Efficient quantum key distribution using Fibonacci-number coding with a biased basis choice

In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

It is shown that period, rank and order of the ) , ( b a -Fibonacci sequence remain invariant by extension to the generalized ) , , , ( d c b a -Lucas sequence provided a specific determinant is relatively prime to the modulus.Given an odd prime modulus p and the 1  p exceptional ) , ( b a -Lucas sequences, for which p divides the discriminant of these sequences, we determine the corresponding periods and relate them to those of the ) , ( b a -Fibonacci sequence.The results can be viewed as extended versions of classical results by Wall in case ) 1 , 1 ( ) , (  b a .

The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematics, combining the modular approach used in the proof of Fermat’s Last Theorem with Baker’s Theory. By contrast, using the knowledge of the all perfect powers in the Fibonacci and Lucas sequences, the determination of the perfect powers among the numbers Fn ± 1 is quite elementary.

We present a simple numerical optimization procedure to search for highly entangled states of 2, 3, 4 and 5 qubits. We develop a computationally tractable entanglement measure based on the negative partial transpose criterion, which can be applied to quantum systems of an arbitrary number of qubits. The search algorithm attempts to optimize this entanglement cost function to find the maximal entanglement in a quantum system. We present highly entangled 4-qubit and 5-qubit states discovered by this search. We show that the 4-qubit state is not quite as entangled, according to two separate measures, as the conjectured maximally entangled Higuchi–Sudbery state. Using this measure, these states are more highly entangled than the 4-qubit and 5-qubit GHZ states. We also present a conjecture about the NPT measure, inspired by some of our numerical results, that the single-qubit reduced states of maximally entangled states are all totally mixed.

It is well known that the companion sequence of the Fibonacci sequence is Lucas's sequence. For the generalized Fibonacci sequences, the companion sequence is not unique. Several authors proposed different definitions, and they are in a certain sense all good. Our purpose is to introduce a family of companion sequences for some generalized Fibonacci sequence: the $r$-Fibonacci sequence. We evaluate the generating functions and give some applications, and we exhibit convolution relations that generalize some known identities such as Cassini's. Afterwards, we calculate the sums of their terms using matrix methods. Next, we propose a $q$-analogue and extend the definition to negative $n$s. Also, we define the incomplete associated sequences using a Euler--Seidel-like approach.

Quantum key distribution (QKD) is an innovative solution in the cryptography world to prevent the information leakage that can sometimes be deliberate. Several QKD protocols were recently presented for building a secure shared key, of which the BB84 protocol is one of those interesting protocols. The authentication between the communicating parties is one of the issues that cause a huge argument. Furthermore, the current well-known QKD protocols are not yet ready to realize the personality of either the sender or the receiver, although the QKD protocol is already protected by the rules of physics and quantum mechanics to detect any interruption. This paper introduces a new QKD protocol that utilizes two quantum channels to provide authenticated communications for legitimate parties. Moreover, the proposed QKD protocol uses two type of physical behaviors, entanglement and superposition states. The entangled states are utilized to confirm the authentication between the end users, while the superposition states carry the secret key that will be shared between the users.

The Horadam sequence Hn(a,b;p,q) unifies a number of well-known sequences, such as Fibonacci and Lucas sequences. We use the context-free grammars as a new tool to study Horadam sequences. By introducing a set of auxiliary basis polynomials (v1,v2,v3) and using the formal derivative associated with the Horadam grammar, we solve the convolution coefficients and provide a unified method to discover convolution formulas associated with binomial coefficients. These results are extended to subsequences with indices kn through a parameterized grammar Gk. Using the modified grammar Gk˜, we derive convolution formulas involving the weighting term (−q)n−i. Furthermore, applying the proposed framework to (p,q)-Fibonacci and (p,q)-Lucas sequences, we derive explicit convolution formulas with parameters (p,q). The framework is also applied to derive specific identities for Pell and Pell–Lucas numbers, as well as for Fermat and Fermat–Lucas numbers.

A system of two two-level atoms interacting with a squeezed vacuum field can exhibit stationary entanglement associated with nonclassical two-photon correlations characteristic of the squeezed vacuum field. The amount of entanglement present in the system is quantified by the well known measure of entanglement called concurrence. We find analytical formulas describing the concurrence for two identical and nonidentical atoms and show that it is possible to obtain a large degree of steady-state entanglement in the system. Necessary conditions for the entanglement are nonclassical two-photon correlations and nonzero collective decay. It is shown that nonidentical atoms are a better source of stationary entanglement than identical atoms. We discuss the optimal physical conditions for creating entanglement in the system, in particular, it is shown that there is an optimal and rather small value of the mean photon number required for creating entanglement.

On the spacings between C-nomial coefficients

In this paper, we give explicit formulas for elements of the Fibonacci, and Lucas Pascal triangles. The structure of these objects and Pascal's original triangle coincide. Keeping the rule of addition, we replace both legs of the Pascal triangle by the Fibonacci sequence, and the Lucas sequence, respectively. At the end of the study we describe how to determine such a formula for any binary recurrence $\{G_n\}^{\infty}_{n=0}$ satisfying $G_n=G_{n−1}+G_{n−2}$. Other scattered results are also presented.

We find that multidimensional determinants "hyperdeterminants", related to entanglement measures (the so-called concurrence or 3-tangle for the 2 or 3 qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the $n \geq 4$ qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.

Language understanding is multimodal. During human communication, messages are conveyed not only by words in textual form, but also through speech patterns, gestures or facial emotions of the speakers. Therefore, it is crucial to fuse information from different modalities to achieve a joint comprehension. With the rapid progress in the deep learning field, neural networks have emerged as the most popular approach for addressing multimodal data fusion [1, 6, 7, 12]. While these models can effectively combine multimodal features by learning from data, they nevertheless lack an explicit exhibition of how different modalities are related to each other, due to the inherent low interpretability of neural networks [2]. In the meantime, Quantum Theory (QT) has given rise to principled approaches for incorporating interactions between textual features into a holistic textual representation [3, 5, 8, 10], where the concepts of superposition andentanglement have been universally exploited to formulate interactions. The advantages of those models in capturing complicated correlations between textual features have been observed. We hereby propose the research on quantum-inspired multimodal data fusion, claiming that the limitation of multimodal data fusion can be tackled by quantum-driven models. In particular, we propose to employ superposition to formulate intra-modal interactions while the interplay between different modalities is expected to be captured by entanglement measures. By doing so, the interactions within multimodal data may be rendered explicitly in a unified quantum formalism, increasing the performance and interpretability for concrete multimodal tasks. It will also expand the application domains of quantum theory to multimodal tasks where only preliminary efforts have been made [11]. We therefore aim at answering the following research question: RQ. Can we fuse multimodal data with quantum-inspired models? To answer this question, we propose to fuse multimodal data with complex-valued neural networks, motivated by the theoretical link between neural networks and quantum theory [4] and advances in complex-valued neural networks [9]. Our model begins with a separate complex-valued embedding learned for each unimodal data based on the existing works [5, 10] which inherently assumes superposition between intra-modal features. Then we construct a many-body system in entangled state for multimodal data, where cross-modality interactions are naturally reflected by entanglement measures. Quantum measurement operators are applied to the entanglement state to address a concrete multimodal task at hand. The whole process is instrumented by a complex-valued neural network, which is able to learn how multimodal features are combined from data, and at the same time explain the combination by means of quantum superposition and entanglement measures. We plan to examine our proposed models on CMU-MOSI [12] and CMU-MOSEI [1] which are benchmarking multimodal sentiment analysis datasets. The dataset targets at classifying sentiment into 2, 5 or 7 classes with the input of textual, visual and acoustic features. We expect to see comparable effectiveness to state-of-the-art models, and we will explore superposition and entanglement measures to better understand the inter-modal interactions.

There is interest in using photon entanglement in biomedical applications. In one application, polarization entangled photons pass through brain tissue. The effect of the brain tissue on the photon entanglement is measured via the decoherence that is imparted on the entangled state. Our current method to obtain a measure of the decoherence involves quantum state tomography, where a minimum of 16 measurements are used in conjunction with tomographic optimization to obtain the density matrix representing the state of the photons. In this work we report on a method to avoid tomographic optimization on behalf of a direct measurement of the elements of the density matrix. We make preliminary comparisons between the two methods.

Deterministic sources of high-fidelity entangled qubit pairs encoded in the dual-rail photonic basis, i.e., presence of a single photon in one of two orthogonal modes, are a key enabling technology of many applications of quantum information processing, including high-rate, high-fidelity quantum communications over long distances. The most popular and mature sources of such photonic entanglement, e.g., those that leverage spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing, generate an entangled (so-called continuous-variable) quantum state that contains contributions from high-order photon terms that lie outside the span of the dual-rail basis, which is detrimental to most applications. One often uses low pump power to mitigate the effects of those high-order terms. However, that reduces the pair generation rate, and the source becomes inherently probabilistic. We investigate a cascaded source that performs a linear-optical entanglement swap between two SPDC sources, to generate a heralded photonic entangled state that has a higher fidelity (to the ideal Bell state) compared to a free-running SPDC source. Furthermore, with the Bell swap providing a heralding trigger, we show how to build a multiplexed source, which despite reasonable switching losses and detector loss and noise, yields a fidelity versus success probability trade-off of a high-efficiency source of high-fidelity dual-rail photonic entanglement. We find, however, that there is a threshold of $1.5$ dB of loss per switch, beyond which multiplexing hurts the fidelity versus success probability trade-off.

We investigate the structural aspects of genuine multipartite entanglement in Gaussian states of continuous variable systems. Generalizing the results of Adesso and Illuminati [Phys. Rev. Lett. 99, 150501 (2007)], we analyze whether the entanglement shared by blocks of modes distributes according to a strong monogamy law. This property, once established, allows us to quantify the genuine $N$-partite entanglement not encoded into $2,\dots{},K,\dots{},(N\ensuremath{-}1)$-partite quantum correlations. Strong monogamy is numerically verified, and the explicit expression of the measure of residual genuine multipartite entanglement is analytically derived, by a recursive formula, for a subclass of Gaussian states. These are fully symmetric (permutation-invariant) states that are multipartitioned into blocks, each consisting of an arbitrarily assigned number of modes. We compute the genuine multipartite entanglement shared by the blocks of modes and investigate its scaling properties with the number and size of the blocks, the total number of modes, the global mixedness of the state, and the squeezed resources needed for state engineering. To achieve the exact computation of the block entanglement, we introduce and prove a general result of symplectic analysis: Correlations among $K$ blocks in $N$-mode multisymmetric and multipartite Gaussian states, which are locally invariant under permutation of modes within each block, can be transformed by a local (with respect to the partition) unitary operation into correlations shared by $K$ single modes, one per block, in effective nonsymmetric states where $N\ensuremath{-}K$ modes are completely uncorrelated. Due to this theorem, the above results, such as the derivation of the explicit expression for the residual multipartite entanglement, its nonnegativity, and its scaling properties, extend to the subclass of non-symmetric Gaussian states that are obtained by the unitary localization of the multipartite entanglement of symmetric states. These findings provide strong numerical evidence that the distributed Gaussian entanglement is strongly monogamous under and possibly beyond specific symmetry constraints, and that the residual continuous-variable tangle is a proper measure of genuine multipartite entanglement for permutation-invariant Gaussian states under any multipartition of the modes.