We derive universal codes for simultaneous transmission of classical messages and entanglement through quantum channels, possibly under attack of a malignant third party. These codes are robust to different kinds of channel uncertainty. To construct such universal codes, we invoke and generalize properties of random codes for classical and quantum message transmission through quantum channels. We show these codes to be optimal by giving a multi-letter characterization of regions corresponding to capacity of compound quantum channels for simultaneously transmitting and generating entanglement with classical messages. Also, we give dichotomy statements in which we characterize the capacity of arbitrarily varying quantum channels for simultaneous transmission of classical messages and entanglement. These include cases where the malignant jammer present in the arbitrarily varying channel model is classical (chooses channel states of product form) and fully quantum (is capable of general attacks not necessarily of product form).
In real world communication using quantum or classical systems, the parameter determining the channel in use may belong to an uncertainty set, rendering the protocols that assume the channel to be perfectly known practically obsolete. Given such uncertainty, when using the channel many times, as done in Shannon theoretic information processing tasks, assuming the channel to be memoryless or fully stationary is not realistic. In this paper, we consider three models that include channel uncertainty without attempting to reduce it via techniques such as channel identification or tomography. We refer to these models as the compound, arbitrarily varying and fully quantum arbitrarily varying channel models. Each of these models are considered here for transmission of entanglement and classical messages simultaneously between a sender and receiver.
Informally, the first two channel models consist of a set of quantum channels known to the communicating parties. In the compound model, communication is done under the assumption that asymptotically, one of the channels from this set (unknown to the parties) is used in a memoryless fashion. The codes used in this model therefore have to be reliable for the whole family of memoryless channels for large enough values of .
In the arbitrarily varying model, given a number of channel uses , an adversarial party chooses the sequence unknown to the communication parties, to yield the channel . The adversary may choose this sequence knowing the encoding procedure used by the sender. The code in use therefore has to be reliable for the whole family of memoryless channels. Finally, in the third channel model, namely that of the fully quantum arbitrarily varying, the assumption of memoryless communication is dropped. Here, the adversary may choose channel states that are not necessarily of the product form mentioned in the previous model.
The quantum channel has different capacities for information transmission. One may consider the capacity of the channel for public ([18, 28]) or private ([15, 13]) classical message transmission, entanglement transmission or entanglement generation () to name a few. These communication scenarios have been considered subsequently under channel uncertainty ([7, 24, 6, 11, 1, 3]). Simultaneous transmission of classical and quantum messages, the subject of this work, has also been of interest(). This includes scenarios where the communication parties would like to enhance their classical message transmission by sharing quantum information primarily at their disposal or vice versa([5, 19, 20]). The body of research in this area is clearly interesting, when regions beyond those achieved by simple time-sharing between established classical message and quantum information transmission codes are reached.
Simultaneous transmission of classical messages and entanglement is a nontrivial problem even if capacity achieving codes for the corresponding univariate transmission goals are at hand. It was already observed in 
for perfectly known quantum channels that the naive time sharing strategy is generally insufficient to achieve the full capacity region. Examples of channels where coding beyond time-sharing is indispensable does not depend on constructing pathologies. They are readily found even within the standard arsenal of qubit quantum channels, e.g. the dephasing qubit channels.
We derive codes for simultaneous transmission of classical messages and entanglement that are robust to the three types of uncertainty mentioned above. The codes used here for the compound model, are different from those used for the point to point communication in  when considering the special case of . Given that the input state approximation techniques used therein prove insufficient in presence of channel state uncertainty, in the present work we use the decoupling approach first established in . We combine robust random codes for classical message transmission from  and a generalization of (decoupling based) entanglement transmission codes from  to construct appropriate simultaneous codes for compound quantum channels under the maximal error criterion. We show that these codes are optimal by giving a multi-letter characterization of the capacity of compound quantum channels with no assumption on, the size of the underlying uncertainty set. We use the asymptotic equivalence of the two tasks of entanglement transmission and entanglement generation to include the capacity region corresponding to simultaneous transmission of classical messages and generation of entanglement between the two parties.
Next, we convert the codes derived for the compound channel, using Ahlswede’s robustification and elimination techniques () to derive suitable codes for arbitrarily varying quantum channels. This is possible given that the error functions associated with codes corresponding to the compound model decay to zero exponentially. We derive a dichotomy statement (), for the simultaneous classical message and entanglement transmission through AVQCs under the average error criterion. This dichotomy is observed when considering two scenarios where the communicating parties do and do not have access to unlimited common randomness, yielding the common-randomness and deterministic capacity regions of the channel model respectively. Therefore, we show that firstly, the common-randomness capacity region of the arbitrarily varying channel is equal to that of the compound channel , namely the compound channel generated by the convex hull of the uncertainty set of channels . Secondly, if the deterministic capacity of the arbitrarily varying channel is not the point , it is equal to the common-randomness capacity of the channel.
We give a necessary and sufficient condition for the deterministic capacity region to be be the point . This condition is known as symmetrizablity of the channel (see  and ). Finally, we show that the codes derived here, can be used for fully quantum AVCs where the jammer is not restricted to product states, but can use general quantum states to parametrize the channel used many times. This model has been introduced in Section 8 along with the main result and related work for fully quantum AVCs and hence here, we avoid further explanation of the techniques used there.
The task of simultaneous transmission of classical messages and entanglement was first considered by Devetak and Shor in 
in case of a memoryless quantum channel under assumption that the channels state is perfectly known to its users. The authors derived a multi-letter characterization of the capacity region in this setting which also classified the naïve time-sharing approach as being suboptimal for simultaneous transmission. A code construction sufficient to achieve also the rate pairs lying outside the time-sharing region was derived using a ”piggy-backing” technique. A specialized construction introduced in allows to encode the identity of the classical message into the coding states of an underlying entanglement transmission code. The mentioned strategy to optimally combine different communication tasks in quantum channel coding was afterwards used and further developed in different directions. We explicitly mention subsequent research activity by Hsieh and Wilde [19, 20] where the idea of ”piggy backing” classical messages onto quantum codes was extended to include entanglement assistance. The resulting code construction being sufficient to achieve each point in the three-dimensional rate region for entanglement-assisted classical/quantum simultaneous transmission leads to a full (multi-letter) characterization of the ”Quantum dynamic capacity” of a (perfectly known) quantum channel  (see the textbook  for an up-to-date pedagocial presentation of the mentioned results).
In order to derive classically enhanced quantum codes being robust against channel uncertainty, we refine the construction entanglement transmission codes for compound quantum channels from [7, 8] instead of elaborating on the usual approach building up on codes from . In fact, it was noticed earlier that deriving entanglement generation codes from secure classical message transmission codes (the strategy which the arguments in  follow) seems to be not suitable when the channel is a compound quantum channel.
In the first section following this introduction, we introduce the notation used in this work. Precise definitions of the channel models, codes used in different scenarios along with capacity regions and finally the main results in form of Theorem 5 and Theorem 12, are given in Section 4. In Section 5, we present preliminary coding results for entanglement transmission (Section 5.1) and classical message transmission (Section 5.2). The entanglement transmission codes introduced in this section are a generalization of the random codes in  and  to accommodate conditional typicality of the input on words from many copies of an alphabet. The classical message transmission codes are those from  that prove sufficient for our simultaneous coding purposes.
Equipped with these results, we move on to Section 6, to prove the coding results for the compound channel model. In this section, after proving a converse for the capacity region in Theorem 5, we prove the direct part in two steps. In the first step, we show that capacity regions that correspond to the case where the sender is restricted to inputting maximally entangled pure states are achieved. In the second step, we prove achievablity of capacity regions corresponding to general inputs, using elementary methods that are less involved that the usual BSST type results used for this generalization in  and .
In Section 7, after proving a converse for the capacity region under the arbitrarily varying channel model, we prove coding results in this model by converting the compound channel model codes using Ahlswede’s robustification method. This, assumes unlimited common randomness available to the legal parties. We then use an instance of elimination to show that if the deterministic capacity region is not the point , negligible amount of common randomness per use of the channel is sufficient to achieve the same capacity region. Also in this section, we prove necessity and sufficiency of symmetrizablity condition for the case where the deterministic capacity region is the point . Finally, in Section 8, we generalize these results to the case of quantum jammer by proving Theorem 31.
3 Notations and conventions
All Hilbert spaces are assumed to have finite dimensions and are over the field . All alphabets are also assumed to have finite dimensions. We denote the set of states by . Pure states are given by projections onto one-dimensional subspaces. To each subspace , we can associate unique projection whose range is the subspace and we write for the maximally mixed state on , i.e.
The set of completely positive trace preserving (CPTP) maps between the operator spaces and
is denoted by . Thus , plays the role of the input Hilbert space to the channel (traditionally
owned by Alice) and is channel’s output Hilbert space (usually in Bob’s possession). stands for the set of completely positive trace decreasing maps between and . will denote in what follows, the group of unitary operators acting on . For a Hilbert space , we will always identify with a subgroup of . For any projection we set .
Each projection defines a completely positive trace decreasing map given by for all . In a similar fashion, any defines a by for . We use the base two logarithm which is denoted by . The von Neumann entropy of a state is given by
The coherent information for and is defined by
where is an arbitrary purification of the state . We also use to denote entropy exchange. A useful equivalent definition of is given in terms of and any complementary channel where denotes the Hilbert space of the environment. Due to Stinespring’s dilation theorem, can be represented as
for where is a linear isometry. The complementary channel of is given by
The coherent information can then be written as
This quantity can also be defined in terms of the bipartite state with
where is the marginal state given by and we have the identity
As a measure of closeness between two states , we may use the fidelity . The fidelity is symmetric in the input and for a pure state , we have . A closely related quantity is the entanglement fidelity, which for and , is given by
with an arbitrary purification of the state .
Another quantity that will be significant in the present work is the quantum mutual information (see e.g ). For a state , the quantum mutual information is defined as
where and are marginal states of .
For the approximation of arbitrary compound channels (introduced in the next section) by finite ones we use the diamond norm , given for any by
where is the identity channel. We state the following facts about (see e.g ). First, for all . Thus, , where denotes the unit sphere of the normed
space . Moreover, for arbitrary linear maps . Throughout this work we have made use of the idea of nets to approximate arbitrary compound quantum channels using ones with finite uncertainty sets. This idea is presented in Appendix A by Definition 38 and proceeding two lemmas.
We use exponentially as or we say approaches (goes to) zero exponentially, if is a strictly positive constant. For and both approaching zero exponentially, we use if . We use to denote the closure of set and finally, we use to denote the group of permutations on elements such that for each and .
4 Basic definitions and main results
We consider two channel models of compound and arbitrarily varying quantum channels. They are both generated by an uncertainty set of CPTP maps. For the purposes of the present work, when considering the arbitrarily varying channel model, we assume finiteness of the generating uncertainty set. This assumption is absent in the case of the compound channel model.
4.1 The compound quantum channel
Here, we consider quantum compound channels. Let be a set of CPTP maps. The compound quantum channel generated by is given by family .
In other words, using instances of the compound channel is equivalent to using instances of one of the channels from the uncertainty set. The users of this channel may or may not have access to the Channel State Information (CSI). We will often use the set to index members of . A compound channel is used times by the sender Alice, to convey classical messages from a set to a receiver Bob. At the same time, the parties would like to communicate quantum information. Here, we consider two scenarios in which quantum information can be communicated between the parties.
Classically Enhanced Entanglement Transmission (CET): While transmitting classical messages using instances of the compound channel, the sender wishes to transmit the maximally entangled state in her control to the receiver. The subspace with and , quantifies the amount of quantum information transmitted. More precisely:
An CET code for , is a family with
We remark that as defined above, for each we have a entanglement transmission code for .
For every and , we define the following performance function for this communication scenario when instances of the channel have been used,
where is a maximally entangled state on and
Classically Enhanced Entanglement Generation (CEG): In this scenario, while transmitting classical messages, Alice wishes to establish a pure state shared between her and Bob. As the maximally entangled pure state shared between the parties is an instance of such a pure state, it can be proven that the previous task achieved in CET, achieves the task laid out by this one, but the opposite is not necessarily true. More precisely:
An CEG code for , is a family , where is a pure state on and
The relevant performance functions for this task, for every and , are
with maximally entangled on .
Averaging over the message set , will give us the corresponding average performance functions for each ,
for . For each scenario, we define the achievable rates.
Let A pair of non-negative numbers is called an achievable X rate for the compound channel , if for each exists a number , such that for each we find and X code such that
are simultaneously fulfilled. We also define X ”average-error-rates” by averaging the performance functions in the last condition over . We define the X capacity region of by
Also the capacity region corresponding to average error criteria is defined as
Moreover, let be an alphabet, , and be a pure state for all . Given the state
we introduce the following set,
with denoting collectively. We will also use
The following statement is the first main result of this paper.
Let be any compound quantum channel. Then
This theorem is proven in the following steps. In Section 6.1, we prove that is a subset of the set on the rightmost set in the above equalities. In Section 6.2, we prove that the rightmost set is a subset of . Together with the operational inclusions
for , we conclude the equalities in the statement of the theorem.
4.2 The arbitrarily varying quantum channel
The arbitrarily varying quantum channel generated by a set of CPTP maps with input Hilbert space and output Hilbert space , is given by family of CPTP maps , where
We use to denote the AVQC generated by . To avoid further technicalities, we always assume for the AVQC generating sets appearing in this paper. Most of the results in this paper may be generalized to the case of general sets by clever use of approximation techniques from convex analysis together with continuity properties of the entropic quantities which appear in the capacity characterizations (see ).
An random CET code for is a probability measure
is a probability measureon , where
The sigma-algebra is chosen such that the function
is measurable with respect to , for all . In (6), is a maximally entangled state on and .
We further require that contains all the singleton sets. The case where is deterministic, namely is equal to unity on a singleton set and zero otherwise, gives us a deterministic CET codes for . Abusing the terminology, we also refer to the singleton sets as deterministic codes.
A non-negative pair of real numbers is called an achievable CET rate pair for with random codes and average error criterion, if there exists a random CET code for with members of singleton sets notified by such that
The random CET capacity region with average error criterion of is defined by
A non-negative pair of real numbers is called an achievable deterministic CET rate for with average error criterion, if there exists a deterministic CET code for with
Correspondingly we define the following capacity region,
The deterministic CET codes defined here, are entanglement transmission codes for each . More precisely we have the following definition.
An , , entanglement transmission code for AVQC is a pair with with and . The corresponding performance function for this task is
Essential to the statement of our results is the concept of symmetrizablity defined in the following.
Let with be an AVQC.
is called -symmetrizable for , if for each finite set with , there is a map such that for all
We call symmetrizable if it is -symmetrizable for all .
We prove the following result to be the second main result of this paper.
Let with be an AVQC. The following hold.
where is the CET capacity of compound channel with average error criterion defined in the previous section and
if and only if is symmetrizable.
5 Universal random codes for quantum channels
In this section we prove universal random coding results for entanglement transmission
and classical message transmission over quantum channels. Most of the statements below, are
implicitly contained in the literature. We state some properties of these codes that stem from their random nature and prove useful when deriving CET codes stated in Section 6.
Before proceeding with the following two sections in which we introduce appropriate entanglement transmission and classical message transmission coding results and for the reader’s convenience, we present briefly the concept of types used in the remainder of this section. For more information on the concept of types, see e.g. .
For , the word that is a string of letters and the state with spectral decomposition , we define the -typical (frequency typical) projection
where is the set of -typical sequences in , defined by
where is the number of occurrences of letter in word .
For each , we consider the set of types over alphabet , defined as
5.1 Entanglement transmission codes
In this section, we prove universal entanglement transmission coding results that are to be combined with suitable classical message transmission codes introduced in the next section. The following lemma is a generalization of random entanglement transmission codes obtained in  and , where a in turn generalization of the decoupling lemma from 
has been obtained. As stated in the following lemma, there are two points to be remarked about these codes. First, the random nature of these codes gives us an encoding state (outcome of the random encoding operation) with a tensor product structure, that is of interest for the present work. Therefore at this stage, we skip the de-randomization step that seemed natural in the original work. Secondly, the integration over unitary groups with respect to the normalized Haar measure done in the random encoding operation therein, is replaced here by an average over the elements of discrete and finite subsets of representations of the unitary group known as unitary designs (see e.g.).
The product structure of the encoding state can be used for an instance of channel coding stated later on. This becomes clear when the tensor product structure of the average state is used to accommodate typicality. For where is some finite alphabet, and , we introduce the following notation. For the tuple where for , we define
where and clearly, . Then denotes the maximally mixed state on (correspondingly denotes the maximally mixed state on for ), a purification of (correspondingly denotes a purification of ) and is a unitary design (see Theorem 16) for . The following lemma reduces to Theorem 5 of  when .
The ingredients to prove this lemma are presented here in form of two lemmas prior to the main proof. The following two lemmas reduce to Lemma 5 and 6 from 111see Lemmas 44 and 45 for the statements. when . Following these lemmas, we state Theorem 16 based on which we replace the integration with respect to Haar measure, with an average over a subset of the unitary groups called unitary designs. In short, the entanglement transmission codes in  were derived given a number and subspace . Here, we derive codes for a subspace , with a tensor product structure determined by word (see the description above Lemma 13).
Let be a probability distribution with on an alphabet . For and , there exist a real number , functions , with and and an orthogonal projection satisfying
The last inequality implies
Let for each , be the frequency typical projection associated with state in terms of Lemma 44. We show that the projection operator has the properties listed in the statement above. We have