On zero-error communication via quantum channels
in the presence of noiseless feedback
We initiate the study of zero-error communication via quantum channels when the receiver and sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon’s zero-error communication theory with instantaneous feedback.
We first show that this capacity is a function only of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub “non-commutative bipartite graph”. Then we go on to show that the feedback-assisted capacity is non-zero (allowing for a constant amount of activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the “conclusive exclusion” of quantum states [Pusey/Barrett/Rudolph, Nature Phys. 8(6):475-478, 2012].
We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede.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 entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedback-assisted 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 feedback-assisted zero-error 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 classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.
- I Zero-error communication assisted by noiseless quantum feedback
- II Characterization of vanishing capacity
- III Shannon theoretic upper bound on
- IV Conclusion
- A equals the adversarial entanglement-assisted capacity
- B A Constrained Post-Selection Lemma
I Zero-error 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 well-known 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 zero-error 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 no-signalling 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 zero-error 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 above-mentioned two models, two of us in (24) (see also (25)) have developed a theory of zero-error communication over memoryless quantum channels assisted by quantum no-signalling 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 zero-error 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 finite-dimensional Hilbert spaces), where denotes the linear operators (i.e. matrices) on , with Choi-Kraus and Stinespring representations
for linear operators such that , and an isometry , respectively. The linear span of the Choi-Kraus 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 feedback-assisted code consists of a state (w.l.o.g. pure) and for each message isometries for encoding and feedback decoding
for and appropriate local quantum systems (Alice) and (Bob), as well the feedback-carrying 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 feedback-assisted 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 feedback-assisted code a zero-error code if there is a measurement on that distinguishes Bob’s output states , with certainty, where the sum is over the states
which are the output states given a specific sequence of Kraus operators. [Note that here and below, for convenience, we use to represent right-to-left 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 Choi-Kraus 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 zero-error and entanglement-assisted zero-error codes in terms of the “non-commutative graph” (18); (22); (23), and of no-signalling assisted zero-error codes in terms of the “non-commutative bipartite graph” (24). Thus we have proved
A quantum feedback-assisted code for a channel being zero-error is a property solely of the Choi-Kraus space . The maximum number of messages in a feedback-assisted zero-error code is denoted . Hence, the quantum feedback-assisted zero-error 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 feedback-assisted zero-error 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 non-commutative bipartite graph) (24), much as was advocated in (23) as a quantum generalisation of an undirected graph.
This number appears also in other zero-error communication problems, namely as the zero-error capacity of the channel assisted by no-signalling 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 non-commutative 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 zero-error list decoding of (with arbitrary but constant list size) is exactly . Thus a coding scheme for with feedback would consist of a zero-error 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 zero-error 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):
For any non-commutative bipartite graph , the feedback-assisted zero-error capacity of vanishes, , if and only if the associated non-commutative graph is complete, i.e. , which is equivalent to vanishing entanglement-assisted zero-error capacity, .
Clearly since on the right hand side we simply do not use feedback, but any code is still a feedback-assisted 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 Hilbert-Schmidt 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)
After that, the channel action consists in one of the Choi-Kraus 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 feedback-assisted 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 zero-error 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 feedback-assisted zero-error capacity
Then the above Proposition 2 can be rephrased as
motivating our focusing on from now on
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 entanglement-assisted capacity over quantum channels consistent with the given non-commutative bipartite graph. represents the best known upper bound on the feedback-assisted zero-error 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.
If the non-commutative 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 .
(“trivial zero capacity”) We show the stronger statement for all and . Indeed, as the zero-error 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 cq-channels first – Subsection II.1 for pure state cq-channels, Subsection II.2 for a mixed state example and Subsection II.3 for general cq-channels –, before completing the proof for general channels in Subsection II.4.
ii.1 Pure state cq-channels
For a given orthonormal basis of the input space , and pure states in the output space, consider the cq-channel
with Kraus subspace
We shall demonstrate first the following result:
For a pure state cq-channel, is always positive unless is trivial, which is equivalent to all being collinear, i.e. for some pure state .
If is trivial, then the above proof of the sufficiency of triviality in Theorem 3 shows .
Conversely, if is non-trivial, 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 pure-state 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 zero-error capacity , but assisted by feedback and a finite number of activating noiseless bits, it is (45). We conclude that .
ii.2 Mixed state cq-channel
To generalize the previous treatment to mixed states, let us first look at a specific simple example: Let () be three mutually distinct but non-orthogonal states in , and define a cq-channel with three inputs , mapping
and the most general channel consistent with this is a cq-channel of the form
We shall show how to construct a zero-error scheme with feedback, achieving positive rate, at least for that are sufficiently close to being orthogonal. For the zero-error 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 zero-error 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 cq-channels
The above examples rely on measuring the output states of the cq-channel 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 far-reaching generalization of the Pusey/Barrett/Rudolph result (41). The version we need can be stated as follows; it is adapted to a cq-channel with -dimensional input space and output states (), whose support projectors are and supports , so that the non-commutative graph is
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 non-commutative bipartite graph with support projection onto the Choi-Jamiołkowski range , where is the maximally entangled state:
For the cq-channel case, , this simplifies to
In particular, for the cq-graph induced by projections in Proposition 5, we have
Let be a finite set of quantum states with supports , and let be the associated non-commutative bipartite graph . Then the following are equivalent:
For sufficiently large and a suitable type , the set can be conclusively excluded.
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,
v. vi. Note that the non-commutative 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
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 .
(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