On zeroerror communication via quantum channels
in the presence of noiseless feedback^{1}
Abstract
We initiate the study of zeroerror communication via quantum channels when the receiver and sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon’s zeroerror communication theory with instantaneous feedback.
We first show that this capacity is a function only of the linear span of ChoiKraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub “noncommutative bipartite graph”. Then we go on to show that the feedbackassisted capacity is nonzero (allowing for a constant amount of activating noiseless communication) if and only if the noncommutative bipartite graph is nontrivial, and give a number of equivalent characterizations. This result involves a farreaching extension of the “conclusive exclusion” of quantum states [Pusey/Barrett/Rudolph, Nature Phys. 8(6):475478, 2012].
We then present an upper bound on the feedbackassisted zeroerror capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate this bound to have many good properties, including being additive and given by a minimax formula. We also prove a coding theorem showing that this quantity is the entanglementassisted capacity against an adversarially chosen channel from the set of all channels with the same ChoiKraus span, which can also be interpreted as the feedbackassisted unambiguous capacity. The proof relies on a generalization of the “Postselection Lemma” (de Finetti reduction) [Christandl/König/Renner, Phys. Rev. Lett. 102:020504, 2009] that allows to reflect additional constraints, and which we believe to be of independent interest. This capacity is a relaxation of the feedbackassisted zeroerror capacity; however, we have to leave open the question of whether they coincide in general.
We illustrate our ideas with a number of examples, including classicalquantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.
Contents:
I Zeroerror communication assisted by noiseless quantum feedback
In information theory it is customary to consider not only asymptotically long messages but also asymptotically vanishing, but nonzero error probabilities, which leads to a probabilistic theory of communication characterized by entropic capacity formulas (44); (14). It is wellknown that when communicating by block codes over a discrete memoryless channel at rate below the capacity, the error probability goes to zero exponentially in the block length, and while it is one of the major open problems of information theory to characterize the tradeoff between rate and error exponent in general, we have by now a fairly good understanding of it. However, if the error probability is required to vanish faster than exponential, or equivalently is required to be zero exactly (at least in the case of finite alphabets), we enter the strange and much less understood realm of zeroerror information theory (45); (37), which concerns asymptotic combinatorial problems, most of which are unsolved and are considered very difficult. There are a couple of exceptions to this rather depressing state of affairs, one having been already identified by Shannon in his founding paper (45), namely the discrete memoryless channel assisted by instantaneous noiseless feedback, whose capacity is given by the fractional packing number of a bipartite graph representing the possible transitions . The other one is the the recently considered assistance by nosignalling correlations (20), which is also completely solved in terms the fractional packing number of the same bipartite graph .
Recent years have seen attempts to create a theory of quantum zeroerror information theory (40), identifying some rather strange phenomena there such as superactivation (18); (22) or entanglement advantage for classical channels (19); (39), but resulting also in some general structural progress such as a quantum channel version of the Lovász number (23). Motivated by the success in the abovementioned two models, two of us in (24) (see also (25)) have developed a theory of zeroerror communication over memoryless quantum channels assisted by quantum nosignalling correlations, which largely (if not completely) mirrors the classical channel case; in particular, it yielded the first capacity interpretation of the Lovász number of a graph. Some of the techniques and insights developed in (24) will play a central role also in the present paper.
In the present paper, we take as our point of departure the other successful case, Shannon’s theory of zeroerror communication assisted by noiseless instantaneous feedback. In detail, consider a quantum channel , i.e. a completely positive and trace preserving (cptp) linear map from the operators on to those of (both finitedimensional Hilbert spaces), where denotes the linear operators (i.e. matrices) on , with ChoiKraus and Stinespring representations
for linear operators such that , and an isometry , respectively. The linear span of the ChoiKraus operators is denoted by
where “” means that is a subspace of , the linear operators (i.e. martrices) mapping to . We will discuss a model of communication where Alice uses the channel times in succession, allowing Bob after each round to send her back an arbitrary quantum system. They may also share an entangled state prior to the first round (if not, they can have it anyway from the second round on, since Bob could use the first feedback to create an arbitrary entangled state). Their goal is to allow Alice to send one of messages down the channel uses such that Bob is able to distinguish them perfectly. More formally, the most general quantum feedbackassisted code consists of a state (w.l.o.g. pure) and for each message isometries for encoding and feedback decoding
(1) 
for and appropriate local quantum systems (Alice) and (Bob), as well the feedbackcarrying systems ; see Fig. 1. For consistency (and w.l.o.g.), are trivial. Note that Bob can use the feedback channel to create any entangled state with Alice for later use before they actually send messages. We use isometries, rather than general cptp maps, to represent encoders and decoders in the feedbackassisted communication scheme, because by the Stinespring dilation (48), all local cptp maps can be “purified” to local isometries. Thus every seemingly more general protocol involving cptp maps can be purified to one of the above form. We will find this form convenient in the later analysis as it allows us to reason on the level of Hilbert space vectors.
We call this quantum feedbackassisted code a zeroerror code if there is a measurement on that distinguishes Bob’s output states , with certainty, where the sum is over the states
(2) 
which are the output states given a specific sequence of Kraus operators. [Note that here and below, for convenience, we use to represent righttoleft multiplications of operators , namely .] In other words, these states have to have mutually orthogonal supports, i.e. for all , all , and all ,
By linearity, we see that this condition depends only on the linear span of the ChoiKraus operator space , in fact it can evidently be expressed as the orthogonality of a tensor defined as a function of , the and , to the subspace – cf. similar albeit simpler characterizations of zeroerror and entanglementassisted zeroerror codes in terms of the “noncommutative graph” (18); (22); (23), and of nosignalling assisted zeroerror codes in terms of the “noncommutative bipartite graph” (24). Thus we have proved
Proposition 1
A quantum feedbackassisted code for a channel being zeroerror is a property solely of the ChoiKraus space . The maximum number of messages in a feedbackassisted zeroerror code is denoted . Hence, the quantum feedbackassisted zeroerror capacity of ,
is a function only of .
In the case of a classical channel with transition probabilities , assisted by classical noiseless feedback, the above problem was first studied – and completely solved – by Shannon (45). To be precise, his model has noiseless instantaneous feedback of the channel output back to the encoder; it is clear that any protocol with general actions (noisy channel acting on the output) by the receiver can be simulated by the receiver storing the output and the encoder getting a copy of the channel output, if shared randomness is available. Our model differs from this only by the additional availability of entanglement; that this does not increase further the capacity follows from (20), see our comments below.
Following Shannon, we introduce the (bipartite) equivocation graph on , which has an edge iff , i.e. the adjacency matrix is ; furthermore the confusability graph on , with an edge iff there exists a such that , i.e., iff the neighbourhoods of and in intersect. The feedbackassisted zeroerror capacity of the channel can be seen to depend only on .
Note that for (the quantum realisation of) a classical channel, i.e.
the corresponding subspace is given by
so should really be understood as the quantum generalisation of the equivocation graph (a noncommutative bipartite graph) (24), much as was advocated in (23) as a quantum generalisation of an undirected graph.
Shannon proved
(3) 
Here, is the socalled fractional packing number of , defined as a linear programme, whose dual linear programme is the fractional covering number (45); (42):
(4) 
This number appears also in other zeroerror communication problems, namely as the zeroerror capacity of the channel assisted by nosignalling correlations (20). There, it is also shown to be the asymptotic simulation cost of a channel with bipartite graph in the presence of shared randomness. This shows that for a classical channel with bipartite graph , interpreted as a quantum channel with noncommutative bipartite graph , .
The first case in eq. (3) of a complete graph is easy to understand: whatever the parties do, and regardless of the use of feedback, any two inputs may lead to the same output sequence, so not a single bit can be transmitted with certainty. In either case, Shannon showed that only some arbitrarily small rate of perfect communication (actually a constant amount, dependent only on ) is sufficient to achieve what we might call the activated capacity , which is always equal to . This was understood better in the work of Elias (28) who showed that the capacity of zeroerror list decoding of (with arbitrary but constant list size) is exactly . Thus a coding scheme for with feedback would consist of a zeroerror list code with list size and rate for uses of the channel , followed by feedback in which Bob lets Alice know the list of items in which he now knows the message falls, followed by a noiseless transmission of bits of Alice to resolve the remaining ambiguity. Shannon’s scheme (45) is based on a similar idea, but whittles down the list by a constant factor in each round, so Bob needs to update Alice on the remaining list after each channel use. The constant noiseless communication at the end of this protocol can be transmitted using an unassisted zeroerror code via the given channel (at most uses), or via an activating noiseless channel.
The dichotomy in eq. (3) has the following quantum channel analogue (in fact, generalization):
Proposition 2
For any noncommutative bipartite graph , the feedbackassisted zeroerror capacity of vanishes, , if and only if the associated noncommutative graph is complete, i.e. , which is equivalent to vanishing entanglementassisted zeroerror capacity, .
Proof.
Clearly since on the right hand side we simply do not use feedback, but any code is still a feedbackassisted code. Hence, if the latter is positive then so is the former. It is well known that if , then , in fact each channel use can transmit at least one bit (22); (23).
Conversely, let us assume that , i.e. . We will show by induction on that for any two distinct messages, w.l.o.g. , Bob’s output states after rounds, on , cannot be orthogonally supported, meaning . Here,
This is clearly true for since at that point Alice and Bob share only , hence . For , let Bob after rounds have one of the states ; by the induction hypothesis, – by a slight abuse of notation meaning that the supports are not orthogonal, or equivalently that the operators are not orthogonal with respect to the HilbertSchmidt inner product. This means that there are indices and such that
This can be expressed equivalently as
Now, in the th round, Alice applies the isometry to the and registers of , hence for (as we do not touch the register)
(5) 
After that, the channel action consists in one of the ChoiKraus operators . Let us assume, with the aim of establishing a contradiction, that Bob’s states after the channel action were orthogonal, i.e. for all and ,
In other words, for all , and operators on ,
But since is arbitrary and the span , this would imply , contradicting (5).
Thus, applying now also the isometry , we find that there exist and such that
and so finally , proving the induction step.
Motivated by of a classical channel (45), see above, we define also feedbackassisted codes with channel uses and up to noiseless classical bits of forward communication. The setup is the same as in eq. (1) and Fig. 1 with rounds, of which feature the isometric dilation of , and the isometry () corresponding to the noiseless bit channel . It is clear that the output states can be written in a way similar to eq. (2), and that the maximum number of messages in a zeroerror code depends only on , and , which we denote . Clearly, and in general, . Furthermore, it can easily be verified that
hence we can define the activated feedbackassisted zeroerror capacity
The rest of the present paper is organized as follows: In Section II we start with a concrete example showing the importance of measurements “conclusively excluding” hypotheses from a list of options, and go on to show several concise characterizations of nontrivial channels, i.e. those for which . In Section III we first review a characterization of the fractional packing number in terms of the Shannon capacity minimized over a set of channels, which then motivates the definition of obtained as a minimization of the entanglementassisted capacity over quantum channels consistent with the given noncommutative bipartite graph. represents the best known upper bound on the feedbackassisted zeroerror capacity. We illustrate the bound by showing how it allows us to determine for Weyl diagonal channels, i.e. spanned by discrete Weyl unitaries. We also show that is the ordinary (small error) capacity of the system assisted by entanglement, against an adversarial choice of the channel (proof in Appendix A, based on a novel Constrained Postselection Lemma, aka “de Finetti reduction”, in Appendix B). After that, we conclude in Section IV with a discussion of open questions and future work.
Ii Characterization of vanishing capacity
In this section, we will prove the following result.
Theorem 3
If the noncommutative bipartite graph contains a subspace with a state vector , meaning that the constant channel has , then ; we call such trivial.
Conversely, if is nontrivial, then .
Proof.
(“trivial zero capacity”) We show the stronger statement for all and . Indeed, as the zeroerror condition is only a property of , we may assume a concrete constant channel with . The outputs of the copies of in the feedback code do not matter at all as they are going to be , which Bob can create himself. Hence the only information arriving at Bob’s from Alice is in the classical bits in the course of the protocol. But even assisted by entanglement and feedback, Alice can convey at most noiseless bits in this way, due to the Quantum Reverse Shannon Theorem (5).
The opposite implication (“nontrivial positive capacity”) will be the subject of the remainder of this section. We will start by looking at cqchannels first – Subsection II.1 for pure state cqchannels, Subsection II.2 for a mixed state example and Subsection II.3 for general cqchannels –, before completing the proof for general channels in Subsection II.4.
ii.1 Pure state cqchannels
For a given orthonormal basis of the input space , and pure states in the output space, consider the cqchannel
with Kraus subspace
We shall demonstrate first the following result:
Proposition 4
For a pure state cqchannel, is always positive unless is trivial, which is equivalent to all being collinear, i.e. for some pure state .
Proof.
If is trivial, then the above proof of the sufficiency of triviality in Theorem 3 shows .
Conversely, if is nontrivial, then there are two output vectors, denoted and , that are not collinear, and we shall simply restrict the channel to the corresponding inputs and . I.e., we focus only on , and the corresponding channel
Consider using it three times, inputting only the code words , and . This gives rise to output states
which have the property that their pairwise inner products are all equal: . By using the channel times, Alice can prepare the states
whose pairwise inner products are all equal and indeed , i.e. arbitrarily close to . Now, if is large enough (so that ), there is a cptp map that Bob can apply to transform
(This follows from well known results on purestate transformations, see e.g. (11).) By now it may be clear where this is going: Bob measures the computational basis and overall we obtain a classical channel with exactly one entry in each row and column:
which has zeroerror capacity , but assisted by feedback and a finite number of activating noiseless bits, it is (45). We conclude that .
ii.2 Mixed state cqchannel
To generalize the previous treatment to mixed states, let us first look at a specific simple example: Let () be three mutually distinct but nonorthogonal states in , and define a cqchannel with three inputs , mapping
(7)  
Thus,
and the most general channel consistent with this is a cqchannel of the form
We shall show how to construct a zeroerror scheme with feedback, achieving positive rate, at least for that are sufficiently close to being orthogonal. For the zeroerror properties, we may as well focus on , which is easier to reason with. For the following, it may be helpful to think of eq. (7) in a partly classical way: any input is mapped to a random , subject to , so that for two uses of the channel, each pair is mapped randomly to one of four , with , . Of course, vice versa each of these nine vectors is reached from exactly four inputs.
Now, assuming that the pairwise inner products of the are small enough, i.e.
to guarantee that there is a deterministic pure state transformation (by cptp map) (11), where
On these states, Bob performs a measurement in the computational basis of the , and we get an effective classical channel mapping randomly to some , subject to the constraint
which means that each is reached from at most eight out of the nine pairs . In fact, the observation of excludes at least two out of nine input symbols, namely and , meaning that this classical channel has zeroerror capacity (plus feedback plus a finite number of noiseless bits) of . In conclusion, we achieve for , and hence for any with , a rate of .
ii.3 General cqchannels
The above examples rely on measuring the output states of the cqchannel by a POVM such that the resulting classical(!) channel with has an equivocation graph with , because then . For this, cf. eq. (4), it is necessary and sufficient that each outcome excludes at least one input , i.e. , or equivalently . A POVM with this property is said to “conclusively exclude” the set of states (41); (4). It is clearly only a property of the support projections of , and w.l.o.g. the POVM is indexed by the same ’s, i.e. such that for all , as well as and .
Our approach in the following will be to characterize when a set of states, or one of its tensor powers , can be conclusively excluded. For instance, Pusey, Barrett and Rudolph (41) showed that for any two linearly independent pure states and , it is always possible to find an integer and a outcome POVM such that
I.e. we can design a quantum measurement that can conclusively exclude the fold states with bit strings as outcomes, even when and are not orthogonal.
We will employ the powerful techniques developed in the proof of ((24), Prop. 14), allowing us to show a farreaching generalization of the Pusey/Barrett/Rudolph result (41). The version we need can be stated as follows; it is adapted to a cqchannel with dimensional input space and output states (), whose support projectors are and supports , so that the noncommutative graph is
Proposition 5
Let be projectors on a Hilbert space , with a transitive group action by unitary conjugation on the , i.e. we have a finite group acting transitively on the labels , and a unitary representation such that for .
Before we prove it, we use it to derive the following general result. To state it, we need some notation: For a set of states, let
The strings are classified according to type (17), which is the empirical distribution of the letters , . There are only many different types. The subset of corresponding to type is denoted
We also recall the definition of the semidefinite packing number (24) of a noncommutative bipartite graph with support projection onto the ChoiJamiołkowski range , where is the maximally entangled state:
(8) 
For the cqchannel case, , this simplifies to
(9) 
In particular, for the cqgraph induced by projections in Proposition 5, we have
Theorem 6
Let be a finite set of quantum states with supports , and let be the associated noncommutative bipartite graph . Then the following are equivalent:

;

is nontrivial;

;

;

;

For sufficiently large and a suitable type , the set can be conclusively excluded.
Proof.
i. ii. has been shown in the first part (necessity) of Theorem 3, at the start of this section.
ii. iii. if and only if .
iii. iv. with equality if and only if there is a common eigenvector with eigenvalue for all of the , i.e. .
iv. v. We check that is feasible for ; indeed,
thus .
v. vi. Note that the noncommutative bipartite graph corresponding to is . Let’s denote the graph of by . In (24) it is shown that is multiplicative, ; indeed, for an optimal assignment of weights feasible for , is feasible (and optimal) for . Hence, there exists a type such that
(10) 
On the other hand, the symmetric group acts transitively by permutation on the strings of type , and equivalently by permutation of the tensor factors of . This representation is well known to have only irreps, each of which has multiplicity . Thus, from eq. (10), we deduce that for sufficiently large , , which by Proposition 5 implies that the set can be conclusively excluded.
vi. i. By sending signals and measuring the output states with a conclusively excluding POVM , we simulate a classical channel whose bipartite equivocation graph has , hence .
Proof.
(of Proposition 5) Assume that we have a feasible () for such that . Concretely, this means that .
We will show that a desired POVM can be found, such that for all and . The problem of finding the POVM then becomes equivalent to finding such that