The invalidity of a strong capacity for a quantum channel with memory

The full channel coding theorem provides a limit on the rate at which a sender can communicate an encoded message to a receiver, such that the probability of a decoding error at the receiver’s side decays exponentially in the number of channel uses. The theorem is comprised of two parts: the direct part of the theorem, which refers to the construction of the code, and the converse to the theorem. The direct part of the quantum channel coding theorem states that using $n$ copies of the channel, we can code with an exponentially small probability of error at a rate $R=\frac{1}{n}\log|\mathcal{M}|$, provided $R<C$ in the asymptotic limit, where $\cal M$ denotes the set of possible codewords to be transmitted over the channel and $C$ denotes the capacity of the channel. If the rate at which classical information is transmitted over a quantum channel exceeds the capacity of the channel, i.e. if $R>C$, then the probability of decoding the information correctly goes to zero in the number of channel uses. The latter is known as the strong converse to the channel coding theorem. The weak converse, on the other hand, states that if $R>C$, then the probability of decoding the information correctly is bounded away from $1$, i.e. the error probability does not tend to zero, whatever encoding/decoding scheme is used.
Shannon Shannon48 first proposed the theorem for classical discrete memoryless channels and the first rigorous proof of the direct part of the theorem was provided by Feinstein Feinstein and the strong converse by Wolfowitz Wolfowitz .
However, it was observed that the existence of the strong converse, and therefore strong capacity, for other types of classical channels does not always hold WolfowitzNoCap . See Ahlswede Ahlswede06 for a more complete discussion of converse results for various types of classical channels.
The strong converse to the channel coding theorem for memoryless classical-quantum channels with product state inputs was determined independently by Winter Winter99 and by Ogawa and Nagaoka ON99 . Their result implies that every memoryless discrete classical-quantum channel has a strong capacity which provides a sharp upper-bound on the rate at which classical information can be transmitted over this type of channel using product states.
Recent results include a proof by Bjelaković and Boche BB08 ; BB09 of a full coding theorem for the discrete memoryless compound classical-quantum channel. Wehner and König KW09 proved the fully general strong converse theorem for a family of channels, that is, they proved that the strong converse theorem holds for a family of quantum channels even in the case when entangled state inputs are allowed.
In this article we relax the assumption that the communication channel in question is memoryless and we concentrate on a particular quantum channel with memory, that is, a channel with correlations between successive channel uses. In our case the correlations between successive uses of the channel can be described by a Markov chain. Communication channels with memory are widely considered to be more realistic than memoryless channels since real-world channels may not exhibit independence between successive errors and correlations are common. Noise correlations are also necessary for certain models of quantum communication Bose03 . See for example Kretschmann and Werner KW05 and Mancini Mancini06 for models of quantum memory channels.
The article is organised as follows. We introduce notation, necessary definitions and define the quantum periodic channel in Section II. In Section III we prove that the periodic channel does not have a strong capacity. The observation relies on a result which is proved in Appendix A, involving a particular instance of a periodic channel and consequently the strong converse does not hold in general for the periodic channel. In Section V we remark on a scale of capacities for the random channel. We then state and prove the main result involving a scale of capacities for the channel.
Note that $\log$ is understood to be taken to the base $2$ throughout the article.

We begin by introducing some notation. A memoryless channel is given by a completely positive trace-preserving (CPT) map ${\Phi}:{\mathcal{B}}({\cal H})\to{\cal B}({\cal K})$, where ${\cal B}({\cal H})$ and ${\cal B}({\cal K})$ denote the states on the input and output Hilbert spaces ${\cal H}$ and $\cal K$, respectively.
Equivalently, we can describe a classical-quantum channel, here also denoted $\Phi$, as a mapping from the classical message to the output state of the channel on $\mathcal{B}(\mathcal{K})$ as follows,

where the message is first encoded into a sequence belonging the set $\mathcal{X}^{n}$, where $\cal X$ represents the input alphabet.
We can combine the two mapping descriptions as follows. We wish to send classical information in the form of quantum states over a quantum channel $\Phi$. A (discrete) memoryless quantum channel, $\Phi$, carrying classical information can be thought of as a map from a (finite) set, or alphabet, $\cal X$ into $\cal B(\cal K)$, taking each $x\in\cal X$ to $\Phi_{x}=\Phi(\rho_{x})$, where the input state to the channel is given by $\{\rho_{x}\}_{x\in\cal{X}}$ and each $\rho_{x}\in\cal B(\cal H)$. Let $d=\dim(\cal H)$ and $a=|\cal X|$.
For a probability distribution $P$ on the input alphabet $\mathcal{X}$, the average output state of a channel $\Phi$ is given by

The conditional von Neumann entropy of $\Phi$ given $P$ is defined by

and the mutual information between the probability distribution $P$ and the channel $\Phi$ is defined as follows,

An $n$-block code for a quantum channel $\Phi$ is a pair $(C^{n},E^{n})$, where $C^{n}$ is a mapping from a finite set of messages $\cal M$, of length $n$, into $\mathcal{X}^{n}$, i.e. a sequence $x^{n}\in\cal X$ is assigned to each of the $|\cal M|$ messages, and $E^{n}$ is a POVM, i.e. a quantum measurement, on the output space $\mathcal{K}^{\otimes n}$ of the channel $\Phi_{x^{n}}^{n}$. The maximum error probability of the code $(C^{n},E^{n})$ is defined as

The code $(C^{n},E^{n})$ is called an $(n,\lambda)$-code, if $p_{e}(C^{n},E^{n})\leq\lambda$. The maximum size $|\cal M|$ of an $(n,\lambda)$-code is denoted $N(n,\lambda)$. Define an finite alphabet $\cal X$ and sequences $x^{n}=x_{1},\dots,x_{n}\in\mathcal{X}^{n}$ and let

for $x\in\mathcal{X}$. The type of the sequence $x^{n}$ is given by the empirical distribution $P_{x^{n}}$ on $\cal X$ such that

Clearly, the number of types is upper bounded by $(n+1)^{a}$, where $a=\big{|}\cal X\big{|}$.

The strong converse for the periodic quantum channel does not hold in general because the following inequality holds

where,

The strict inequality above can be shown explicitly for a periodic channel consisting of two branches of qubit amplitude-damping channels (see Appendix A below for detailed proof). On the other hand, equality for expression (3.19) can be shown to hold for a periodic channel with depolarising channel branches DM09 .

Let us now investigate whether we can prove a full coding theorem for rates $R$ such that

We first define the average probability of error as follows

where $p_{e}^{i}$ denotes the probability of error for the $i$-th channel branch.

Our coding strategy is as follows. We choose a code i.e. a particular encoding and decoding scheme, suitable for a particular channel branch labeled by the index $i\in\{0,\dots,L-1\}$. Here a ‘branch’ is defined as one term in the sum (2.14) i.e. $\Phi_{i}^{(n)}=\Phi_{i}\otimes\Phi_{i+1}\otimes\dots\otimes\Phi_{i+n-1}$. According to the coding theorem for memoryless channels, there is a code with error probability tending to zero for this branch with rate $R$. Indeed, for each $j$ there exists a probability distribution $P_{j}$ of states optimising $\chi_{j}^{*}=\sup_{P}I(P;\Phi_{j})$ and we can choose states from a typical subspace for these distributions, which can be interlaced at the positions $j-i+kL$, where $k\in[\frac{n}{L}-1]$. The probability of choosing a particular branch correctly is given by $\frac{1}{L}$ and therefore the probability of error approaches

We thus have a $\lambda$-code for all $\lambda>1-\frac{1}{L}$. In particular, the error probability is bounded away from 1, and the strong converse does not hold.

On the other hand the strong converse does hold for $R>\overline{C_{p}}$. Indeed, the codewords can be decomposed into sub-codewords corresponding to the different stages of a period: $x^{n}=(x_{0}^{n},\dots,x_{L-1}^{n})$, where the components of the $x_{i}^{n}$ are understood to be interlaced in $x^{n}$. We distinguish types $P_{0},\dots,P_{L-1}$ for the sub-codewords. Then we have an analogue of the single-type strong converse given by Lemma II.1:

Clearly, for the complete channel, it follows that the number of codewords such that the sub-codewords are of types $P_{0},\dots,P_{L-1}$, satisfies

Summing over the types, we obtain the strong converse.

We can conclude that the strong converse holds for rates $R>\overline{C_{p}}$.

The above obviously raises the question if smaller error probabilities can be attained for smaller rates, but still above $C_{p}$. For this, we define a ‘pair capacity’ $C_{p}^{\,(2)}$ as follows:

Suppose the maximum is attained at a certain pair $(i_{1},i_{2})$. With probability $2/L$, one of the two branches $i_{1}$ or $i_{2}$ is chosen. We attach a preamble to the code as in the proof of the product-state capacity of the periodic channel (2.16). If, for example, the branch $i_{1}$ is selected by the channel, the receiver can determine that this is the case by measuring the preamble, and can then choose states for each value of $k$ from the typical space corresponding to the maximising distribution $P_{k}$ for the CPT map $\Phi_{i_{1}+k}$. This constitutes an encoding with rate given by the average of the mutual informations $I(P;\Phi_{i_{1}+k})$ for $k=0,\dots,L-1$, which is greater than or equal to the pair capacity $C^{\,(2)}$ given by Equation (4.26). We have thus constructed a $\lambda$-code for $\lambda>1-\frac{2}{L}$.

On the other hand, let $R>C^{\,(2)}_{p}$ and suppose that $(C^{n},E^{n})$ is a sequence of $(n,\lambda)$-codes with $\lambda<1-\frac{1}{L}$, and assume that

First note that we may assume that the number of codewords with sub-codewords of types $P_{0},\dots P_{L-1}$ is bounded by

for some fixed $i=0,\dots,L-1$. Indeed, otherwise, by Lemma III.1, $p^{i}_{e}>\lambda$ for all $i$ and hence $p_{e}>\lambda$.

We now claim that for every other $j\neq i$, and $\epsilon>0$ small enough,

If this were not the case then the pair capacity for a single type $P_{k}$ can be written as

and hence

Summing over the types $P_{0},\dots,P_{L-1}$ leads to a contradiction with (4.27).

Now, expression (4.29) implies with Lemma III.1 that, if the $j$-th branch is selected by the channel, then the error probability $p^{j}_{e}>1-\eta$ for any $\eta>0$. Since with probability $1-\frac{1}{L}$ one of the branches $j$ other than $i$ is selected, we conclude that the error probability $p_{e}>\left(1-\frac{1}{L}\right)(1-\eta)>\lambda$ if $\eta<1-\frac{1}{L}-\lambda$ is small enough.

It is now clear that this argument can be generalised to prove:

The situation for the random channel is similar, but more complicated due to the fact that different branches can have different probabilities $q_{i}$. We can in general distinguish break points at values of the error probability given by

We have an analogue of the detailed theorem for periodic channels above:

The situation is less clear-cut than it seems, however. In fact, not every $q(\Delta)$ is necessarily a point of discontinuity for the capacity, because $C_{p}^{\Delta}$ is in general not monotonic in the probabilities $q(\Delta)$!

One of the most surprising and interesting results which has emerged from Shannon Theory is the observation that the strong information-carrying capacity of a memoryless channel is independent of the upper bound on the maximum error probability of that channel, usually denoted $\lambda$. The independence of the parameter $\lambda$ is crucial to the existence of a so-called strong capacity for the channel Wolfowitz .
The dependency of some channel capacities on this parameter $\lambda$, including non-stationary discrete memoryless classical channels, led to the definition of a capacity function Ahlswede06 . Note that recently Ahlswede Ahlswede10 proved that the capacity functions can now be thought of as so-called capacity-sequences.
For the case of the quantum periodic channel, and also the random channel, we have shown that an analogous parameter-dependent capacity can be defined, which takes the form of a scale of capacities applicable for various ranges of the error parameter.

Note. It appears that similar results to ours were obtained by Datta, Hsieh and Brandão DHB11 , using different methods.