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communication of information requires an encoding of the information into a physical system . the laws of physics therefore govern the limits on processing and communication of information . by modelling real world noise in terms of simpler models , the maximum rate for information transfer
may be obtained .
the seminal work of shannon @xcite showed that a memoryless noisy channel , where the noise acts independently on each symbol sent through the channel , can be parameterized by a single quantity , the capacity of the channel @xmath0 .
shannon defined the capacity as the maximum rate that information may be sent through a channel , and he also showed that there exist codes that asymptotically achieve this rate with a vanishing probability of error .
the work of shannon has been extended to include channels with `` memory '' , where the noise is no longer independent of past channel uses @xcite , or where the channel consists of an arbitrary transformation of the input states @xcite .
encoding classical information into quantum states of physical systems gives a physical implementation of the constructs of information theory .
the majority of research into quantum communication channels has focused on the memoryless case , although there have been a number of important results obtained for quantum channels with correlated noise operators , or more general quantum channels @xcite .
recently a model for quantum channels with memory has been proposed that can consistently define quantum channels with markovian correlated noise @xcite .
the model also extends to describe channels that act on transmitted states in such a way that there is no requirement for interaction with an environment within the model .    in this paper a single upper bound on the capacity of arbitrary indecomposable quantum channels is derived .
a quantum channel is defined as a completely positive , trace preserving map from the set of density operators to itself .
any such map may be represented as a unitary operation between the system state and an environment with a known initial state @xcite .
memoryless quantum channels act on each input state independently of the previous input or output states .
for a single channel use the output state is given by , @xmath1\ ] ] with @xmath2 the input state , @xmath3 the initial state of the environment , @xmath4 a unitary operation between the state @xmath5 and environment @xmath6 , and @xmath7 the output state . for a sequence of transmissions through a memoryless channel , the output state is given by , @xmath8 \nonumber \\ & = \big(\lambda_{n } \otimes ... \otimes \lambda_{1 } \big ) \rho_q \label{eqn : memoryless_model}\end{aligned}\ ] ] where the state @xmath2 now represents a ( possibly entangled ) input state across the @xmath9 channel uses , the unitary operations @xmath10 are all identical , and the environment state is a product state @xmath11 .
one model of a quantum memory channel is where each state going through the channel acts with a unitary interaction _ on the same channel memory state _ , as well as an independent environment .
the backaction of the channel state on the message state therefore gives a memory to the channel .
the general model thus includes a channel memory @xmath12 , and the independent environments for each qubit @xmath13 .
hence , @xmath14 \nonumber \\ & = \mathrm{tr}_{m } \big [ \lambda_{n , m } ... \lambda_{1,m }   \big(\rho_q \otimes \omega_m \big ) \big ] \label{eqn : memory_model}\end{aligned}\ ] ] where @xmath15 is the initial memory state , @xmath2 and @xmath16 are the input and the output states of the channel , respectively , and the trace over the environment is over all environment states .
figure [ fig : memory_channel ] illustrates the action of the unitary operators on the input , memory and environment states representing the channel .
if the unitaries factor into independent unitaries acting on the memory and the combined state and environment , that is , @xmath17 , then the memory traces out and we have a memoryless channel .
if the unitaries reduce to @xmath18 , we can call it a _ perfect _ memory channel , as no information is lost to the environment .
the mapping of the memory state under the unitary operation also corresponds to a quantum channel on the memory state .
memory channel representations where the action on the memory state is independent of the input state are termed _
point _ memory channels , as the memory state will have a fixed point under the action of the represetation @xmath19\omega_m = \omega_m$ ] for all @xmath2 . fixed 
point channels may also be seen to be symbol independent ( si ) , as the previous input states do not affect the action of the channel on the current input state .
this is opposed to channels with intersymbol interference ( isi ) , where the previous input state affects the action of the channel on the current input .
an extreme example of an isi channel is the quantum shift channel , where each input state is replaced by the previous input state .
there exist examples of si channels that are not fixed point channels .
the `` classical '' memory channel defined in terms of unitary kraus error operators by , @xmath20 for @xmath21 the transition probabilities of a markov chain , is not a fixed point channel , but displays no isi . in this case , however , the channel may equivalently be defined by , @xmath22 where the environment states @xmath23 are orthogonal .
this results in a fixed  point channel with the equivalent output as obtained from ( [ eqn : markovmem ] ) .
if the representations of a given memory channel are treated as an equivalence class under the input  output action of the channel , then for every channel of the above type , there exists a fixed  point representation of the channel . whether this is true for arbitrary isi channels is not known
. we may also conjecture that the types of memory channels that display _ only _ isi may be represented as perfect memory channels .     and
environment @xmath13 .
the correlations between the error operators on each state @xmath24 are determined by the unitary operation @xmath25 and the memory state at each stage of the channel evolution.,width=240 ]
the fano inequality , combined with the holevo upper bound on the accessible information , provides an entropic upper bound on the classical information capacity of any quantum channel . utilizing classical  quantum states of the form , @xmath26 where the @xmath27 form an orthonormal set .
it is possible to derive the holevo bound from the von neumann mutual information bound over these classical  quantum states . to view the upper bound in
a more physically motivated setting we can note that any separable state @xmath28 can be extended , in a larger hilbert space , to a state @xmath29 in the form of ( [ eqn : class_quant1 ] ) . to show this note that any separable state may be written in the form , @xmath30 and each @xmath31 may be purified into a direct sum of hilbert spaces @xmath32 with orthogonal support , such that , @xmath33 where @xmath34 = \rho^j_r$ ] . due to the monotonicity of the von neumann mutual information @xmath35
, the upper bound may be expressed as , @xmath36 where @xmath37 is the set of all separable states .
the maximum amount of mutual information that may be generated through a quantum channel @xmath38 is therefore bounded by the maximum amount of `` classical '' correlation that may be shared by states through the channel .
the upper bound on the classical capacity of a finite memory channel , which includes the classically correlated channels , is given by @xcite , @xmath39q ) \label{eqn : mem_holevo_upperbound}\ ] ] where @xmath40q = \lambda^{(n)}[\omega_m]\rho_q$ ] is the action of the channel on the input state , with initial memory state @xmath15 , and @xmath37 the set of separable states .
this bound has been shown to be attainable whenever the channel is described by ( [ eqn : markovmem2 ] ) with unitary kraus operators @xmath41 , and the initial error probabilities are equal to the steady state probabilities for the regular markov chain @xcite .
an indecomposable channel is one where the long - term behavior of the channel is independent of the initial channel state .
memory channels with markov correlated noise are an example of indecomposable channels .
we begin by defining the _ trace distance _ of both probability distributions and density operators . for probability distributions
the trace distance is defined as @xmath42 , for distributions @xmath43 and @xmath44 , where both distributions share the index set @xmath45 @xcite . for density operators
the trace distance is defined as @xcite , @xmath46 where @xmath47 , taking the positive square root .
a finite - memory quantum channel is _ indecomposable _ if for any input state @xmath48 and @xmath49 there exists an @xmath50 such that for @xmath51 , @xmath52 where @xmath53 and @xmath54 are the memory states after @xmath9 uses of the channel for the intial memory states @xmath15 and @xmath55 respectively .
the long term behavior of an indecomposable channel is therefore independent of the initial memory state . fixed point channels for which the map on the memory state is a strictly contractive mapping @xmath56 , are automatically indecomposable .
the behavior of the memory state is of fundamental importance due to the fact that memory channels that may be represented in the form of ( [ eqn : memory_model ] ) are _ memory continuous_. a channel is memory continuous if for any @xmath49 there exist a @xmath57 , such that , @xmath58\rho_q - \lambda[\sigma_m]\rho_q\| \leq \epsilon .
\label{eqn : mem_cont}\end{aligned}\ ] ] note that the trace distance is monotonic @xmath59 for any trace preserving quantum operation @xmath60 .
hence , for @xmath60 the partial trace operation @xmath61 .
furthermore , due to the unitary invariance of the trace distance , we find , @xmath62 \rho_q - \lambda [ \sigma_m]\rho_q \| \nonumber \\ & \leq \big\|u_{qme } \big(\rho_q \otimes \omega_m \otimes |0_{e } \rangle \langle 0_{e}| \nonumber \\ & \phantom{=}\qquad\qquad\qquad\:- \rho_q \otimes \sigma_m \otimes |0_e\rangle \langle 0_e| \big ) u_{qme}^{\dag } \big\| \\ & = \big\|\,\rho_q \otimes \omega_m \otimes |0_{e } \rangle \langle 0_{e}| - \rho_q \otimes \sigma_m \otimes |0_e\rangle \langle 0_e| \big\| \\ & = \| \,\omega_m - \sigma_m \|\end{aligned}\ ] ] and ( [ eqn : mem_cont ] ) is satisfied for all memory channels of the form of ( [ eqn : memory_model ] ) , by simply making @xmath63 .
following gallager s derivation for classical finite state channels ( fsc ) @xcite , two classical capacities may be defined .
the lower capacity @xmath64 and the upper capacity @xmath65 are defined as , @xmath66 q \big ) \nonumber \\ \overline{c } & = \lim_{n\rightarrow \infty } \frac{1}{n } \max_{\omega_m } \max_{\rho_{rq}\in \mathcal{d } } s\big ( r:\lambda^{(n)}[\omega_m ] q \big ) \ ; .\end{aligned}\ ] ] it is obvious from the definitions that @xmath67 , and we wish to determine channels for which equality holds . in order to bound the difference in entropies for the output states of channels with different initial memory states , we utilize fannes inequality @xcite , @xmath68 where @xmath69 is the dimension of the hilbert space for the states @xmath70 and @xmath71 . from this
we can see that , @xmath72\rho\big ) - s\big(\lambda[\sigma ] \rho\big ) \big| \nonumber \\ & \leq \frac{1}{n}\bigg [ \big\| \lambda[\omega]\rho - \lambda[\sigma ] \rho \big\| \log d^{\ , n } + \frac{\log e}{e } \bigg ] \\ & = \big\| \lambda[\omega]\rho - \lambda[\sigma ] \rho \big\| \log d + \frac{\log e}{ne}\end{aligned}\ ] ] and by showing the trace distance of the output states may be made arbitrarily small for any input state @xmath48 , the entropies of the output states must converge asymptotically .
given any @xmath49 then there exists an @xmath50 such that the trace distance between the memory states is less than @xmath73 .
thus for @xmath74 we have , @xmath72\rho\big ) - s\big(\lambda[\sigma ] \rho\big ) \big| \nonumber \\ & \leq \frac{1}{n}\bigg [ \big| s\big(\lambda[\omega(n)]\rho\big ) - s\big(\lambda[\sigma(n ) ] \rho\big ) \big| + n(\epsilon ) \log d \bigg ] \nonumber \\ & \leq \frac{1}{n}\bigg [ ( n - n(\epsilon))\big\|\lambda[\omega(n)]\rho - \lambda[\sigma(n ) ] \rho\ , \big\| \log d \nonumber \\ & \phantom{=}\qquad + \frac{\log e}{e } + n(\epsilon ) \log d \bigg ] \nonumber \\ & \leq \epsilon \log d + \frac{\log e}{ne } + \frac{n(\epsilon ) \log d}{n}(1-\epsilon ) \label{eqn : ave_ent_bound}\end{aligned}\ ] ] where @xmath75 $ ] and @xmath76 $ ] denote the channels with initial memories @xmath15 and @xmath55 acting on input states @xmath77 to @xmath9 , and the maximum difference in entropy for the first @xmath50 output states is bounded by @xmath78 .
therefore , taking @xmath79 in ( [ eqn : ave_ent_bound ] ) , the average entropies converge asymptotically for any state @xmath48 .
the difference between the upper bounds @xmath65 and @xmath64 must therefore also converge asymptotically .
a single upper bound on the capacity for the class of indecomposable quantum memory channels has been derived .
the maximum rates for information transmission through noisy quantum channels has primarily been developed for memoryless channels , where the noise on each transmitted state is treated as independent . 
 many real world communication channels experience noise which is modelled better by errors that are correlated between separate channel uses . in this paper , upper bounds on the classical information capacities of a class of quantum memory channels are derived . 
 the class of channels consists of indecomposable quantum memory channels , a generalization of classical indecomposable finite  state channels .