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the quest for a quantum theory of gravity is now several decades old and has been studied with approaches ranging from string theory and non - commutative geometry to loop quantum gravity , spin foams , and discrete models of planck - scale spacetime . most of these take different interpretations of what a quantum theory of gravity should achieve , but with one thing in common : that it should be some merger of general relativity and quantum theory . the final goal has not yet been achieved , but all these approaches have brought to us new understandings of the problem .    we come to the motivation for this paper by putting together a consistent list of features , selectively chosen from among the various indications of what a quantum theory of gravity may be . such a list is by no means unique , but we will find that it is fruitful ; it leads to a new notion of quantum cosmology and a new formulation of evolution in a planck - scale theory . our list is the following : 1 ) at energies close to the planck scale , the description of spacetime as a continuous , @xmath0-dimensional manifold breaks down ; this is the usual idea for regularization of quantum field theory . 2 ) causality is still present in planck - scale geometry . 3 ) quantum theory is valid in planck - scale physics . 4 ) in a fundamental theory of the universe , there should be a finite number of degrees of freedom in any finite region of the universe ; this is supported by black hole entropy calculations from both string theory and loop quantum gravity , and bekenstein s physical arguments . 5 ) the theory should be background independent . quantum causal histories ( qch ) were introduced in @xcite as a formalism for quantum cosmology that realizes most of the above assumptions . the idea is to use a causal set ( locally finite , partially ordered set ) as the description of the causal structure of the universe . quantum theory is present through the assignment of finite - dimensional hilbert spaces to these events . they can be regarded as representing the fundamental building blocks of the universe at the planck scale . the composite hilbert space for two spacelike separated events is the tensor product of the individual hilbert spaces . evolution is implemented by unitary operators between complete pairs , special cases of sets of spacelike separated events which can be understood as isolated planck - scale systems , or the discrete equivalent of successive cauchy surfaces of an isolated component of spacetime .    clearly , this implements the first four properties above . the fifth is more difficult ( and will not be resolved here ) . because it involves a fixed causal structure , a qch is not a completely background independent description . nevertheless , all other degrees of freedom are quantum , and with the algebraic treatment presented here this may be a step toward a truly background independent model . quantum causal histories can also describe causal spin foam models ( see the review @xcite ) . these are pre - spacetime models ; that is , gravity and the familiar spacetime manifold are meant to be derived as the low - energy continuum approximation of such models . a particular example of a qch is a causal spin network @xcite and this was a principal motivation for the introduction of qch s . a variation on the idea of a qch was applied to a spin foam model in @xcite .    however , as was already discussed in @xcite , the definition of qch there does not adequately take into account the causal structure of the underlying causal set . in particular , the unitary evolution operators in that definition are indifferent to the individual relations interpolating between the ingoing and outgoing events . the same unitary evolution is admitted regardless of these individual relations . this means that the quantum evolution does not respect the underlying causal structure . further , given two related events , there might not be a complete pair containing them , in which case we can not say anything about the relation between states at the two events . these issues are discussed in detail in the next section .    in this paper we address all of the above issues by giving a revised definition for a quantum causal history in which the causal structure is faithfully reflected in the quantum causal history . we find that doing so requires working with algebras of operators , instead of hilbert spaces , on the events . following the above idea of local finiteness at the fundamental level , these algebras are simple matrix c@xmath1-algebras . we find that a causal relation in the causal set corresponds to a completely positive map in the qch . this is not surprising , since this is the most general kind of physical transformation allowed by quantum theory . we also show that these maps can be regarded as fundamental and the unitary operators of the original definition can be reconstructed from them with no loss of information .    as a quantum cosmology , a qch is most naturally interpreted as describing the universe as a very large collection of open quantum mechanical building blocks . unitary evolution arises only where there is a complete pair , which is the discrete analogue of an isolated system . this is much more general than a wheeler - dewitt type of quantum cosmology . that `` canonical '' approach to quantum gravity is strongly dependent on global hyperbolicity and cauchy surfaces . in contrast , qch s allow that hyperbolicity may dissolve in the quantum foam . this seems more likely to describe the apparent non - unitarity of black hole evaporation than does the more traditional approach . a qch is also especially suited for describing the evolution of localized quantities without having to refer to a path integral or hamiltonian constraint for the entire universe . aside from cosmology , we see with hindsight that there are several further ways of interpreting a qch ; we discuss these in section [ qft ] and the conclusions . concretely , the present paper is an in - depth analysis of qch s and opens up several new possibilities for the concrete implementation of models of that type . for example , it is interesting , and potentially practical , that evolution in a qch is almost identical to the quantum information theoretic description of quantum noise . also , the continuous time analogue of the completely positive evolution in a qch , is a master equation , already extensively used in quantum cosmology . we comment on these connections in the conclusions . the outline of this paper is as follows . in section [ qch ] , we review the original definition of quantum causal histories given in @xcite , as well as the definitions of the basic structures in a causal set that are needed for the rest of the paper . one of our main results is that the relations in a causal set correspond to completely positive maps in the qch . in order to give the reader some intuition about these maps before they have to face the precise definition of a qch and the following proof that unitary evolution can be reconstructed from them , we give in section [ cpmaps ] a basic introduction to completely positive maps and a simple example . section [ ach ] is the revised , complete definition of a qch . although this was not a motivation for this work , this definition can be understood as an algebraic quantum field theory on a causal set , with the operator algebras being simple matrix c@xmath1-algebras . we explain this in section [ qft ] . in section [ proof ] , we show that nothing is lost in going from the unitary operators of the original definition to the completely positive maps , since the former can be reconstructed from the latter , up to irrelevant phase factors . we discuss the interpretation of our results and future directions in the conclusions . in this section we review quantum causal histories . they were defined in @xcite as a general formalism for causal spin foams in quantum gravity @xcite . an important and distinguishing feature of the geometry of spacetime ( as opposed to space ) is causal structure . if a spacetime is time - orientable and has no closed timelike curves , then its causal structure can be completely described as a partial order relation on the events ( points ) of the spacetime . the definition of the relation is that @xmath2 ( `` @xmath3 precedes @xmath4 '' ) if there exists a future - directed non - spacelike curve from @xmath3 to @xmath4 . this relation is transitive , i.e. , if @xmath2 and @xmath5 then @xmath6 , because future - directed curves can be concatenated . the assumption that there exist no closed timelike curves means precisely that @xmath2 and @xmath7 if and only if @xmath8 . these are the defining properties of a _ partial order relation_.    we are concerned here with a discrete analogue of a smooth spacetime . causal set _ @xmath9 is a locally finite , partially ordered set . that is , for any two events @xmath10 , there exist ( at most ) finitely many events @xmath11 such that @xmath12 . if either @xmath2 or @xmath7 , then we say that the events @xmath3 and @xmath4 are _ related_. if two events are not related , then we say that they are _ spacelike separated _ and write @xmath13 .    an _ acausal set _ is a subset @xmath14 such that all events in @xmath15 are spacelike separated to one another , i.e. , @xmath16 . acausal sets play the role of spacelike hypersurfaces in causal sets . a causal set can usefully represented by the ( directed ) graph of elementary relations ( fig . [ cpairs ] ) . the set of vertices is simply @xmath9 . there is an edge @xmath17 if @xmath2 but there does not exist any @xmath11 such that @xmath12 . there is a simple analogue of a future - directed non - spacelike curve in a causal set . a _ future - directed path _ is a directed path through the graph , that is , a sequence of events such that there exists an edge from each event to the next . a future - directed path is _ future ( past ) inextendible _ if there exists no event in @xmath9 which is in the future ( past ) of the entire path . note that this graph generates the partial ordering : @xmath2 if and only if there exists a future directed path from @xmath3 to @xmath4 .    from this , we can define complete future and past . an acausal set @xmath15 is a _ complete future _ of an event @xmath3 if @xmath15 intersects any future - inextendible future - directed path that starts at @xmath3 . similarly , an acausal set @xmath15 is a _ complete past _ of an event @xmath3 if @xmath15 intersects any past - inextendible future - directed path that ends at @xmath3 .    in order to describe a local quantum structure on our causal set , we would like a ( finite - dimensional ) hilbert space @xmath18 for every event @xmath19 . for two spacelike separated events @xmath20 , the composite state space for @xmath21 is @xmath22 . more generally , for any acausal set @xmath23 , @xmath24 .    in quantum mechanics , time evolution is unitary evolution between the hilbert spaces at different times . when should we expect unitary maps between the hilbert spaces for different acausal sets @xmath25 ? unitarity can be interpreted as meaning that no information is lost or added . we can only expect that no information is lost in evolving from @xmath15 to @xmath26 if @xmath26 is a complete future of @xmath15 ; otherwise , information coming from @xmath15 might `` miss '' @xmath26 . likewise , we can only expect a state at @xmath26 to be completely determined by one at @xmath15 if @xmath15 is a complete past of @xmath26 . when both of these conditions are satisfied @xmath15 is a complete past of @xmath26 and @xmath26 is a complete future of @xmath15  we call @xmath15 and @xmath26 a complete pair and write @xmath27 . we should expect a unitary map @xmath28 if and only if @xmath27 . one can think of a complete pair as successive cauchy surfaces of an isolated component of spacetime , or of all spacetime ( see fig . [ cpairs ] ) . this leads to the definition of a quantum causal history given in @xcite . it consists of a causal set @xmath9 , a finite - dimensional hilbert space @xmath18 for every @xmath19 , and a unitary map @xmath29 for any complete pair @xmath27 , such that for @xmath30 , @xmath31    with this definition , the unitary maps on the complete pairs tell us very little about the causal relations between the two acausal sets . there can be many configurations of the interpolating causal relations in a complete pair , for example :        the rules of a quantum causal history tell us that there is a unitary map @xmath32 , but not how it should depend on the interpolating causal relations . this is insufficient . consider for example the complete pair :        if we start with an unentangled pure state @xmath33 , then this will evolve to @xmath34 . the state at @xmath35 is described by a density matrix , @xmath36 . with this causal structure , the state at @xmath35 should be completely determined by that at @xmath4 . in other words @xmath37 should be independent of @xmath38 . however , with the above definition of a quantum causal history , there is no restriction on @xmath39 that would prevent @xmath37 from depending upon @xmath38 . another problem with this definition is the reliance on complete pairs . given two related events @xmath2 , there may not exist a complete pair containing these events . if so , with this definition of a quantum causal history , we do not know anything about the relation between states at @xmath3 and @xmath4 . there should be some relation , and the concept of quantum causal history must be refined to provide this .    in this paper we address all of the above issues by giving a revised definition of a quantum causal history in which the causal structure of @xmath9 is faithfully reflected in the quantum causal history . we will define the qch as an assignment of completely positive maps to pairs of causally related events . these maps relate the local observables at the causally related events . in the next section we introduce the definition of a completely positive map on open quantum mechanical systems , in order to provide some intuition for the evolution in qch before giving the new definition and results in sections [ ach ] and [ proof ] . an ordinary quantum mechanical system may be isolated , or it may be coupled to some environment . the states of the system can be described with density matrices and the most general evolution that quantum mechanics allows for such a system is a linear , trace - preserving , completely positive ( cp ) map . in this section we define cp maps and discuss their physical interpretation . consider a system @xmath40 in some mixed state given by @xmath41 , a density matrix on its hilbert space . let @xmath40 interact with an environment @xmath42 , also in a mixed state @xmath43 . the evolution of the joint system is given by a unitary operator @xmath44 . the process could be such that it is not feasible to extract the system @xmath40 again , but instead we extract @xmath45 , which has a density matrix @xmath46 . see fig.[cpex ] . the transformation from @xmath41 to @xmath46 is given by a map @xmath47 of the form @xmath48.\ ] ]    such generic transformations from a quantum system to another are maps from the set of density matrices on @xmath40 to the set of density matrices on @xmath45 that satisfy the following conditions :    _ linear on the set of density matrices : _ a probabilistic mixture of inputs leads to a probabilistic mixture of outputs , i.e. , for probabilities @xmath49 , @xmath50    _ trace - preserving : _ this simply means that the normalization is preserved for any density matrix . the total probability for _ anything _ to happen is , and remains , @xmath52 .    _ positive : _ probabilities are never negative , and so a density matrix has to be a positive operator ( i.e. , self - adjoint with only non - negative eigenvalues ) in order to describe a physical system . this is preserved by @xmath53 : if @xmath41 is positive , then @xmath54 is also positive . note that this implies that @xmath55 for any matrix @xmath56 . _ completely positive : _ it turns out that positivity is not enough for @xmath53 to describe a physical process . suppose that we couple the system @xmath40 to another system @xmath42 that has trivial dynamics , @xmath57 . for all such trivial extensions , @xmath58 should also be positive . this is the condition of complete positivity . complete positivity ensures that the evolution of @xmath40 is extendible in a trivial way to a physical evolution in a larger system @xmath59 . it is useful because it covers the case where @xmath40 is initially entangled with some distant isolated system . a completely positive map on density matrices is equivalent to a completely positive map of observable operators in the opposite direction ( in this case , backwards in time ) . a trace - preserving cp map of density matrices is dual to a unital . ] cp map of observables . the most general setting for completely positive maps is between c@xmath1-algebras ( see @xcite ) . in that context , there are many important tools for working with cp maps . in particular , the stinespring construction shows that any unital cp map from a c@xmath1-algebra to the algebra of bounded operators on a hilbert space can be written in a universal form , as a representation of the c@xmath1-algebra on a larger hilbert space , composed with the projection onto the smaller hilbert space . a quantum causal history can be reexpressed in terms of matrix algebras . nothing is lost but trivial phase factors .    to any event @xmath60 , we associate the algebra , @xmath61 \mbox,\ ] ] of matrices acting on @xmath18 . likewise , for any acausal @xmath62 there is the algebra , @xmath63 = \bigotimes_{x\in\xi } { \mathcal{a}}(x ) \mbox.\ ] ] the inner products on the hilbert spaces lead to involutions on the algebras . in this way , they can be regarded as c@xmath1-algebras . what do the unitary maps @xmath64 give us ? for a complete pair @xmath27 , @xmath64 determines a @xmath65-isomorphism , @xmath66 if we are only interested in states ( and not phase - factors ) , then @xmath53 does not lose any important information from @xmath64 . note that the maps @xmath53 are directed backward in time . this is because we are regarding the matrix algebras as consisting of observable operators acting on the hilbert spaces of local states . this is equivalent to a map of states ( density matrices ) in the other direction . indeed , states should evolve into the future . however , because the unitary maps we are starting with are invertible , everything that we are doing can be expressed equivalently with time reversed .    what does @xmath64 give us more locally ? if @xmath67 , then @xmath68 can be naturally thought of as a unital subalgebra of @xmath69 . thus , when @xmath27 and @xmath67 , there is a unital @xmath65-homomorphism , @xmath70 , given by restricting @xmath71 to @xmath68 . however , this is as far as homomorphisms will go . for @xmath72 and @xmath67 the naturally induced map @xmath73 is not a homomorphism ; it is only completely positive . see fig.[maps1 ] .    before describing the construction , we should note some properties of the algebras that we are working with . they are simple , finite - dimensional matrix algebras defined on hilbert spaces . as such , they can be viewed in several ways and have convenient structures that are not available with most infinite - dimensional algebras . we have already noted that they are c@xmath1-algebras . they are also vonneumann algebras , and because they are simple , they are ( type i ) vonneumann factors . we will make use of the trace on such an algebra ; however , we will take advantage of the finiteness and use the normalized trace , given by the formula , @xmath74 the advantage of this will be to maintain the time - reversal symmetry of the formalism as explicitly as possible ; the disadvantage is a slightly unorthodox normalization for density matrices ( @xmath75 is correctly normalized ) .    using the trace and the involution , we can construct an inner product on a matrix algebra , @xmath76 making the algebra into a hilbert space as well . since our algebras are hilbert spaces , the linear maps among them have adjoints . in particular , if @xmath77 is a linear map respecting the involution @xmath65 , the definition of its adjoint , @xmath78 reduces to the property that for any @xmath79 and @xmath80 , @xmath81=\tau_b[v(a)b]$ ] . we denote the adjoint of a map with a dagger to avoid confusion with the c@xmath1-algebra involution @xmath82 . the adjoint of a homomorphism will not generally be a homomorphism , however , it will be a completely positive ( cp ) map . the adjoint of any cp map is cp . if @xmath77 is cp , and @xmath83 and @xmath84 are subalgebras , then define the _ reduction _ of @xmath85 to @xmath86 to be the cp map @xmath87 , where @xmath88 and @xmath89 are the inclusions . @xmath90 a quantum causal history thus leads to cp maps for pairs of events . specifically , if @xmath27 , @xmath72 , and @xmath67 , then @xmath91 is the reduction of @xmath71 . maps for pairs of related events give a more local description of a quantum causal history than maps for complete pairs of acausal sets . in a local description , we should regard the maps @xmath92 as fundamental , and specify consistency conditions on them . this leads to two questions , do local consistency conditions exist , and is any information lost in going from global homomorphisms to local cp maps ? the answers , we will show , are yes and no .    there is one circumstance in which the adjoint of a homomorphism is another homomorphism .    [ unitary ] if @xmath93 is an isomorphism , then @xmath94 is its inverse ( also an isomorphism ) . homomorphisms in this category are trace - preserving . so , for @xmath95 and @xmath80 , @xmath96 = \tau_b\left[v(a\,v^{-1}(b))\right ]      = \tau_a[a\,v^{-1}(b ) ]      \mbox.\ ] ] thus @xmath97 . so , for @xmath27 , @xmath71 and @xmath98 are both isomorphisms . there is thus a time - reversal covariance ; this should be reflected in our axioms . states on matrix algebras are all given by density matrices . specifically , a density matrix is a positive matrix @xmath41 with @xmath99 ; the expectation value of a matrix @xmath56 is @xmath100 . the map of algebras @xmath101 induces a pullback map from states at @xmath3 to states at @xmath4 . this is simply the adjoint @xmath102 applied to density matrices . when the map of algebras is a homomorphism , the map of states will take pure states to pure states . suppose that @xmath15 and @xmath26 are acausal sets , not necessarily a complete pair , but that @xmath26 is a complete future of @xmath15 . in this case , no information should be lost in going from @xmath15 to @xmath26 , and pure states should map to pure states . thus , there should be a homomorphism @xmath103 . conversely , if instead @xmath15 is a complete past of @xmath26 , then there should exist @xmath103 such that @xmath98 is a homomorphism . suppose that @xmath104 and that @xmath15 is a complete past of both . the homomorphisms @xmath105 and @xmath106 are restrictions of @xmath107 . so , for @xmath108 and @xmath109 , @xmath110 thus the images of @xmath105 and @xmath106 must commute . if such @xmath53 s exist , and the @xmath111 s are reductions of them , then they will satisfy the following properties . a quantum causal history consists of a simple matrix c@xmath1-algebra @xmath112 for every event @xmath19 and a completely positive map @xmath113 for every pair of related events @xmath2 , satisfying the following axioms . _ axiom 1 : _ ( extension ) for any @xmath114 and @xmath62 a complete past of @xmath4 , there exists a homomorphism @xmath115 such that for each @xmath72 , the reduction of @xmath116 to @xmath117 is @xmath111 . likewise , for any @xmath118 a complete future of @xmath4 , there exists a map @xmath119 such that @xmath120 is a homomorphism and for any @xmath67 , the reduction ( restriction ) of @xmath121 to @xmath122 is @xmath123 . see fig.[ax1 ]    _ axiom 2 : _ ( spacelike commutativity ) if @xmath124 and @xmath14 is a complete past of @xmath4 and @xmath35 , then the images of @xmath125 and @xmath116 ( in @xmath126 ) commute . likewise , if @xmath118 is a complete future of @xmath127 , then the images of @xmath128 and @xmath129 commute . see fig.[ax2 ] . _ axiom 3 : _ ( composition ) if @xmath26 is a complete future of @xmath3 and a complete past of @xmath4 , then @xmath130 . this notion of quantum causal history can also be motivated from the ideas of algebraic quantum field theory . algebraic quantum field theory ( see @xcite ) is a general approach to quantum field theory based on algebras of local observables , the relations among them , and their representations . the correct choice of axioms is still a matter of research , but for our purposes , these subtleties are largely irrelevant . an algebraic quantum field theory associates a vonneumann algebra to each causally complete region of spacetime . this generalizes easily to causal sets . the following definitions are exactly the same as for continuous spacetime . for any subset @xmath131 , define the _ causal complement _ as @xmath132 the set of events which are spacelike to all of @xmath133 . the _ causal completion _ of @xmath133 is @xmath134 , and @xmath133 is _ causally complete _ if @xmath135 . a causal complement is always causally complete ( i.e. , @xmath136 ) .    in the most restrictive axiomatic formulation of algebraic quantum field theory there is a vonneumann algebra @xmath137 for every causally complete region ; these all share a common hilbert space . whenever @xmath138 , @xmath139 . for any causally complete region @xmath133 , @xmath140 is @xmath141 , the commutant of @xmath137 . the algebra associated to the causal completion of @xmath142 is generated by @xmath137 and @xmath143 . some of the standard arguments about the properties of the local vonneumann algebras are valid for causal sets ; some are not . the algebras should all be simple ( i.e. , vonneumann factors ) because the theory would otherwise have local superselection sectors . for continuous spacetime it is believed that the local algebras should be type iii@xmath144 hyperfinite factors ; however , the reasoning involves the assumption that there exists a good ultraviolet scaling limit . this does not apply here ; the small - scale structure of a causal set is discrete and not self similar at all . instead , it seems reasonable ( following property 4 in the introduction ) that only a finite amount of structure should be entrusted to each event . in other words , each vonneumann algebra should be a finite - dimensional matrix algebra . in vonneumann algebra terms , these are finite type i factors . consider the causal completion @xmath145 of a finite acausal set @xmath15 ; @xmath146 should be generated by the algebras @xmath112 for @xmath72 . in fact , these algebras should commute , and therefore @xmath147 evolution by homomorphisms is sometimes possible . if @xmath15 and @xmath26 are acausal sets such that @xmath148 , then @xmath149 and so @xmath150 . this inclusion is the evolution homomorphism when @xmath15 is a complete past of @xmath26 . not surprisingly , simple matrix algebras are much easier to work with than type iii vonneumann factors . using the ( unique ) normalized trace , any state is given by a density matrix . recall that the adjoint maps @xmath102 in a quantum causal history are the induced maps on density matrices . suppose that @xmath151 is a density matrix at @xmath3 . the density matrix @xmath152 is the best approximation to @xmath41 among density matrices at @xmath4 , in the sense that it minimizes the trace norm of the difference . the trace norm metric on density matrices is equal to the metric on states induced by the operator norm on observables . so , we see that the obvious notion of an algebraic quantum field theory on a causal set , with the physically reasonable assumption of finite algebras on events , gives the structure of a qch . the two concepts intersect , but are not equivalent . one could ignore the arguments about entropy _ et cetera _ and consider a field theory with infinite algebras on events ; however , depending upon the choice of algebras , even this might be described by some suitable generalization of a qch . likewise , a given qch can not necessarily be derived from an algebraic quantum field theory . this does mean that the structure of a qch encompasses a reasonable notion of a quantum field theory , and hence is capable of describing matter degrees of freedom . it also indicates how quantum fields on curved spacetime might be obtained as a limit of some quantum gravity model based on qch s . finally , this gives the skeptical reader another possible motivation for considering quantum causal histories . in this section we show that , if we start with the cp maps on the causal relations as the fundamental evolution operators , we can recover the unitary operators between the complete pairs . the main result is that the axioms in the definition of a quantum causal history imply the following theorem :    [ main ] for any acausal sets @xmath153 , if @xmath26 is a complete future of @xmath15 or @xmath15 is a complete past of @xmath26 then there exists a unique map @xmath154 such that    1 .   for any @xmath72 and @xmath67 , the reduction of @xmath71 to @xmath73 is @xmath92 . if @xmath15 is a complete past of @xmath26 , then @xmath71 is a homomorphism . if @xmath26 is a complete future of @xmath15 , then @xmath98 is a homomorphism . if @xmath27 is a complete pair , then @xmath71 is an isomorphism . 5 .   if @xmath30 , then @xmath155 .    in order to prove this , we first need to prove the uniqueness aspect . let @xmath156 , _ et cetera _ be ( finitely many ) simple , finite - dimensional , matrix algebras . let @xmath157 . any unital homomorphism @xmath158 has a reduction @xmath159 for each @xmath160 . we need to show that two distinct homomorphisms will have distinct reductions . let @xmath161 be some unital homomorphism . any other unital homomorphism can be obtained from this by composing it with an automorphism of @xmath162 . all homomorphisms in this category are injective , so to simplify notation , regard @xmath163 as a subalgebra of @xmath162 and @xmath164 as the natural inclusion . first we prove the result at the lie algebra level . denote the canonical inclusions by @xmath165 . the reductions of @xmath164 are simply @xmath166 . for any @xmath167 , define @xmath168 by @xmath169_-$ ] . [ infinitessimal ] for any @xmath167 , if @xmath170 for all @xmath160 , then @xmath171 . let @xmath172 be the subspace spanned by the @xmath173 s . by the hypothesis , for any @xmath79 , @xmath174_-$ ] must be orthogonal as a hilbert space . ] to every @xmath173 , and thus to @xmath175 . hence for any @xmath176 , @xmath177_- ) = \tau ( \beta [ a , c]_- ) \mbox.\ ] ] so @xmath178^\perp$ ] . we need to show that @xmath179^\perp\subseteq a'$ ] ( the commutant of @xmath163 ) . the orthogonal complement of @xmath180 is @xmath181 $ ] , so we simply need to prove that @xmath181\subseteq [ a , c]$ ] . @xmath175 is closed as a lie algebra and acts unitarily on @xmath162 by @xmath182 . @xmath175 generates @xmath162 as an algebra , therefore the only part of @xmath162 commuting with @xmath175 is the center , @xmath183 . this is thus the only trivial @xmath175-subrepresentation in @xmath162 , and so @xmath184=[b , b]$ ] . @xmath179=\operatorname{ad}_c(a)$ ] is the sum of the nontrivial irreducible @xmath175-representations which intersect with @xmath163 . thus indeed @xmath181\subseteq [ a , c]$ ] , and the claim is proven . [ reconstruction ] any homomorphism distinct from @xmath164 will have reductions distinct from those of @xmath164 . let @xmath185 be the subgroup that leaves the reductions of @xmath164 invariant and @xmath186 the subalgebra of @xmath187-invariant elements . @xmath188 leaves @xmath164 invariant , so @xmath189 and @xmath190 . @xmath187 is compact , so we can construct the orthogonal projection @xmath191 by averaging over @xmath187 . each map @xmath166 factors through @xmath192 , @xmath193    let @xmath194 be any element commuting with @xmath186 . for any @xmath79 and @xmath195 , @xmath196_-)=\tau(a[b,\beta]_-)=0 \mbox,\ ] ] so @xmath197 . thus @xmath198 therefore @xmath199 . this implies that @xmath200 . any other homomorphism can be written as @xmath201 with @xmath202 . if this is distinct from @xmath164 then @xmath203 , but if @xmath201 has the same reductions as @xmath164 then @xmath204 . we would then have @xmath187 larger than just @xmath205 and thus @xmath186 would have to be a _ proper _ subalgebra of @xmath163 . lemma [ reconstruction ] implies the uniqueness of the @xmath206 in axiom 1 . if @xmath15 is a complete past of @xmath26 , then if a homomorphism @xmath207 exists with the @xmath125 s ( for @xmath67 ) as its reductions , it can be reconstructed by multiplying the @xmath125 s together , so @xmath208 is unique . conversely , because of axiom 2 , we _ can _ multiply together the @xmath209 s to define @xmath208 . analogously , for @xmath26 a complete future of @xmath15 , a unique map @xmath210 exists , satisfying the requirements of property 3 . suppose that both of these are true . that is , @xmath27 is a complete pair . we need to show that @xmath211 and is an isomorphism . because all homomorphisms in this category are injective , the existence of homomorphisms in both directions between @xmath126 and @xmath69 shows that they are of equal dimension and therefore isomorphic . these homomorphisms must therefore be isomorphisms . so , @xmath212 is an isomorphism , and by theorem  [ unitary ] , @xmath210 is its inverse . the reductions of @xmath210 are the same as for @xmath208 and as it is a homomorphism , the above uniqueness implies @xmath213 . now suppose that @xmath30 . let @xmath72 and @xmath214 . by what we have proven so far , @xmath215 is a reduction of @xmath216 . by axiom 3 , @xmath217 . so , @xmath111 is a reduction of @xmath218 . by lemma [ reconstruction ] , since the reductions are the same , we must have that @xmath219 . using lemma [ reconstruction ] again , this shows that @xmath155 . we have then shown that the system of isomorphisms associated to complete pairs can be reconstructed from the local information of cp maps associated to pairs of events . because @xmath9 is assumed to be locally finite , there is no problem in going from these isomorphisms of algebras to a system of unitary maps satisfying the original definition of a quantum causal history ; the only arbitrariness is a choice of irrelevant phase factors .    on the other hand , the local formulation is restrictive . not every choice of unitary maps for complete pairs can be expressed in terms of cp maps in this way . this formulation enforces local causality . in this paper , we have provided a complete definition of a quantum causal history , a pre - spacetime framework for a quantum theory of gravity in which there is a simple matrix algebra of operators for every event in the universe ( represented by a causal set ) and a completely positive evolution map for every causally related pair of events . these represent the transfer of quantum information between events in the quantum spacetime . we showed that the axioms in the definition allow us to reconstruct the unitary evolution between complete pairs , a necessary step since complete pairs correspond to isolated systems .    in the picture of the universe as a qch , it can be thought of as a very large collection of quantum - mechanical building blocks . everything is ( locally ) finite , both the dimensionality of these systems as well as the causal set . we regard this as a simple way to model the idea that there is a finite number of fundamental degrees of freedom in a finite region of the universe . although there is certainly no hard evidence for this , such ideas are supported by bekenstein s arguments , by the black hole entropy calculations from both string theory and loop quantum gravity , and is related to holographic ideas and the arguments for a many - hilbert space m - theory in @xcite . even independent of such motivations , finiteness allows significant simplifications which we have been able to exploit in this paper .    it may be possible to generalize a qch to use some infinite - dimensional algebras . theorem [ main ] may extend to type ii@xmath144 vonneumann factors , which have finite traces . however , we have not investigated this possibility in detail because the physical motivations for using matrix algebras are much stronger . it is interesting to note the multiple ways that quantum causal histories can be motivated , including :    1 .   the original definition considering them as the general formalism for causal spin networks @xcite . 2 .   as an algebraic quantum field theory on a causal set ( section [ qft ] ) . 3 .   as an alternative to the usual quantum cosmology of a wavefunction of the universe evolving according to the wheeler - dewitt equation . a qch is a framework for quantum cosmology that is not restricted to globally hyperbolic spacetimes , as is the case with a wdw - type quantum cosmology . this is discussed in more detail in @xcite . the local evolution in a qch is well - suited for addressing the important problem of identifying localized propagating degrees of freedom in a spin foam , without having to use the entire partition function . precisely the same objects that we used ( matrix algebras and cp maps ) are basic tools in quantum information theory . conceptually , this suggests that the universe modeled as a qch is nothing but a background - independent quantum computer . more practically , the analogue of the much sought - after propagating `` particle '' in a spin foam , in quantum information theoretic language , is a quantum channel . tools such as the operator - sum representation of cp maps , and even ideas from the theory of error - correction in quantum computation , might be used to identify such objects . there is an important clarification we should make . if quantum causal histories are motivated via 2 above , then this is not gravity , but a kind of quantum field theory on a peculiar background . this is related to the issue of the _ given _ causal set that a qch is built on . if the procedure is simply `` given a causal set @xmath9 , a qch is constructed via the procedure of section [ ach ] , '' then this can not have much to do with a dynamical theory of space - time . unless , somehow , all the gravitation degrees of freedom are in the algebras and @xmath9 is some auxiliary background , but we are not aware of such a scheme . in the quantum causal histories motivated from spin foams in quantum gravity ( 4 above ) , there is an further step . the partition function for a ( causal ) spin foam model includes a sum over all causal sets . this is the usual path integral formulation of quantum gravity , with each term in the path integral being a qch . it would be very interesting to understand if there is a way for a qch to be a dynamical quantum space - time without the summation of a path integral . for this , one would need the causal set and the algebras to `` interact '' with each other . the reason why there are so many ways to understand quantum causal histories is partly the general nature of a cp map . it is , after all , the most general physical transformation allowed by quantum theory . completely positive maps already appear in several places in quantum cosmology and we close this paper by referring to a few examples .    in the context of continuous time evolution , master equations are the familiar tools used to describe quantum noise , for example in quantum optics . these are differential equations , governing the non - unitary evolution of the density matrices of dissipative quantum systems ( e.g. , @xcite ) . when a master equation is integrated , the evolution of the density matrix for a finite time - step is given by a completely positive map . another interesting comparison is between our cp maps and decoherence . the term decoherence is used in the literature both to describe quantum noise , and the more specific process of phase damping that is believed to play a role in the emergence of classicality ( see , for example , @xcite ) . using decoherence to mean the second , we note that it is possible to incorporate processes of that type in a quantum causal history . in @xcite , an interesting scheme for describing decoherence was proposed that is closely related to quantum causal histories .    in this new formulation , with the completely positive maps as the fundamental evolution operators , we obtain a quantum cosmology in which the universe is simply a very large collection of finite quantum systems . the information gets shuffled around , according to the causal structure . local observers can lose information simply because , by causality , it is not available to them . this is much like information loss due to the presence of a horizon . finally , we note that there are several tools that can be used for the practical implementation of cp maps in a specific model . these include the operator - sum representation that is frequently used in quantum information theory ( see @xcite , chapter 8 and @xcite for excellent reviews , and , for example , @xcite for the case of dimensionality change from input to output ) , and the stinespring construction in operator algebras @xcite . we are very grateful to the jesse phillips foundation for providing the funds that brought us together in berlin . fm would like to thank barbara terhal for very useful pointers to the relevant literature in quantum information theory , mike mosca and david poulin for discussions on this subject , lee smolin and olaf dreyer for their comments on this work , and the albert einstein institute , golm , where part of this work was carried out . part of this work was supported by nsf grant phy-0090091 .
we provide a precise definition and analysis of quantum causal histories ( qch s ) . 
 a qch consists of a discrete , locally finite , causal pre - spacetime with matrix algebras encoding the quantum structure at each event . 
 the evolution of quantum states and observables is described by completely positive maps between the algebras at causally related events . 
 we show that this local description of evolution is sufficient and that unitary evolution can be recovered wherever it should actually be expected . 
 this formalism may describe a quantum cosmology without an assumption of global hyperbolicity ; it is thus more general than the wheeler - dewitt approach . 
 the structure of a qch is also closely related to quantum information theory and algebraic quantum field theory on a causal set . 
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