On the group theoretic structure of a class of quantum dialogue protocols

Intrinsic symmetry of the existing protocols of quantum dialogue are explored. It is shown that if we have a set of mutually orthogonal $n$-qubit states {\normalsize $\{|\phi_{0}>,|\phi_{1}>,....,|\phi_{i}\}$ and a set of $m-qubit$ ($m\leq n$) unitary operators $\{U_{0},U_{2},...,U_{2^{n}-1}\}:U_{i}|\phi_{0}>=|\phi_{i}>$ and $\{U_{0},U_{2},...,U_{2^{n}-1}\}$ forms a group under multiplication then it would be sufficient to construct a quantum dialogue protocol using this set of quantum states and this group of unitary operators}. The sufficiency condition is used to provide a generalized protocol of quantum dialogue. Further the basic concepts of group theory and quantum mechanics are used here to systematically generate several examples of possible groups of unitary operators that may be used for implementation of quantum dialogue. A large number of examples of quantum states that may be used to implement the generalized quantum dialogue protocol using these groups of unitary operators are also obtained. For example, it is shown that GHZ state, GHZ-like state, W state, 4 and 5 qubit Cluster states, Omega state, Brown state, $Q_{4}$ state and $Q_{5}$ state can be used for implementation of quantum dialogue protocol. The security and efficiency of the proposed protocol is appropriately analyzed. It is also shown that if a group of unitary operators and a set of mutually orthogonal states are found to be suitable for quantum dialogue then they can be used to provide solutions of socialist millionaire problem.Comment: 15 page

On the Communication Complexity of Secure Computation

Information theoretically secure multi-party computation (MPC) is a central primitive of modern cryptography. However, relatively little is known about the communication complexity of this primitive. In this work, we develop powerful information theoretic tools to prove lower bounds on the communication complexity of MPC. We restrict ourselves to a 3-party setting in order to bring out the power of these tools without introducing too many complications. Our techniques include the use of a data processing inequality for residual information - i.e., the gap between mutual information and G\'acs-K\"orner common information, a new information inequality for 3-party protocols, and the idea of distribution switching by which lower bounds computed under certain worst-case scenarios can be shown to apply for the general case. Using these techniques we obtain tight bounds on communication complexity by MPC protocols for various interesting functions. In particular, we show concrete functions that have "communication-ideal" protocols, which achieve the minimum communication simultaneously on all links in the network. Also, we obtain the first explicit example of a function that incurs a higher communication cost than the input length in the secure computation model of Feige, Kilian and Naor (1994), who had shown that such functions exist. We also show that our communication bounds imply tight lower bounds on the amount of randomness required by MPC protocols for many interesting functions.Comment: 37 page

New lattice-based protocols for proving correctness of a shuffle

In an electronic voting procedure, mixing networks are used to ensure anonymity of the casted votes. Each node of the network re-encrypts the input and randomly permutes it in a process named shuffle, and must prove that the process was applied honestly. State-of-the-art classical proofs achieve logarithmic communication complexity on N (the number of votes to be shuffled) but they are based on assumptions which are weak against quantum computers. To maintain security in a post-quantum scenario, new proofs are based on different mathematical assumptions, such as lattice-based problems. Nonetheless, the best lattice-based protocols to ensure verifiable shuffling have linear communication complexity on N. In this thesis we propose the first sub-linear post-quantum proof for the correctness of a shuffe, for which we have mainly used two ideas: arithmetic circuit satisfiability and Benes networks to model a permutation of N elements

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes

Limitations of semidefinite programs for separable states and entangled games

Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no $\omega(1)$-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the $2 \rightarrow 4$ norm of a matrix, and the set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state. In both cases no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Watrous and Aaronson et al. These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelson-type bounds to restrict quantum correlations, as well as the SDP hierarchies of Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published versio

An Operational Road towards Understanding Causal Indefiniteness within Post-Quantum Theories

A theory, whatever it does, must correlate data. We commit ourselves to operational methodology, as a means towards studying the space of Generalised Probability Theories that are compatible with Indefinite Causality. Such a space of theories that combines the radical aspects of Quantum Theory: its probabilistic nature and of Relativity: its dynamic causal structure, is expected to house Quantum Gravity. In this thesis, we ask how we may understand Indefinite Causality. In the first half, we explore the consequences of (Indefinite) Causality on Communication tasks. Motivated by the game with one way signalling that provided the Causal Inequality, we study competing two-way signalling and provide protocols for Bidirectional Teleportation and Bidirectional Dense-Coding. Further, we provide a theorem for when tensor products of processes are valid. This result has consequences for setting up of a theory of process communication. In the second half, we revisit the Causaloid Framework by Hardy, a framework that studies this space of theories and prescribes how to recover the correlations within a theory from operational data to calculate probabilities through three levels of physical compression -- Tomographic, Compositional and Meta. We present a diagrammatic representation for the Causaloid Framework and leverage it to study Meta-Compression through which we characterise a Hierarchy of theories. The rungs of this Hierarchy are differentiated by the nature of the Causal Structure of the theory. We apply the Causaloid Framework to the space of Generalised Probability Theories pertaining to circuits, through the Duotensor Framework and show that finite dimensional Quantum Theory as well as Classical Probability Theory belong to its second rung. To summarise, we work towards a better understanding of communication tasks when the underlying causal structure is indefinite, and characterise the space of Generalised Probability Theories with Indefinite Causality, through the nature of their causal structures

Scheduling of space to ground quantum key distribution

Satellite-based platforms are currently the only feasible way of achieving intercontinental range for quantum communication, enabling thus the future global quantum internet. Recent demonstrations by the Chinese spacecraft Micius have spurred an international space race and enormous interest in the development of both scientific and commercial systems. Research efforts so far have concentrated upon in-orbit demonstrations involving a single satellite and one or two ground stations. Ultimately satellite quantum key distribution should enable secure network communication between multiple nodes, which requires efficient scheduling of communication with the set of ground stations. Here we present a study of how satellite quantum key distribution can service many ground stations taking into account realistic constraints such as geography, operational hours, and most importantly, weather conditions. The objective is to maximise the number of keys a set of ground stations located in the United Kingdom could share while simultaneously reflecting the communication needs of each node and its relevance in the network. The problem is formulated as a mixed-integer linear optimisation program and solved to a desired optimality gap using a state of the art solver. The approach is presented using a simulation run throughout six years to investigate the total number of keys that can be sent to ground stations