An important part of the information theory folklore had been about the output statistics of codes that achieve the capacity and how the empirical distributions compare to the output distributions induced by the optimal input in the channel capacity problem. Results for a variety of such empirical output distributions of good codes have been known in the literature, such as the comparison of the output distribution of the code to the optimal output distribution in vanishing and non-vanishing error probability cases. Motivated by these, we aim to achieve similar results for the quantum codes that are used for classical communication, that is the setting in which the classical messages are communicated through quantum codewords that pass through a noisy quantum channel. We first show the uniqueness of the optimal output distribution, to be able to talk more concretely about the optimal output distribution. Then, we extend the vanishing error probability results to the quantum case, by using techniques that are close in spirit to the classical case. We also extend non-vanishing error probability results to the quantum case on block codes, by using the second-order converses for such codes based on hypercontractivity results for the quantum generalized depolarizing semi-groups.

This work investigates the use of quantum resources in distributed storage systems. Consider an $(n,k,d)$ distributed storage system in which a file is stored across $n$ nodes such that any $k$ nodes suffice to reconstruct the file. When a node fails, any $d$ helper nodes transmit information to a newcomer to rebuild the system. In contrast to the classical repair, where helper nodes transmit classical bits, we allow them to send classical information over quantum channels to the newcomer. The newcomer then generates its storage by performing appropriate measurements on the received quantum states. In this setting, we fully characterize the fundamental tradeoff between storage and repair bandwidth (total communication cost). Compared to classical systems, the optimal storage--bandwidth tradeoff can be significantly improved with the enhancement of quantum entanglement shared only among the surviving nodes, particularly at the minimum-storage regenerating point. Remarkably, we show that when $d \geq 2k-2$, there exists an operating point at which \textit{both storage and repair bandwidth are simultaneously minimized}. This phenomenon breaks the tradeoff in the classical setting and reveals a fundamentally new regime enabled by quantum communication.

We consider the storage problem in an asymmetric $X$-secure private information retrieval (A-XPIR) setting. The A-XPIR setting considers the $X$-secure PIR problem (XPIR) when a given arbitrary set of servers is communicating. We focus on the trade-off region between the average storage at the servers and the average download cost. In the case of $N=4$ servers and two non-overlapping sets of communicating servers with $K=2$ messages, we characterize the achievable region and show that the three main inequalities compared to the no-security case collapse to two inequalities in the asymmetric security case. In the general case, we derive bounds that need to be satisfied for the general achievable region for an arbitrary number of servers and messages. In addition, we provide the storage and retrieval scheme for the case of $N=4$ servers with $K=2$ messages and two non-overlapping sets of communicating servers, such that the messages are not replicated (in the sense of a coded version of each symbol) and at the same time achieve the optimal achievable rate for the case of replication. Finally, we derive the exact capacity for the case of asymmetric security and asymmetric collusion for $N=4$ servers, with the communication links $\{1,2\}$ and $\{3,4\}$, which splits the servers into two groups, i.e., $g=2$, and with the collusion links $\{1,3\}$, $\{2,4\}$, as $C=\frac{1}{3}$. More generally, we derive a capacity result for a certain family of asymmetric collusion and asymmetric security cases.

We consider the quantumsymmetricprivate information retrieval (QSPIR) problem in a system withNdatabases andKmessages, withUunresponsive servers,T-colluding servers, andX-security parameter, under several fundamental threat models. In the first model, there areE1eavesdropped links in the uplink direction (the direction from the user to theNservers),E2eavesdropped links in the downlink direction (the direction from the servers to the user), where |E1|, |E2| ≤E; we coin this eavesdropper setting asdynamiceavesdroppers. We show that super-dense coding gain can be achieved for some regimes. In the second model, we consider the case with Byzantine servers, i.e., servers that can coordinate to devise a plan to harm the privacy and security of the system together with static eavesdroppers, by listening to the same links in both uplink and downlink directions. It is important to note the considerable difference between the two threat models, since the eavesdroppers can take huge advantage of the presence of the Byzantine servers. Unlike the previous works in SPIR with Byzantine servers, that assume that the Byzantine servers can send only random symbols independent of the stored messages, we follow the definition of Byzantine servers in [1], where the Byzantine servers can send symbols that can be functions of the storage, queries, as well as the random symbols in a way that can produce worse harm to the system. In the third and the most novel threat model, we consider the presence of Byzantine servers and dynamic eavesdroppers together. We show that having dynamic eavesdroppers along with Byzantine servers in the same system model creates more threats to the system than having static eavesdroppers with Byzantine servers. This is the first work that considers the quantum version of unresponsive and eavesdropped threat model. In addition, this is the first work, classical or quantum, that considers the presence of Byzantine servers together with eavesdroppers in the same system model (static or dynamic eavesdroppers). Another layer of difficulty that we handle in this work stems from the symmetric privacy requirement in the presence of Byzantine servers, which by itself has never been studied before in classical or quantum variations; here the Byzantine servers may attempt to leak information about undesired messages to the user, which is not allowed in SPIR.

In a classification task, counterfactual explanations provide the minimum change needed for an input to be classified into a favorable class. We consider the problem of privately retrieving the exact closest counterfactual from a database of accepted samples while enforcing that certain features of the input sample cannot be changed, i.e., they are immutable. An applicant (user) whose feature vector is rejected by a machine learning model wants to retrieve the sample closest to them in the database without altering a private subset of their features, which constitutes the immutable set. While doing this, the user should keep their feature vector, immutable set and the resulting counterfactual index information-theoretically private from the institution. We refer to this as immutable private counterfactual retrieval (I-PCR) problem which generalizes PCR to a more practical setting. In this paper, we propose two I-PCR schemes by leveraging techniques from private information retrieval (PIR) and characterize their communication costs. Further, we quantify the information that the user learns about the database and compare it for the proposed schemes.

We consider the problem of finding the asymptotic capacity of symmetric private information retrieval (SPIR) with$B$Byzantine servers. Prior to finding the capacity, a definition for the Byzantine servers is needed since in the literature there are two different definitions. In [1], where it was first defined, the Byzantine servers can send any symbol from the storage, their received queries and some independent random symbols. In [2], Byzantine servers send any random symbol independently of their storage and queries. It is clear that these definitions are not identical, especially when symmetric privacy is required. To that end, we define Byzantine servers, inspired by [1], as the servers that can share everything, before and after the scheme initiation. In this setting, we find an upper bound, for an infinite number of messages case, that should be satisfied for all schemes that protect against this setting and develop a scheme that achieves this upper bound. Hence, we identify the capacity of the problem.

We study a linear computation problem over a quantum multiple access channel (LC-QMAC), where S servers share an entangled state and separately store classical data streams W1,⋯,WSover a finite field ${\mathbb{F}_d}$. A user aims to compute K linear combinations of these data streams, represented as $Y = {{\mathbf{V}}_1}{W_1} + {{\mathbf{V}}_2}{W_2} + \cdot s + {{\mathbf{V}}_S}{W_S} \in \mathbb{F}_d^{K \times 1}$. To this end, each server encodes its classical information into its local quantum subsystem and transmits it to the user, who retrieves the desired computations via quantum measurements. In this work, we propose an achievable scheme for LC-QMAC based on the stabilizer formalism and the ideas from entanglement-assisted quantum error–correcting codes (EAQECC). Specifically, given any linear computation matrix, we construct a self-orthogonal matrix that can be implemented using the stabilizer formalism. Also, we apply precoding matrices to minimize the number of auxiliary qudits required. Our scheme achieves more computations per qudit, i.e., a higher computation rate, compared to the best-known methods in the literature, and attains the capacity in certain cases.

We consider both classical and quantum variations ofX-secure,E-eavesdropped andT-colluding symmetric private information retrieval (SPIR). This is the first work to study SPIR withX-security in classical or quantum variations. We first develop a scheme for classicalX-secure,E-eavesdropped andT-colluding SPIR (XSETSPIR) based on a modified version of cross subspace alignment (CSA), which achieves a rate of$R= 1 - \frac {X+\max (T,E)}{N}$. The modified scheme achieves the same rate as the scheme used forX-secure PIR with the extra benefit of symmetric privacy, i.e., user-privacy as well as database-privacy. Next, we extend this scheme to its quantum counterpart based on theN-sum box abstraction. This is the first work to consider the presence of eavesdroppers in quantum private information retrieval (QPIR). In the quantum variation, the eavesdroppers have better access to information over the quantum channel compared to the classical channel due to the over-the-air decodability. To that end, we develop two different schemes for quantumX-secure,E-eavesdropped andT-colluding SPIR (QXSETSPIR) with secure over-the-air decoding. The first scheme achieves the highest possible super-dense coding gain, i.e.,$R_{Q} = \min \left \{{{ 1, 2\left ({{1-\frac {X+\max (T,E)}{N}}}\right)}}\right \}$, which requires additional uploads from the user. The second scheme on the other hand requires no extra uploads. However, it does not achieve the super-dense coding gain in some cases based on the relation between the number of eavesdropped links and the number of interference terms. The second scheme is based on the idea that there exist some special entanglement states that can be used to hide the contents of the user-required messages from the eavesdroppers using the interference symbols.

We consider the problem of private set membership aggregation of$N$parties by using an entangled quantum state. In this setting, the$N$parties, which share an entangled state, aim to privately know the number of times each element (message) is repeated among the$N$parties, with respect to a universal set$\mathcal{K}$. This problem has applications in private comparison, ranking, voting, etc. We propose an encoding algorithm that maps the classical information into distinguishable quantum states, along with a decoding algorithm that exploits the distinguishability of the mapped states. The proposed scheme can also be used to calculate the$N$party private summation modulo$P$.

We consider the problems arising from the presence of Byzantine servers in a quantum private information retrieval (QPIR) setting. This is the first work to precisely define what the capabilities of Byzantine servers could be in a QPIR context. We show that quantum Byzantine servers have more capabilities than their classical counterparts due to the possibilities created by quantum encoding procedures. We focus on quantum Byzantine servers that can apply any reversible operation on their individual qudits. In this case, Byzantine servers can generate any error, i.e., this covers all possible single qudit operations that can be applied by Byzantine servers on their qudits. We design a scheme based on cross-subspace alignment (CSA) and we show that this scheme achieves superdense coding gain in some cases.

Scheduling in 5G networks is a challenging task due to the heterogeneous Quality of Service (QoS) requirements of traffic sources. In this paper, we consider the problem of uplink scheduling in 5G networks for mixed traffic that includes Ultra-Reliable Low Latency Communications (URLLC) devices and enhanced Mobile Broad-Band (eMBB) users. For this purpose, a mathematical model for Grant Free (GF) services is derived for the k-repetitions Hybrid Automatic Repeat reQuest (HARQ). We formulate the scheduling problem as a mixed-integer non-linear programming optimization problem. We introduce a complete system model that includes grant-free and grant-based subsystems. We then introduce our proposed solution to the scheduling problem that addresses the two traffic types. Different scheduling techniques are then compared and a performance upper bound is added as a reference. The results show that the proposed technique provides near-optimal results and outperforms other scheduling techniques with a significant complexity reduction.