arXiv subjects

Christian Deppe

Publications and source records attributed to Christian Deppe.

At least 73 records · Page 4Linked to original sources

arXiv2024

On Effective Convergence in Fekete's Lemma and Related Combinatorial Problems in Information Theory

Fekete's lemma is a well known result from combinatorial mathematics that shows the existence of a limit value related to super- and subadditive sequences of real numbers. In this paper, we analyze Fekete's lemma in view of the arithmetical hierarchy of real numbers by Zheng and Weihrauch and fit the results into an information-theoretic context. We introduce special sets associated to super- and subadditive sequences and prove their effective equivalence to $Σ_1$ and $Π_1$. Using methods from the theory established by Zheng and Weihrauch, we then show that the limit value emerging from Fekete's lemma is, in general, not a computable number. Given a sequence that additionally satisfies non-negativity, we characterize under which conditions the associated limit value can be computed effectively and investigate the corresponding modulus of convergence. Subsidiarily, we prove a theorem concerning the structural differences between computable sequences of computable numbers and computable sequences of rational numbers. We close the paper by a discussion on how our findings affect common problems from information theory.

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arXiv2024

An Achievable Rate-Distortion Region of Joint Identification and Sensing for Multiple Access Channels

In contrast to Shannon transmission codes, the size of identification (ID) codes for discrete memoryless channels (DMCs) experiences doubly exponential growth with the block length when randomized encoding is used. Additional enhancements within the ID paradigm can be realized through supplementary resources such as quantum entanglement, common randomness (CR), and feedback. Joint transmission and sensing demonstrate significant benefits over separation-based methods. Inspired by the significant impact of feedback on the ID capacity, our work delves into the realm of joint ID and sensing (JIDAS) for state-dependent multiple access channels (SD-MACs) with noiseless strictly casual feedback. Here, the senders aim to convey ID messages to the receiver while simultaneously sensing the channel states. We establish a lower bound on the capacity-distortion region of the SD-MACs. An example shows that JIDAS outperforms the separation-based approach.

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arXiv2024

Identification via Gaussian Multiple Access Channels in the Presence of Feedback

We investigate message identification over a K-sender Gaussian multiple access channel (K-GMAC). Unlike conventional Shannon transmission codes, the size of randomized identification (ID) codes experiences a doubly exponential growth in the code length. Improvements in the ID approach can be attained through additional resources such as quantum entanglement, common randomness (CR), and feedback. It has been demonstrated that an infinite capacity can be attained for a single-user Gaussian channel with noiseless feedback, irrespective of the chosen rate scaling. We establish the capacity region of both the K-sender Gaussian multiple access channel (K-GMAC) and the K-sender state-dependent Gaussian multiple access channel (K-SD-GMAC) when strictly causal noiseless feedback is available.

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arXiv2024

Quantum optimization of coherent chaotic systems: A case for buses of Kathmandu

In this paper, we propose a novel quantum computing approach to solve the real-world problem of optimizing transportation in bustling Kathmandu city. The transportation system in Kathmandu is chaotic, with no central authority controlling the transportation. We leverage this chaotic feature in our quantum optimization procedure. The quantum chaos theory's Wigner-Dyson distribution surfaced as the most effective bus spacing distribution for a bus driver to maximize their profit. We investigate the statistical properties of the buses with real-time GPS bus location data and optimize bus spacing and interval distribution around the 27 km circular ring road in Kathmandu. Using tools like quantum simulation, eigenvalue distributions, and output wave function analysis, we show that such optimal bus spacing distribution could be achieved.

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arXiv2024

Second 6G life Workshop on Post Shannon Theory

The one-day workshop, held prior to the "ZIF Workshop on Information Theory and Related Fields", provided an excellent opportunity for in-depth discussions on several topics within the field of post-Shannon theory. The agenda covered deterministic and randomized identification, focusing on various methods and algorithms for identifying data or signals deterministically and through randomized processes. It explored the theoretical foundations and practical applications of these techniques. The session on resources for increasing identification capacity examined the different resources and strategies that can be utilized to boost the capacity for identifying information. This included discussions on both hardware and software solutions, as well as innovative approaches to resource allocation and optimization. Participants delved into common randomness generation, essential for various cryptographic protocols and communication systems. The session highlighted recent advancements and practical implementations of common randomness in secure communications. The workshop concluded with a detailed look at the development and practical deployment of identification codes. Experts shared insights on code construction techniques, implementation challenges, and real-world applications in various communication systems. We extend our thanks to the esteemed speakers for their valuable contributions: Caspar von Lengerke, Wafa Labidi, Ilya Vorobyev, Johannes Rosenberger, Jonathan Huffmann, and Pau Colomer. Their presentations and insights significantly enriched the workshop. Additionally, we are grateful to all the participants whose active engagement, constructive comments, and stimulating discussions made the event a success. Your involvement was crucial in fostering a collaborative and intellectually vibrant environment.

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arXiv2024

Optimal Signal Processing for Common Randomness Generation over MIMO Gaussian Channels with Applications in Identification

Common randomness (CR), as a resource, is not commonly exploited in existing practical communication systems. In the CR generation framework, both the sender and receiver aim to generate a common random variable observable to both, ideally with low error probability. The availability of this CR allows us to implement correlated random protocols that can lead to faster and more efficient algorithms. Previous work focused on CR generation over perfect channels with limited capacity. In our work, we consider the problem of CR generation from independent and identically distributed (i.i.d.) samples of a correlated finite source with one-way communication over a Gaussian channel. We first derive the CR capacity for single-input single-output (SISO) Gaussian channels. This result is then used for the derivation of the CR capacity in the multiple-input multiple-output (MIMO) case. CR plays a key role in the identification scheme since it may allow a significant increase in the identification capacity of channels. In the identification framework, the decoder is interested in knowing \emph{whether} a specific message of special interest to him has been sent or not, rather than knowing \emph{what} the received message is. In many new applications, such as several machine-to-machine and human-to-machine systems and the tactile internet, this post-Shannon scheme is more efficient than classical transmission. In our work, we also consider a CR-assisted secure identification scheme and develop a lower bound on the corresponding secure identification capacity.

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arXiv2024

Secure Identification for Gaussian Channels and Identification for Multi-Antenna Gaussian Channels

New applications in modern communications are demanding robust and ultra-reliable low latency information exchange such as machine-to-machine and human-to-machine communications. For many of these applications, the identification approach of Ahlswede and Dueck is much more efficient than the classical message transmission scheme proposed by Shannon. Previous studies concentrate mainly on identification over discrete channels. For discrete channels, it was proved that identification is robust under channels uncertainty. Furthermore, optimal identification schemes that are secure and robust against jamming attacks have been considered. However, no results for continuous channels have yet been established. That is why we focus on the continuous case: the Gaussian channel for its known practical relevance. We deal with secure identification over Gaussian channels. Provable secure communication is of a high interest for future communication systems. A key technique for implementing secure communication is the physical layer security based on information theoretic security. We model this with the wiretap channel. In particular, we provide a suitable coding scheme for the Gaussian wiretap channel (GWC) and determine the corresponding secure identification capacity. We also consider Multiple-Input Multiple-Output (MIMO) Gaussian channels and provide an efficient signal-processing scheme. This scheme allows a separation of signal-processing and Gaussian coding.

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arXiv2024

Computability of the Zero-Error Capacity of Noisy Channels

Zero-error capacity plays an important role in a whole range of operational tasks, in addition to the fact that it is necessary for practical applications. Due to the importance of zero-error capacity, it is necessary to investigate its algorithmic computability, as there has been no known closed formula for the zero-error capacity until now. We show that the zero-error capacity of noisy channels is not Banach-Mazur computable and therefore not Borel-Turing computable. We also investigate the relationship between the zero-error capacity of discrete memoryless channels, the Shannon capacity of graphs, and Ahlswede's characterization of the zero-error-capacity of noisy channels with respect to the maximum error capacity of 0-1-arbitrarily varying channels. We will show that important questions regarding semi-decidability are equivalent for all three capacities. So far, the Borel-Turing computability of the Shannon capacity of graphs is completely open. This is why the coupling with semi-decidability is interesting.

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arXiv2024

Computability of the Channel Reliability Function and Related Bounds

The channel reliability function is an important tool that characterizes the reliable transmission of messages over communication channels. For many channels, only upper and lower bounds of the function are known. We analyze the computability of the reliability function and its related functions. We show that the reliability function is not a Turing computable performance function. The same also applies to the functions related to the sphere packing bound and the expurgation bound. Furthermore, we consider the $R_\infty$ function and the zero-error feedback capacity, since they play an important role in the context of the reliability function. Both the $R_\infty$ function and the zero-error feedback capacity are not Banach Mazur computable.

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arXiv2024

Common Randomness Generation from Sources with Infinite Polish Alphabet

We investigate the problem of common randomness (CR) generation in the basic two-party communication setting in which a sender and a receiver aim to agree on a common random variable with high probability. The terminals observe independent and identically distributed (i.i.d.) samples of sources with an arbitrary distribution defined on a Polish alphabet and are allowed to communicate as little as possible over a noisy, memoryless channel. We establish single-letter upper and lower bounds on the CR capacity for the specified model. The derived bounds hold with equality except for at most countably many points where discontinuity issues might arise.

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arXiv2024

Optimal depth and a novel approach to variational quantum process tomography

In this work, we present two new methods for Variational Quantum Circuit (VQC) Process Tomography onto $n$ qubits systems: PT_VQC and U-VQSVD. Compared to the state of the art, PT_VQC halves in each run the required amount of qubits for process tomography and decreases the required state initializations from $4^{n}$ to just $2^{n}$, all while ensuring high-fidelity reconstruction of the targeted unitary channel $U$. It is worth noting that, for a fixed reconstruction accuracy, PT_VQC achieves faster convergence per iteration step compared to Quantum Deep Neural Network (QDNN) and tensor network schemes. The novel U-VQSVD algorithm utilizes variational singular value decomposition to extract eigenvectors (up to a global phase) and their associated eigenvalues from an unknown unitary representing a general channel. We assess the performance of U-VQSVD by executing an attack on a non-unitary channel Quantum Physical Unclonable Function (QPUF). U-VQSVD outperforms an uninformed impersonation attack (using randomly generated input states) by a factor of 2 to 5, depending on the qubit dimension. For the two presented methods, we propose a new approach to calculate the complexity of the displayed VQC, based on what we denote as optimal depth.

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arXiv2024

Function Computation and Identification over Locally Homomorphic Multiple-Access Channels

We develop the notion of a locally homomorphic channel and prove an approximate equivalence between those and codes for computing functions. Further, we derive decomposition properties of locally homomorphic channels which we use to analyze and construct codes where two messages must be encoded independently. This leads to new results for identification and K-identification when all messages are sent over multiple-access channels, which yield surprising rate improvements compared to naive code constructions. In particular, we demonstrate that for the example of identification with deterministic encoders, both encoders can be constructed independently.

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arXiv2024

Krylov Complexity of Fermionic and Bosonic Gaussian States

The concept of \emph{complexity} has become pivotal in multiple disciplines, including quantum information, where it serves as an alternative metric for gauging the chaotic evolution of a quantum state. This paper focuses on \emph{Krylov complexity}, a specialized form of quantum complexity that offers an unambiguous and intrinsically meaningful assessment of the spread of a quantum state over all possible orthogonal bases. Our study is situated in the context of Gaussian quantum states, which are fundamental to both Bosonic and Fermionic systems and can be fully described by a covariance matrix. We show that while the covariance matrix is essential, it is insufficient alone for calculating Krylov complexity due to its lack of relative phase information. Our findings suggest that the relative covariance matrix can provide an upper bound for Krylov complexity for Gaussian quantum states. We also explore the implications of Krylov complexity for theories proposing complexity as a candidate for holographic duality by computing Krylov complexity for the thermofield double States (TFD) and Dirac field.

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arXiv2024

Identification Capacity of the Discrete-Time Poisson Channel

Numerous applications in the field of molecular communications (MC) such as healthcare systems are often event-driven. The conventional Shannon capacity may not be the appropriate metric for assessing performance in such cases. We propose the identification (ID) capacity as an alternative metric. Particularly, we consider randomized identification (RI) over the discrete-time Poisson channel (DTPC), which is typically used as a model for MC systems that utilize molecule-counting receivers. In the ID paradigm, the receiver's focus is not on decoding the message sent. However, he wants to determine whether a message of particular significance to him has been sent or not. In contrast to Shannon transmission codes, the size of ID codes for a Discrete Memoryless Channel (DMC) grows doubly exponentially fast with the blocklength, if randomized encoding is used. In this paper, we derive the capacity formula for RI over the DTPC subject to some peak and average power constraints. Furthermore, we analyze the case of state-dependent DTPC.

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arXiv2024

Common Randomness Generation from Finite Compound Sources

We investigate the problem of generating common randomness (CR) from finite compound sources aided by unidirectional communication over rate-limited perfect channels. The two communicating parties, often referred to as terminals, observe independent and identically distributed (i.i.d.) samples of a finite compound source and aim to agree on a common random variable with a high probability for every possible realization of the source state. Both parties know the set of source states as well as their statistics. However, they are unaware of the actual realization of the source state. We establish a single-letter lower and upper bound on the compound CR capacity for the specified model. Furthermore, we present two special scenarios where the established bounds coincide.

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arXiv2023

Information-Theoretically Secret Reed-Muller Identification with Affine Designs

We consider the problem of information-theoretic secrecy in identification schemes rather than transmission schemes. In identification, large identities are encoded into small challenges sent with the sole goal of allowing at the receiver reliable verification of whether the challenge could have been generated by a (possibly different) identity of his choice. One of the reasons to consider identification is that it trades decoding for an exponentially larger rate, however this may come with such encoding complexity and latency that it can render this advantage unusable. Identification still bears one unique advantage over transmission in that practical implementation of information-theoretic secrecy becomes possible, even considering that the information-theoretic secrecy definition needed in identification is that of semantic secrecy. Here, we implement a family of encryption schemes, recently shown to achieve semantic-secrecy capacity, and apply it to a recently-studied family of identification codes, confirming that, indeed, adding secrecy to identification comes at essentially no cost. While this is still within the one-way communication scenario, it is a necessary step into implementing semantic secrecy with two-way communication, where the information-theoretic assumptions are more realistic.

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arXiv2023

Reed-Muller Identification

Ahlswede and Dueck identification has the potential of exponentially reducing traffic or exponentially increasing rates in applications where a full decoding of the message is not necessary and, instead, a simple verification of the message of interest suffices. However, the proposed constructions can suffer from exponential increase in the computational load at the sender and receiver, rendering these advantages unusable. This has been shown in particular to be the case for a construction achieving identification capacity based on concatenated Reed-Solomon codes. Here, we consider the natural generalization of identification based on Reed-Muller codes and we show that, although without achieving identification capacity, they allow to achieve the exponentially large rates mentioned above without the computational penalty increasing too much the latency with respect to transmission.

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arXiv2023

Testing of Hybrid Quantum-Classical K-Means for Nonlinear Noise Mitigation

Nearest-neighbour clustering is a powerful set of heuristic algorithms that find natural application in the decoding of signals transmitted using the M-Quadrature Amplitude Modulation (M-QAM) protocol. Lloyd et al. proposed a quantum version of the algorithm that promised an exponential speed-up. We analyse the performance of this algorithm by simulating the use of a hybrid quantum-classical implementation of it upon 16-QAM and experimental 64-QAM data. We then benchmark the implementation against the classical k-means clustering algorithm. The choice of quantum encoding of the classical data plays a significant role in the performance, as it would for the hybrid quantum-classical implementation of any quantum machine learning algorithm. In this work, we use the popular angle embedding method for data embedding and the swap test for overlap estimation. The algorithm is emulated in software using Qiskit and tested on simulated and real-world experimental data. The discrepancy in accuracy from the perspective of the induced metric of the angle embedding method is discussed, and a thorough analysis regarding the angle embedding method in the context of distance estimation is provided. We detail an experimental optic fibre setup as well, from which we collect 64-QAM data. This is the dataset upon which the algorithms are benchmarked. Finally, some promising current and future directions for further research are discussed.

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