Message Transmission over Classical Quantum Channels with a Jammer with Side Information: Correlation as Resource, Common Randomness Generation

TL;DR

This paper investigates the capacity of arbitrarily varying classical-quantum channels with a jammer having side information, establishing a single-letter formula for correlation-assisted capacity and exploring its implications for common randomness generation.

Contribution

It derives a single-letter formula for correlation-assisted capacity in AVCQCs with an informed jammer and analyzes the computability of the associated capacities.

Findings

Derived a single-letter formula for correlation-assisted capacity.

Showed that common randomness generation capacity can be non-Turing computable without public communication.

Established that capacity evaluation can be impossible algorithmically even without known deterministic capacity formulas.

Abstract

In this paper we analyze the capacity of a general model for arbitrarily varying classical-quantum channels (AVCQCs) when the sender and the receiver use correlation as a resource. In this general model, a jammer has side information about the channel input. We determine a single letter formula for the correlation assisted capacity. As an application of our main result, we determine the correlation assisted common randomness generation capacity. In this scenario the two channel users have access to correlation as a resource,and further use an AVCQC with an informed jammer for additional discussion. The goal is to create common randomness between the two channel users. We also analyze these capacity formulas when only a small number of signals from the correlation are available. For the correlation assisted common randomness generation capacity, we show an additional interesting property: For a sufficient amount of "public communication", common randomness generation capacity is Turing computable, however without this public communication constraint, the correlation assisted common randomness generation capacity is, in general, not Turing computable. Furthermore, we show that even without knowing the capacity formula of the deterministic capacity using maximal error criterion, we can show that it is impossible to evaluate the performance algorithmically on any current or future digital computer.

Figures

Equations

$$\mathcal{S}(\mathcal{H}):=\{\rho \in \mathcal{L}(\mathcal{H}):\rho \text{ is Hermitian, }\rho \ge 0_{\mathcal{H}}\text{ , }\mathrm{tr}(\rho )=1\}\text{ ,}$$

$$S(\rho )=-\mathrm{tr}(\rho \log \rho )\text{ .}$$

$$S(\mathcal{W}|P):=\sum _{x\in \mathcal{X}}P(x)S(\mathcal{W}(x))\text{ .}$$

$$\chi (Q;\Phi ):=S\left(\sum _{x\in \mathbf{A}}Q(x){\rho }_x\right)-\sum _{x\in \mathbf{A}}Q(x)S\left({\rho }_x\right)\text{ .}$$

$${\rho }^{\otimes n}(\mathbf{x},\mathbf{s}):=\rho (x_1,s_1)\otimes \rho (x_2,s_2)\otimes \dots ,\otimes \rho (x_n,s_n)$$

$$1-\frac{1}{J_n}\sum _{j=1}^{J_n}\mathrm{tr}\left({\rho }^{\otimes n}(u^n)D_j\right)<ϵ\text{ .}$$

$$\underset{{\mathbf{s}}^n(\cdot )}{\max }{P}_e(\mathcal{C},{\mathbf{s}}^n(\cdot ))<ϵ\text{ ,}$$

$$P_e(\mathcal{C},{\mathbf{s}}^n(\cdot )):=1-\frac{1}{J_n}\sum _{j=1}^{J_n}\mathrm{tr}(\rho (u^n(j),{\mathbf{s}}^n(u^n(j)))D_j^n)\text{ .}$$

$$P_e(\Gamma )=\underset{\mathbf{s}}{\max }\mathbb{E}tr[{\rho }^{\otimes n}(\mathbf{u}(J,K),\mathbf{s}(\mathbf{u}(J,K)))({\mathbb{I}}_{\mathcal{H}}-\mathcal{D}(J,K))]$$

$$=\underset{\mathbf{s}}{\max }\frac{1}{|\mathcal{J}|}\sum _{j\in \mathcal{J}}\sum _{k\in \mathcal{K}}Pr\{K=k\}tr[{\rho }^{\otimes n}(\mathbf{u}(j,k),\mathbf{s}(\mathbf{u}(j,k)))({\mathbb{I}}_{\mathcal{H}}-\mathcal{D}(j,k))].$$

$$P_e(\Gamma )=\sum _{\mathbf{x}}Pr\{\mathbf{u}(J,K)=\mathbf{x}\}\underset{\mathbf{s}\in {\mathcal{S}}^n}{\max }\mathbb{E}\{tr[{\rho }^{\otimes n}(\mathbf{u}(J,K),\mathbf{s})({\mathbb{I}}_{\mathcal{H}}-\mathcal{D}(J,K))]|\mathbf{u}(J,K)=\mathbf{x}\}.$$

$$\underset{{\mathbf{s}}^n(\cdot )}{\max }\sum _{{v^{\prime }}^n\in {{\mathcal{V}}^{\prime }}^n}\sum _{v^n\in {\mathcal{V}}^n}p({v^{\prime }}^n,v^n)P_e(\mathcal{C}({v^{\prime }}^n,v^n),{\mathbf{s}}^n(\cdot ))<ϵ\text{ ,}$$

$$P_e(\mathcal{C}({v^{\prime }}^n,v^n),{\mathbf{s}}^n(\cdot )):=1-\frac{1}{J_n}\sum _{j=1}^{J_n}\mathrm{tr}(\rho (u_{{v^{\prime }}^n}(j),{\mathbf{s}}^n(u_{{v^{\prime }}^n}(j)))D_j^{(v^n)})\text{ .}$$

$$\underset{{\mathbf{s}}^n(\cdot )}{\max }\sum _{{v^{\prime }}^{l_n}\in {{\mathcal{V}}^{\prime }}^{l_n}}\sum _{v^{l_n}\in {\mathcal{V}}^{l_n}}p({v^{\prime }}^{l_n},v^{l_n})P_e(\mathcal{C}({v^{\prime }}^{l_n},v^{l_n}),{\mathbf{s}}^n(\cdot ))<ϵ\text{ ,}$$

$$P_e(\mathcal{C}({v^{\prime }}^{l_n},v^{l_n}),{\mathbf{s}}^n(\cdot )):=1-\frac{1}{J_n}\sum _{j=1}^{J_n}\mathrm{tr}(\rho (u_{{v^{\prime }}^{l_n}}(j),{\mathbf{s}}^n(u_{{v^{\prime }}^{l_n}}(j)))D_j^{(v^{l_n})})\text{ .}$$

$$\nu (v|g,Q):=P_V(v)P_{V^{\prime }|V}(g^{-1}(x)|v)\sum _{s}Q(s|x)\rho \left(x,s\right),$$

$$\vec{\nu }(g,Q):={\left(\nu (v|g,Q)\right)}_{v\in V},$$

$$\mathcal{A}(g):=\{\vec{\nu }(g,Q):Q\in \mathcal{P}(\mathcal{S}|\mathcal{X})\}.$$

$$\bar{\bar{{\cal W}}}:=\{\{\bar{\bar{\rho}}_{Q}(x):=\sum_{s}Q(s|x)\rho(x,s),x\in{\cal X}\}:\mbox{ for all $Q:{\cal X}\rightarrow{\cal S}$}\}.$$

$$\mathcal{A}(g_0)\cap \mathcal{A}(g_1)=\varnothing .$$

$$P_{V^{\prime }}^n(g_0^{-1}(x))=P_{V^{\prime }}^n(g_1^{-1}(x)),$$

$$Pr\left(g_0(u^{\iota })=x(m)\right)=Pr\left(u^{\iota }:g_1(u^{\iota })=x(m)\right)$$

$$\sum _v{P}_V(v)|v⟩⟨v|\otimes P_{V^{\prime }\mid V}\left(g_i^{-1}(x)\mid v\right)\sum _{s}Q(s\mid x)\rho (x,s)\text{ ,}$$

$$P_V^{\iota }(v^{\iota })\sum _x{P}_{V^{\prime }|V}^{\iota }(g_0^{-1}(x)|v^{\iota })\sum _s{Q}_1(s|x)\rho (x,s)=P_V^{\iota }(v^{\iota })\sum _{x^{\prime }}{P}_{V^{\prime }|V}^{\iota }(g_1^{-1}(x^{\prime })|v^{\iota })\sum _{s^{\prime }}{Q}_0(s^{\prime }|x^{\prime })\rho (x^{\prime },s^{\prime }),$$

$$P_V^{\iota }(v^{\iota })\sum _{u^{\iota }}{P}_{V^{\prime }|V}^{\iota }(u^{\iota }|v^{\iota })\sum _s{Q}_1(s|g_0(u^{\iota }))\rho (g_0(u^{\iota }),s)=P_V^{\iota }(v^{\iota })\sum _{u^{\iota }}{P}_{V^{\prime }|V}^{\iota }(u^{\iota }|v^{\iota })\sum _{s^{\prime }}{Q}_0(s^{\prime }|g_1(u^{\iota }))\rho (g_1(u^{\iota }),s^{\prime }),$$

$$\sum _{u^{\iota }}{P}_{V^{\prime }|V}^{\iota }(u^{\iota }|v^{\iota })[\sum _s{Q}_1(s|g_0(u^{\iota }))\rho (g_0(u^{\iota }),s)-\sum _{s^{\prime }}{Q}_0(s^{\prime }|g_1(u^{\iota }))\rho (y|g_1(u^{\iota }),s^{\prime })]=0^{\mathcal{H}},$$

$$\left({\mathfrak{P}}_{\iota }\otimes i{d}^{\mathcal{H}}\right)\vec{w}=0^{\mathcal{H}}.$$

$$\sum _s{Q}_1(s|g_0(u^{\iota }))\rho (g_0(u^{\iota }),s)=\sum _{s^{\prime }}{Q}_0(s^{\prime }|g_1(u^{\iota }))\rho (g_1(u^{\iota }),s^{\prime }),$$

$$\sum _s{Q}_1(s|x(0))\rho (x(0),s)=\sum _{s^{\prime }}{Q}_0(s^{\prime }|x(m))\rho (x(m),s^{\prime })$$

$$\mathcal{A}(g_0)\cap \mathcal{A}(g_1)=\varnothing .$$

$$\mathrm{tr}\left((A^{\prime }-b\cdot \mathrm{id})(\vec{v}(g_0,s))\right)>0$$

Full text

As an application of our main result, we determine the correlation assisted common randomness generation capacity. In this scenario the two channel users have access to correlation as a resource, and further use an AVCQC with an informed jammer for additional discussion. The goal is to create common randomness between the two channel users. We also analyze these capacity formulas when only a small number of signals from the correlation are available.

For the correlation assisted common randomness generation capacity, we show an additional interesting property: For a sufficient amount of “public communication”, common randomness generation capacity is Turing computable, however without this public communication constraint, the correlation assisted common randomness generation capacity is, in general, not Turing computable. Furthermore, we show that even without knowing the capacity formula of the deterministic capacity using maximal error criterion, we can show that it is impossible to evaluate the performance algorithmically on any current or future digital computer.

I Introduction

Quantum information theory is a new field that allows us to exploit new possibilities, while at the same time impose fundamental limitations. In this work we consider classical-quantum channels. The capacity of classical-quantum channels has been determined in Ho , Sch/Ni , and Sch/Wes .

The arbitrarily varying channel (AVC) was introduced in Bl/Br/Th2 . This model describes communication including a jammer who tries to disturb the channel users’ communication by changing his input in every channel use (cf. Figure 1). This model completely captures all possible jamming attacks which depend on the knowledge of the jammer. In Ahl/Bli and Bj/Bo/Ja/No , the capacity of arbitrarily varying classical-quantum channels is analyzed.

It is understood that the sender and the receiver have to select their coding scheme first, and that this coding scheme is known by the jammer, who chooses the most advantaged jamming attacking strategy depending on this knowledge. In the conventional model in all above-mentioned works, it is assumed that the jammer has no knowledge about the codeword which the sender sends.

Resources shared by users play an important role in the coding theorems for the AVC (cf. Figure 2). For example, in wireless communication, the communication service may send some signals via satellite to its users. In 1978 Ahlswede demonstrated in Ahl1 the importance of the resources (of shared randomness) in a very clear form by showing the surprising result that either the deterministic capacity of an arbitrarily varying channel is zero, or it is equal to its randomness assisted capacity (Ahlswede dichotomy).

In Bo/No , a classification of various resources is given. A distinction is made between two extremal cases: randomness and correlation. Randomness is the strongest resource, it requires a perfect copy of the outcome of a random experiment, and thus we should assume an additional perfect channel to generate this kind of resource. On the other hand, the weakest resource is defined as follows. Assume that a bipartite source, modeled by an i.i.d. (independent and identically distributed) random variable $(V^{\prime},V)$ with values in a finite product set ${\mathbf{V}}^{\prime}\times{\mathbf{V}}$, is observed by the sender and receiver. The sender has access to the random variable $V^{\prime}$ and the receiver to $V$. We call $(V^{\prime},V)$ a correlation.

The work Bo/No showed that the common randomness is a stronger resource than the correlation in the following sense: a sufficiently large amount of common randomness allows the sender and receiver to asymptotically simulate the statistics of any correlation. To the contrary, an example is given where not even a finite amount of common randomness can be extracted from a given correlation without further communication.

As already mentioned, in the above-mentioned works it is assumed that the jammer knows the coding scheme, but has no side information about the codeword. In many applications, especially for the secure communications, it is too optimistic to assume this. In Bo/Ca/Ca it has been shown that the jammer can benefit from his knowledge about the sent codeword, i.e, he may have a better jamming strategy. Thus in our previous paper Bo/Ca/Ca we considered the scenario when the jammer knows both the coding scheme and the input codeword.

This work is an extension of our previous paper Bo/Ca/Ca , where we determined the randomness assisted capacity of AVCQCs with a jammer knowing the channel input (cf. Figure 3). However, as mentioned above, common randomness is a very “costly” resource. A promising result of this work is that the much “cheaper” resource, the correlation, is also an equally powerful resource (cf. Figure 4). Furthermore, a correlation $(V^{\prime},V)$ does not have to be “very good” to be helpful in achieving a positive secrecy capacity, since $(V^{\prime},V)$ is a helpful resource even if $I(V^{\prime},V)$ is only slightly larger than zero. We also show that the same capacity can been achieved using a smaller amount (as compared to the number of channel uses) of correlation.

As an application of our results, we turn to the question: How much common randomness can an AVCQC with an informed jammer generate, using the correlation as a resource? In this scenario, the two channels users have, as a resource, access to a correlation which is characterized by a random variable $(V^{\prime},V)$, with values in a finite product set ${\mathbf{V}}^{\prime}\times{\mathbf{V}}$. The sender has access to the output of $V^{\prime}$ and the receiver to $V$. In addition, the sender is allowed to send helper messages via an AVCQC with an informed jammer. The goal is to create shared random variables with negligibly small errors. The amount of shared random variable per channel use is denoted by the common randomness generation capacity. Capacities of common randomness generation over classical perfect channels and over classical noisy channels have been determined in Ahl/Cs . In this work, we deliver the common randomness generation capacity with an informed jammer using correlation as the resource. We also analyze the case when only a smaller amount (as compared to the number of channel uses) of correlation is used.

In 1912, Borel attempted to express concepts of computability in Bo . In Tu1 and Tu2 , Turing introduced the concept of computable numbers and computable functions on computable real numbers. Computable numbers are real numbers that are computable by a Turing machine, which is a mathematical model of an abstract machine that manipulates symbols on a strip of tape and can simulate any given algorithm. Problems which are not computable by a Turing machine cannot be solved or even algorithmically approximated by any current or future digital computer. Banach and Mazur defined in Ba/Ma a notion of computability for functions on real numbers, using computable sequences.

The Turing computability of a capacity is a necessary condition for evaluating the performance of the corresponding system. Otherwise, it is impossible to algorithmically evaluate or even approximate it by computable continuous functions. Bo/Sch/Ba/Po considered secret key capacities and secure authentication capacities over several classical channel network models, and determined whether they are computable, i.e. if they can be algorithmically solved with the help of Turing machines. As a further application of our results, we now analyze some capacity formulas of this work and determine whether they are Turing computable. We show that whether the common randomness generation capacity with informed jammer is Turing computable depends on the amount of correlation the channel users have access to. We also analyze the Turing computability of a capacity whose formula is still unknown.

II Definitions and Communication Models

II.1 Basic Notations

Throughout the paper, the random variables will be denoted by capital letters, e. g. $S,X,Y,$. Their realizations (or values) and domains (or alphabets) will be denoted by the corresponding lower case letters, e.g. $s,x,y,$ and script letters, e.g. ${\cal S},{\cal X},{\cal Y}$, respectively. Random sequences will be denoted by the capital bold-face letters, whose lengths are understood by the context, e.g. ${\bf S}=(S_{1},S_{2},\ldots,S_{n})$ and ${\bf X}=(X_{1},X_{2},\ldots,X_{n})$, and deterministic sequences will be written as lower case bold-face letters, e.g. ${\bf s}=(s_{1},s_{2},\ldots,s_{n}),{\bf x}=(x_{1},x_{2},\ldots,x_{n})$.

$P_{X}$ is the distribution of the random variable $X$. Joint distributions and conditional distributions of random variables $X$ and $S$ will be written as $P_{SX}$, etc. and $P_{S|X}$ etc., respectively, and $P_{XS}^{n}$ and $P_{S|X}^{n}$ will be their product distributions, i.e. $P_{XS}^{n}({\bf x},{\bf s}):=\prod_{t=1}^{n}P_{XS}(x_{t},s_{t})$, and $P_{S|X}^{n}({\bf s}|{\bf x}):=\prod_{t=1}^{n}P_{S|X}(s_{t}|x_{t})$.

Throughout the paper, dimensions of all Hilbert spaces are finite. For a finite-dimensional complex Hilbert space ${\cal H}$, we denote the (convex) set of density operators on ${\cal H}$ by

[TABLE]

where $\mathcal{L}({\cal H})$ is the set of linear operators on ${\cal H}$, and $0_{{\cal H}}$ is the null matrix on ${\cal H}$. Note that any operator in $\mathcal{S}({\cal H})$ is bounded.

Throughout the paper, the logarithm base is $2$. For a discrete random variable $X$ on a finite set ${\cal X}$ and a discrete random variable $Y$ on a finite set ${\cal Y}$, we denote the Shannon entropy of $X$ by $H(X)=-\sum_{x\in\mathcal{X}}P_{X}(x)\log P_{X}(x)$ and the mutual information between $X$ and $Y$ by $I(X;Y)=\sum_{x\in\mathcal{X}}\sum_{y\in\mathcal{Y}}P_{X,Y}(x,y)\log{\left(\frac{P_{X,Y}(x,y)}{P_{X}(x)P_{Y}(y)}\right)}$. Here $P_{X,Y}$ is the joint probability distribution function of $X$ and $Y$, and $P_{X}$ and $P_{Y}$ are the marginal probability distribution functions of $X$ and $Y$, respectively.

If the sender wants to transmit a classical message set to the receiver using a quantum channel, his encoding procedure will include a classical-to-quantum encoder to prepare a quantum state $\rho\in\mathcal{S}({\cal H})$ suitable as an input for the channel. In view of this, we have the following definition.

Definition 1

Let $\mathcal{H}$ be a finite-dimensional complex Hilbert space. A classical-quantum channel is a mapping ${\cal W}:\mathcal{X}\rightarrow\mathcal{S}({\cal H})$, specified by a set of quantum states $\{\rho(x),x\in{\cal X}\}$ $\subset\mathcal{S}({\cal H})$, indexed by “input letters” $x$ in a finite set ${\cal X}$. ${\cal X}$ and ${\cal H}$ are called input alphabet and output space, respectively. We define the $n$-th extension of the classical-quantum channel ${\cal W}$ as follows. The channel outputs a quantum state $\rho^{\otimes n}({\bf x}):=\rho(x_{1})\otimes\rho(x_{2})\otimes\ldots,\otimes\rho(x_{n})$ in the $n$th tensor power ${\cal H}^{\otimes n}$ of the output space ${\cal H}$ when an input codeword ${\bf x}=(x_{1},x_{2},\ldots,x_{n})\in{\cal X}^{n}$ of length $n$ is input into the channel.

For a quantum state $\rho\in\mathcal{S}(H)$ we denote the von Neumann entropy of $\rho$ by

[TABLE]

Let ${\cal W}$: $\mathcal{X}\rightarrow\mathcal{S}({\cal H})$ be a classical-quantum channel. For $P\in P(\mathcal{X})$, the conditional entropy of the channel for ${\cal W}$ with input distribution $P$ is presented by

[TABLE]

Let $\Phi:=\{\rho_{x}:x\in\mathcal{X}\}$ be a classical-quantum channel, i.e. a set of quantum states labeled by elements of $\mathcal{X}$. For a probability distribution $Q$ on $\mathcal{X}$, the Holevo $\chi$ quantity is defined as

[TABLE]

II.2 Communication Scenarios

In this subsection we introduce our communication concept and some resource models whose capacity results (well-known from previous works) we need for this work.

Definition 2

An arbitrarily varying classical-quantum channel (AVCQC) ${\cal W}$ is specified by a set $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ of classical-quantum channels with a common input alphabet ${\cal X}$ and output space ${\cal H}$ which are indexed by elements $s$ in a finite set ${\cal S}$. Elements $s\in{\cal S}$ are usually called the states of the channel. ${\cal W}$ outputs a quantum state

[TABLE]

if an input codeword ${\bf x}=(x_{1},x_{2},\ldots,x_{n})$ is input into the channel and the channel is governed by a state sequence ${\bf s}=(s_{1},s_{2},\ldots,s_{n})$, while the state varies from symbol to symbol in an arbitrary manner.

We assume that the channel state $s$ is in control of the jammer. Without loss of generality, we also assume that the jammer always chooses the most advantageous attacking strategy, i.e. he wants the channel users to transmit a least reliable message. This is important for the applications of our result to other channel models, e.g. compound channels. The conventional AVCQC model when the jammer has information about the channel input is shown in Figure 1.

Definition 3

An $(n,J_{n})$ code $\mathcal{C}$ consists of an encoder $u^{n}:\{1,\cdots,J_{n}\}\rightarrow{{\cal X}}^{n}$, and a set of collections of positive-semidefinite operators $\{D_{j}^{n}:j=1,\cdots,J_{n}\}$ on ${\cal H}^{\otimes n}$ which fulfills $\sum_{j=1}^{J_{n}}D_{j}^{n}=\mathrm{id}_{{\cal H}^{\otimes n}}$.

Definition 4

A non-negative number $R$ is an achievable rate for a classical-quantum channel ${\rho}(x)$ if for every $\epsilon>0$, $\delta>0$, and sufficiently large $n$ there exists an $(n,J_{n})$ code \mathcal{C}=\bigl{(}u^{n},\{D_{j}:j=1,\cdots J_{n}\}\bigr{)}, such that $\frac{\log J_{n}}{n}>R-\delta$, and

[TABLE]

The supremum on achievable deterministic rates of ${\rho}(x)$ is called the capacity of ${\rho}(x)$, denoted by $C({\rho})$.

In the context of the arbitrarily varying channels, randomness can be an important resource for a reliable communication over an arbitrarily varying channel. The message transmission task over AVCQC with common randomness between the sender and the receiver is illustrated in Figure 2. Ahlswede showed in Ahl1 (cf. also Ahl2 and Ahl3 ), the surprising result that either the deterministic capacity of an arbitrarily varying channel is zero, or it equals its randomness assisted capacity (Ahlswede Dichotomy). Bo/Ca/De shows there are indeed arbitrarily varying classical-quantum channels which have zero deterministic capacity and positive random capacity. Therefore randomness is indeed a very helpful resource for message transmission (and secure message transmission) through an arbitrarily varying classical-quantum channel. Having some resource is particularly essential for the scenario we consider in this work (see the discussion below).

Most of the previous works in AVCQCs consider the case when the jammer knows the coding scheme, but has no side information about the codeword of the transmitters. However, Bo/Ca/Ca shows that the jammer can strictly reduce the capacity when he knows the codeword. Thus we concentrate on message transmission over classical-quantum channels with a jammer with additional side information about the codeword. We assume that the jammer chooses the most advantageous attacking strategy according to his side information. The corresponding message transmission task is illustrated in Figure 3.

Definition 5

Let $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ be an AVCQC. A non-negative number $R$ is an achievable deterministic rate with an informed jammer (under the average error criterion) for $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$, if for every $\epsilon>0$, $\delta>0$, and every sufficiently large $n$ there exists a code \mathcal{C}=\biggl{\{}u^{n},\{D_{j}^{n}:j\in\{1,\cdots,J_{n}\}\}\biggr{\}} such that $\frac{\log J_{n}}{n}>R-\delta$, and

[TABLE]

where $P_{e}(\mathcal{C},{\bf s}^{n})$ is defined as

[TABLE]

Here the maximum $\max_{{\bf s}^{n}(\cdot)}$ is taken over all functions ${\cal X}^{n}\rightarrow{\cal S}^{n}$.

The supremum on achievable deterministic rates of $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ with an informed jammer under the average error criterion is called the deterministic capacity of $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ with an informed jammer, denoted by $C(\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\})$.

Throughout the paper, all rates are defined, unless otherwise stated, under the average error criterion.

Our scenario (when the jammer knows the input codeword) is already a challenging topic for classical arbitrarily varying channels. This has been analyzed by Sarwate in Sar , where only the randomness assisted capacity has been determined. The deterministic capacity formula, i.e. without additional resource, is even in the classical case an open problem. It has been shown by Ahlswede in Ahl0 that the classical capacity under the maximal error criterion in this scenario contains the zero-error capacity of related discrete memoryless channels as a special case. A deterministic capacity formula for this is still unknown. In particular, Bo/Ca/Ca shows a violation of the Ahlswede dichotomy in our scenario.

Coding for AVCQC with an informed jammer is even harder. Due to the non-commutativity of quantum operators, many techniques, concepts and methods of classical information theory may not be extended to quantum information theory. For instance, it is still unknown how to extend list decoding, which has been used in Sar ’s proof, and non-standard decoder, to quantum information theory. In Bo/Ca/Ca we determined the randomness assisted capacities of AVCQCs when the jammer has access to the channel input.

Definition 6

A randomness assisted code $\Gamma$ for an AVCQC ${\cal W}$ is a uniformly distributed random variable, taking values in a set of codes $\{({{\cal V}^{\prime}}(k),\{{\cal D}(j,k),j\in{\cal J}\}),k\in{\cal K}\}$ with a common message set ${\cal J}$, where ${{\cal V}^{\prime}}(k)=\{{\bf u}(j,k),j\in{\cal J}\}$ and $\{{\cal D}(j,k),j\in{\cal J}\}$ are the code book and decoding measurement of the $k$th code in the set, respectively. $|{\cal K}|$ depends on the length of the codes in this set, i.e. it is a function of $n$. Particularly, for a fixed $n$, $|{\cal K}|$ is finite.

Definition 7

By assuming that the random message $J$ is uniformly distributed, we define

[TABLE]

*This can be also rewritten as *

[TABLE]

A non-negative number $R$ is an achievable rate for the arbitrarily varying classical-quantum channel ${\cal W}$ under randomness assisted coding with an informed jammer, if for every $\delta>0$ and $\epsilon>0$ and every sufficiently large $n$ there is a randomness assisted code $\Gamma$ of length $n$, such that $\frac{\log|{\cal J}|}{n}>R-\delta$ and $P_{e}(\Gamma)<\epsilon$.

The supremum on achievable rate under randomness assisted coding of ${\cal W}$ with an informed jammer is called the randomness assisted capacity of ${\cal W}$ with an informed jammer, denoted by $C^{*}({\cal W})$.

II.3 Code Concepts and Resources

In this subsection we introduce the relevant code concepts and resource models for this work.

A correlated source is a discrete memoryless source (DMS) $(({V^{\prime}}^{n},V^{n}))_{n}$ observed by the sender and receiver, modeled by $n$ independent copies of a random variable $(V^{\prime},V)$, with values in some finite set ${\cal V}^{\prime}\times{\cal V}$. The sender has access to the random variable $V^{\prime}$, and the receiver to $V$. The corresponding message transmission task over AVCQC with correlation between the sender and the receiver is illustrated in Figure 4. We call \Bigl{(}({V^{\prime}}^{n},V^{n})\Bigr{)}_{n} a correlated source, or a correlation. Since \Bigl{(}({V^{\prime}}^{n},V^{n})\Bigr{)}_{n} is memoryless, we also say $(V^{\prime},V)$ instead of \Bigl{(}({V^{\prime}}^{n},V^{n})\Bigr{)}_{n}. Without loss of generality, we assume that $(V^{\prime},V)$ is binary (since one can easily reduce a non-binary $(\bar{V^{\prime}},\bar{V})$ with $I(\bar{V^{\prime}},\bar{V})>0$ to some $(V^{\prime},V)$ with $I(V^{\prime},V)>0$). The only exception is Section IV, where $(V^{\prime},V)$ may be not binary. It has been shown in Ahl/Cai that this is a helpful resource for information transmission through an arbitrarily varying classical channel. The use of mere correlation already allows one to transmit messages at any rate that is achievable using the optimal form of shared randomness. The capacity of an arbitrarily varying quantum channel assisted by correlated shared randomness as a resource has been discussed in Bo/No , where equivalent results were found.

Our previous work Bo/Ca/Ca determined the randomness assisted capacity with an informed jammer, where we used randomness as a resource. However, as Bo/No showed, common randomness is a very “costly” resource, we have to require that the sender and the receiver each obtain pairs of identical signals that a random experiment outputs. Thus in this work we consider the correlation as a resource, which is a much “cheaper” resource in the sense that we can simulate any $(V^{\prime},V)$ correlation by common randomness asymptotically, however, there exists a class of sequences of bipartite distributions which cannot model common randomness (cf. Bo/No ).

Now we consider the correlation assisted code.

Definition 8

We assume that the transmitters have access to an arbitrary correlated source $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$. A $(V^{\prime},V)$-correlation assisted $(n,J_{n})$ code $\mathcal{C}(V^{\prime},V)$ for the arbitrarily varying classical-quantum channel $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ consists of a set of encoders $\left\{u_{{v^{\prime}}^{n}}:\{1,\cdots,J_{n}\}\rightarrow{{\cal X}}^{n}:{v^{\prime}}^{n}\in{{\cal V}^{\prime}}^{n}\right\}$ and a set of collections of positive-semidefinite operators \Bigl{\{}\{D_{j}^{(v^{n})}:j=1,\cdots,J_{n}\}:v^{n}\in{\cal V}^{n}\Bigr{\}} on ${\cal H}^{\otimes n}$, which fulfills $\sum_{j=1}^{J_{n}}D_{j}^{(v^{n})}=\mathrm{id}_{{\cal H}^{\otimes n}}$ for every ${v}^{n}\in{\cal V}^{n}$.

Definition 9

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source. A non-negative number $R$ is an achievable $(V^{\prime},V)$-correlation assisted rate with an informed jammer for the AVCQC $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$, if for every $\epsilon>0$, $\delta>0$ and sufficiently large $n$ there exists a $(V^{\prime},V)$-correlation assisted $(n,J_{n})$ code \mathcal{C}(V^{\prime},V)=\biggl{\{}\Bigl{(}u_{{v^{\prime}}^{n}},\{D_{j}^{({v}^{n})}:j\in\{1,\cdots,J_{n}\}\}\Bigr{)}:{v^{\prime}}^{n}\in{{\cal V}^{\prime}}^{n},\text{ }v^{n}\in{\cal V}^{n}\biggr{\}} such that $\frac{\log J_{n}}{n}>R-\delta$ and

[TABLE]

where $P_{e}(\mathcal{C}({v^{\prime}}^{n},v^{n}),{\bf s}^{n}(\cdot))$ is defined as

[TABLE]

For a given correlated source $(V^{\prime},V)$, the supremum on achievable $(V^{\prime},V)$-correlation assisted rates of $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ with an informed jammer is called the $(V^{\prime},V)$-correlation assisted capacity with an informed jammer, denoted by $C(\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\};corr(V^{\prime},V))$. Notice that by definition, $C(\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\};corr(V^{\prime},V))$ is a function of $(V^{\prime},V)$.

In Definition 8 and Definition 9 we assume that each channel user obtains one part of a realization of $(V^{\prime},V)$ for every channel use. Now we consider the scenario when we constrain the amount of resources we use.

Definition 10

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source. For a sequence of natural numbers $(l_{n})_{n\in\mathbb{N}}$, a $({V^{\prime}},V)$-correlation assisted $(n,l_{n},J_{n})$ code $\mathcal{C}({V^{\prime}}^{l_{n}},V^{l_{n}})$ for the arbitrarily varying classical-quantum channel $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ consists of a set of encoders $\left\{u_{{v^{\prime}}^{l_{n}}}:\{1,\cdots,J_{n}\}\rightarrow{{\cal X}}^{n}:{v^{\prime}}^{l_{n}}\in{{\cal V}^{\prime}}^{l_{n}}\right\}$ and a set of collections of positive-semidefinite operators \Bigl{\{}\{D_{j}^{(v^{l_{n}})}:j=1,\cdots,J_{n}\}:v^{l_{n}}\in{\cal V}^{l_{n}}\Bigr{\}} on ${\cal H}^{\otimes n}$, which fulfills $\sum_{j=1}^{J_{n}}D_{j}^{(v^{l_{n}})}=\mathrm{id}_{{\cal H}^{\otimes n}}$ for every ${v}^{l_{n}}\in{\cal V}^{l_{n}}$.

Definition 11

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source. A non-negative number $R$ is an achievable $((V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$-correlation assisted rate with an informed jammer under the average error criterion for the AVCQC $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$, if for every $\epsilon>0$, $\delta>0$ and sufficiently large $n$ there exists a $(V^{\prime},V)$-correlation assisted $(n,l_{n},J_{n})$ code \mathcal{C}(V^{\prime},V)=\biggl{\{}\Bigl{(}u_{{v^{\prime}}^{l_{n}}},\{D_{j}^{({v}^{l_{n}})}:j\in\{1,\cdots,J_{n}\}\}\Bigr{)}:{v^{\prime}}^{l_{n}}\in{{\cal V}^{\prime}}^{l_{n}},\text{ }v^{l_{n}}\in{\cal V}^{l_{n}}\biggr{\}} such that $\frac{\log J_{n}}{n}>R-\delta$, and

[TABLE]

where $P_{e}(\mathcal{C}({v^{\prime}}^{l_{n}},v^{l_{n}}),{\bf s}^{n}(\cdot))$ is defined as

[TABLE]

The supremum on achievable $((V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$-correlation assisted rates of $\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ with an informed jammer under the average error criterion is called the $((V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$-correlation assisted capacity with an informed jammer, denoted by $C(\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\};corr(V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$.

III Main Results and Proofs

III.1 Quantum Version of

Kiefer and Wolfowitz’s Results for Classical Channels

At first, we will introduce an important tool for our main result. That tool implies the positive correlation assisted capacity of an AVCQC when this AVCQC has positive randomness assisted capacity and we have access to some correlated source $(V^{\prime},V)$, such that $I(V^{\prime},V)>0$.

III.1.1 Proof Concepts

Before we go on with our proof for the positivity, we would at first like to introduce two well-known approaches of using correlation to achieve positive capacity for a classical AVC, and explain why they will not work for our model. The first approach works only for AVC (classical or classical-quantum) when the jammer has no side information, but not for AVC when the jammer has side information. The second approach works only for classical AVC when the jammer has side information, but not for AVCQC when the jammer has side information.

For classical AVCs, Ahl/Cai showed the equality of correlation assisted capacity and randomness assisted capacity (under the average error criterion) for any correlated source $(V^{\prime},V)$ (with $I(V^{\prime},V)>0$) when the jammer had no side information. The idea of the proof was at first to show that the correlation assisted capacity satisfies the positivity conditions of Cs/Na . Then the channel users can create a sufficient amount of common randomness using codewords of negligible length. For this proof it is essential that the randomness is uniformly distributed.

However, when the jammer has side information about the channel input, the positivity conditions of Cs/Na cannot be applied since there is no Ahlswede Dichotomy (cf. Bo/Ca/Ca ). For classical AVCs with an informed jammer, we need to find another positivity condition.

In KW62 , Kiefer and Wolfowitz delivered another condition for the positivity of a classical AVC (under the maximal error criterion). This is when we can conclude the classical channel outputs into two disjunct convex compact sets in the real vector space. When this condition is fulfilled, then there exists a hyperplane separating the classical channel outputs into two parts in their real vector space. The authors of KW62 showed that if there exists such a hyperplane, one can construct a classical binary point to point channel with positive capacity.

One of the main difficulties is that we can not apply the classical results of Kiefer and Wolfowitz for correlations directly on the set of quantum states since they do not form a real vector space. Thus we have to find a new approach to show a quantum version of the classical positivity condition.

Our idea is concluded in the following two approaches:

• •

At first we show that if the the channel users share a source $(V^{\prime},V)$ such that $I(V^{\prime},V)>0$, then there exists a hyperplane separating the quantum outputs into two parts.

• •

Similar to the classical proposal of KW62 , we will show that when this condition is satisfied, then the correlation assisted capacity of an AVCQC is positive, when this AVCQC has positive randomness assisted capacity.

III.1.2 A Positivity Condition

for AVCQC

At first we need same notations.

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source, and $\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC with input alphabet ${\cal X}$ and output space ${\cal H}$. For a mapping $g:{\cal V}^{\prime}\rightarrow{\cal X}$ and a conditional probability distribution $Q\in{\cal P}({\cal S}|{\cal X})$, where ${\cal P}({\cal S}|{\cal X})$ is the set of conditional probability distributions from ${\cal X}$ to the state set ${\cal S}$ of ${\cal W}$, we define

[TABLE]

[TABLE]

and

[TABLE]

For a given AVCQC ${\cal W}=\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ with set of state ${\cal S}$, let

[TABLE]

In the following Lemma 1, we show the existence of a hyperplane separating the quantum outputs, if the the channel users share a source $(V^{\prime},V)$ such that $I(V^{\prime},V)>0$.

Lemma 1

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source, and $\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC with input alphabet ${\cal X}$ and $d$-dimensional output space ${\cal H}$. Suppose $I(V^{\prime};V)>0$ and $\min_{\rho\in\bar{\bar{{\cal W}}}}C(\rho)>0$ hold. Then we can find $g_{0}$ and $g_{1}$ $:{\cal V}^{\prime}\rightarrow{\cal X}$, such that

[TABLE]

Proof: Let $I(V^{\prime};V)>0$, $\min_{\rho\in\bar{\bar{{\cal W}}}}C(\rho)>0$, ${\cal V}^{\prime}={\cal V}=\{0,1\}$ and $|{\cal X}|=\alpha$. We label the input letters in ${\cal X}$ as $x(0),x(1),\ldots,x(\alpha-1)$. Our proof is based on constructions of two functions $g_{0}:{{\cal V}^{\prime}}^{\iota}\rightarrow{\cal X}$ and $g_{1}:{{\cal V}^{\prime}}^{\iota}\rightarrow{\cal X}$ (for a properly defined $\iota$ in the next paragraph), satisfying (8) and

[TABLE]

for all $x\in{\cal X}$. Notice that we will need property (9) for the proof of Theorem 1.

Let $\iota$ be the smallest integer $\kappa$ such that $\sum_{\tau=0}^{\kappa}\lfloor\frac{1}{2}{\kappa\choose\tau}\rfloor\geq\alpha$. We shall construct $g_{0}$ and $g_{1}$ satisfying (8). To this end we shall group the sequences in ${{\cal V}^{\prime}}^{\iota}$. But first we need to label them in following way.

$\bullet$ For $h=0,1,\ldots\iota$, divide the sequences in ${{\cal V}^{\prime}}^{\iota}$ with Hamming weight $h$ into two parts with equal sizes $\lfloor\frac{1}{2}{\iota\choose h}\rfloor$ and label them as $u^{\iota}(h,1a),u^{\iota}(h,2a),\ldots,u^{\iota}(h,\lfloor\frac{1}{2}{\iota\choose h}\rfloor_{a})$ and $u^{\iota}(h,1b),u^{\iota}(h,2b),\ldots,u^{\iota}(h,\lfloor\frac{1}{2}{\iota\choose h}\rfloor_{b})$, respectively. When ${h\choose\tau}$ is odd, we denote the remaining sequence by $u^{\iota}(h^{*})$.

$\bullet$ Order labels $(h,k)$, $h=0,1,2,\ldots,\iota,k=1,2,\ldots,\lfloor\frac{1}{2}{\iota\choose h}\rfloor$ by lexicographic order, as $m=1,2,\ldots,\sum_{\tau=0}^{\iota}\lfloor\frac{1}{2}{\iota\choose\tau}\rfloor$ and rewrite $u^{\iota}(h,k_{a})$ and $u^{\iota}(h,k_{b})$ to $u^{\iota}(m_{a})$ and $u^{\iota}(m_{b})$ respectively, if $(h,k)$ is the $m$th label in the order. That is, for $u^{\iota}(m_{a})=u^{\iota}(h,k_{a})$ and $u^{\iota}(m^{\prime}_{a})=u^{\iota}(h^{\prime},k^{\prime}_{a})$, we have $m_{a}<m^{\prime}_{a}$ if and only if $h<h^{\prime}$ or $h=h^{\prime}$ and $k<k^{\prime}$; for $u^{\iota}(m_{b})=u^{\iota}(h,k_{b})$ and $u^{\iota}(m^{\prime}_{b})=u^{\iota}(h^{\prime},k^{\prime}_{b})$, we have $m_{b}<m_{b}^{\prime}$ if and only if $h<h^{\prime}$ or $h=h^{\prime}$ and $k_{a}<k^{\prime}_{a}$.

Next we assign values of $g_{0}$ and $g_{1}$ to the sequences in ${{\cal V}^{\prime}}^{\iota}$ according to three groups:

Group 1: For $m=1,2,\ldots,\alpha-1$, we let $g_{0}(u^{\iota}(m_{a}))=g_{1}(u^{\iota}(m_{b}))=x(0)$, $g_{0}(u^{\iota}(m_{b}))=x(m)$ and $g_{1}(u^{\iota}(m_{a}))=x(m)$, respectively. Notice that as $u^{\iota}(m_{a})$ and $u^{\iota}(m_{b})$ have the same Hamming weight for all $m$, for every $u^{\iota}\in{{\cal V}^{\prime}}^{\iota}$, we have

[TABLE]

for $m=0,1,\ldots,\alpha-1$ (i. e. for all $x(m)\in{\cal X}$), in the assignment to the members in group 1.

Group 2: For all $m=\alpha,\alpha+1,\ldots,\sum_{\tau=0}^{\iota}\lfloor\frac{1}{2}{\iota\choose\tau}\rfloor$, we arbitrarily choose $0\leq\zeta_{0}(m))<\zeta_{1}(m)\leq\alpha-1$ and let $g_{0}(u^{\iota}(m_{a}))=g_{1}(u^{\iota}(m_{b}))=x(\zeta_{0}(m))$ and $g_{0}(u^{\iota}(m_{b}))=g_{1}(u^{\iota}(m_{a}))=x(\zeta_{1}(m))$. Again, because $u^{\iota}(m_{a})$ and $u^{\iota}(m_{b})$ have the same Hamming weight, (10) also holds for the assignment to the members of group 2.

Group 3: Finally for each $u^{\iota}(h^{*})$, we arbitrarily choose a letter in the alphabet ${\cal X}$, say $x(i_{h})$, and let $g_{0}(u^{\iota}(h^{*}))=g_{1}(u^{\iota}(h^{*}))=x(i_{h})$. Obviously (10) also holds for the assignment of group 3.

Notice that the property (10) of the assignments to the 3 groups yields (9).

Let $\{|v\rangle:v\in{\cal V}\}$ be an orthonormal basis of $\mathbb{C}^{\otimes|{\cal V}|}$. Then the output $\vec{v}(g;Q)$ of the channel can be presented as a classical-quantum state in the $d|{\cal V}|$-dimensional complex Hilbert space $\mathbb{C}^{\otimes|{\cal V}|}\otimes{\cal H}$:

[TABLE]

when one applies the function $g_{i}$, $i=0,1$ to the input of the channel according to the output of the source $V^{\prime}$. Consequently, in the notation, ${\cal A}(g_{i})$ is presented as $\{\sum_{v}P_{V}(v)|v\rangle\langle v|\otimes P_{V^{\prime}\mid V}\left(g_{i}^{-1}(x)\mid v\right)\sum_{s}Q(s\mid x)\rho(x,s):Q\in{\cal P}({\cal S}\mid{\cal X})\}$.

Next we shall show (8) by assuming a contradiction that (8) will not hold. That is, there exist $Q_{0},Q_{1}\in{\cal P}({\cal S}|{\cal X})$ such that, $\vec{\nu}(g_{0},Q_{1})=\vec{\nu}(g_{1},Q_{0})$ or

[TABLE]

for all $v^{\iota}$, which can be rewritten as

[TABLE]

by re-arranging the terms. That is,

[TABLE]

for all $v^{\iota}$, where $0^{\mathcal{H}}$ is the zero ensemble on $\mathcal{H}$.

Denote by $\mathfrak{P}_{t}$, the $|{\cal V}^{t}|\times|{{\cal V}^{\prime}}^{t}|$ matrix whose $(v,u)$th entry is $P_{V^{\prime}|V}^{t}(u^{t}|v^{t})$ for all $t$. We write $\mathfrak{P}_{1}=\mathfrak{P}$. Then we observe that for every positive integer $t$, we have $\mathfrak{P}_{t}=\mathfrak{P}^{\otimes t}$, i. e. $\mathfrak{P}_{t}$ is $t$th-sensor power of $\mathfrak{P}$. Recall that we have assumed that $I(V^{\prime};V)>0$, which implies that $\frac{P_{V^{\prime}|V}(0|0)}{P_{V^{\prime}|V}(0|1)}\not=\frac{P_{V^{\prime}|V}(1|0)}{P_{V^{\prime}|V}(1|1)}$, or $\det(\mathfrak{P})\not=0$. Therefore $\mathfrak{P}$ is a full rank matrix, and hence so is $\mathfrak{P}_{t}$ for all $t$. Next let $\vec{w}$ be the $d|{{\cal V}^{\prime}}^{\iota}|\times d$ matrix, where $d:=\dim\mathcal{H}$, whose components are $\sum_{s}Q_{1}(s|g_{0}(u^{\iota}))\rho(g_{0}(u^{\iota}),s)-\sum_{s^{\prime}}Q_{0}(s^{\prime}|g_{1}(u^{\iota}))\rho(g_{1}(u^{\iota}),s^{\prime}),u^{\iota}\in{{\cal V}^{\prime}}^{\iota}$, in a proper order. Then (12) can be rewritten as

[TABLE]

Because $\mathfrak{P}_{\iota}$ is full rank, the linear function $\mathfrak{P}_{\iota}z^{|{{\cal V}^{\prime}}|^{\iota}}=0$ with respect to $z^{|{{\cal V}^{\prime}}|^{\iota}}$ has no non-zero solution. This implies that $\vec{w}={0^{\mathcal{H}}}^{\otimes|{{\cal V}^{\prime}}|^{\iota}}$, or

[TABLE]

for all $u^{\iota}$. Now we substitute $u^{\iota}(m_{a})$ for $m=1,2\ldots,\alpha-1$ in the group 1 to (13) and then have that

[TABLE]

for $m=1,2,\ldots\alpha-1$. By choosing $(\sum_{s}Q_{1}(s|x(0))\rho(x(0),s))$ and $(\sum_{s^{\prime}}Q_{0}(s^{\prime}|x(m))\rho(x(m),s^{\prime}))$ for $m=1,2,\ldots,\alpha-1$, we have a channel in $\bar{\bar{{\cal W}}}$ with an identity row, which contradicts the assumption $\min_{\rho\in\bar{\bar{{\cal W}}}}C(\rho)>0$, and therefore (8) is proven.

$\Box$

In the following Lemma, we show that this condition is indeed a “positivity condition”, i.e. this implies a positive correlation assisted capacity.

Lemma 2

There is a positive $r$ such that for all $\epsilon,\lambda>0$, there is an $(n,2^{nr})$ correlation assisted code for a sufficiently large $n$, with maximum probability of error smaller than $\lambda$ and rate $r^{\prime}>r-\epsilon$, if there exist two mappings $g_{k},k=0,1$ from $V^{\prime}$ to ${\cal X}$ with

[TABLE]

Proof: We show the lemma in the same way as in KW62 .

The set $P(\mathcal{S}|\mathcal{X})$ is bounded and equal to $\overline{P(\mathcal{S}|\mathcal{X})}$ and thus convex and compact. For a fixed $g$, $\vec{v}(g,\cdot)$ $:P(\mathcal{S}|\mathcal{X})\rightarrow\mathcal{H}^{|\tilde{\mathcal{V}}|}$ is linear, thus for every given $g$, $\mathcal{A}(g)$ is also a convex compact set.

${\cal A}(g_{k}),k=0,1$ are compact convex sets in the $|{\cal V}||\mathcal{X}|$-dimensional real space. When $\mathcal{A}(g_{0})\cap\mathcal{A}(g_{1})=\emptyset$ holds, then there is a hyperplane $\grave{H}$ which separates $\mathcal{A}(g_{0})$ and $\mathcal{A}(g_{1})$. Let $b$ be a real number such that $\grave{H}-b$ is a subspace. By Riesz’s representation theorem we can find a self-adjoint operator $A^{\prime}$ =\Bigl{(}a(v)\Bigr{)}_{v\in{\cal V}} on ${\cal H}^{d|\tilde{\mathcal{V}}|}$ such that

[TABLE]

for all $\vec{v}(g_{0},s)\in\mathcal{A}(g_{0})$ and

[TABLE]

for all $\vec{v}(g_{1},s)\in\mathcal{A}(g_{1})$.

We define the the self-adjoint operator $A$ on ${\cal H}^{d|\tilde{\mathcal{V}}|}$ by $A:=A^{\prime}-b\cdot\mathrm{id}$. Suppose $A$ has a spectral decomposition $A=\sum_{l=1}^{d|\mathcal{V}|}a_{l}A_{l}$. Similar to Ahl/Bj/Bo/No , we define for every $m\in\mathbb{N}$ and $l^{m}=(l_{1},\cdots,l_{m})$

[TABLE]

where $A_{l^{m}}:=\bigotimes_{i=1}^{m}A_{l_{i}}$. We have $\mathsf{P}_{0}+\mathsf{P}_{1}=\mathrm{id}^{m}$.

We define

[TABLE]

Now we consider every term in $\sum_{v^{m}}P_{V}^{m}(v^{m})\sum_{x^{m}}P_{V^{\prime}|V}^{m}((g_{0}^{-1})^{m}(x^{m})|v^{m})\sum_{s^{m}}Q^{m}(s^{m}|x^{m})\rho^{\otimes m}(x^{m},{s}^{m})$ $=\sum_{v^{m}}P_{V}^{m}(v^{m})$ $\sum_{u^{m}}P_{V^{\prime}|V}^{m}(u^{m}|v^{m})$ $\sum_{s^{m}}Q^{m}(s^{m}|g_{0}^{m}(u^{m}))$ $\rho^{\otimes m}(g_{0}^{m}(u^{m}),{s}^{m})$. We define

[TABLE]

For every fixed $u\in\mathcal{U}$, we have $\{\mathbf{s}(g_{0}(u)),\mathbf{s}(\cdot)\in\mathbf{S}\}$ $=\mathcal{S}$, thus similar to the proof in Ahl/Bj/Bo/No we may apply Chebyshev’s inequality for every $s^{m}\in\mathcal{S}$ to show

[TABLE]

where $N(s|s^{m})$ the number of occurrences of the symbol $s$ in $s^{m}$.

Similarly,

[TABLE]

Using the idea of Ahl/Bj/Bo/No we can now define a classical binary AVC by

[TABLE]

and

[TABLE]

for $i=0,1$,

[TABLE]

$\hat{\cal W}$ (with input alphabet $\{g_{0}^{m},g_{1}^{m}\}$) is a binary AVC such that $\hat{W}(0|g_{0},\hat{s})>1-\eta$ and $\hat{W}(0|g_{1},\hat{s^{\prime}})<\eta$ for all $\hat{s},\hat{s^{\prime}}\in\hat{\cal S}$. Every deterministic code for $\hat{W}$ also defines a correlation assisted code for $\rho$.

Now we apply a lemma from Ahlswede and Wolfowitz AW70 .

Lemma 3

Let $W^{bc}:=\left\{\{W(j\mid i,s),i,j\in\{0,1\}\},s\in{\cal S}\right\}$ be a binary classical AVC. Then its deterministic code capacity is equal to

[TABLE]

if for all $s,s^{\prime}$, we have $W(0\mid 0,s)+W(1\mid 1,s^{\prime})>1$.

Applying Lemma 3 we can construct a binary point to point channel with positive capacity. This shows Lemma 2. $\Box$

III.1.3 An Alternative Proof

In Section III.1.1, we point out that the concept of KW62 would work for classical AVC with an informed jammer, but not for AVCQC with an informed jammer. In this section we would like to introduce a “trick” to apply the concept of KW62 on AVCQC. This delivers an alternative proof for Lemma 2.

At first, observe that as mentioned in Section III.1.1, the reason why we can not apply the classical results of KW62 on the set of quantum states $\{\nu(v|g,Q)\}$ directly is that the set of the probability matrices do not form a real vector space over $\mathbb{C}$. However, there is an isomorphism which maps every set of Hermitian complex $m\times m$ matrices to a $m^{2}$-dimensional subspace of the $2m^{2}$-dimensional vector space of complex $m\times m$ matrices over $\mathbb{R}$. Thus alternatively, we can prove Lemma 2 as follows.

Let $\{|v\rangle:v\in{\cal V}\}$ be an orthonormal basis of ${\cal H}^{\otimes|{\cal V}|}$. As mentioned in the proof of Lemma 1 above, the channel can be presented as a classical-quantum state in the $d|{\cal V}|$-dimensional complex Hilbert space:

[TABLE]

when one applies the function $g_{i}$, $i=0,1$ to the input of the channel according to the output of the source $V^{\prime}$. ${\cal A}(g_{i})$, presented as $\{\sigma_{Q,g_{i}}:Q\in{\cal P}({\cal S}\mid{\cal X})\}$, is a compact convex subset in the real vector space $\mathfrak{S}$ formed by $d|{\cal V}|\times d|{\cal V}|$ Hermitian matrices, a $(d|{\cal V}|)^{2}$-dimensional real Hilbert subspace of $2(d|{\cal V}|)^{2}$-dimensional space of complex $d|{\cal V}|\times d|{\cal V}|$ matrices (with the inner product of $A$ and $B$, $\langle A|B\rangle=tr(A*B)$).

Let $\phi$ be a linear isomorphic mapping from $\mathfrak{S}$ to the $(d|{\cal V}|)^{2}$-dimensional real linear vector space $\mathfrak{V}$, (keeping the inner product unchanged $\langle A|B\rangle=tr(A*B)$). Then $\phi(A(g_{i})):=\{\phi(\sigma_{Q,g_{i}}):Q\in{\cal P}({\cal S}\mid{\cal X})\}$ for $i=0,1$ are compact convex subsets in $\mathfrak{V}$. By Lemma 1 we have

[TABLE]

Then we have

[TABLE]

as well, by the isomorphism. Thus, $\phi({\cal A}(g_{0}))$ and $\phi({\cal A}(g_{0}))$ can be separated by a hyperplane. Namely, there is a $(d|{\cal V}|)^{2}$-dimensional real vector $\vec{\mathbf{a}}$ and a real number $b$ such that

[TABLE]

for all $\phi(\sigma_{Q_{i},g_{i}})\in\phi(A(g_{i}))$, $i=0,1$. Let $A$ be the inverse image of $\vec{\mathbf{a}}$ under the mapping $\phi$. Then by the isomorphism again, we have that

[TABLE]

for all $\sigma_{Q_{i},g_{i}}\in A(g_{i})$, $i=0,1$. (Notice, we have that $\sigma_{Q_{i},g_{i}}*=\sigma_{Q_{i},g_{i}}$ and $A*=A$ here). We can now apply the classical results of Kiefer and Wolfowitz to construct a binary point to point classical channel with positive capacity. This delivers an alternative proof for Lemma 2.

III.2 Correlation Assisted Capacity Formula

Our next step is creating a sufficient amount of common randomness, similar to the technique in Bj/Bo/Ja/No and Bo/Ca/De2 . In our previous work Bo/Ca/Ca , we delivered the randomness assisted capacity when the jammer has side information about the channel input. For this proof, only a negligible amount of randomness was needed. This, together with our last step, demonstrates the equality of correlation assisted capacity and randomness assisted capacity for AVCQCs. For our proof it is essential that the randomness we create in the last step must be uniformly distributed.

Theorem 1

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source and ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC. When $I(V^{\prime},V)>0$ holds, then

[TABLE]

Here $\bar{\bar{{\cal W}}}$ is defined as in (7), i.e. \{\{\bar{\bar{\rho}}_{Q}(x):=\sum_{s}Q(s|x)\rho(x,s),x\in{\cal X}\}:\mbox{ for all Q:{\cal X}\rightarrow{\cal S}}\}.

Our model is a generalization of the standard AVC model, since the standard ACV model is limited to the case when the jammer has no side information about the codeword. Furthermore, our model is more complicated than the standard AVC model. Since the jammer can choose his jamming strategy according to a block of the channel input, the size of the eavesdropper’s possible output will be a double-exponential of the code length. Notice that when we have no access to any resource, then the (deterministic) capacity formula for arbitrarily varying channels with an informed jammer is still an open problem, even for classical arbitrarily varying channels. In classical information theory it is known that zero error capacity can be reduced to a special case (cf. Bo/Ca/Ca and Ahl0 ). However, to determine zero error capacity is one of the hardest well-known open problems of more than six decades, since 1956 (cf. Sha ). Theorem 1 delivers a single letter characterization of the correlation assisted capacity with an informed jammer. This is particularity interesting and promising, since it shows that correlation, the weakest form of resource, is already powerful enough to protect against such a mighty jamming strategy, and is as helpful as common randomness, the “costly” resource.

Proof: In Bo/Ca/Ca it has been shown that the randomness assisted capacity of ${\cal W}$ under the average error criterion with an informed jammer is equal to $\min_{\bar{\bar{\rho}}(\cdot)\in\bar{\bar{W}}}\chi(P_{X},\bar{\bar{\rho}}(\cdot))$. In this proof a we use a random variable uniformly distributed on a finite set $\mathcal{K}$ such that

[TABLE]

where $c_{k}$ is a positive constant depending on $|{\cal X}|$ and $|{\cal S}|$.

Since the $(V^{\prime},V)$-correlation assisted capacity cannot exceed the randomness assisted capacity, the converse is trivial.

When $\min_{\bar{\bar{\rho}}(\cdot)\in\bar{\bar{W}}}\chi(P_{X},\bar{\bar{\rho}}(\cdot))$ $=0$ holds, then the randomness assisted capacity of ${\cal W}$ is equal to zero and thus the $(V^{\prime},V)$-correlation assisted capacity of ${\cal W}$ is also equal to zero. This case is trivial. Now we assume that both $I(V^{\prime},V)>0$ and $\min_{\bar{\bar{\rho}}(\cdot)\in\bar{\bar{W}}}\chi(P_{X},\bar{\bar{\rho}}(\cdot))$ $>0$ hold. Our idea is to build a two-part code word, the first part is used to create the common randomness for the sender and the receiver, the second is used to transmit the message to the receiver.

Our idea is at first to build a pre-code, which is a $(V^{\prime},V)$-correlation assisted code, generating the random variable uniformly distributed on a finite set $\mathcal{K}$ we used for the randomness assisted code in Bo/Ca/Ca . The next step is to apply the result of Bo/Ca/Ca , i.e. a second code for message transmission, which is a randomness assisted code using the random variable we generated. Thus the codeword we use at the end is a two-part codeword.

**Definition of pre-code **

Let $\mathcal{K}$ be a finite set such that $\left|\mathcal{K}\right|$ is of polynomially size of $n$. We denote $\nu(n):=\frac{3}{r}\log n$, where $r$ is defined as in Lemma 2. Recall that $\left|\mathcal{K}\right|=c_{k}n^{2}<2^{\nu(n)r}$.

By Lemma 2 we can apply the coding theorem of AVCs with a binary output. Every deterministic code for $\hat{W}$ also defines a correlation assisted code for $\rho$. Thus by Lemma 2 there exists a $(V^{\prime},V)$-correlation assisted code \Bigl{(}\left(u_{{v^{\prime}}^{\nu(n)}}(k)\right)_{k=1,\cdots,\left|\mathcal{K}\right|},\{D_{k}^{\nu(n)}:k=1,\cdots,\left|\mathcal{K}\right|\}\Bigr{)} with deterministic encoder $u_{{v^{\prime}}^{\nu(n)}}(k)\in\{g_{0}^{m},g_{1}^{m}\}^{\nu(n)}$ for ${\cal W}$ of length $\nu(n)$ (cf. KW62 ), such that

[TABLE]

Here we use the property (9) in the proof of Lemma 1, that $P_{V^{\prime}}^{n}(g_{0}^{-1}(x))=P_{V^{\prime}}^{n}(g_{1}^{-1}(x))$ for all $x\in{\cal X}$, to show that the randomness we built is a uniformly distributed random variable.

When the jammer knew the shared randomness, he could render the shared randomness completely useless (cf. Bo/Ca/De3 ). Thus we have to guarantee that the jammer, knowing the code word, has no access to the randomness. Notice that the sender sends the input $(g_{{v^{\prime}}_{1}},g_{{v^{\prime}}_{2}},\ldots,g_{{v^{\prime}}_{\nu(n)}})$ of the AVCQC ${\cal W}$, if he wants to send a binary input $({v^{\prime}}_{1}{v^{\prime}}_{2}\ldots{v^{\prime}}_{\nu(n)})$ to the binary classical AVC $\hat{W}$. Since $P_{V^{\prime}}^{n}(g_{0}^{-1}(x))=P_{V^{\prime}}^{n}(g_{1}^{-1}(x))$ holds, for any $x^{\nu(n)}\in\mathcal{X}^{\nu(n)}$ we have

[TABLE]

for every ${v_{1}^{\prime}}^{\nu(n)}$, ${v_{2}^{\prime}}^{\nu(n)}$ $\in{\tilde{\cal V^{\prime}}}^{\nu(n)}$. This means that the jammer, knowing the code word, has no access to the randomness.

**Definition of the two-part code **

By Bo/Ca/Ca there is an $(n,J_{n})$ random code \Bigl{\{}u(j,k),D_{k,j}^{n}\Bigr{\}}, such that

[TABLE]

Now we can construct a $(V^{\prime},V)$-correlation assisted $(\nu(n)+n,J_{n})$ random code \mathcal{C}(V^{\prime},V)=\biggl{\{}\Bigl{(}u_{{v^{\prime}}^{\nu(n)+n}},\{D_{j}^{v^{\nu(n)+n}}:j\in\{1,\cdots,J_{n}\}\}\Bigr{)}:{v^{\prime}}^{\nu(n)+n}\in{\mathcal{V}^{\prime}}^{\nu(n)+n},v^{\nu(n)+n}\in\mathcal{V}^{\nu(n)+n}\biggr{\}}, where we set

[TABLE]

and

[TABLE]

Remark 1

Notice that the first part of this two-part codeword does not depend on the message, while the second part does not depend on the correlation. Thus our $(n,J_{n})$ code $\mathcal{C}(V^{\prime},V)$ is actually a $(\nu(n)+n,J_{n},\nu(n))$ code of $\nu(n)+n$ length.

By (20), for any $s^{\nu(n)+n}(\cdot)$ we have

[TABLE]

$\Box$

In Theorem 1 we consider that for every channel use, the channel users receive one signal per channel use from the correlation. Since the correlation, which can be regard as some “public communication” may have a cost, we want to analyze now the scenario when, instead of one signal per channel use, only a small number of signals from the correlation for mutable channel uses are available.

Corollary 1

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source and ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC. There is a positive $r^{\prime\prime}$ such that for any sequence of natural numbers $(l_{n})_{n\in\mathbb{N}}$ such that $r^{\prime\prime}<$ $\liminf_{n\rightarrow\infty}\frac{l_{n}}{\log n}$ $\leq\limsup_{n\rightarrow\infty}\frac{l_{n}}{\log n}$ $<\infty$, when $I(V^{\prime},V)>0$ holds, we have

[TABLE]

Proof: We define $r^{\prime\prime}:=\frac{3}{r}$, where $r$ is defined as in Lemma 2. Let $(l_{n})_{n\in\mathbb{N}}$ be a sequence such that $\frac{\nu(n)}{\log n}$ $=r^{\prime\prime}$ $<\liminf_{n\rightarrow\infty}\frac{l_{n}}{\log n}$ $\leq\limsup_{n\rightarrow\infty}\frac{l_{n}}{\log n}$ $<\infty$. We have: $l_{n}\geq\nu(n)$ for all $n$. By Remark 1, for any positive $\delta$, $\epsilon$ and sufficiently large $n$ there is a $(l_{n}+n,J_{n},l_{n})$ code, where $J_{n}=2^{n(\min_{\bar{\bar{\rho}}(\cdot)\in\bar{\bar{W}}}\chi(P_{X},\bar{\bar{\rho}}(\cdot))-\delta)}$, such that

[TABLE]

Since $2^{l_{n}}$ is in polynomial order of $n$, for any positive $\varepsilon$, if $n$ is large enough, we have $\frac{1}{n}\log J_{n}-\frac{1}{l_{n}+n}\log J_{n}\leq\varepsilon$. Thus when $I(V^{\prime},V)>0$,

[TABLE]

$\Box$

In Theorem 1 we consider the scenario when the positivity condition of Lemma 1 is fulfilled. What remains is to analyze the scenario when this positivity condition does not hold.

Lemma 4

Let ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC. When for every $n\in\mathbb{N}$ and every $x_{1}$, $x_{2}\in{\cal X}^{n}$,

[TABLE]

holds, then the deterministic capacity of ${\cal W}$ with an informed jammer is zero.

Proof: When for every $n\in\mathbb{N}$ and every $x_{1}$, $x_{2}\in{\cal X}^{n}$,

[TABLE]

holds, by Ahl/Bj/Bo/No the deterministic capacity of ${\cal W}$ with an uninformed jammer is zero. Since the deterministic capacity of ${\cal W}$ with an informed jammer cannot exceed the deterministic capacity of ${\cal W}$ with an uninformed jammer, the lemma has been shown.

$\Box$

Since classical arbitrarily varying channels can be regard as special cases of AVCQCs, the following Corollary 2 (the classical capacity formula for classical AVC with informed jammer) is a direct consequence of Theorem 1.

Corollary 2

The $(V^{\prime},V)$-correlation assisted capacity of a classical AVC ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ is equal to

[TABLE]

when $I(V^{\prime},V)>0$. Here $\bar{\bar{{\cal W}}}$ $:=\{\{\sum_{s}Q(s|x)\rho(x,s),x\in{\cal X}\}:\forall Q:{\cal X}\rightarrow{\cal S}\}$.

Instead of deducing Corollary 2 from Theorem 1, we can also show this Corollary using a classical technique, when we apply the classical results of Kiefer and Wolfowitz in KW62 directly to the real vector space of the channel output, as mentioned in Section III.1.1. We would like to skip a lengthy proof and leave the extended proof of Corollary 2 to the readers as an exercise. We give just a sketch of the proof of Corollary 2 by means of the classical results of Kiefer and Wolfowitz as follows.

At first, similar to Lemma 1, we demonstrate that there exists a hyperplane separating the classical channel outputs into two parts for any correlated source $(V^{\prime},V)$ with $I(V^{\prime},V)>0$. Then, we apply the classical results of Kiefer and Wolfowitz in KW62 to construct a binary point to point classical channel with positive capacity, as in Section III.1.3. Finally, to show the equality of correlation assisted capacity and randomness assisted capacity, we create common randomness with a pre-code with negligible length, as in the proof of Theorem 1. With this approach we can show the coding theorem for the arbitrarily varying classical channel.

IV Applications

Common randomness generation plays a fundamental role in various problems of cryptography and information theory. Here the channel users want to calculate a shared random variable using an AVCQC and correlation as a resource. This can be used, for instance, as a strong resource for a randomized protocol. Obviously, the message transmission capacity of any channel is upper bounded by its common randomness capacity. Furthermore, the common randomness capacity of any channel is upper bounded by its identification capacity.

Common randomness generating over a classical arbitrarily varying channel using correlation as a resource was first introduced in Ahl/Cs , where it was assumed that the jammer had no side information about the input codeword. As an application of our results in Section III, we now want to analyze the common randomness generation using correlation as resource for our scenario in Section III, i.e. we assume that the jammer knows the input codeword. We assume that the sender and the receiver use a correlation assisted code to generate a pair of random variables $(J,L)$, both distributed on a finite set ${\cal J}$, so that the probability that $J\not=L$ can be kept arbitrarily small.

As a second application of our results, we analyze the capacity formulas of this work and some of our previous works to determine whether they are Turing computable.

IV.1 Correlation Assisted

Common Randomness Generation Capacity with an Informed Jammer

In this Section, we consider how much common randomness an AVCQC with an informed jammer can generate, using correlation as resource. We consider the two scenarios: In one scenario the channel users receive one signal from the correlation for every channel use, while in the other scenario only a small number of signals from the correlation for mutable channel uses are available (cf. the discussion in Section III.2).

IV.1.1 Capacity Definition and Communication Scenario

Definition 12

Let ${\cal W}$ $=\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ be an AVCQC and $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source. A non-negative number $R$ is an achievable $(V^{\prime},V)$-correlation assisted common randomness generation rate with an informed jammer for ${\cal W}$, if for every $\epsilon>0$, $\delta>0$, and sufficiently large $n$ there exists a random variable $J$ distributed on a finite set ${\cal J}$, a set of encoders $\left\{u_{{v^{\prime}}^{n}}:{\cal J}\rightarrow{{\cal X}}^{n}:{v^{\prime}}^{n}\in{{\cal V}^{\prime}}^{n}\right\}$ and a set of measurements $\left\{L_{{v}^{n}}:\mathcal{S}({\cal H}^{\otimes n})\rightarrow{\cal J}:{v}^{n}\in{{\cal V}}^{n}\right\}$ such that $\frac{1}{n}H(J)>R-\delta$ and

[TABLE]

The supremum on achievable $(V^{\prime},V)$-correlation assisted common randomness generation rate with an informed jammer of ${\cal W}$ is called the $(V^{\prime},V)$-correlation assisted common randomness generation capacity with an informed jammer for ${\cal W}$, denoted by $\tilde{C}({\cal W})$.

Definition 13

Let ${\cal W}$ $=\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ be an AVCQC and $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source. A non-negative number $R$ is an achievable $((V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$-correlation assisted common randomness generation rate with an informed jammer for ${\cal W}$, if for every $\epsilon>0$, $\delta>0$, and sufficiently large $n$ there exists an encoder $\left\{u_{{v^{\prime}}^{l_{n}}}:{\cal J}\rightarrow{{\cal X}}^{n}:{v^{\prime}}^{l_{n}}\in{{\cal V}^{\prime}}^{l_{n}}\right\}$, and a set of measurements $\left\{L_{{v}^{l_{n}}}:\mathcal{S}({\cal H}^{\otimes n})\rightarrow P({\cal J}):{v}^{l_{n}}\in{{\cal V}}^{l_{n}}\right\}$, such that $\frac{1}{n}H(J)>R-\delta$ and

[TABLE]

The supremum on achievable on $((V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$-correlation assisted common randomness generation rate with an informed jammer for ${\cal W}$ is called the $((V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$-correlation assisted common randomness generation capacity with an informed jammer for ${\cal W}$, denoted by $\tilde{C}({\cal W},(l_{n})_{n\in\mathbb{N}})$.

IV.1.2 Capacity Results

Corollary 3

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source and ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC. There is a positive $r^{\prime\prime}$ such that for any sequence of natural numbers $(l_{n})_{n\in\mathbb{N}}$ such that $r^{\prime\prime}<$ $\liminf_{n\rightarrow\infty}\frac{l_{n}}{\log n}$ $\leq\limsup_{n\rightarrow\infty}\frac{l_{n}}{\log n}$ $<\infty$, when $I(V^{\prime},V)>0$ holds, then

[TABLE]

Proof: We define $r^{\prime\prime}:=\frac{3}{r}$, where $r$ is defined as in Lemma 2. We fix a $P$ and define

[TABLE]

for an arbitrary positive $\delta$. Now the sender chooses a random variable uniformly distributed on $\{1,\cdots J_{n}\}$. Let $(l_{n})_{n\in\mathbb{N}}$ be a sequence such that $\frac{\nu(n)}{\log n}$ $=r^{\prime\prime}$ $<\liminf_{n\rightarrow\infty}\frac{l_{n}}{\log n}$ $\leq\limsup_{n\rightarrow\infty}\frac{l_{n}}{\log n}$ $<\infty$. By Corollary 1 he can send the output to the receiver using an $(n,l_{n},J_{n})$ code. When the receiver fails to decode the output, he randomly chooses one element in $\{1,\cdots J_{n}\}$. By Corollary 1, when $I(V^{\prime},V)>0$, the probability of failure with an informed jammer can be kept arbitrarily small when $n$ is sufficiently large.

We define $L_{l_{n}}:=\lfloor 2^{l_{n}r}\rfloor$. Now by classical common randomness generation technique in Ahl/Cs we can define a set of $L_{l_{n}}$ deterministic codes $\{{\cal C}^{\hat{W}}_{l}:l\in\{1,\cdots,L_{l_{n}}\}\}$ for $\hat{W}$, which is the classical channel we defined in Lemma 2, such that $L_{l_{n}}$ messages can be sent, and furthermore, for ever random variable $K_{s}$ distributed on $\{{\cal C}^{\hat{W}}_{l}:l\in\{1,\cdots,L_{l_{n}}\}\}$, the receiver can generate random variable $K_{r}$ distributed on $\{1,\cdots,L_{l_{n}}\}$, such that when $l_{n}$ is sufficiently large, $Pr\{K_{s}\not=K_{r}\}<\epsilon$ for any positive $\epsilon$.

Let $\mathcal{K}$ be the set in the proof of Theorem 1, on which the common randomness takes value. $\{{\cal C}^{\hat{W}}_{l}:l\in\{1,\cdots,L_{l_{n}}\}\}$ defines a $(V^{\prime},V)$-correlation assisted code

[TABLE]

with deterministic encoder $u_{{v^{\prime}}^{\nu(n)}}(k)\in\{g_{0}^{m},g_{1}^{m}\}^{\nu(n)}$ for ${\cal W}$ of length $\nu(n)$ such that when $l_{n}$ is sufficiently large, $L_{l_{n}}$ messages can be sent, and the receiver can generate a random variable $K_{r}$ on $\{1,\cdots,L_{l_{n}}\}$, with $Pr\{K_{s}\not=K_{r}\}<\epsilon$ for any positive $\epsilon$.

We choose

[TABLE]

Now we can contract an $(n-l_{n},l_{n},J_{n})$ two-part code as in the proof for Theorem 1, where the second part is a common randomness assisted code sending the random output of a variable uniformly distributed on $\{1,\cdots J_{n}\}$, and the first part is used both to send $L_{l_{n}}$ messages (which are used as common randomness for the second part), and to generate random variables $K_{s}$ and $K_{r}$, both uniformly distributed on $\{1,\cdots,L_{l_{n}}\}$, with $Pr\{K_{s}\not=K_{r}\}<\epsilon$ for any positive $\epsilon$. $\Box$

Corollary 4

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrary correlated source, and ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC.

1. B.1

When

[TABLE]

holds, then

[TABLE] 2. B.2

When

[TABLE]

holds, then

[TABLE]

Proof: Our proof is based on the approach in Ahl/Cs for correlation assisted common randomness of classical channels.

At first we assume that $I(V^{\prime},V)\leq\max_{P}\min_{\bar{\bar{\rho}}(\cdot)\in\bar{\bar{W}}}\chi(P,\bar{\bar{\rho}}(\cdot))$ holds.

We fix a $P$ and define

[TABLE]

for an arbitrary positive $\delta$. Our idea is to have the transmitters share a random variable uniformly distributed on $\{1,\cdots J_{n}^{\prime}\}$ by means of ${\cal W}$.

Let $\mathcal{K}$ be the set in the proof of Theorem 1 on which the common randomness takes value. We choose a $|{\cal V}|$ dimensional Hilbert Space ${\cal H}^{|{\cal V}|}$ and a set of pairwise orthogonal pure quantum states $\{\sigma_{v}:v\in V^{\prime}\}\in\mathcal{S}({\cal H}^{|{\cal V}|})$. We further define a map $g:{\cal V}\rightarrow\{\sigma_{v}:v\}$ by $g(v)=\sigma_{v}$. We have $\chi(V^{\prime},g(\cdot))=I(V^{\prime},V)$. We now generate

[TABLE]

random variable

[TABLE]

u in ${\cal X}$. Since

[TABLE]

by Bo/Ca/Ca when $n$ is sufficiently large, then with a positive probability according to the joint distribution of $V^{\prime}$, $V$, and the uniform distribution on $\mathcal{K}$, there is a map $g:{\cal V}\rightarrow\{\sigma_{v}:v\}$, a realization $\{u_{k,v^{\prime}}(j):j,k,v^{\prime}\}$ of $\{U_{k,v^{\prime}}(j):j,k,v^{\prime}\}$, and a family of decoding sets

[TABLE]

such that for any positive $\epsilon$ and sufficiently large $n$

[TABLE]

according to the joint distribution of $V^{\prime}$, $V$, and the uniform distribution on $\mathcal{K}$ with an informed jammer.

Now we can contract an $(n,J_{n}^{\prime})$ two-part code. By Corollary 3 we can define a deterministic code of negligible length such that $\left|\mathcal{K}\right|$ messages can be sent. The first part of the codeword sends $\left|\mathcal{K}\right|$ messages as common randomness for the second part. The second part is the randomness assisted code defined in (27), sending the random output of a variable distributed on $\{1,\cdots J_{n}^{\prime}\}$. This shows the direct part for this case.

For the converse, we suppose that after the message transmission the sender and the receiver share a random variable $M$ which is independent of $V^{\prime}$ and $V$. We now consider the Markov chain $M\rightarrow PV^{\prime}\rightarrow\rho(\cdot)V\rightarrow M$. By the data processing inequality and the capacity formula for $\rho$, with an informed jammer in Bo/Ca/Ca , we have

[TABLE]

Now we assume that $I(V^{\prime},V)>\max_{P}\min_{\bar{\bar{\rho}}(\cdot)\in\bar{\bar{W}}}\chi(P,\bar{\bar{\rho}}(\cdot))$ holds.

Similar to above, we fix a $P$ and define

[TABLE]

for an arbitrary positive $\delta$ and an $U\rightarrow V^{\prime}\rightarrow V$ such that $I(U;V^{\prime})-I(U;V)\leq\chi(P,\bar{\bar{\rho}}(\cdot))$.

We now generate

[TABLE]

random variable

[TABLE]

Similar to above, we want to have the transmitters share a random variable uniformly distributed on $\{1,\cdots J_{n}^{\prime\prime}\}$ by means of ${\cal W}$.

When $U\rightarrow V^{\prime}\rightarrow V$ holds, then it also holds that $U\times P\rightarrow V^{\prime}\times P\rightarrow g(V)\otimes\rho(P)$. Since

[TABLE]

when $n$ is sufficiently large, then with a positive probability according to the joint distribution of $V^{\prime}$, $V$, and the uniform distribution on $\mathcal{K}$, there is a map $g:{\cal V}\rightarrow\{\sigma_{v}:v\}$, a realization $\{u_{k,v^{\prime}}(j):j,k,v^{\prime}\}$ of $\{U_{k,v^{\prime}}(j):j,k,v^{\prime}\}$, and a family of decoding sets

[TABLE]

such that for any positive $\epsilon$ and sufficiently large $n$

[TABLE]

according to the joint distribution of $V^{\prime}$, $V$, and the uniform distribution on $\mathcal{K}$.

Similar to above, we can contract an $(n,J_{n}^{\prime\prime})$ two-part code, where the first part is used to send $\left|\mathcal{K}\right|$ messages and the second part is a randomness assisted code sending the random output of a variable distributed on $\{1,\cdots J_{n}^{\prime\prime}\}$.

For the converse, we consider a $U$ with $U\rightarrow V^{\prime}\rightarrow V$. Let $Y$ be the classical random outcome of the decoding measurement. By the data processing inequality and the capacity formula for $\rho$ with an informed jammer in Bo/Ca/Ca , we have

[TABLE]

We now apply the results for common randomness capacity via classical channel in Ahl/Cs on the resulting classical arbitrarily varying channel with an informed jammer $X\rightarrow Y$. We have

[TABLE]

$\Box$

IV.2 Computability

As an application, we now consider whether the capacity formulas in Section III and Section IV.1 can be calculated algorithmically in finite time, or whether they are computable.

IV.2.1 Capacity Definition and Communication Scenario

Definition 14

A sequence of rational numbers $\{r_{n}:n\in\mathbb{N}\}$ is called a computable sequence if there exist recursive functions $a$, $b$, and $s$ $:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $n\in\mathbb{N}$ we have $b(n)\not=0$ and

[TABLE]

Definition 15

A function $f$ : $\mathbb{R}_{c}\rightarrow\mathbb{R}_{c}$ is called Banach-Mazur computable if it maps any computable sequence $\{r_{n}:n\in\mathbb{N}\}$ of real numbers into a computable sequence $\{f(r_{n}):n\in\mathbb{N}\}$ of real numbers. Here $\mathbb{R}_{c}$, the set of computable numbers, is defined as the set of real numbers that are computable by Turing machines.

Definition 16

A rapidly converging Cauchy representation of a computable real $x$ is a sequence of real numbers $\{x_{n}:n\in\mathbb{N}\}$ that converges to $x$ when $n\rightarrow\infty$ rapidly, i.e. for every $i$ and $j\geq i$ it holds $|x_{j}-x_{i}|<2^{-i}$.

Definition 17

A function $f$ : $\mathbb{R}_{c}\rightarrow\mathbb{R}_{c}$ is called Borel computable if there is an algorithm that transforms each given rapidly converging Cauchy representation of a computable real $x$ into a corresponding representation for $f(x)$.

Notice that Borel computability implies Banach-Mazur computability.

Definition 18

An AVCQC ${\cal W}$ with input alphabet ${\cal X}$ and output space ${\cal H}$ is computable if it maps every letter in ${\cal X}$ quantum state in $\mathcal{S}({\cal H})$ with computable coefficient in $\mathbb{C}$.

Definition 19

Assume we have two random variables $(V^{\prime},V)$ and $(\dot{V}^{\prime},\dot{V})$ both taking values in a finite set ${\cal V}^{\prime}\times{\cal V}$ with joint distributions $P_{V^{\prime},V}$ and $P_{\dot{V}^{\prime},\dot{V}}$, respectively. We define their distance by

[TABLE]

Assume we have two AVCQCs, ${\cal W}$ $=\{\{\rho(x,s),x\in{\cal X}\},s\in{\cal S}\}$ and $\dot{{\cal W}}$ $=\{\{\dot{\rho}(x,s),x\in{\cal X}\},s\in{\cal S}\}$, with correlated sources $(V^{\prime},V)$ and $(\dot{V}^{\prime},\dot{V})$ with alphabets $({\cal V}^{\prime},{\cal V})$, respectively. We define a distance of \Bigl{(}{\cal W},(V^{\prime},V)\Bigr{)} and \Bigl{(}\dot{{\cal W}},(\dot{V}^{\prime},\dot{V})\Bigr{)} by

[TABLE]

Here

[TABLE]

IV.2.2 Capacity Results

As an application of our capacity results, in this section we want to analyze whether the capacity formulas determined in Section III and in Section IV.1 are computable functions of the channel parameters on a Turing machine or not.

Our computability analysis is novel because, according to our knowledge, most of the capacity results, e.g. correlation assisted common randomness generation capacity with an informed jammer, do not yet exist for classical channels. Furthermore, the deterministic capacity formula using maximal error criterion of arbitrarily varying channels with an informed jammer is still unknown, since it contains Shannon’s zero-error capacity, which is still an open problem as a special case (cf. Bo/Ca/Ca ). Corollary 6 shows that even if we do not know the capacity formula, we can predict that no computable formula can ever be found (cf. Remark 3).

We would also like to point out that, since classical arbitrarily varying channels can be regard as special cases of AVCQCs, the computability of a classical-quantum capacity formula implies the computability of its corresponding classical capacity formula, while the non-computability of a classical capacity formula implies the non-computability of its corresponding classical-quantum capacity formula. Thus, for any classical-quantum capacity formula, which is proved to be computable in this section, it is clear that the corresponding classical capacity formula is automatically computable also. On the other hand, to show that a classical-quantum capacity formula is not computable, it is sufficient to show that the corresponding classical capacity formula is not computable.

Corollary 5

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrarily correlated source, and ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC. When $I(V^{\prime},V)>0$ holds, then $C({\cal W};corr(V^{\prime},V))$ is Turing computable.

Proof: By Theorem 1, when $I(V^{\prime},V)>0$ holds, then we have

[TABLE]

$\chi(P,\bar{\bar{\rho}}(\cdot))$ is a Turing computable continuous function, and the minimum of a computable continuous function on a computable set is also a computable continuous function. Thus $C({\cal W};corr(V^{\prime},V))$ is Turing computable. $\Box$

Corollary 6

Let $(V^{\prime},V)$ with alphabets $({\cal V}^{\prime},{\cal V})$ be an arbitrarily correlated source, and ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ be an AVCQC. When $I(V^{\prime},V)=0$ holds, then $C({\cal W};corr(V^{\prime},V))$ using maximal error criterion on the message set is, in general, not Turing computable.

Proof: We show Corollary 6 by a counterexample for a classical AVC, since classical arbitrarily varying channels can be regarded as special cases of AVCQCs.

Let ${\cal X}^{\prime}={\cal Y}^{\prime}=\{a,0,1,2\}$ and ${\cal S}=\{s_{0},s_{1}\}$. We define a classical arbitrarily varying channel ${\cal W}^{\prime}$ such that $W^{\prime}(a|a,s_{0})=W^{\prime}(a|a,s_{1})=1$, $W^{\prime}(y|x;s_{i})=1$ if $y=x+i~{}(mod~{}3)$ for $x,y\in\{0,1,2\}$. That is, the transmission matrices in ${\cal W}^{\prime}$ are

[TABLE]

In our previous work Bo/Ca/Ca , we have shown that the randomness assisted capacity of ${\cal W}^{\prime}$ with an informed jammer using maximal error criterion is equal to $\log\frac{5}{2}$. By Theorem 1, when we have a random variable pair $B$ and $B^{\prime}$ with $I(B,B^{\prime})>0$, then $C({\cal W}^{\prime};corr(B,B^{\prime}))$ using maximal error criterion is also equal to $\log\frac{5}{2}$. Furthermore, in Bo/Ca/Ca we have shown that the deterministic capacity (i.e. with no resource) of ${\cal W}^{\prime}$ with an informed jammer using maximum error probability is equal to $2$.

Let $r\in[0,1]$. Let ${\cal X}^{\prime\prime}={\cal Y}^{\prime\prime}=\{0,1\}$. We define a classical channel ${\cal W}_{r}^{\prime\prime}$ such that $W_{r}^{\prime\prime}(0|0)=W_{r}^{\prime\prime}(1|1)=1$ and $W_{r}^{\prime\prime}(0|1)=W_{r}^{\prime\prime}(1|0)=1-r$. It is easy to see that the capacity of ${\cal W}_{r}^{\prime\prime}$ is zero if and only if $r=0$.

We now define ${\cal X}={\cal Y}:={\cal X}^{\prime}\times{\cal X}^{\prime\prime}$, and ${\cal S}=\{s_{0},s_{1}\}$. We define a classical arbitrarily varying channel ${\cal W}_{r}:={\cal W}_{r}^{\prime\prime}\otimes{\cal W}^{\prime}$, that is $W_{r}((y,y^{\prime})|(x,x^{\prime});s_{i})=W_{r}^{\prime\prime}(y|x)\cdot W^{\prime}(y^{\prime}|x^{\prime};s_{i})$.

When $r>0$ holds, the capacity of ${\cal W}_{r}^{\prime\prime}$ is positive, thus there is a distribution $B^{\prime}$ on ${\cal X}^{\prime\prime}$ such that $I(B^{\prime},{\cal W}_{r}^{\prime\prime}(B^{\prime}))>0$. It follows directly that there is a distribution $B$ on ${\cal X}$ such that $I(B,W_{r}(B))>0$. Similar to the proof of Theorem 1, we may use a negligible amount of bits to create common randomness. Notice that since we only demand maximal error criterion on the message set, for the common randomness we may use the average error criterion. This means that for any random variable pair $V^{\prime}$ and $V$, even when $I(V^{\prime},V)=0$ holds, $C({\cal W}_{r};corr(V^{\prime},V))$ using maximal error criterion is always larger or equal to $\log\frac{5}{2}$.

However, when $r=0$, the capacity of $\cal V$ is zero. In this case we have $C({\cal W}_{0};corr(V^{\prime},V))$ $=C({\cal W}^{\prime};corr(V^{\prime},V))$. When $I(V^{\prime},V)=0$ holds, we have no resource, then $C({\cal W}_{0};corr(V^{\prime},V))$ using maximal error criterion is equal to $2$.

Thus $C({\cal W};corr(V^{\prime},V))$ using maximal error criterion is not a continuous function of the channel parameters when $I(V^{\prime},V)=0$. Since a Banach-Mazur computable function is continuous on the computable sets, $C({\cal W};corr(V^{\prime},V))$ using maximal error criterion is not Banach-Mazur computable and thus is not Turing computable (cf. Bo/Sch/Ba/Po ). $\Box$

Remark 2

Usually the capacity formulas of arbitrarily varying channels are discontinuous in zero, i.e. on the discontinuity points, positive capacity sinks rapidly to zero. Corollary 6 is a rare example where the discontinuity point is positive. Here, when $r\rightarrow 0$, the capacity has a fluctuation, but remains positive.

Remark 3

Notice that the capacity formula when $I(V^{\prime},V)=0$, i.e. the deterministic capacity formula using maximal error criterion of arbitrarily varying channels with an informed jammer is still an open problem, even for classical channels, since it contains the zero-error capacity of related discrete memoryless channels as a special case (cf. Bo/Ca/Ca ). Corollary 6 shows that we can make a statement about the computability, even when we do not have a formula for this quantity.

Theorem 2

If the condition B.1 of Corollary 4 is satisfied, then $\tilde{C}({\cal W})$ is Banach-Mazur computable and Turing computable.

Proof: We define

[TABLE]

If B.1 is satisfied, then by Corollary 4 we have

[TABLE]

The two expressions $\chi(P,\bar{\bar{\rho}}(\cdot))$ and $I(V^{\prime},V)$ are both Turing computable continuous functions. Since the minimum of a computable continuous function on a computable set is also a computable continuous function, $\max_{P}\min_{\bar{\bar{\rho}}(\cdot)\in\bar{\bar{W}}}\chi(P,\bar{\bar{\rho}}(\cdot))$ is Turing computable. $\Phi({\cal W},P_{(V^{\prime},V)})$ is the sum of two computable functions, and thus also a computable function. $\Box$

Theorem 3

$\tilde{C}({\cal W},(l_{n})_{n\in\mathbb{N}})$* is in general not Banach-Mazur computable and thus not Turing computable.*

Proof:

Let ${\cal V}^{\prime}={\cal V}=\{1,2\}$. For $n\in\mathbb{N}$ we consider a correlated source $(V_{n}^{\prime},V_{n})$ with alphabets $({\cal V}^{\prime},{\cal V})$ and joint distributions $P_{V_{n}^{\prime},V_{n}}$

[TABLE]

We have $I(V_{n}^{\prime},V_{n})>0$ for all $n>1$. Let $(V^{\prime},V)$ be the correlated source on $({\cal V}^{\prime},{\cal V})$ with joint distributions $P_{V^{\prime},V}$

[TABLE]

We have

[TABLE]

We consider an AVCQC ${\cal W}=\{\rho(x,s):x\in{\cal X},s\in{\cal S}\}$ such that for all $s\in{\cal S}$ and all $x\in{\cal X}$ we have

[TABLE]

for a fixed quantum state $\delta\in\mathcal{S}({\cal H})$. It holds

[TABLE]

The correlation assisted message transmission capacity is always zero, even when the jammer has no side information about the input. Furthermore, any classical channels which arise from ${\cal W}$ also have zero capacity. Thus the $(V_{n}^{\prime},V_{n})$-correlation assisted common randomness capacity of ${\cal W}$ with an informed jammer is equal to the $(V_{n}^{\prime},V_{n})$-correlation assisted common randomness capacity of a useless classical arbitrary varying channel without jamming attack. Here, useless classical channel means a classical channel with zero message transmission capacity. By Wit , the $(V_{n}^{\prime},V_{n})$-correlation assisted common randomness capacity of any useless classical arbitrary varying channel without jamming attack is equal to zero, thus $\tilde{C}({\cal W},(V_{n}^{\prime},V_{n}))=0$ for $n>1$. Furthermore, it is not hard to see that $\tilde{C}({\cal W},(V^{\prime},V))=1$. Thus $\tilde{C}({\cal W},(V^{\prime},V))$ and $\tilde{C}({\cal W},(V_{n}^{\prime},V_{n}))$ for all $n>1$ lie in the set of computable numbers. When we let $n$ tend to infinity, the correlation assisted common randomness capacity is discontinuous on $({\cal W},(V^{\prime},V))$. Since a Banach-Mazur computable function is continuous on the computable sets, the correlation assisted common randomness capacity is not Banach-Mazur computable and thus not Turing computable (cf. Bo/Sch/Ba/Po ).

$\Box$

Corollary 7

The $((V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$-correlation assisted common randomness generation capacity of an AVCQC ${\cal W}$ with an uninformed jammer, i.e. when the jammer has no information about the input code word, is in general, not Turing computable.

Proof: In the no-computable example which we give in the proof of Corollary 3, the correlation assisted message transmission capacity is always zero, even when the jammer has no side information about the input. Thus the $(V_{n}^{\prime},V_{n})$-correlation assisted common randomness capacity of ${\cal W}$ with an uninformed jammer is equal to the $(V_{n}^{\prime},V_{n})$-correlation assisted common randomness generation capacity of an arbitrary useless classical channel without jamming attack. By the proof of Corollary 3, the $(V_{n}^{\prime},V_{n})$-correlation assisted common randomness generation capacity is not Turing computable even when the jammer has no side information about the input.

$\Box$

Corollary 8

The $((V^{\prime},V),(l_{n})_{n\in\mathbb{N}})$-correlation assisted common randomness generation capacity of a classical-quantum channel $\rho$ with no jammer is, in general, not Turing computable.

Proof: In the no-computable example which we give in the proof of Corollary 3, we consider that there is no jamming attack. Thus $(V_{n}^{\prime},V_{n})$-correlation assisted randomness generation over classical-quantum channels with no jammer contains this example as a special case.

$\Box$

Acknowledgment

The work of H. Boche and N. Cai was supported by the Gottfried Wilhelm Leibniz program of the German Research Foundation (DFG) via Grant BO 1734/20-1 and partly supported by the German Research Foundation (DFG) under Germany s Excellence Strategy EXC-2111-390814868. The work of M. Cai was supported by the Bundesministerium für Bildung und Forschung (BMBF) via Grant 16KIS0118K and partly supported by the national research initiative on quantum technologies of Bundesministerium für Bildung und Forschung (BMBF), within the project QuaDiQua 16KIS0948. The research direction on jamming and active attacks on communication systems was initiated by the German Research Foundation (DFG) under Grants BO 1734/24-3 and BO 1734/25-1.

References