 to demonstrate the incompleteness of quantum mechanics , struck schrdinger with the concept of ` steering ' .however , only recently , wiseman _ et al . _ have formalized the concept of steering in the form of a task .the task of steering can be seen as one s inability to construct a _ local hidden variable- local hidden state _( lhv - lhs ) model that reproduces a given bipartite correlation . the work of wiseman __ has generated an immense interest in the study of this steering phenomenon .on the other hand , the concept of steering has been extended for multipartite case and the idea of n - partite genuine mutipartite steering have been explored . unlike the two well studied non classical correlations , namely _nonlocality _ and _ entanglement _ , there is an inherent asymmetry in the task of steering .this is because in case of steering , on one subsystem ( which is being ` steered ' ) the statistics must arise out of a valid measurement on a valid quantum state but no such constraint is required for the other subsystem . here in addition to the simplicity of bell s assumptions of _ local causality _one must also perform trusted measurements on one subsystem , whereas the other subsystem need not be trusted .apart form the foundational interest , the study of steering also finds applications in semi device independent scenario where only one party has trust on his / her quantum device but the other party s device is untrusted . as a concrete exampleit has been shown that steering allows for secure quantum key distribution ( qkd ) when one of the parties can not trust their devices .one big advantage in this direction is that such scenarios are experimentally less demanding than fully device - independent protocols ( where both of the parties distrust their devices ) and , at the same time , require less assumptions than standard quantum cryptographic scenarios .  in 1964, bell sought a way to demonstrate that certain correlations appearing in quantum mechanics are incompatible with the notions of _ locality _ and _ reality _ aka _ local - realism _ ( lr ) , through an inequality involving measurement statistics .a violation of such inequality implies the usefulness of correlations for epr argument . in 1969 , clauser - horne - shimony - holt ( chsh ) proposed a set of simple bell inequalities which are easy to realize experimentally . in the same spirit of bell s inequality in non - locality , several steering inequalities ( si )have been proposed , so that a violation of any such steering inequalities can render a correlation to be steerable .but an unavoidable hindrance to formalize such sis follows from the fact that steering scenario is device - independent only on one - side .recently cavalcanti _ et al ._ have proposed a chsh - like inequality for quantum steering .they have derived an epr - steering inequality that is necessary and sufficient for a set of correlations in the simplest two - party scenario involving two measurement settings per site and two outcomes per measurement , with mutually unbiased measurements at trusted party . in this article we derived the tight optimal quantum violation of the epr - steering inequality proposed in .we have also studied the violation amount of this inequality for different well - known classes of -qubit states .interestingly , we find that for different -qubit pure entangled states the optimal violation amount of this inequality differ .note that , for all other existing sis such difference is not observed .the organization of this article goes as follows : we first briefly review few existing steering inequalities along with the newly proposed chsh like steering inequality . then we show that the optimal violation of this chsh like steering inequality in quantum theory is restricted to . then we study how the optimal violation amount of the chsh like inequality varies for different entangled states .to test epr steering reid first proposed a testable formulation for continuous - variable systems based on position - momentum uncertainty relation which was experimentally tested by ou et al .developed a general construction of experimental epr - steering criteria based on the assumption of existence of lhs model .importantly this general construction is applicable to both the discrete as well as the continuous - variable observables and reid s criterion appears as a special case this general formulation . on the other hand , walborn _ et al ._ formulated a steering inequality based on bialynicki - birula and mycielski s entropic position - momentum uncertainty relation .as the entropic uncertainty relation implies heisenberg s uncertainty relation , hence the set of states violating walborn _s steering inequality contains all the states violating reid s inequality .thus walborn _ et al . _s steering criterion is more powerful than reid s one .however this is true for continuous variable case only , not for the discrete case . in all instancesthe violation of the aforementioned inequalities by different pure entangled states are the same .all of these steering inequalities have been proved to be as sufficient conditions for witnessing steering in bipartite quantum systems . but none of these condition is supposed to be a necessary and sufficient condition for steering .search for such an necessary and sufficient condition has been culminated in a recent development by cavalcanti _they have proposed an chsh like epr - steering inequality that is necessary and sufficient for the set of correlations in the simplest scenario involving two settings and two outcomes per setting , with mutually unbiased measurements at trusted end . at this pointit is interesting to ask the following questions :  * what is the optimal violation of this newly proposed chsh like steering inequality in quantum theory ? * how the violation amount of the concerned inequality depends on the state ?we provide a definite answer for the first question and study the second one for some classes of states .let us first briefly review the steering scenario as introduced by wiseman _given a pair of systems at alice and bob , denote and the sets of observables in the hilbert space of alice s and bob s system , respectively .an element of is denoted by , with a set of outcomes labeled by , and similarly for bob .the joint state of the system is steerable by alice iff it is not the case that for all , , , , the joint probability distributions can be written in the form where denotes an arbitrary probability distribution and denotes the quantum probability of outcome given measurement on state . in other wordsthe state will be called steerable if it does not satisfy a lhv - lhs model .note that , if for a given measurement strategy the correlations have a lhv lhs model , this does not imply that the underlying state is not steerable , since there could be another strategy that does not . in the simplest scenario where alice and bob each have a choice between two dichotomic measurements to perform : , , and outcomes of are labeled and similarly for the other measurements , the authors of ref. derived a necessary and sufficient criterion for steering which reads as : we know that , in the simplest bell scenario which involves two observers with dichotomic measurements per site , the set of local correlations lies in a polytope ( lhv polytope ) with chsh inequality providing the non trivial facets of the lhv polytope . in the steering scenario with similar settings the set of correlations having lhv - lhs description also form a polytope .one of the authors of this article , along with other collaborators , have shown that measurement incompatibility limits the bell - chsh inequality violation in quantum theory to cirelson bound . adopting similar approach and using a recently established connection between measurement incompatibility and steering , we derive the optimal quantum violation of the steering inequality ( [ eq_nec_suf ] ) . before establishing this resultwe first briefly review the concept of measurement incompatibility and the concept of unsharp measurement in quantum theory .  _measurement incompatibility : _ in the case of projective measurements , compatibility is uniquely captured by the notion of commutativity .non commutative projective measurements in quantum mechanics do not admit unambiguous joint measurement . with the introduction of the generalized measurement i.e. positive operator - valued measures ( povms ) , it was shown that observables which do not admit perfect joint measurement , may allow joint measurement if the measurements are made sufficiently fuzzy .therefore , for general measurements there is no unique notion of compatibility . in this article measurement incompatibilityis captured by non joint measurability .mathematically , a povm consists of a collection of operators which are positive , , and sum up to the identity , . here denotes measurement outcome and denotes measurement choices .physically , any povm can be realized by first letting the physical system interact with an auxiliary system and then measuring an ordinary observable on the auxiliary system . a set of povms is called jointly measurable if there exists a measurement with outcome  ] is known as unsharpness parameter " and the fraction is called degree of reality " .it may happen that the observables and do not allow any joint measurement , but with introduction of sufficient amount of unsharpness , there unsharp versions and may allow joint measurement . in ref. , the authors have proved that given any d - dimensional quantum system , joint measurement for unsharp versions of any two dichotomous observables and of the system is possible with the largest allowed value of the unsharpness parameter .note that the expectation value of an unsharp observable over some quantum state is related to the expectation value of its sharp version in the following manner , similarly , if alice performs unsharp measurement on her part and bob performs sharp measurement on his part of a bipartite shared state than we have ,   except from quantum entanglement , another necessary ingredient which is necessary for study of quantum nonlocality is the existence of incompatible set of measurements . in the simplest bipartite scenario wolf __ showed that any set of two incompatible povms with binary outcomes can always lead to violation of the chsh - bell inequality .but , recently in refs. the authors have proved that this result does not hold in the general scenario where numbers of povms and outcomes are arbitrary .however in this general settings the authors of have established a connection between measurement incompatibility and a weaker form of quantum nonlocality i.e. , epr - schrdinger steering .they have shown that for any set of incompatible povms ( i.e. not jointly measurable ) , one can find an entangled state , such that the resulting statistics violates a steering inequality .please note that , one of the authors of this article has recently proved that connection between measurement incomparability and steering holds for a more general class of tensor product theories rather than just hilbert space quantum theory .let alice performs a measurement assemblage on her part of a bipartite shared quantum state .upon performing measurement , and obtaining outcome , the ( un - normalized ) state held by bob is given by .the normalized state on bob side is given by .also we have , which actually ensure no signaling from alics to bob .the state assemblage is un - steerable iff it admits a decomposition of the form where .existence of such decomposition for state assemblage on bob s side ensures that the statistics obtained from the state admits a combined lhv - lhs model of the form of eq.([steer ] ) .the authors in refs. have shown that the assemblage , with , is un - steerable for any state acting on if and only if the set of povms acting on is jointly measurable . as a corollary of this result we can say that  _ corollary 1 _ : the assemblage , with and , is unsteerable for any state acting in if and only if the set of povms acting on is jointly measurable .  at this stage , we are now in a position to prove our main result , which is described in the following theorem .  _ theorem 2 _ : consider a composite quantum system composed of two subsystem with state spaces and , respectively . for any pair of dichotomic observables , for the first system and the mutually unbiased dichotomic observables , for the second system and the joint state acting on , we have the following inequality : where is the optimal unsharpness parameter that allows joint measurement for any two dichotomic quantum observables .  _ proof_ : let us consider two arbitrary dichotomic observables on alice side , and .these two observables in general may not allow joint measurement .however , introduction of unsharpness makes it possible to measure the unsharp version of these two observables jointly .let the optimal unsharpness is which allows joint measurement for any two dichotomic observables .  now according to corollary , as far as observables on alice s side are jointly measurable, they will not violate any steering inequality and hence the steering inequality ( [ eq_nec_suf ] ) .thus we have    now using the expressing from eq.([eta ] ) we get ,    the value of in quantum theory is proved to be  .therefore the upper bound of the steering inequality ( [ eq_nec_suf ] ) in quantum theory is , i.e. , , which is numerically equal to the celebrated cirelson value .naturally the question arises whether this value is tight or not . in the following section we answer this question affirmatively by showing that sharing a maximally two qubit entangled state with suitable choice of measurement, this value can be achieved in quantum theory .in this section we study the optimal violation of the steering inequality ( [ eq_nec_suf ] ) for different given entangled states .  _ observation 1 _ : is different for different -qubit pure entangled states .consider that an arbitrary 2-qubit pure entangled state is shared between alice and bob .alice performs measurements and on her part of the entangled particle where .similarly bob performs measurements and on his part of the entangled particle where and , i.e. are mutually unbiased qubit measurements .varying over the measurement directions of alice and bob we have numerically found the optimal violation amount of the steering inequality ( [ eq_nec_suf ] ) for a given -qubit pure entangled state and in fig.[fig1 ] we have plotted it with respect to state parameter .  ) for different -qubit pure entangled states.,width=264,height=170 ]  from fig.[fig1 ] it is clear that all -qubit pure entangled states violate the steering inequality ( [ eq_nec_suf ] ) . it is important to note that the optimal violation amount is different for different states with maximally entangled state providing the maximum violation where alice performs measurement in the directions and and bob s mub measurements look like and .please note that except from the steering inequality considered here , for all other existing steering inequalities the violation amount does not change with respect to different pure entangled states ._ observation 2 _ : for -qubit werner class of states .consider that alice and bob share the werner state ; where is the singlet state . likewise in the previous case ,varying over the measurement setup of alice and bob we find the optimal violation amount of the steering inequality ( [ eq_nec_suf ] ) for different werner state and plot this values in fig.[fig2 ] .  ) for -qubit werner class of states . ]  from fig.[fig2 ] it is clear that werner states violate the steering inequality for and the violation amount increase with the parameter .there have been several attempts to quantify quantum steering .two most recent instances are ` steerable weight ' , ` steering robustness ' and ` relative entropy of steering ' .but all of these proposed quantifiers assign same value for all pure entangled states . at this pointour observation becomes interesting .since the amount of violation can also be shown to be non increasing under steering non - increasing operations ( snios ) , one can take this amount of violation to be a valid quantifier of quantum steering .our study stipulates further studies whether some semi device independent protocol(s ) can be deigned whose payoff scales with violation amount of the steering inequality considered here . in such case different violation amount of the steering inequality by different pure entangles states will have an operational explanation .we also find the violation of the chsh like steering inequality by -qubit werner states .the inequality is violated by werner states if and beyond this value it follows a trait- the more entangled the state , more is the violation . we also find that the violation of this steering inequality in qm is tightly upper bounded by , the well known cirelson quantity .  * acknowledgment * : it is a pleasure to thank prof .guruprasad kar for various simulating discussions and useful suggestions .we also thank dr .ashutosh rai for useful discussions .am acknowledge support from the csir project 09/093(0148)/2012-emr - i .n. mb thankfully acknowledges discussions with prof .sibasish ghosh during recent visit to the institute of mathematical sciences , chennai , india .bernhard wittmann , sven ramelow , fabian steinlechner , nathan k. langford , nicolas brunner , howard wiseman , rupert ursin , anton zeilinger , http://dx.doi.org/10.1088/1367-2630/14/5/053030[_new.j . phys . * 14 * , 053030 ( 2012 ) _ ]

Abstract: we study a recently proposed einstein - podolsky - rosen steering inequality [ http://arxiv.org / abs/1412.8178[_arxiv- 1412.8178 ( 2014 ) _ ] ] . analogous to clauser - horne - shimony - holt ( chsh ) inequality for bell nonlocality , in the simplest scenario , i.e. , 2 parties , 2 measurements per party and 2 outcomes per measurement , this newly proposed inequality has been proved to be necessary and sufficient for steering . in this article , using an equivalence between measurement incompatibility ( non joint measurability ) and steering , we find the optimal violation amount of this inequality in quantum theory . interestingly , the optimal violation amount matches with optimal quantum violation of chsh inequality , i.e. , cirelson quantity . we further study the optimal violation of this inequality for different bipartite quantum states . to our surprise we find that optimal violation amount is different for different -qubit pure entangled states , which is not the case for all other existing steering inequalities .


this paper is concerned with the axelrod model  for the dissemination of cultures , probably the most popular stochastic model of culture dynamics .the model includes explicit space in the form of local interactions using the framework of interacting particle systems : individuals are located on the set of vertices of a graph whose edges represent potential dyadic interactions .this work focuses on the one - dimensional lattice where each individual can only interact with her nearest left and right neighbors .individuals are characterized by their culture , a vector of opinions about various cultural features , rather than a single opinion like in the voter model  .specifically , the model is a continuous - time markov chain whose state at time  is a function with the integers  and  denoting respectively the number of cultural features and the common number of possible opinions per cultural feature .the dynamics is dictated by what has been identified as the two most important social mechanisms :  * * homophily * which is defined as the tendency of individuals to interact more frequently with individuals who are more similar and * * social influence * which is defined as the tendency of individuals to become more similar as the result of their interactions .note that the set of cultures is equipped with a natural distance : the function that counts the number of disagreements between two cultures .this distance is the key to modeling both homophily and social influence : homophily by assuming that neighbors interact at a rate that decreases with the distance between their cultures and social influence by assuming that the result of an interaction is to decrease the cultural distance between the neighbors .specifically , in the axelrod model , pairs of nearest neighbors interact at a rate equal to the fraction of cultural features for which they agree and , as a result of an interaction , one of the two neighbors chosen at random mimics the other neighbor for one of the cultural features for which they disagree ( if any ) .in particular , given that two neighbors disagree about exactly  cultural features , one given neighbor mimics the other one for one given cultural feature for which they disagree at rate the left - hand side is the probability that one given neighbor rather than the other one updates her culture times the probability that any of the  cultural features for which they disagree is the one chosen for update times the rate at which both individuals indeed interact , while the right - hand side is simply a convention based on the fact that , when both neighbors already agree on all cultural features , the interaction has no effect .therefore , letting and denoting the hamming distance by the dynamics is described by the transition rates for all  and opinion  .  *previous mathematical results * the axelrod model has been extensively studied both heuristically and through numerical simulations in the past 15 years while its analytical study has been initiated more recently in  .the main question that has been explored mathematically is the dichotomy between fluctuation and fixation , namely whether individuals change their culture infinitely often or a finite number of times .more precisely , we say that the system note that whether fluctuation or fixation occurs is very sensitive to the initial distribution .also , to fix the ideas , we assume from now on that the system starts from the most natural distribution : the product measure in which all the cultures are equally likely , i.e. , the first mathematical result about the axelrod model , established in  , states that the two - feature two - opinion system fluctuates and clusters : the proof is based on a coupling between the cultural dynamics and systems of annihilating random walks , such coupling being obtained by putting particles between neighbors to keep track of their disagreements . using this coupling together with delicate symmetry arguments modeled after the construction given in  , fluctuation and clusteringhave been extended in  to the process with any finite number of cultural features but again only two opinions per cultural feature .increasing the number of opinions , the particles that keep track of the disagreements can not only annihilate but also coalesce .in addition , regardless of the number of opinions , there are active particles that jump at a positive rate and mark the boundary between neighbors that can interact , and frozen particles that can not jump and mark the boundaries between neighbors that can not interact .the main result in  states that the system fixates whenever which is proved by counting and comparing the initial numbers of active and frozen particles in a large interval .some basic algebra shows that  holds whenever refining some of the arguments to obtain , it is also proved that the system with two cultural features and three opinions per feature fixates which , together with  , gives a complete picture of the model with two features .see figure [ fig : diagram ] for a summary of these results .the process has also been studied on the two - dimensional torus in  when both the number of cultural features and number of opinions per feature are large .there , it is proved that , for all , the system reaches a consensus with probability close to one on a giant connected component whenever that is , there is a connected component that covers a positive fraction of the graph and in which all the vertices share the same culture eventually .* variable number of opinions * this work is motivated by the simple observation that the assumption on the fixed number of opinions across cultural features seems unrealistic .just to give a concrete example , if there are two cultural features called politics and religion , there is no reason for the number of candidates at the next election to be equal to the number of possible religious beliefs . to define a more general model with a variable number of opinions, we simply assume that the state space of the process is given by the set of functions since the transition rates  do not depend on  , they again describe the dynamics of the model with a variable number of opinions . in particular , we now consider the model with state space and local transition rates and , and starting from the product measure with in this more general setting , the particles that keep track of the disagreements between neighbors again evolve like annihilating - coalescing random walks where active particles jump at rates that depend on the number of disagreements .the techniques we develop to study the generalized model requires active particles to all jump at the same rate , which is the case only when there are two cultural features . in this case , the results from  give a complete picture of the long - term behavior when the number of opinions per feature is constant : the system our main result extends the fixation region .   [ th : fixation ] assume that  and  .then , the system fixates .note that our theorem together with  gives a complete picture of the general system with two features and a variable number of opinions except when  .* structure of the proof * the rest of the paper is devoted to the proof of the theorem .first , we explain how to construct the process from a so - called graphical representation and give a rigorous definition of the system of particles that keeps track of the disagreements between neighbors .this system of particles being coupled with the cultural model , it can be constructed from the same graphical representation , which is also used to prove that it evolves according to a certain system of annihilating - coalescing random walks .the dynamics of this system of random walks has already been described in  but only heuristically .in contrast , we give a rigorous proof for each of the transition rates .in addition to the coupling between the axelrod model and annihilating - coalescing random walks , there are two key ingredients to prove our fixation result :  * the first ingredient is a construction due to bramson and griffeath  based on duality - like techniques to obtain an implicit condition for fixation in terms of the initial number of active and frozen particles in a large interval .their argument has been extended in  to systems where the state at each vertex is a vector , which is the result we use .* the second ingredient is a monotonicity relationship between the random number of collisions among the particles that keep track of the disagreements for some cultural feature and the number of possible opinions for this feature .this relationship is irrelevant to understand the standard model with a fixed number of opinions across cultural features but it is crucial in our general context where the number of opinions is variable .  having these two ingredients in hands , the rest of the proof is mostly technical .first , combining these ingredients with the law of large numbers , we deduce a weak version of our theorem : the one - dimensional process fixates when  .the estimates obtained to prove this weak version only account for the initial distribution assuming a worst case scenario for the realization of the system of particles .these estimates are then improved by also accounting for specific collision events which , together with the ergodic theorem , gives the full fixation result .the axelrod model falls into the general class of interacting systems considered in  therefore the process is well - defined and can be constructed starting from any initial configuration using a so - called graphical representation . in the case of the axelrod model ,this graphical representation consists of a random graph involving independent poisson processes marking the times of potential interactions and additional collections of independent bernoulli random variables and uniform random variables to determine the outcome of each interaction .more precisely , + for all pairs vertex - cultural feature  ,  * we let be independent rate one poisson processes , * we denote by the arrival time : , * we let be collections of independent bernoulli variables with * and we let be collections of independent .  then , at each time , we draw an arrow and call this arrow * active * if and only if in words , arrows in the graphical representation mark the times of potential interactions whereas active arrows correspond to the random subset of these arrows that indeed result in an interaction and an update of the system . in particular , the one - dimensional axelrod model can be constructed from the graphical representation above by setting which indeed produces the desired local transition rates in .the first key ingredient , introduced in  and improved in  , to study the one - dimensional axelrod model is a coupling between the cultural dynamics and a certain system of annihilating - coalescing random walks that keeps track of the disagreements between nearest neighbors . in this section ,we define this coupling and give a rigorous proof of the evolution rules of the system of random walks that has been described heuristically in  . to begin with , we visualize the culture of each individual as a column of  dots where the dot at level  can have  different colors corresponding to the  possible states for the  cultural feature .since the particles that keep track of the disagreements between nearest neighbors evolve on the set of edges rather than the set of vertices , it is convenient to identify edges with their midpoint and to define translations on the set of edges and vertices as follows : to keep track of the disagreements between neighbors , we then set this defines a spin system that we visualize by putting a particle at each pair edge - level which is in state 1 and we refer to figure [ fig : coupling ] for a picture of this coupling for the stochastic process with four cultural features .the number of particles per edge , defined as is a key quantity to understand the dynamics of this spin system since it counts the number of disagreements between neighbors which , in turn , is related to the rate at which these individuals interact . the next four lemmas give together a full description of the dynamics induced by our coupling on the spin system . to state these lemmas ,we introduce the notations the first lemma gives an expression of the probability that an arrow in the graphical representation is active using directly the spin system rather than the axelrod model .  [ lem : active ] for each pair edge - level  ,   first , we observe that hence the arrow  is active if and only if in particular , using that the poisson processes and random variables in the graphical representation are independent and that the graphical representation after time  is independent of the configuration at time  , we deduce that this completes the proof .   +  + in order to describe the dynamics of the spin system , the next step is to understand the effect of an active arrow on the particles .first , we note that which , in terms of the spin system , becomes in particular , we only need to determine whether the pairs  and  are empty or occupied just after the interaction .recall from  that , given the active arrow in the statement of the previous lemma , the pair  is occupied just before the interaction .the next lemma shows that the effect of this active arrow is to make the pair empty with probability one .  [ lem : vanish ] for each pair edge - level  ,   in view of the effect  of an active arrow , we have this completes the proof .+  + now , to determine whether the pair  is empty or occupied just after the occurrence of the active arrow , we distinguish two cases depending on whether this pair is empty or occupied just before the interaction . in the next lemma , we show that , in case the pair is empty just before the interaction , it becomes occupied .this , together with the previous lemma , indicates that , in this case , there is a jump of a particle in the direction of the active arrow .  [ lem : jump ] for each pair edge - level  ,   using , as in the previous lemma , the effect  of an active arrow together with the fact that the simultaneous occurrence of arrows in the graphical representation is a negligible event , we deduce that , given the conditioning in the statement of the lemma , with probability one , since in particular , it follows that which proves the lemma .+  + the last step is to determine whether the pair  is empty or occupied just after the interaction given that this pair is occupied by a particle just before the interaction , which we interpret respectively as a jump of a particle in the direction of the active arrow and a collision with another particle that may cause both particles to either * annihilate * or * coalesce*. the answer is simple when we know the background configuration of the axelrod model and we have but the problem is made challenging by the fact that the configuration of the spin system only gives us a partial knowledge of the configuration of the axelrod model .however , using duality - like techniques and the fact that the initial states are independent , we can prove that successive collisions result independently in either annihilation or coalescence with some probabilities that can be computed explicitly , which is done in the next lemma .for an illustration of some of the arguments in the proof , we refer the reader to figure [ fig : paths ] .  [ lem : outcome ] for each pair edge - level  ,   this is similar to the proof of ( * ? ? ?* lemma 3 ) . to begin with, we define active paths in order to keep track of the origin of an opinion going backwards in time : we say that there exists an * active -path * from point  to point  whenever there exist such that the following two conditions hold :  1 .  for all ,there is an active arrow .2 .  for all , and , there is no active arrow .we write this event  and observe that in addition , recalling  and using a simple induction , we have so we write  and call  the * ancestor * of  for the  cultural feature . to prove the lemma using the concept of active path ,the first ingredient is to observe that , due to one - dimensional nearest neighbor interactions , active paths at the same level  can not cross each other , so the dynamics preserve the order of the ancestors at each level : moreover , given the conditioning in the statement of the lemma , there is one particle on each side of vertex  at level  just before time  from which it follows that this implies that the inequalities in  are strict : showing in particular that all three ancestors are different . combining  and using that the initial states are independent anduniformly distributed imply that the second ingredient is to observe that , given again the conditioning in the statement of the lemma and using the same argument as in the proof of lemma [ lem : jump ] , we have from  , it follows that whether the pair  is empty or occupied at time  is an event independent of the realization of the spin system up to time  and that this completes the proof .+  + in conclusion , combining lemmas [ lem : active][lem : outcome ] and obvious symmetry implying that each of these lemmas extends to arrows directed to the left rather than to the right , we obtain the following description of the spin system  which basically consists of the superposition of non - independent systems of one - dimensional annihilating - coalescing symmetric random walks :  * the particles at edge  jump independently at the same rate  one unit to the left or one unit to the right . in particular , * * in case there are  particles at  , they can not jump so + we call these particles * frozen * particles and the edge a * blockade * at time  .* * in case there are less than  particles at  , they jump at a positive rate so + we call these particles * active * particles and the edge a * live edge * at time  .* when a particle jumps onto a pair edge - level  which is already occupied , both particles annihilate or coalesce independently of the past with respective probabilities   see figure [ fig : dynamics ] for an illustration of the dynamics .as previously mentioned , in addition to the coupling with annihilating - coalescing random walks , the first ingredient to prove the theorem is a construction due to bramson and griffeath  based on duality - like techniques to obtain an implicit condition for fixation in terms of the initial distribution of active and frozen particles in a large interval . in this section , we briefly recall their construction and derive a condition for fixation of the axelrod model following  .  [ lem : fixation ] for all , let then , the system fixates whenever   this follows exactly the proof of ( * ? ? ? * lemma 4 ) .+  + to make this condition for fixation more explicit , the idea is to study the connection between the initial configuration of the system and the key event that appears in .more precisely , we prove that , on the event  , there is an arbitrarily large random interval such that all the blockades initially in this interval must have been destroyed by either active particles initially in this interval or active particles that result from the destruction of these blockades . to make this construction precise , we let be the first time an active -path that originates from the interval hits the origin and observe that the event  can be written as note that , though active paths at the same level can not cross each other , active paths at different levels can . to construct an interval where the initial blockades can not be destroyed by active particles that originate from outside this interval, we need to extend our definition : we say that there is a * generalized active path * from  to  whenever there exist such that the following two conditions hold :  1 .  for all ,there exists  such that .2 .  for all , and , there is no active arrow .we write this event  and observe that , due to one - dimensional nearest neighbor interactions , generalized active paths can not cross each other . to construct the random interval mentioned above given the event  , we define the random variables and set  .this construction implies that  * all the blockades initially in  must break , i.e. , each pile of frozen particles is turned into a smaller pile of active particles due to an annihilating event , by time  . *the active particles initially outside  can not jump inside the space - time region delimited by the two generalized active paths defined implicitly in  since , due to one - dimensional nearest neighbor interactions , generalized active paths can not cross each other .in particular , given the event  , all the blockades initially in the interval  must have been destroyed by either active particles initially in this interval or active particles that result from the destruction of these blockades . to keep track of and count these particles ,we attribute a weight to each edge based on the number of particles it carries initially . to begin with , we give an arbitrary weight , say weight  , to each particle initially active by setting to define the weight of a blockade , we observe that , before the blockade breaks due to an annihilating event , which occurs almost surely on the event  , a random number of active particles have disappeared due to successive coalescing events with the blockade .moreover , the destruction of this blockade results in  frozen particle becoming active so we let and define the weight of a blockade initially at  as the fact that the occurrence of  implies that all the blockades initially in  must have been destroyed by either active particles initially in this interval or active particles that result from the destruction of these blockades can then be written as which will be used in the next sections to prove fixation .starting from this section , we focus on the case  where the process is referred to as the two - feature axelrod model .motivated by the results of the previous section , the main objective is to find a stochastic lower bound for the number of collisions of an active particle with a blockade before this one breaks , and therefore a lower bound for the weight of a blockade . to begin with ,we observe that , according to lemma [ lem : outcome ] , the number of collisions different particles undergo before they annihilate are independent geometric random variables whose success parameter only depends on the number of states at the corresponding level . in particular , assuming that  is initially a blockade and letting  be defined as in , we have for  where for , are independent , and where  means stochastically larger than . to improve the lower bound, the idea is to show that the density of active particles , and by translation invariance the number of collisions per edge per unit of time , at level 1 is always above the density of active particles at level 2 . from this key result, we will deduce that the number of collisions to break a blockade is stochastically larger than a certain convex combination of the geometric random variables  and  . to make the argument rigorous, we let and observe that , since the initial distribution and the graphical representation are translation invariant , these functions do not depend on the choice of  and are well - defined . the next lemma gives a monotonicity property between the densities and the number of states .   [ lem : density ] assume that . then , for all .  since there is a frozen particle at  at one level if and only if there is a frozen particle at  at the other level , the density of frozen particles is the same at both levels at all times : in view of the initial distribution of the system , we also have which , together with , implies that the inequality to be proved holds at time 0 .now , assume by contradiction that this inequality is not true at some time  .since  is nonincreasing because the particles can only coalesce or annihilate , and since for all times  small the function  is continuous . in particular, it follows from our assumption and the intermediate value theorem that there exists a time  such that in words , the density of particles at time  is the same at both levels , therefore the density of active particles is the same at both levels and the density of frozen particles is the same at both levels according to .since in addition all the active particles jump at the same rate when there are only two levels , the expected number of collisions per unit of time at a given edge is also the same at each level . butaccording to lemma [ lem : outcome ] , because  , the collisions at level 1 are less likely to result in annihilation and more likely to result in coalescence , and therefore remove in average less particles , than the ones at level 2 .in particular , there exists  such that in contradiction with . in conclusion ,the density of particles is always larger at level 1 than at level 2 and the lemma follows by using  once more .+  + combining lemmas [ lem : outcome ] and [ lem : density ] , we can now prove the main result of this section , which improves the stochastic lower bound in  when the number of states are different : .  [ lem : collisions ] assume that and . then ,   first , we let be the number of collisions the particle originally at undergoes before it annihilates . in view of lemma [lem : outcome ] , each collision of two particles results independently in their annihilation with probability  therefore in distribution : in addition , since the density of active particles at level 1 is larger than the density of active particles at level 2 according to lemma [ lem : density ] , that the active particles all jump at the same rate , and that the distribution of particles is translation invariant , the number of collisions with the blockade at  before it breaks is stochastically larger at level 1 than at level 2 , that is , the lemma directly follows from .in this section , we combine lemma [ lem : collisions ] and  to prove a weak version of the theorem . another key to obtaining an explicit condition for fixationis the use of the law of large numbers in order to show that the asymptotic probabilities of the event in  can be studied by simply looking at whether the expected value of the contribution of a typical edge is positive or negative .this idea will be used again in the next section together with the ergodic theorem as well as additional arguments to prove the full theorem . to begin with ,we let be independent . recalling , we deduce from lemma [ lem : collisions ] that  * almost surely when and * in distribution when .this , together with , implies that where for all  and  to state our next lemma , we also introduce the probabilities   [ lem : lln ] we have the convergence   in view of the initial distribution , we have from which it follows that the initial number of blockades in a given finite interval is a binomial random variable with success probability  .therefore , by the law of large numbers , similarly , the initial number of active particles in a given finite interval is a binomial random variable with success probability given by therefore the law of large numbers implies that since the geometric random variables  are independent , the law of large numbers again applies , from which it follows that we have the almost sure convergence combining and  and using that the geometric random variables attached to a given edge are independent of the event that this edge is initially a blockade , we deduce finally , we combine  and  to conclude almost surely .this completes the proof .  [ lem : weak - fixation ] the system fixates whenever  .assume that  .then , in particular , applying lemma [ lem : lln ] , we get almost surely , which implies that this , together with lemma [ lem : fixation ] and , implies the lemma .this last section is devoted to proving the theorem .to explain the idea behind the proof , note that our definition of weight is based on the worst case scenario where active particles _ do their best _ to destroy the blockades and turn as many frozen particles as possible into active particles .this assumption ignores in particular annihilating events involving two active particles , coalescing events involving two active particles , and events where two active particles at different levels move towards each other to form a blockade . to prove the full theorem, we now take these events into account .more precisely , we define the following collections of events : the next lemma gives an improvement of  taking into account these events .note that the random variable is the one on the right - hand side of , which is obtained based on the worst case scenario where all the active particles first hit a blockade rather than coalescing or annihilating with another active particle or forming a blockade with another active particle .the two sums in  are correction terms taking these events into account . to quantify these corrections , note that in particular , the random variables are stochastic lower bounds for the fraction of blockade formations and the fraction of collisions of active particles that occur in the space - time region delimited by the two generalized active paths defined implicitly in .since , when two active particles form a blockade , their total weight can be replaced by the weight of a blockade , the correction term due to blockade formations is in distribution .in addition , when two active particles collide , their total weight can be decreased by the weight of either one or two active particles depending on whether the collision results in a coalescing or an annihilating event , respectively .recalling also from lemma [ lem : outcome ] that the geometric random variable  counts the number of collisions at level  until the first annihilating event occurs , we deduce that the correction term due to collisions of active particles at level  is in distribution .the lemma directly follows from  and .   +  + in the next two lemmas , we compute the limits as  of the random variables on the right - hand side of inequality . these lemmas can be seen as the analog of lemma [ lem : lln ] . to state these two lemmas ,we introduce the probabilities    for all  and  , we define the events which are measurable with respect to the initial configuration of the system , as well as the events measurable with respect to the graphical representation where time  is the first time one of the following eight arrows occurs in the graphical representation .note that note also that the events in  attached to adjacent edges are not independent so the law of large numbers no longer applies .however , since the initial distribution is the uniform product measure and since the poisson processes in the graphical representation are independent , we can apply the ergodic theorem and the same argument as in the proof of lemma [ lem : lln ] to get almost surely for  .in addition , in view of lemma [ lem : outcome ] and since the graphical representation is independent of the initial distribution , for each edge  , the random variables this , together with and  , implies that almost surely .similarly , we prove that the additional factor  is the probability that edge  is initially empty while the  comes from the requirement in the graphical representation : after one of the two active particles jumps , we still need the next update to be the other active particle moving to edge  , which occurs if and only if one among eight arrows occurs first .the lemma follows by adding .   using the events introduced in  and applying again the ergodic theorem as in the previous lemma , we first prove that almost surely for  . using again and instead of  together with the independence of the random variables, we deduce that almost surely , and for the same reasons as in lemma [ lem : ergodic - frozen ] , almost surely .the lemma follows from .    using the same argument as in the proof of lemma [ lem : weak - fixation ] together with lemmas [ lem : improve][lem : ergodic - active ] , we deduce that the system with parameters  and  fixates whenever but a straightforward calculation shows that , when  , this shows the lemma and completes the proof of the theorem .

Abstract: the axelrod model is a spatial stochastic model for the dynamics of cultures that includes two key social mechanisms : homophily and social influence , respectively defined as the tendency of individuals to interact more frequently with individuals who are more similar and the tendency of individuals to become more similar when they interact . the original model assumes that individuals are located on the vertex set of an interaction network and are characterized by their culture , a vector of opinions about  cultural features , each of which offering the same number  of alternatives . pairs of neighbors interact at a rate proportional to the number of cultural features for which they agree , which results in one more agreement between the two neighbors . in this article , we study a more general and more realistic version of the standard axelrod model that allows for a variable number of opinions across cultural features , say  possible alternatives for the cultural feature . our main result shows that the one - dimensional system with two cultural features fixates when  .