dynamical decoupling is a highly successful strategy to protect quantum systems from decoherence .its particular strength is that it is applicable even if the details of the system - environment coupling are unknown .historically dynamical decoupling dates back to pioneering work in nuclear magnetic resonance ( nmr ) by u. haeberlen and j. s. waugh . in order to increase the resolution in nmr spectroscopy, pulse sequences were developed that coherently average out unwanted interactions .prominent examples are spin - echo techniques , such as the famous hahn echo , allowing us to measure relaxation times through applying a sequence of rotations on a spin and detecting the echo signal . in the context of suppressing decoherence and quantum information theory ,the theoretical framework was developed by l. viola and s. lloyd in the late 90 s . over the yearsthe efficiency of various decoupling schemes was studied and improved for several environmental models in .many experiments , such as , demonstrate the applicability of dynamical decoupling in an impressive way by prolonging coherence times a few orders of magnitude . additionally dynamical decoupling can be combined with the implementation of quantum gates , which makes it a viable option to error correction . however , when it comes to infinite - dimensional quantum systems , such as quantum harmonic oscillators , a general framework for dynamical decoupling is missing in the literature .a first step towards this direction was done in by investigating a specific system - environment model and identifying an operation that allows to suppress decoherence .  in this articlewe consider a broader class of continuous variable systems by investigating dynamical decoupling for systems that are described by hamiltonians quadratic in the operators and  .these hamiltonians are of particular importance since they describe a wide range of continuous variable systems and their main sources of decoherence . for instance , quadratic hamiltonians describe linear quantum optical systems with applications in optical quantum computing and quantum metrology , the vibrational modes of an ion chain with nearest - neighbour interactions and in general harmonic crystals .moreover , many continuous variable systems can be described by quadratic hamiltonians in a certain approximate regime , for example opto - mechanical systems and nano - mechanical oscillators by linearizing interactions .our article is organized as follows .we start with the question of the existence of decoupling in finite and infinite dimensions .we then introduce dynamical decoupling for quadratic hamiltonians and arrive at procedures we call _ homogenization _ and _ decoherence suppression for environment - coupling_. the desired dynamics is achieved by rapidly swapping coordinates or rapidly rotating the system , respectively , see the blue arrows in figure [ fig : oscillators ] .again we discuss their existence for given quantum systems .finally we introduce a randomized scheme for these procedures and derive explicit analytic approximations and bounds for the gate error , i.e. , the discrepancy from the idealized time - evolution of infinitely fast operations .the proofs are very technical and therefore deferred to the appendix .we illustrate and confirm the usefulness of these formulae with typical numerical examples .illustration of the homogenization procedure of a system of coupled oscillators ( a c , see theorem [ th3 ] ) and suppression of decoherence of a single oscillator in a heat bath ( d f , see theorem [ th4 ] ) . ]before we develop dynamical decoupling for quadratic hamiltonians in infinite - dimensional quantum systems , we first review the concept of dynamical decoupling for finite - dimensional quantum systems , focussing on the group - based approach .consider a finite - dimensional quantum system , say of dimension , with hilbert space and hamiltonian .the idea of dynamical decoupling is to rapidly rotate the quantum system by means of classical fields in order to average the system - environment coupling in to zero .this can be achieved by modifying the dynamics applying unitary _ decoupling operations _ instantaneously in time steps , which are taken from a decoupling set .such a _ decoupling set _ is a finite group of unitary matrices such that , for every , with some depending on .it can easily be seen that .in particular , for traceless we have .these decoupling operations can be applied according to a fixed deterministic scheme or randomly from the set .let us look for simplicity at the fixed scheme .we then get the modified dynamics of time - evolution up to time : where denotes the standard operator norm on .condition guarantees that and ensures the cancellation of the modified unitary time - evolution operator in first order in , while higher orders can be neglected under this assumption .therefore , for , the modified time evolution becomes up to a global phase effectively the identity .we refer to for a detailed definition of dynamical decoupling .the first question here is when a decoupling set can actually be found .[ th1 ] for every quantum system of finite dimension , there exists a decoupling set , for example the group of unitary matrices with entries in .notice that a decoupling set for a given quantum system is not unique , in general .we first notice from that , for every , lies in the commutant of , which follows from the group property and finiteness of .thus if acts irreducibly on then this commutant consists of alone .we therefore have to construct that acts irreducibly .we claim that does this .the elements in are those unitary matrices which have exactly one non - zero entry per row and per column , and this entry is either or .it is easily seen that forms a finite group .given an arbitrary , with -th component non - zero , we can take the diagonal matrix with on all diagonal entries except for at the -th position. then and , where is the -th canonical basis vector .all permutation matrices of ( matrices that permute coordinate entries w.r.t. the canonical basis of ) are contained in , and so applying a suitable permutation matrix to , we can get a multiple of an arbitrary basis vector , , hence every vector in through linear combination .this shows that acts irreducibly and transitively on , and thus proves our claim .let us look at some simple examples .for a single qubit such a decoupling set may be chosen more explicitly as the pauli group , whereas for a -qubit system , it may be chosen as combinations of the pauli spin operators on the tensor factors .the size of the decoupling set scales exponentially with the number of qubits of the considered system .this makes deterministic decoupling schemes inefficient for large quantum systems , since the modified dynamics is obtained through taking the decoupling operations one after the other deterministically from .clearly , in order to suppress decoherence that is induced by a generic system - environment hamiltonian with and being hermitian system and environmental operators respectively , it is enough to act with on the system alone . for traceless system operators , in the limit of infinitely fast decoupling we obtained a dynamics that is decoupled from the environment , i.e. .this makes it possible to fully suppress decoherence , independently of the specific form of the system - environment interaction that is present .as long as the system is finite - dimensional , we note that this is even true in the case of infinite - dimensional environments described by some unbounded operators and as in , if we make certain plausible domain assumptions .the question arises if , similarly to the finite - dimensional case , in an arbitrary infinite - dimensional setting decoherence can always be suppressed through dynamical decoupling .before we address this question , we begin with an example , which is , to the best of our knowledge , the only study in the literature where dynamical decoupling is investigated for a specific infinite - dimensional system .the model that was considered in consists of a single harmonic oscillator that interacts with an environment of harmonic oscillators .the system - environment interaction is given by where and are bosonic creation and annihilation operators of the system and the environmental oscillators respectively and are real coupling constants determining the strength of the interaction with each environmental oscillator .we note that can be brought into the form by expressing the annihilation and creation operators through and .it was pointed out in that through a decoupling operation , which corresponds to a phase space rotation of the system oscillator around , the sign in front of can be reversed .this makes it possible to suppress such a system - environment interaction if we apply the decoupling operation reasonably fast .the efficiency of this procedure was studied in detail in in terms of spectral properties of the environment . for our purposeswhat is important is the observation that such an interaction can always be suppressed with a single operation .we then may ask if we can identify a decoupling set that allows us to suppress decoherence for arbitrary infinite - dimensional systems .the most natural infinite - dimensional meaning of decoupling would be the following : for an infinite - dimensional quantum system with hilbert space and bounded operators , a decoupling set is a finite subgroup such that with some depending on , is satisfied for all .[ th0 ] an infinite - dimensional quantum system has no decoupling set .consider an infinite - dimensional system with hilbert space and suppose there is a decoupling set . then considering a rank - one projection , we see that has rank 1 again , and hence that the left - hand side of has rank at least 1 ( because sum of positive elements ) and at most ( because each of the summands has rank 1 ) . on the other hand ,the right - hand side has either rank 0 ( if ) or .this is a contradiction , so no such exists .let us make a few remarks :  * without contradicting theorem [ th0 ] , it could still be possible to achieve for specific hamiltonians rather than all .* it can be shown that interactions containing system operators with an unbounded positive spectrum can never be suppressed through applying unitary decoupling operations infinitely fast . *it seems worth studying an infinite compact version of decoupling sets , which is work in progress .  as dynamical decouplingdoes not work for generic infinite - dimensional quantum systems , we decided to investigate in more detail the specific class of quadratic hamiltonians and to adjust the decoupling condition to suit that setting in a meaningful way .this allows us to represent the dynamics by a symplectic transformation on a _ finite - dimensional _ space .additionally this has the advantage that we can avoid the mathematical subtleties arising in infinite - dimensional spaces and the related problems in the characterization of the relevant time - scales .we consider an -mode bosonic system described by pairs of quadrature operators and acting on an infinite - dimensional hilbert space and satisfying the canonical commutation relation =i\delta_{i , j}{\mathds{1}}_{\mathcal{h}} ] where is the symplectic form .we note that a suitable basis change leads to but we will use the former choice of basis except otherwise mentioned .we are interested in quantum systems that are described by a quadratic hamiltonian of the form with being a real and symmetric matrix .the corresponding unitary time - evolution operations are the so - called gaussian operations since they preserve the gaussian character of quantum states .if we consider the heisenberg evolution of the quadrature operator we obtain where belongs to the symplectic group . restricting to quadratic hamiltonians, we see that there is a one - to - one correspondence between the time evolution operators acting on an infinite - dimensional hilbert space and the finite - dimensional matrices , which allows us to reduce to a finite - dimensional setup . for further details regarding quadratic hamiltonians and symplectic transformationswe refer to .now we want to introduce dynamical decoupling within the framework of symplectic transformations .while usually dynamical decoupling is introduced within quantum control theory using averaged hamiltonians we consider here a decoupling sequence that captures the main aspects of dynamical decoupling .we remark that a formulation within averaged hamiltonian theory in terms of symplectic transformation is straightforward .here we instantaneously apply symplectic decoupling operations from a finite group , in time steps , with being the number of elements in .if this is done deterministically in a fixed periodic cycle running through all of , call it with , then the correction operation applied instantaneously at time is , and the modified dynamics at becomes if we repeat this cycle times and shrink the time steps by a factor , we obtain using a generalized trotter formula , we get analogously to the unitary case we can define a tentative symplectic decoupling condition for : with some depending on .however , using the relation , valid for every , we can rewrite as considering e.g. a positive ( hence nonzero ) in this equation , we see that the left - hand side being a sum of positive operators must be positive again , whereas the right - hand side is antisymmetric , which is impossible .thus the tentative decoupling condition can not be realized .it reflects for exmaple the fact that a system of non - interacting harmonic oscillators can not be stopped rotating .as already pointed out in the previous section , dynamical decoupling in the usual sense can not work for every infinite - dimensional system , and apparently also for quadratic hamiltonians this is not possible in this strict sense .however , maybe we are demanding too much .since we can not decouple harmonic oscillators , a natural relaxation of condition may be written as follows :  a finite subgroup is called a _homogenization set _ if , for every symmetric , we have or equivalently with some depending on .  in words : instead of requiring that the system does not evolve anymore , we now require that a set of harmonic oscillators do not interact with each other but rotate all with the same frequency after we have applied symplectic operations infinitely fast .we call this processwe note here that the homogenization procedure is similar to symmetrization of quantum states developed in .it remains to identify a set of symplectic operations that satisfies the homogenization condition .[ th3 ] for every , a homogenization set for exists , for example   first of all , it is clear that and commute , so , for every , and .thus .we decompose according to the product into four blocks , where and .we can then write as . \end{aligned}\ ] ] now we recall from the proof of theorem [ th1 ] that the group acts irreducibly on , so it forms a decoupling set for in the sense of . applying to each block, we find which completes the proof and we identify as an averaged frequency .the elements in are signed permutation matrices " . in words: through rapidly swapping the coordinates of the oscillators , we can map a set of interacting harmonic oscillators onto non - interacting oscillators rotating with an averaged frequency .so far , the correction operations are assumed to be applied in the form of infinitely strong instantaneous pulses .this is certainly idealised and not very physical , yet generally taken for granted when discussing the theory of dynamical decoupling as it is a reasonable approximation facilitating most computations .if we explicitly want to work with finite - energy operations , a typical alternative would be so - called _ eulerian dynamical decoupling _ . in that case , the decoupling operations are implemented by a continuous path in the automorphism group of the system rather than a discontinuous path which jumps between the elements .let us describe the idea briefly , following but adapted to the context of homogenization .we refer to future work for details of the construction , proofs , and further consequences .let be a homogenization set , i.e. , a finite subgroup of as above , and a minimal generating set for . consider the cayley graph , which has as vertices and as directed edges .this means that every vertex has incoming and outgoing edges .an eulerian cycle is a cycle through this graph which travels precisely once through each edge according to its orientation , and hence times through each vertex , and with total length .one can prove that such a cycle exists .let us fix a total time and write for the time difference between consecutive operations , i.e. , between consecutive vertices .let now be such an eulerian cycle , which we write as an ordered tuple and we extend this periodically to , so . then at time , the operation is applied instantaneously , so that the total time evolution in becomes   so far , we have just modified the order of the correction operations , not their energy .now instead suppose that the operation is not applied instantaneously at time but rather obtained as a continuous curve from to .such a map is constructed as follows : let us recall , e.g. from , that every element can be decomposed as with certain .then we can define { \rightarrow}{\operatorname{sp}}(2n,{\mathbb{r}}),\quad \tilde{\gamma}(s ) = \left\ { \begin{array}{l@{\ : : \:}l }  e^ { sy } & s\in[0,\tau/2]\\ e^{(s-\frac{\tau}{2})x } e^{\frac{\tau}{2}y } & s\in(\tau/2,\tau ] .\end{array } \right.\ ] ] and we have and . then changes to a time - ordered product , and in first order expansion this becomes with some possibly depending on everything in the second line , as follows from the homogenization condition for .  to summarize : if is a homogenization set generated by a minimal set then the modified procedure using a continuous implementation of eulerian cycles of length yields homogenization again but with finite - energy correction operations .now we want to come back to our initial motivation , the suppression of decoherence induced by generic quadratic system - environment interactions .the previous homogenization condition describes a mapping of interacting harmonic oscillators onto non - interacting oscillators that rotate with an averaged frequency .hence in that condition the suppression of the interactions between the oscillators is included and it can be achieved by applying the operations infinitely fast to the whole system . in generalwe have no access to the environment and therefore we now want to formulate a condition that allows us to suppress the system - environment interactions if we act on the system alone .we first partition the total system into the system of interest ( s ) with oscillators and the environment ( e ) with oscillators , so , noticing that for symplectic dynamics we have a direct sum structure of the underlying space .switching to the basis in , we thus write where , are symmetric matrices describing the uncoupled dynamics of s and e , and describes the interactions between system and environment .now , if we apply the decoupling operations only to the system , namely of the form , we obtain in the limit of infinitely fast decoupling a dynamics governed by obviously , in order to suppress the interaction with the environment , we need a group satisfying the simplest such group we can imagine is given by , and we notice that it leaves the system dynamics invariant . thus , i.e. , .the two operations in correspond to `` no - rotation '' and a global -rotation of the system oscillators .it shows that the operation from , introduced in the beginning , allows us to decouple arbitrary quadratic system - environment interactions too .this is not really surprising , since in the unitary picture we can always reverse the sign in front of interaction parts of the form and by applying . herehowever we want to emphasize two things .first of all , in contrast to finite - dimensional systems , the system can always be decoupled from the environment using two operations , independent of how big the system or the environment is .second , for finite - dimensional systems , on the one hand the irreducible action of the decoupling set suppresses all interactions with the environment , while on the other it modifies the system dynamics in such a way that it is ( up to a global phase ) given by the identity .for continuous variable systems described by quadratic hamiltonians instead we can always suppress the interaction with the environment without disturbing the system dynamics at the same time .let us summarize this as follows :  [ th4 ] suppose a system of oscillators couples to an environment of oscillators and is such that the total time - evolution is described by a quadratic hamiltonian .then the interaction can be decoupled completely without influencing the system s internal dynamics , by choosing the decoupling set   remark : since the homogenization set satisfies eq .we can combine both procedures .the system can be homogenized while suppressing the interaction with the environment .moreover , this can be adjusted again to eulerian cycles and finite - strength correction operations .up to now we have discussed how we can achieve homogenization and the suppression of decoherence for quadratic hamiltonians in the limit of infinitely fast operations .clearly , in practice this limit is not attainable meaning that non - zero orders in enter the dynamics . throughout this article denotes the standard operator norm on ( or some other operator algebra depending on the context ) . in the following we provide bounds , characterizing how well dynamical decoupling works for continuous variable systems if the decoupling operations are applied reasonably though not infinitely fast .we discuss here the case of homogenization , while decoherence suppression can be treated analogously .typically error estimates for dynamical decoupling are obtained by estimating the higher orders of the magnus expansion , the dyson series or the trotter formula .here we consider a random dynamical decoupling scheme and use a central limit theorem developed in in order to obtain the description of the modified time evolution as a stochastic process in .the idealized though impossible time evolution would result from ; it would be given by the ( non - random ) function while has to be replaced by in if we want to study decoherence suppression instead .we would then like to find an upper bound for the expectation value of the gate error .more precisely , we consider the case in which symplectic decoupling operations at time , with , are taken independently and uniformly random from as in such that the dynamics is modified according to at time .again , and would have to be replaced by and if we want to study decoherence suppression rather than homogenization .the dynamics is now described by a random walk on the symplectic group . in order to apply analytical tools, we would like to approximate this by a continuous - time stochastic process .this idea has been realized in for the case of dynamical decoupling .a similar treatment for homogenization or decoherence suppression , as shown in appendix [ app ] , leads to a limit stochastic process if then this is a very good approximation of the actual time evolution .the quantity we would like to investigate then is the expectation of the _ gate error _ : where denotes the hilbert schmidt norm and the idealized time evolution given by .we find :  [ th5 ] for , i.e. , if homogenization or decoupling operations are applied sufficiently fast , the expected gate error for homogenization behaves as \approx  2\tau t \| a-\pi(a)\|_2 ^2 \le 16\tau t n \|a\|_\infty^2 .\end{aligned}\ ] ] in the case of decoherence suppression we get \approx 2\tau t \| a-\tilde{\pi}(a)\|_2 ^ 2 \le 16\tau t n_{s}n_{b } k^{2 } , \end{aligned}\ ] ] with being the greatest absolute value of system - environment coupling entries in the matrix in .  for a proof and more general treatment of bounds we refer to appendix [ app ] .moreover , it would be interesting to study an adaption of the random procedure to eulerian cycles as introduced in the context of homogenization , and to obtain explicit error bounds . although we expect this to be feasible and meaningful, it goes far beyond the scope of the present article and provides interesting future work .\(a ) +  numerical simulation of the averaged gate error for random dynamical decoupling . for homogenization ( a ) as function of the temporal spacing ( on an inverse double logarithmic scale ) between the decoupling operations and for the suppression of the system - environment interaction ( b ) as a function of the number of environmental oscillators for fixed .the average was taken over 20 trajectories for a total time .( a ) : for a system of 4 interacting harmonic oscillators described by a randomly chosen matrix with entries bounded from above by ( black triangles ) and by ( blue squares ) .the solid lines represents the corresponding analytic approximation depending on given by .( b ) : for a system of two interacting harmonic oscillators coupled to an environment consisting of oscillators ( black triangles ) .the matrix describing the total system was chosen randomly with entries between 0 and 0.1 .the solid lines show the more precise approximation depending on ( black line ) and the more general upper bound ( grey line ) in , with .,title="fig : " ] + ( b ) +  numerical simulation of the averaged gate error for random dynamical decoupling . for homogenization ( a ) as function of the temporal spacing ( on an inverse double logarithmic scale ) between the decoupling operations and for the suppression of the system - environment interaction ( b ) as a function of the number of environmental oscillators for fixed .the average was taken over 20 trajectories for a total time . (a ) : for a system of 4 interacting harmonic oscillators described by a randomly chosen matrix with entries bounded from above by ( black triangles ) and by ( blue squares ) .the solid lines represents the corresponding analytic approximation depending on given by .( b ) : for a system of two interacting harmonic oscillators coupled to an environment consisting of oscillators ( black triangles ) .the matrix describing the total system was chosen randomly with entries between 0 and 0.1 .the solid lines show the more precise approximation depending on ( black line ) and the more general upper bound ( grey line ) in , with .,title="fig : " ] +  in order to illustrate and confirm the meaningfulness of our results , especially of theorem [ th4 ] , let us now look at simulations for some standard models .  in figure[ fig : decouplingbounds ] we studied the validity of our analytic approximation in theorem [ th4 ] by numerically evaluating the gate error for homogenization ( a ) and for decoherence suppression of the system - environment interactions ( b ) . in both plots we evaluated the gate error by taking the average over 20 trajectories that were obtained according to for a total time . for ( a )the decoupling operations were taken independently and uniformly random from given by and the expectation of the gate error was studied as a function of .we investigated homogenization for interacting harmonic oscillators described by a randomly chosen matrix with entries between and ( blue squares ) and and ( black triangles ) . in figure [ fig : decouplingbounds ] b ) we studied the suppression of the system - environment interaction for a system of interacting harmonic oscillators as a function of the number of environmental oscillators for fixed and taking the decoupling operations uniformly random from .the matrix describing the total system was chosen randomly with entries between and . in both figuresthe solid lines represent the corresponding analytic expressions and .remarkably , even for they describe very well the efficiency of the random decoupling scheme in terms of the temporal separation of the decoupling operations , the total evolution time and the entries of .exemplarily , a rough estimate can be given for optomechanical microresonators . herethe frequency of the mechanical oscillator and the cavity mode are in the mhz regime and couplings of the order of khz can be achieved between the mechanical oscillator and the cavity mode .the bound suggests that for such systems is required for homogenization of the motion of the mechanical oscillator and the cavity mode for a total time .a time - scale analysis for decoherence suppression , for instance for a mechanical oscillator interacting with a bath of phonons , requires a microscopic model of the underlying decoherence mechanism .particularly the order of the interaction between the mechanical oscillator and its environment or the spectral density of the environment need to be known .we have studied dynamical decoupling for continuous variable systems that are described by quadratic hamiltonians .we first proved that dynamical decoupling can not work for every infinite - dimensional system .thus , in contrast to finite - dimensional systems , not every infinite - dimensional system can be protected from decoherence using dynamical decoupling .using the framework of symplectic transformations we afterwards investigated in more detail dynamical decoupling for quadratic hamiltonians .we identified a condition and a set of operations that allows us to map a set of interacting harmonic oscillators onto non - interacting oscillators rotating with an averaged frequency .we called this process homogenization .moreover we showed that every quadratic system environment - interaction can be suppressed with two simple operations without modifying the system dynamics at the same time . using a random dynamical decoupling scheme we developed bounds characterizing how efficient both schemes are .we found that the efficiency depends on the temporal spacing of the decoupling operations , the total evolution time and the energy constants characterizing the considered system .numerical simulations confirm the meaningfulness of the developed bounds .our results pave the way for protecting an infinite - dimensional quantum system from decoherence which is induced by quadratic interactions .the reduction of noise in such systems is likely to find several applications .for instance , dynamical decoupling has the potential to decrease the environmentally induced errors in optical quantum computing and quantum metrology .moreover , dynamical decoupling for continuous variables might assist in verifying collapse models in the macroscopic superposition regime . in particular , the reduction of noise caused by the interaction with the environment through dynamical decoupling might make the small derivations from the usual schrdinger dynamics more visible .the proof of theorem [ th4 ] is similar in spirit to the diffusion limit theorem in , and we refer to that article for notation , motivation and procedure , in order to keep the present exposition succinct .we focus on the homogenization , and comment on decoherence suppression , which is treated analogously , towards the end .  to start with, we have to define the distribution of the increments in our random walk .a suitable choice is this constitutes a family of measures on the lie group with standard borel -algebra , such that is the increment distribution of the actual time evolution operator to , resulting from the instantaneous random homogenization operations as in .we then consider the measures : an application of ( * ? ? ?* th.3 ) , cf .also , shows that they converge to a normal distribution on as , and converges to a gaussian process on , with distribution such that .the corresponding ( dual ) contraction semigroup has infinitesimal generator the process can not be expressed explicitly , but the expectation values of its matrix elements and higher moments can , thanks to the expression for : we can apply it to the functions and where forms an orthonormal basis of .more precisely , we get = & { \mathbb{e } } [ \|s_0(t)-s^{(\infty)}(t ) \|_2 ^ 2 \nonumber ] \\ \nonumber = & \sum_{k , l=1}^{2n } { \mathbb{e}}\big [ |\langle e_k , s^{(\infty)}(t)(e_l)\rangle|^2  + |\langle e_k , s_0(t)(e_l)\rangle|^2 \\ \nonumber & \quad -\langle e_k , s^{(\infty)}(t)(e_l)\rangle \overline{\langle e_k , s_0(t)(e_l)\rangle } -\overline{\langle e_k , s^{(\infty)}(t)(e_l)\rangle } \langle e_k , s_0(t)(e_l)\rangle \big]\\ = & \sum_{k , l=1}^{2n } \langle ( e_k\otimes e_l ) , e^{t \hat{l}^{(2)}}(e_l\otimes e_k)\rangle + 2n \\ \nonumber & \quad -\langle e_k , e^{t\hat{l } } ( e_l)\rangle \langle e_l , s_0(-t)(e_k)\rangle -\langle e_l , e^{t\hat{l}^\dagger}(e_k)\rangle \langle e_k , s_0(t)(e_l)\rangle , \nonumber\end{aligned}\ ] ] where and   equation is the precise expression for $ ] , which may actually be used for computer programs if is explicitly known and the dimension is reasonably small . in most other circumstances it makes sense to simplify under the physically realistic assumption that . a first order expansion of all the exponentials in and the fact that and , for all , lead to = & \nonumber \frac{2\tau t}{|g| } \sum_{k , l=1}^{2n}\sum_{g\in g }  \langle e_k , g(a-\pi(a))j g^{-1 } e_l \rangle \langle e_l , ( g(a-\pi(a))j g^{-1})^\dagger e_k \rangle + o(\tau^2 t^2 \|a\|_\infty^4)\\ \approx &  \frac{2\tau t}{|g| } \sum_{g\in g } \|g(a-\pi(a))j g^{-1 } \|_2 ^ 2\\ = & 2\tau t \|a-\pi(a)\|_2 ^ 2.\nonumber\end{aligned}\ ] ] we notice that , so which completes the proof of for homogenization .  in the case of decoherence suppression instead , becomes \approx 2\tau t \|a-\tilde{\pi}(a)\|_2 ^ 2.\ ] ] now we note that the maximum entry of is bounded above by , the maximum entry of instead by , with being the greatest absolute value of system - environment coupling entries in the matrix . thus which proves the last part of theorem [ th4 ] .

Abstract: for finite - dimensional quantum systems , such as qubits , a well established strategy to protect such systems from decoherence is dynamical decoupling . however many promising quantum devices , such as oscillators , are infinite dimensional , for which the question if dynamical decoupling could be applied remained open . here we first show that not every infinite - dimensional system can be protected from decoherence through dynamical decoupling . then we develop dynamical decoupling for continuous variable systems which are described by quadratic hamiltonians . we identify a condition and a set of operations that allow us to map a set of interacting harmonic oscillators onto a set of non - interacting oscillators rotating with an averaged frequency , a procedure we call homogenization . furthermore we show that every quadratic system - environment interaction can be suppressed with two simple operations acting only on the system . using a random dynamical decoupling or homogenization scheme , we develop bounds that characterize how fast we have to work in order to achieve the desired uncoupled dynamics . this allows us to identify how well homogenization can be achieved and decoherence can be suppressed in continuous variable systems .


the causal dynamical triangulations approach to the construction of quantum theories of gravity is patterned on lattice regularization approaches to the construction of quantum theories of fields . while one introduces a lattice regularization to render the quantum theory well - defined , one is ultimately interested in the hypothetical continuum limit in which the regularization is removed in such a manner that physical quantities remain finite .the existence of a continuum limit of the lattice - regularized quantum theory is typically contingent on this theory exhibiting a second order phase transition .the nontriviality of a continuum limit is then contingent on the existence of an ultraviolet fixed point along the second order phase transition .the identification of such a fixed point calls for a renormalization group analysis , which one performs in concert with finite size scaling .there is now compelling evidence for the existence of a second order phase transition within the causal dynamical triangulations of -dimensional einstein gravity .this second order phase transition moreover occurs at the boundary of the phase exhibiting physical quantum geometry , potentially implying that the hypothetical continuum limit also possesses this property .a search for this continuum limit in the form of an ultraviolet fixed point is thus clearly warranted .ambjrn , grlich , jurkiewicz , kreienbuehl , and loll very recently made the first such attempt . under their simplest interpretation of the continuum limit, they did not find evidence for the existence of an ultraviolet fixed point , but , under a more complicated interpretation of the continuum limit , they did find plausible evidence for the existence of an ultraviolet fixed point .i propose a renormalization group scheme that differs significantly from those of these authors .hopefully , my proposal assists in the search for an ultraviolet fixed point within the causal dynamical triangulations of -dimensional einstein gravity .   in paper i of this two - part series ,i considered the formulation and implementation of a renormalization group scheme for lattice - regularized quantum theories of gravity .drawing on the well - established methods for renormalization group analysis of lattice - regularized quantum theories of fields , i proposed a general method for renormalization group analysis of lattice - regularized quantum theories of gravity .i now review my proposal , dividing its procedure into six steps . the first four steps ,although separated in the following for clarify of presentation , are so intimately connected that one typically pursues them all at once .moreover , one should subject these four steps to a comprehensive statistical analysis , only proceeding subsequently with the last two steps .one first selects a model for the continuum limit of the lattice - regularized quantum theory of gravity .a model consists of a progression of theories along a renormalization group trajectory within some truncation of the space of all theories . from numerical measurements interpreted with respect to the model ,one aims to specify this progression of theories by determining the values of the renormalized couplings characterizing each theory .one constructs the renormalization group flows of the model s couplings in this manner .there is of course no guarantee that the lattice - regularized quantum theory of gravity possesses a continuum limit .typically , the existence of a continuum limit requires the existence of a second order phase transition within the quantum theory s phase structure .even if a continuum limit does not exist , the theory might prove effective over a considerable range of scales , so one might still want a continuous description with a finite ultraviolet cutoff .one next selects a finite size scaling _ansatz _ relating discrete quantities within the lattice - regularized quantum theory of gravity to continuous quantities within the chosen model for its continuum limit .this _ ansatz _ should be consistent with the chosen model in that it should be based on a correspondence between a physical observable of the chosen model and the hypothesized discrete analogue of this physical observable .the form of the finite size scaling _ansatz _ may vary across the phase diagram of the lattice - regularized quantum theory of gravity as different scaling regimes are encountered .one delineates renormalization group trajectories as follows . consider a physical observable of the chosen model for the continuum limit whose value generically depends on the length scale which one probes .suppose that one has experimental access to on the interval of scales .a renormalization group transformation effects a change in the interval of scales to which one has access , in general , from to . since a renormalization group transformation does not effect changes in physics , measurements of on the interval of scales and on the interval of scales must agree within the intersection of these two intervals .accordingly , one delineates a renormalization group trajectory as the progression of theories through the space along which the physics of physical observables do not change . the number of physical observables required to delineate a renormalization group trajectorydepends on the chosen model .a physical observable is either dimensionless or dimensionful .the value of a dimensionless physical observable at some scale is simply a real number .the value of a dimensionful physical observable at some scale is also a real number , namely the ratio of the physical observable to a standard unit of measure .this reasoning also applies to the scale at which a physical observable is measured . to measure a dimensionful physical observable, one thus requires an appropriate standard unit of measure .i assume the presence of certain constants of nature , such as the speed of light and the planck constant , that allow for any physical observable s dimension to be expressed in powers of length .  typically , one employs a standard unit of measure defined externally , meaning that the dynamics through which the standard unit of measure is defined are sufficiently decoupled from the dynamics to which the standard unit of measure is applied . in the context of lattice - regularized quantum theories of gravity ,external reference scales are usually lacking , so one must choose a dynamically generated scale itself defined by a particular physical observable to serve as the standard unit of measure .when one implements a renormalization group scheme , one attempts to ascertain whether or not one s chosen standard unit of measure functions as a good standard .when one extracts any dimensionful physical observable from numerical measurements , one necessarily obtains its value in units of the lattice spacing .( the lattice spacing is the only dimensionful quantity present . ) for the chosen standard unit of length , numerical measurements yield .since the lattice spacing is _ a priori _ arbitrary , one sets its value in units of .one takes not as a measurement of in units of but as a definition of in units of .one then measures any other scale , determined from numerical measurements only as , by forming the ratio , which is independent of .in particular , the choice of a standard unit of length makes definite the interval of scales accessible at some point along a renormalization group trajectory .one defines as the smallest scale present the lattice spacing in units of , and one defines as the largest scale present measured in units of .one now determines how to extract the renormalized couplings of the chosen model for the continuum limit .one must perform a set of numerical measurements whose outcomes yield the values of the renormalized couplings when put in correspondence with the chosen model _ via _ the finite size scaling _ ansatz_.   onenext implements a renormalization group transformation .no matter the means by which one effects a renormalization group transformation typically a coarse graining procedure it must integrate out degrees of freedom on the interval of scales while preserving the physics of physical observables .one finally constructs the renormalization group flows of the chosen model s couplings by iterating the renormalization group transformation and extracting their values after each iteration . for a renormalization group transformation that incrementally increases and leaves fixed , one may take the value of as the parameter along the flow .now , in paper ii of this two - part series , i apply this method to the case of causal dynamical triangulations .i review the formalism of causal dynamical triangulations in section [ theory ] , and i review the phenomenology of causal dynamical triangulations in section [ phenomenology ] .as i argued in paper i , the phenomenology of a lattice - regularized quantum theory of gravity forms the basis of a renormalization group scheme .accordingly , i provide a comprehensive discussion , paying particular attention to the phenomenological scales that arise dynamically . drawing on this phenomenology ,i propose a renormalization group scheme for causal dynamical triangulations in section [ renormalization ] .this proposal constitutes the paper s primary novel contribution .there remains an ambiguity in my scheme the choice of a standard unit of length , more than one of which appears to be consistent that i hope its implementation resolves .i conclude in section [ conclusion ] by considering how the observed phase structure of causal dynamical triangulations might mesh with my proposed renormalization group scheme .causal dynamical triangulations is an approach to the quantization of classical metric theories of gravity based on a particular lattice regularization of the formal path integral=\int_{\mathbf{g}|_{\partial\mathscr{m}}=\gamma}\mathrm{d}\mu(\mathbf{g})\,e^{is_{\mathrm{cl}}[\mathbf{g}]/\hbar}.\ ] ] the transition amplitude  ] by integrating over all metric tensors satisfying the boundary condition , weighting each metric tensor by the product of the measure and the exponential /\hbar} ] is the action of the classical theory of gravity that one wishes to quantize .this computation is , however , plagued by severe conceptual and technical difficulties .what precisely is the class of all metric tensors ?which measure on this class should one employ ?how should one regularize the integration s divergences ?  the causal dynamical triangulations approach prescribes a rigorous definition of the path integration in equation that addresses all of these difficulties .one begins by asserting that the path integration in equation is restricted to the subclass of so - called causal spacetimes those admitting a global foliation by spacelike hypersurfaces all of the same topology for a definite choice of this topology .accordingly , a causal spacetime manifold , equipped with a causal metric tensor , possesses the structure , the direct product of a -dimensional spatial manifold and a real temporal interval .one thus defines a quantum theory of gravity for each choice of within the causal dynamical triangulations approach .the relevant path integral for one such choice is then formally =\int_{\substack{\mathscr{m}_{c}\cong\sigma\times\mathcal{i } \\\mathbf{g}_{c}|_{\partial\mathscr{m}_{c}}=\gamma}}\mathrm{d}\mu(\mathbf{g}_{c})\,e^{is_{\mathrm{cl}}[\mathbf{g}_{c}]/\hbar}.\ ] ] one next invokes a lattice regularization of the path integration in equation . in particular , one replaces continuous causal spacetimes by so - called causal triangulations piecewise - minkowski simplicial manifolds possessing a distinguished global foliation by spacelike hypersurfaces all of the chosen topology .one constructs a causal triangulation by appropriately gluing together the types of causal -simplices , each a piece of minkowski spacetime . in figure[ 4simplices ] i depict the four types of -simplices .these -simplices assemble so that each spacelike hypersurface is triangulated by regular spacelike -simplices of squared edge length , and adjacent spacelike hypersurfaces are connected by timelike edges of squared edge length for real parameter .the _ a priori _ arbitrary constant defines the lattice spacing .the connectivities of all -simplices comprising a causal triangulation together with these edge length assignments completely determine its spacetime geometry .this lattice regularization results in the replacement of the path integral by the path sum =\sum_{\substack{\mathcal{t}_{c } \\\mathcal{t}_{c}\cong\sigma\times\mathcal{i } \\\mathcal{t}_{c}|_{\partial\mathcal{t}_{c}}=\gamma}}\mu(\mathcal{t}_{c})\,e^{i\mathcal{s}_{\mathrm{cl}}[\mathcal{t}_{c}]/\hbar}\ ] ] taken over all causal triangulations isomorphic to with boundary .the measure is the inverse of the order of the automorphism group of the causal triangulation , and the action  ] into the regge calculus of causal triangulations .  in most casesthe path sum is analytically intractable , so one studies the quantum theory so defined by numerical methods . in particular, one would like to perform markov chain monte carlo simulations of representative paths contributing to the path sum ; however , such simulations can not handle the complex amplitude /\hbar} ] is the resulting euclidean action .since one can only run markov chain monte carlo simulations of finite causal triangulations , one further specifies a fixed number of spacelike hypersurfaces , introducing a discrete time coordinate to enumerate these time slices , and a fixed number of -simplices .one thus simulates representative paths contributing to the partition function =\sum_{\substack{\mathcal{t}_{c } \\\mathcal{t}_{c}\cong\sigma\times\mathcal{i } \\t(\mathcal{t}_{c})=\bar{t } \\ n_{d+1}(\mathcal{t}_{c})=\bar{n}_{d+1}}}\mu(\mathcal{t}_{c})\,e^{-\mathcal{s}_{\mathrm{cl}}^{(\mathrm{e})}[\mathcal{t}_{c}]/\hbar},\ ] ] related by a legendre transform to the partition function .a markov chain monte carlo simulation outputs an ensemble of causal triangulations for the chosen values of the bare couplings , , of the action  ] with associated eigenvalues . in figure [ eigenvectors ]i display the first four eigenvectors {j} ] of the connected -point function within the central accumulation for an ensemble within phase c characterized by , , , and . ]   of the connected -point function within the central accumulation for an ensemble within phase c characterized by , , , and . ]i have restricted consideration to the central accumulation so that is a matrix .the central accumulation spans time slices . as i demonstrate in section [ model ] ,the eigenvectors {j} ] into its discrete fourier modes , one obtains its dominant frequency or period , again characterized by a dimensionless number . in figure [ eigenvectorspectra ]i display the power spectra of the discrete fourier transforms of the first four eigenvectors {j} ] of the connected -point function within the central accumulation for an ensemble within phase c characterized by , , , and . ]numerical measurements are sensitive to periods neither less than twice the spacing between adjacent time slices nor greater than half the number of time slices .the dominant periods of the eigenvectors {j} ] and associated eigenvalues solving the characteristic equation \nu_{3}=\mu\nu_{3}. -\frac{1}{48\pi^{3}\hbar g_{\mathrm{ren}}\ell_{\mathrm{ds}}^{3}\sqrt{g_{tt}}}\left\{\frac{\mathrm{d}}{\mathrm{d}t}\left[\sec^{3}{\left(\frac{\sqrt{g_{tt}}t}{\ell_{\mathrm{ds}}}\right)}\frac{\mathrm{d}}{\mathrm{d}t}\right]+\frac{4g_{tt}}{\ell_{\mathrm{ds}}^{2}}\sec^{5}{\left(\frac{\sqrt{g_{tt}}t}{\ell_{\mathrm{ds}}}\right)}\right\}\left[\nu_{3}(t)\right]_{j}=\upsilon_{j}\left[\nu_{3}(t)\right]_{j}.\ ] ] the eigenvalues have the dimensions of an effective spacetime energy - momentum density .the -point function thus defines a ladder of scales corresponding to its eigenvalue spectrum .i now connect the eigenvalue spectrum to the numerically measured eigenvalue spectrum again using the finite size scaling _the discrete analogue of equation is }\nonumber\\ & & \qquad\qquad\qquad\qquad+\left(\frac{2}{\bar{s}_{0}\langle n_{4}^{(1,4)}\rangle^{1/4}}\right)^{2}\sec^{5}{\left(\frac{\tau}{\bar{s}_{0}\langle n_{4}^{(1,4)}\rangle^{1/4}}\right)}\bigg\}\left[\eta_{3}^{\mathrm{sl}}(\tau)\right]_{j}=\frac{1}{\omega}_{j}\left[\eta_{3}^{\mathrm{sl}}(\tau)\right]_{j},\end{aligned}\ ] ] for a discretization of the derivative with respect to the discrete time coordinate .the inverse of the eigenvalue appears in equation because the -point function is the inverse of the differential operator .the predicted relation between the eigenvalues and the eigenvalues is thus for a discretization of the differential operator such that there are eigenvalues . in figure [ eigenvectorfits ]i display the fit of the first four eigenfunctions {j} ] for a typical ensemble of causal triangulations within phase c.  _ { j} ] of the connected -point function within the central accumulation for an ensemble within phase c characterized by , , , and . ]the fits in figure [ eigenvectorfits ] do not involve any adjustable parameters : one simply computes the eigenfunctions {j} ] for low .this concurs with the fact that the eigenvectors {j} ] .  from the relation , the definition of , and the expression for the de sitter length in units of the lattice spacing, one can determine in units of the lattice spacing as or in units of the de sitter length as  is of course the square of the planck length , and the ratio is proportional to . with one obtains determinations of in either units .the model of assumes that has a definite value or , equivalently , that does not have a significant renormalization group flow on the applicable scales .accordingly , i average the determinations of .the variance in the determinations of quantifies to some extent the validity of this assumption . since the eigenvectors {j} ] , into which i previously decomposed the ensemble average , represent excitations of the quantum geometry on successively smaller scales . as the renormalization group transformation incrementally integrates out degrees of freedom , i expect to decompose into incrementally fewer eigenvectors {j} ] , i expect to decrease incrementally .what does this expectation imply for the eigenvalues of the eigenvectors {j} ] of continuum limits of the eigenvalues of the eigenvectors {j}$ ] of not physical observables either since explicitly depends on the gauge parameter . does nevertheless contain some physical information : one infers the euclidean de sitter form of fluctuations in the large scale ensemble average quantum geometry by matching the shape of to the shape of .one only gathers this information by first hypothesizing that the discrete time coordinate has as its continuum limit the global time coordinate .the spectral dimension as a function of the diffusion time the continuum limit of a physical observable .even though the chosen model does not describe the dynamical dimensional reduction of on sufficiently small scales , the spectral dimension on these and all other scales is constructed in the manner of physical observables of the chosen model . clearly , is scale - dependent and dimensionless .of the two physical observables that i just identified the spacetime -volume and the spectral dimension are viable for the delineation of a renormalization group trajectory ? as a scale - independent and dimensionful physical observable , is viable given a choice of standard unit of length .the value of in units of should then be constant along a renormalization group trajectory . as a scale - dependent and dimensionless physical observable , which one can moreover measure over essentially the whole interval of scales , is in principle supremely viable .the value of at a particular diffusion time , which corresponds to some particular scale , should be constant along a renormalization group trajectory ( until that scale is no longer accessible ) .currently , one only understands the connection between and for . for these sufficiently large discrete diffusion times , the finite size scaling _ ansatz_ obtains , mapping into _ via _ the identification of with .accordingly , the value of at any particular value of should be constant along a renormalization group trajectory . to complete the prescription for delineating renormalization group trajectories , i now consider the candidates for a standard unit of length in terms of which all other length scales are measured .i explore the consequences for my renormalization group scheme of each such choice .i find that some choices are not viable while other choices are viable .my renormalization group scheme thus contains a residual ambiguity : the choice of a standard unit of length .i hope that an application of my scheme to numerically simulated ensembles of causal triangulations within phase c resolves this ambiguity .more than one choice may prove physically consistent , a possibility potentially indicative of multiple different choices for a standard unit of length being mutually consistent .[ [ lattice - spacing - a - as - the - standard - unit - of - length - ell_mathrmunit ] ] lattice spacing as the standard unit of length + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  consider using the lattice spacing as the standard unit of length .although i have continually advised against making such a choice , i wish to consider its implications if only to reinforce my point .constancy of the spacetime -volume , measured in units of the lattice spacing , dictates _ via _ equation constancy of the product .accordingly , one must compensate for any change in the number of -simplices by a commensurate change in the discrete -volume of an effective -simplex . since depends on the bare couplings and but varies only weakly with , one can not change while holding and fixed .this is the only definite condition on the renormalization group trajectories that follows from this choice for . whether or not one can maintain this condition is a test of this choice .although i can compute and in principle and , i do not know how these quantities should change along a renormalization group trajectory beyond the restrictions imposed by the general expectations of subsection [ guidelines ] .one could impose these expectations as further conditions on the renormalization group trajectories .note that the interval of scales is invariant along a renormalization group trajectory for this choice of standard unit of length . is unity by definition , and is proportional to the fourth root of .there is thus no clear parameter to parametrize the renormalization group flow .[ [ de - sitter - length - ell_mathrmds - as - the - standard - unit - of - length - ell_mathrmunit ] ] de sitter length as the standard unit of length + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  consider using the de sitter length as the standard unit of length .constancy of the spacetime -volume , measured in units of the de sitter length , now holds by definition upon identification of as that of euclidean de sitter space .this choice for the standard unit of length thus trivializes the condition of constant along a renormalization group trajectory . without any further conditions determining the renormalization group trajectories, there is thus no way to define them . within the renormalization group scheme that i propose, this choice of standard unit of length simply does not work in its desired capacity .[ [ planck - length - ell_mathrmp - as - the - standard - unit - of - length - ell_mathrmunit ] ] planck length as the standard unit of length + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  consider using the planck length as the standard unit of length .constancy of the spacetime -volume , measured in units of the planck length , dictates constancy of the ratio upon identification of as that of euclidean de sitter space .equivalently , the dimensionless product remains constant along a renormalization group trajectory . since and are the only couplings of the chosen model , there are no renormalization group flows for this choice of standard unit of length .the ratio determines the magnitude of the fluctuations in the spatial -volume relative to itself .accordingly , the magnitude of the fluctuations in the number of spacelike -simplices relative to itself must also be constant along a renormalization group trajectory .this condition dictates that one can not vary while holding fixed the bare couplings and to generate a renormalization group trajectory .recalling from equation that depends on the values of , , and , all of which depend on the bare couplings , one must make certain that the appropriate combination of , , and remains constant as one changes , , and .this adjustment potentially leaves some freedom in changes of , , and that preserve the value of .one may then bring to bear the general expectations of subsection [ guidelines ] , possibly using these expectations to reduce or eliminate this residual freedom .the variation of , given by the ratio , along a renormalization group trajectory should be in concert with these expectations , with increasing as the renormalization group transformation is iterated .[ [ classical - scale - ell_mathrmcl - as - the - standard - unit - of - length - ell_mathrmunit ] ] classical scale as the standard unit of length + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  consider using the classical scale as the standard unit of length .constancy of the spacetime -volume , measured in units of , now dictates constancy of upon identification of with that of euclidean de sitter space .accordingly , the renormalized cosmological constant , measured in units of , does not flow under renormalization , but the renormalized newton constant , measured in units of may flow under renormalization .this choice of standard unit of length potentially leaves some freedom in changes of , , and that preserve the value of .one may again bring to bear the general expectations of subsection [ guidelines ] , possibly using these expectations to reduce or eliminate this residual freedom .the variation of , given by the ratio , along a renormalization group trajectory should be in concert with these expectations , with increasing as the renormalization group transformation is iterated .this choice of standard unit of length is closest to the original definition of the meter as a standard unit of length namely as a particular metal bar in that this bar establishes a scale of classical proportions , neither quantum - mechanical nor cosmological .[ [ quantum - scale - ell_mathrmqm - as - the - standard - unit - of - length - ell_mathrmunit ] ] quantum scale as the standard unit of length + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  consider the quantum - mechanical scale as the standard unit of length .constancy of the spacetime -volume , measured in units of , dictates constancy of the ratio upon identification of with that of euclidean de sitter space .accordingly , the renormalized cosmological constant , measured in units of , does not flow under renormalization , but the renormalized newton constant , measured in units of , may flow under renormalization .this choice of standard unit of length potentially leaves some freedom in changes of , , and that preserve the value of .one may again bring to bear the general expectations of subsection [ guidelines ] , possibly using these expectations to reduce or eliminate this residual freedom .the variation of , given by the ratio , along a renormalization group trajectory should be in concert with these expectations , with increasing as the renormalization group transformation is iterated .this choice of standard unit of length is closest to the modern definition of the meter as a standard unit of length namely in terms of atomic transitions in that such transitions establish a scale of quantum - mechanical proportions , neither classical nor cosmological .the choice suffers from one obvious drawback : as the scale is incrementally increased , one expects to reach an interval of scales on which the scale is no longer accessible .if one could correlate with , as the modern definition of the meter is correlated with the original definition of the meter , then one could use either or as a standard .when has increased sufficiently that is no longer accessible , one could then switch to using .the possibility of establishing such a correlation depends on the degree of decoupling between the quantum - mechanical and classical regimes .the chosen model has two couplings , the renormalized newton constant and the renormalized cosmological constant .one extracts these couplings from numerical measurements on an ensemble of causal triangulations as explained in subsection [ model ] .one thus obtains in units of as in equation and in unit of as in equation .determining requires measurements of , , , , and .determining requires measurements of and .the value of stems from measurements of , , , and , the value of stems from measurements of , and the value of stems from measurements of .the first four quantities determine and .subsequent fitting of the function to determines .the eigenvalues follow from diagonalization of .alternatively , one can measure and in units , , , or .if one wishes to use either or as the standard unit of length , then one must make numerical measurements to extract these quantities in units of the lattice spacing .determining and requires measurement of followed by the analysis of , which itself calls for the measurement of , , , and .i choose to achieve indirectly the implementation of a renormalization group transformation . to probe different intervals of scales , i propose generating ensembles of causal triangulations characterized by these different intervals of scales .how does one generate an ensemble of causal triangulations characterized by a particular interval of scales ?one lets the dynamics do this job . when generating an ensemble , one selects the number of time slices , the number of -simplices , and the bare couplings and assuming a spacetime manifold of the form . the ensemble average quantum geometry that dynamically results from these chosen values contains information only about a certain interval of scales . how does one then determine this interval of scales ?the dimensionless ultraviolet scale is given by the smallest scale present the lattice spacing to the chosen standard unit of length .the dimensionless infrared scale is given by the largest present scale the de sitter length to the chosen standard unit of length .  to implement a renormalization group transformation ,one thus proceeds as follows .given an ensemble of causal triangulations characterized by the interval of scales and certain values for the physical observables and , one generates another ensemble of causal triangulations characterized by the interval of scales and the same values for the physical observables and .how the interval of scales and the values of the physical observables change with the parameters defining an ensemble is not yet well - established . at least at first , one must work by trial and error ., a coarse graining procedure for causal dynamical triangulations should function to reduce simultaneously the number of -simplices and the number of time slices while preserving the causal nature of the triangulation .undoubtedly , one could design such a coarse graining procedure .its relevance for ensembles of causal triangulations that one can currently simulate is likely quite limited : owing to the relatively small values of obtained , very few iterations of the coarse graining procedure could be implemented .the method for implementation of renormalization group transformations that i have proposed should allow for much more incremental iterations . ]i construct the renormalization group flows of the gravitational couplings by iterating the renormalization group transformation and extracting the renormalized couplings after each iteration . in practice, one proceeds as follows .one first generates a very large number of ensembles of causal triangulations within phase c. one next performs the measurements and analyses described in subsections [ observables ] and [ model ] to extract the values of the physical observables selected above .one then identifies subsets of these ensembles for which the values of the selected physical observables are all the same .one also identifies the interval of scales characterizing each ensemble .one finally strings together the ensembles within each subset according to their intervals of scales .this string represents the renormalization group trajectory from which the renormalization group flows of the couplings then follows .one may take , given by the ratio , as the parameter along the renormalization group flow .as an approach to the construction of quantum theories of gravity based on a lattice regularization of the path integral , causal dynamical triangulations demands a renormalization group analysis . based on the general considerations of paper i , i here proposed a renormalization group scheme for the causal dynamical triangulations of -dimensional einstein gravity .its aim is the extraction of the renormalization group flows of the couplings of the ( hypothetical ) continuum limit of the lattice - regularized quantum theory .the nature of these renormalization group flows should allow for the determination of whether or not a continuum limit exists .to understand how such a determination would work , i return to consideration of the phase structure of causal dynamical triangulations .recall from subsection [ phases ] that the phase diagram of the partition function for the action contains a second order phase transition .the presence of this phase transition raises the prospect of rigorously defining the continuum limit of the quantum theory of gravity so defined .if there exists an ultraviolet fixed point along the phase transition , then one could remove the ultraviolet cutoff at this fixed point .in particular , by tuning the bare couplings to this fixed point , one could keep physical observables finite while letting the number of -simplices diverge and the lattice spacing vanish . equipped with the renormalization group scheme presented in section [ renormalization ] , one would attempt to determine if there exists an ultraviolet fixed point as follows .the renormalization group trajectories delineated by the method of subsection [ formulation ] should exhibit a characteristic behavior in the presence of an ultraviolet fixed point .moving along a renormalization group trajectory from the interior of phase c to the ultraviolet fixed point on the boundary with phase b , one should find that necessarily increases without bound and that necessarily decreases towards zero . that a renormalization group transformation does not effect a change in physics the condition leading to the delineation of the renormalization group trajectories in the first place would necessitate these behaviors of and .on approaching the ultraviolet fixed point , the renormalization group flows of the couplings would dictate the nature of the continuum limit . if all of the couplings parametrizing interactions flow to zero at the ultraviolet fixed point , then the continuum limit is asymptotically free . if a finite number of the couplings parametrizing interactions flow to finite values at the ultraviolet fixed point , and if the rest of the couplings parametrizing interactions flow to zero at the ultraviolet fixed point , then the continuum limit is asymptotically safe .this discussion has assumed the existence of an ultraviolet fixed point .if there does not exist an ultraviolet fixed point along the second order phase transition , then renormalization group trajectories from the interior of phase c may not even be connected to the second order phase transition . in this casethe lattice - regularized quantum theory only defines an effective description valid on length scales larger than that of the regularization .the renormalization group scheme should nevertheless yield the renormalization group flows of the couplings of this effective description .causal dynamical triangulations is not the only approach seeking to define a quantum theory of gravity as a renormalizable quantum theory of fields .the functional renormalization group has emerged as a popular analytic technique complementary to lattice regularization techniques . in this approach oneattempts to solve the exact renormalization group equation describing the scale - dependence of the effective ( average ) gravitational action within some chosen truncation of the space of all theories .this calculation has been successfully performed for an ever - increasing array of truncations , essentially all of which yield evidence for an asymptotically safe ultraviolet fixed point .one would eventually want to compare the results of the functional renormalization group approach to those of causal dynamical triangulations .the obvious point of comparison is the nature of the ultraviolet fixed points identified in each approach .specifically , one would want to compare the critical exponents characterizing each ultraviolet fixed point since the critical exponents determine the universality class of the continuum limit .both the causal dynamical triangulations approach and the functional renormalization group approach aim to achieve nonperturbatively the renormalizability of the quantum theories of gravity that they respectively define . particularly the advent of hoava - lifshitz gravity has reignited interest in the prospect of constructing perturbatively renormalizable quantum theories of gravity .there are some indications that the continuum limit of the causal dynamical triangulations of -dimensional einstein gravity is some version of quantum hoava - lifshitz gravity .the recent renormalization group study of ambjrn , grlich , jurkiewicz , kreienbuehl , and loll potentially provides further evidence for this possibility .if this is indeed the case , then one would expect to find appropriate deviations from the model of subsection [ model ] as one approaches more closely the ultraviolet fixed point along a given renormalization group trajectory .a definitive conclusion awaits a more comprehensive understanding of the physics in the vicinity of the second order phase transition .i wish to thank jan ambjrn , steven carlip , and renate loll for several useful conversations .i also wish to thank renate loll for comments on a draft of this paper .i acknowledge support from the foundation for fundamental research on matter itself supported by the netherlands organization for scientific research . this research was supported in part by the perimeter institute for theoretical physics .research at the perimeter institute is supported by the government of canada through industry cananda and by the province of ontario through the ministry of economic development and innovation .this research was also supported in part by the united states department of energy under grant de - fg02 - 91er40674 at the university of california , davis .                  c. anderson , s. j. carlip , j. h. cooperman , p. hoava , r. k. kommu , and p. r. zulkowski . quantizing hoava - lifshitz gravity _ via _ causal dynamical triangulations . " _ physical review d _ 85 ( 2012 ) 044027 .

Abstract: the causal dynamical triangulations approach aims to construct a quantum theory of gravity as the continuum limit of a lattice - regularized model of dynamical geometry . a renormalization group scheme in concert with finite size scaling analysis is essential to this aim . formulating and implementing such a scheme in the present context raises novel and notable conceptual and technical problems . i explored these problems , and , building on standard techniques , suggested potential solutions in the first paper of this two - part series . as an application of these solutions , i now propose a renormalization group scheme for causal dynamical triangulations . this scheme differs significantly from that studied recently by ambjrn , grlich , jurkiewicz , kreienbuehl , and loll .


collisions in orbit pose a threat to spacecraft , to astronauts and to the global commons of near - earth space .several collisions have already occurred between spacecraft and debris , while the iridium 33/cosmos 2251 collision of january 2009 represented the first documented satellite - satellite collision .unfortunately , the cumulative number of collisions thus far is consistent with the prescient predictions of a runaway chain reaction .  reducing debris from current and future space missions ( debris mitigation )is an important measure .however , even the unrealistic case of no new launches " is inadequate to curb runaway growth .there are two remaining ways to alleviate the problem : debris removal and collision avoidance . for the latter, one needs prediction precision to be sufficiently accurate so as not to imply an unwieldy number of collision avoidance maneuvers .in addition , one needs a means of doing debris - debris collision avoidance since applying collision avoidance only to the subset of conjunctions involving maneuverable spacecraft does not suffice to curb the growth .this paper describes a method to increase the orbit prediction accuracy based on publicly available tles .many satellite owner operations have inadequate ( if any ) processes for conjunction assessment and collision avoidance , since they would have to screen their asset(s ) against all other space objects .the only source of knowledge at their disposal for the majority of other objects is the publicly available two - line element ( tle ) sets . butpredictions based on tles using the associated analytic propagator ( sgp4 ) are not sufficiently accurate to warrant maneuvering to avoid potential collisions : they imply an unacceptably large number of potential collisions per space object , each of which has very low probability .the problem is similar for debris - debris conjunctions except then both objects , not just one , are subject to these imprecisions .  to address this tle / sgp4 accuracy problem, we investigated several methods to improve the propagation errors for non - maneuvering orbital objects whilst using only tles as input data .the following research target was set : `` increase the predictive accuracy for orbital objects , using only historical tle data , such that it enables operational conjunction assessment for collision avoidance '' .  in the following sections , we describe one approach to this target , assess its accuracy , and discuss the requirements for extension to the entire space object catalog .finally , we propose a new method of debris - debris collision avoidance enabled by long - term high - accuracy conjunction assessment .esa uses tles from the publicly available catalog to initially screen their sun - synchronous orbit ( sso ) spacecraft ers2 and envisat for conjunctions .telemetry from their operational spacecraft provide precision orbital ephemeredes ( poes ) for those spacecraft . for screening against all other potentially conjuncting objects ,only tles are used . describe a method to estimate error covariances of tles in order to quantify collision probability assessments .their method provided inspiration for the present work . herewe extend and adapt their approach : based solely on the object s historical tles , we improve the accuracy of the object s predicted position , as opposed to quantifying the accuracy of the object s sgp4 propagation errors .our method is essentially to use tle data as `` pseudo - observations '' and to fit an orbit to these pseudo - observations using a high - precision special perturbations propagator and traditional batch least - squares differential correction .the fitted orbit is then propagated into the future using the same high - precision orbit propagator .the prediction accuracy is assessed by comparison with precision orbital ephemeris ( poe ) data from the international laser ranging service .  for each objectwe wish to analyze we choose a time window with two sections : a fitting period and a subsequent prediction period .the fitting period is initially set to ten days , typical of the period over which u.s .space surveillance network observations are fitted when generating tles for leo objects .section details a more principled approach to determining the fitting period .the prediction period is 30 days .  for a given object ,all tles with epochs within its entire window are obtained .however , only tles with epochs within the fitting period are used for the fitting process ; those in the prediction period are only used for validation and test .  using all tles from the fitting period and interpolating using sgp4, we generate a series of order state vectors ( `` pseudo - observations '' ) equally spaced in time within the fitting period .we initialize the differential correction with a state vector derived from the first tle in the fitting period , and then propagate that using a high precision propagator .we apply batch least - squares differential correction to minimize the rms error in the radial , in track and cross track ( ric ) components of relative positions between the fitted orbit and the pseudo - observations .the trajectory obtained from the converged differential corrector is our fit .the high precision propagator incorporates a egm2008 gravity field including solid earth tides , point masses for the solar and lunar gravity fields and a nrl - msise atmospheric drag model .we assume constant values for solar f and ap equal to their averages over the fitting period .solar radiation pressure is modelled using a simple the biconic approximation with earth and moon as eclipsing bodies .[ figure1 ]   days .the fitting period is days for stella and days for etalon- .all fitting uses only publicly available tles .truth data are only used for plotting .selection of the length of the fitting period is discussed in section 4 .also shown are the updated tles ( no prediction ) in blue ., width=529 ]  we applied our method to four non - manoeuvring spacecraft for which poe data were readily available : stella , starlette , ajisai and etalon- .the summary of basic orbital and physical properties for these spacecraft are shown in table .they were picked in part for their range of orbital altitudes ..key parameters of geodesic spacecraft used [ cols="<,^,^,^,^,^,^ " , ]   [ table1 ][ figure4 ]   runs corresponding to random starting dates in ., width=377 ]  the majority of tracked space objects ( i.e debris ) do not have precision orbital ephemerides they are not tracked by the ilrs and they transmit nothing . many are , however , tracked by the united states space surveillance network ( ssn ) .unfortunately , the only publicly available source of data derived from these ssn observations are , at present , the tles accessible at space-track.org ( and archived e.g. at celestrak.com ) .while there is currently no publicly available source of high - accuracy trajectory for debris objects , we believe the method described above could be applied to refine the predictions of debris objects future positions based only upon comparatively inaccurate tles , facilitating conjunction assessment and collision avoidance .however , there are several issues which complicate the application of our procedure to debris :  the first set of issues arise when tles are the _ only _ source of data - we have no `` truth '' in order to perform validation and test .this is solved by the use of tles from beyond the fitting period as a proxy for truth data .our initial manual experiments suggested that different fitting periods were optimal for different orbit categories .these results are consistent with .we made a more principled investigation , utilizing cross - validation , to optimize the length of the fitting interval on a per - object basis using only tles for both fitting and validation .figure 4 shows how a fitting period of days was determined to be optimal for satellite ajisai .the important point shown in the figure is that a day fitting period is optimal both when validating predictions with truth data _ and _ when validating predictions with future tles . andthe day fitting period is optimal whether predicting , or days into the future .thus it appears `` future '' tles can be used as a proxy for truth data when optimizing the length of the fitting period on a per object basis .the second set of issues is that for debris objects one must also solve for area to mass ratio ( drag ) and validate this as well using only future tles .we are currently performing these experiments .further accuracy improvements could come from a principled analysis of the number of fitting points and their weighting as a function of location within the fitting period , the removal of outlier data ( i.e. filtering ) and other techniques from statistical orbit determination , treating tles as observations . also , recent work suggests that there may be simple global transformations of tle data that can improve tle / sgp accuracy .these systematic corrections might also be further refined on the basis of orbital parameters , resulting in a look - up - table used to correct tles as a function of their orbital regimes .in addition , the scheme could be tested on actual known conjunctions in order to analyse its efficiency for conjunction assessment .these issues will be the subject of a future paper .as an initial assessment of the computational requirements needed to perform conjunction analysis for all space objects in the publicly available catalog , a simple conjunction analysis system was parallelized on the pleiades supercomputer at nasa ames research center . using approximately cpu cores , checking all objects against all objects for conjunctions days into the future takes about a minute .the conjunctions involving spacecraft precisely match the results from socrates .the same system takes about minutes to perform 7 days of all - on - all conjunction assessment of a `` simulated s - band fence '' catalog ( obtained from the nasa orbital debris office ) containing `` pseudo - tles '' for approximately million objects of size cm .this demonstrates the feasibility of doing all - on - all conjunction assessment on the present and expected near - future catalogs .scaling the fitting method discussed above to the future catalog will be the subject of a later paper .an along track of cm / s provides of order 1 km displacement per day .since error growth for our method is m per day , such small manoeuvres might suffice for collision avoidance . for manoeuvres this small one could use radiation pressure to impart the necessary momentum . to verify the conjecture, we increased the flux of radiation by a factor of 10x the solar radiation constant for 10 minutes on a typical debris object ( area to mass ratio m/kg ) in sso .we then propagated that for a further 5 days .the displacement was km / day along track compared to a reference object that was not so irradiated .two cumulative such events , if one ensured appropriate geometry , would thus lead to the necessary along - track displacement .a factor of 10x the solar rational constant could conceivably be applied from the ground .for example , a 10kw class laser attached to a 1 m class telescope with appropriate tracking , could , ( considering diffraction limits only ) provide the necessary radiation pressure for a debris object of area in sso .one would need to perform a detailed engineering analysis of the feasibility of such a device , particularly the viability of maintaining small beam divergence and tracking .debris - debris collision avoidance could eliminate the need for an active debris removal program .the results of imply , to first order , that performing debris - debris collision avoidance for 5 additional carefully selected objects per annum would curb debris field growth .furthermore , any additional debris - debris collision avoidance would start reducing the net debris density , eventually to below a critical level , because debris creating collisions are being nulled whereas atmospheric drag continues to cause objects re - entry .thus this method could provide not just a stop gap measure , but a permanent solution .this fitting and propagation method , based solely on tles , could potentially provide initial conjunction analysis sufficiently accurate for an operationally viable collision avoidance system .if similar improvements are possible for debris objects , as we expect , then it could be used for all - on - all conjunction assessment . however , to ensure that the scheme works in an operational setting , it would be necessary to have additional data : the so - called `` uncorrelated objects '' ( a.k.a . analysts set ) which account for approximately 30 percent of potential conjunctions in leo .further , we show that all - on - all conjunction assessment is possible with moderate computer infrastructure , even with the large increase in size of the catalog of tracked objects that is expected in the next few years .  finally , given accurate predictions, we claim _ debris - debris collision avoidance may be possible _ by externally inducing small manoeuvres using radiation pressure from kw class power density ground - based lasers . if feasible , this could negate the need for a large scale debris removal program .k. alfriend , s. paik , v. boikov , z. khutorovsky , & a. testov .comparison of the russian & us algorithms for catalog maintenance for geosynchronous satellites .adv . astronaut ., 112(2):11571176 , 2002 .j. boers , s. coffey , w. barnds , d. johns , m. davis , & j. seago . accuracy assessment of the naval space command special perturbations cataloging system .spaceflight mechanics 2000 , volume 105 of adv . astronaut ., pages 12911304 , 2000 .s. coffey , h. neal , c. visel , & p. conolty .demonstration of a special - perturbations - based catalog in the naval space command system .spaceflight mechanics 1998 , volume 99 of adv . astronaut ., pages 227248 , 1998 .t. s. kelso & s. alfano .satellite orbital conjunction reports assessing threatening encounters in space ( socrates ) .proceedings of the 15th aas / aiaa space flight mechanics conference .paper aas 05 - 124 , 2005 .p. legendre , r. garmier , g. prat , b. revelin , and s. delavault .improvement of the two - line element accuracy assessment based on a mixture of gaussian laws adv ., 129(3):21892207 , 2008 .

Abstract: the density of orbital space debris constitutes an increasing environmental challenge . there are three ways to alleviate the problem : debris mitigation , debris removal and collision avoidance . this paper addresses collision avoidance , by describing a method that contributes to achieving a requisite increase in orbit prediction accuracy . batch least - squares differential correction is applied to the publicly available two - line element ( tle ) catalog of space objects . using a high - precision numerical propagator , we fit an orbit to state vectors derived from successive tles . we then propagate the fitted orbit further forward in time . these predictions are compared to precision ephemeris data derived from the international laser ranging service ( ilrs ) for several satellites , including objects in the congested sun - synchronous orbital region . the method leads to a predicted range error that increases at a typical rate of 100 meters per day , approximately a 10-fold improvement over tle s propagated with their associated analytic propagator ( sgp4 ) . corresponding improvements for debris trajectories could potentially provide initial conjunction analysis sufficiently accurate for an operationally viable collision avoidance system .   we discuss additional optimization and the computational requirements for applying all - on - all conjunction analysis to the whole tle catalog , present and near future . finally , we outline a scheme for debris - debris collision avoidance that may become practicable given these developments .  space debris , conjunction analysis , orbit determination  = 0.5 cm