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In this paper, we first study the maximally commutative set in prime dimensional systems, which is a set of generalized Pauli matrices, and it can be used to detect the local discrimination of generalized Bell states. We give a simple characterization of prime dimensional maximally commutative sets, that is, a subset of a set of generalized Bell states, whose second subscript is a multiple of the first subscript. Furthermore, some sets of generalized Bell states which can be locally distinguishable by one-way local operation and classical communication (LOCC) are constructed by using the structural characteristics of prime dimensional maximally commutative sets. Based on these distinguishable generalized Bell states, we propose a (t, n)-threshold quantum secret sharing scheme. Compared with the existing quantum secret sharing scheme, it can be found that there are enough distinguishable states to encode classical information in our scheme, the dealer only needs to send entangled particles once to make the participants get their secret share, which makes the secret sharing process more efficient than the existing schemes. Finally, we prove that this protocol is secure under dishonest participant attack, interception-and-resend attack and entangle-and-measure attack. [ABSTRACT FROM AUTHOR]

A new highly symmetrical model of the compact Lie algebra \mathfrak {g}^c_2 is provided as a twisted ring group for the group \mathbb {Z}_2^3 and the ring \mathbb {R}\oplus \mathbb {R}. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in \mathfrak {su}(2) and of the Gell-Mann matrices in \mathfrak {su}(3). As a bonus, the split Lie algebra \mathfrak {g}_2 is also seen as a twisted ring group. [ABSTRACT FROM AUTHOR]

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph C a y (P n , S P n ) of the generalized Pauli group P n on n -qubits; in fact, P n may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set S P n allows us to recognize the structure of central product of graphs in C a y (P n , S P n ). Using this approach, we are able to recursively construct the adjacency matrix of C a y (P n , S P n ) for each n ≥ 1 , and to explicitly describe its spectrum and the associated eigenvectors. It turns out that C a y (P n , S P n ) is a (3 n + 2) -regular bipartite graph on 4 n + 1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P 1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral. [ABSTRACT FROM AUTHOR]

In this paper, we develop an extended space-time geometry that includes an internal space built with division octonions with realization via Pauli matrices and Zorn (vectorial) matrices. Here, we extend a former field theory that used split octonion algebra to a division octonion algebra on a non-Riemannian manifold. The octonionic algebra is the last possible algebra allowed by the Hurwitz theorem. The interpretation of the “octonionic field” in this division octonionic algebra is not straightforward, however it behaves in a similar way to the quaternionic Yang–Mills fields. Former work suggests that this Yang–Mills-like octonionic field is associated with the field governing quarks within nucleons. In this work, we discover that the inclusion of octonionic fields in the geometry of an internal space necessarily excludes the quaternionic (Yang–Mills) fields in an extended non-Riemannian geometry. This is not what is expected from the standpoint of a “unified” field theory, which leads us to propose a different approach to Einstein’s unified field theory. [ABSTRACT FROM AUTHOR]

Motivation Imaging Mueller polarimetry has already proved its potential for biomedicine, remote sensing, and metrology. The real-time applications of this modality require both video rate image acquisition and fast data post-processing algorithms. First, one must check the physical realizability of the experimental Mueller matrices in order to filter out non-physical data, i.e. to test the positive semi-definiteness of the 4 × 4 Hermitian coherency matrix calculated from the elements of corresponding Mueller matrix pixel-wise. For this purpose, we compared the execution time for the calculations of (i) eigenvalues, (ii) Cholesky decomposition, (iii) Sylvester's criterion, and (iv) coefficients of the characteristic polynomial (two different approaches) of the Hermitian coherency matrix, all calculated for the experimental Mueller matrix images (600 pixels × 700 pixels) of mouse uterine cervix. The calculations were performed using C++ and Julia programming languages. Results Our results showed the superiority of the algorithm (iv) based on the simplification via Pauli matrices over other algorithms for our dataset. The sequential implementation of latter algorithm on a single core already satisfies the requirements of real-time polarimetric imaging. This can be further amplified by the proposed parallelization (e.g. we achieve a 5-fold speed up on six cores). Availability and implementation The source codes of the algorithms and experimental data are available at https://github.com/pogudingleb/mueller_matrices. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem to determine a unique quantum state from given distribution of position (density) and momentum density of particles, namely the so called Pauli problem in quantum physics. In the first part, we will review the method of wave function lifting developed in [4,5] to construct a complex wave function ψ ∈ H s ( R d) , s = 1 , 2 , associated to given density and momentum density. The second part is focused on the dynamical version of the wave function lifting, namely we study the relation between solutions to quantum fluid models and wave functions solving the nonlinear Schrödinger equation. The uniqueness of the lifted wave function is essentially related to the structure of vacuum regions of the position density. [ABSTRACT FROM AUTHOR]

We present a new geminal product wave function Ansatz where the geminals are not constrained to be strongly orthogonal or to be of seniority-zero. Instead, we introduce weaker orthogonality constraints between geminals that significantly lower the computational effort without sacrificing the indistinguishability of the electrons. That is to say, the electron pairs corresponding to the geminals are not fully distinguishable, and their product has yet to be antisymmetrized according to the Pauli principle to form a bona fide electronic wave function. Our geometrical constraints translate into simple equations involving the traces of products of our geminal matrices. In the simplest non-trivial model, a set of solutions is given by block-diagonal matrices where each block is 2 × 2 and consists of either a Pauli matrix or a normalized diagonal matrix multiplied by a complex parameter to be optimized. With this simplified Ansatz for geminals, the number of terms in the calculation of the matrix elements of quantum observables is considerably reduced. A proof of principle is reported and confirms that the Ansatz is more accurate than strongly orthogonal geminal products while remaining computationally affordable. [ABSTRACT FROM AUTHOR]

Quantum Computing in the Noisy Intermediate-Scale Quantum (NISQ) era has shown promising applications in machine learning, optimization, and cryptography. Despite these progresses, challenges persist due to system noise, errors, and decoherence. These system noises complicate the simulation of quantum systems. The depolarization channel is a standard tool for simulating a quantum system's noise. However, modeling such noise for practical applications is computationally expensive when we have limited hardware resources, as is the case in the NISQ era. This work proposes a modified representation for a single-qubit depolarization channel. Our modified channel uses two Kraus operators based only on X and Z Pauli matrices. Our approach reduces the computational complexity from six to four matrix multiplications per channel execution. Experiments on a Quantum Machine Learning (QML) model on the Iris dataset across various circuit depths and depolarization rates validate that our approach maintains the model's accuracy while improving efficiency. This simplified noise model enables more scalable simulations of quantum circuits under depolarization, advancing capabilities in the NISQ era. [ABSTRACT FROM AUTHOR]

We introduce the notion of (renormalized) quantum uncertainty and reveal its basic features. In terms of this quantity, we completely characterize the minimum and maximum quantum uncertainty states for qubit systems involving Pauli matrices. It turns out that the minimum quantum uncertainty states consist of both certain pure states and certain mixed states, in sharp contrast to the case of conventional Heisenberg uncertainty relation. The maximum quantum uncertainty states are H -type magic states arising from the stabilizer formalism of quantum computation, and can be obtained from minimum quantum uncertainty states via the T -gate. [ABSTRACT FROM AUTHOR]

In this study, the mean-field approximation (MFA) is applied to the study of the XZ model for spin-1/2 in terms of Pauli spin matrices and their exponentials to obtain the magnetization components for the ferromagnetic (FM) and antiferromagnetic (AFM) cases. The details of obtaining the series expansion for the exponential Pauli spin matrices by using their fundamental properties and commutativity relations are presented. The partition function, free energy, and magnetization are obtained for a given coordination number in a simple form as hyperbolic tangent functions. The thermal variations of the magnetization components in the x- and z-directions are studied to obtain the phase diagrams in possible planes, which only present the second-order phase transitions. The reentrant behavior is also observed in the AFM case. [ABSTRACT FROM AUTHOR]

Under the spin–position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra substitutes the vector–matrix spin model built on the Pauli spin operator. The standard quantum operator‐state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space—isomorphic to the space of Pauli matrices. In the single‐spin case, the novel spin 1/2 representation (1) is Hermitian, (2) shows handedness, (3) yields all the standard results and its modulus equals the total spin angular momentum Stot=3ℏ/2, (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding of the 2D spin space in 3D. Maximally entangled spin pairs (1) are in phase and have opposite handedness, (2) relate by one of the four basic improper rotations in 3D: plane reflections (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one—or two—sided, the formalism appears in two complementary forms, the "spinor" or the "vector" form, respectively. The proposed scheme provides a clear geometric picture of spin correlations and transformations entirely in the 3D physical orientation space. [ABSTRACT FROM AUTHOR]

The mean-field approximation (MFA) of spin-1/2 XYZ model was studied by using the Pauli spin matrices and their exponentials which led to very nice hyperbolic tangent functions in nonlinear form. The magnetization components Mx, My and Mz were obtained for the ferromagnetic (FM) case and then it was modified for the antiferromagnetic (AFM) case by introducing the sublattices. The bilinear exchange interaction parameters Jx, Jy and Jz, and the external magnetic fields Hx, Hy and Hz were considered along the three-dimensions for various coordination numbers q = 3, 4 and 6. The thermal variations of the magnetizations and thus the phase diagrams were obtained to illustrate the behaviours of phase transition lines in the AFM case. [ABSTRACT FROM AUTHOR]

Spinors are components of a complex vector space that can be related to Euclidean space in both geometry and physics. In essence, the forms of usage include quaternions that are equivalent to Pauli spin matrices, which may be produced by thinking of a quaternion matrix as the compound. This study's objective is the spinor structure that forms based on the quaternion algebra. In this work, first, spinors have been mathematically presented. Then, Padovan and Perrin spinors have been defined using the Padovan and Perrin quaternions. Later, we defined the algebraic structure for these spinors. Finally, we have established certain identities such as the Binet formulas and generating functions for Padovan and Perrin spinors. [ABSTRACT FROM AUTHOR]

We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices. This is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calculations. The strength of this strategy is benchmarked against state-of-the-art techniques, showing a remarkable acceleration. As a side product, we provide an optimized method for one key calculus in quantum simulations: the Pauli basis decomposition of Hamiltonians. [ABSTRACT FROM AUTHOR]

In this work, an analogue to the Pauli spin matrices is presented and investigated. The proposed Hermitian spin matrices exhibit four symmetries for spin-1/n particles. The spin projection operators are derived, and the electrodynamics for hypothetical spin-1/2 fermions are explored using the proposed spin matrices. The fermionic quantum Heisenberg model is constructed using the proposed spin matrices, and comparative studies against simulation results using the Pauli spin matrices are conducted. Further analysis of the key findings as well as discussions on extending the proposed spin matrix framework to describe hypothetical bosonic systems (spin-1 particles) are provided. [ABSTRACT FROM AUTHOR]

To address the risk of spoofing the information exchanged by users and reduce the trustworthiness of third party, we propose a quantum information fair exchange protocol based on three-particle GHZ states. This protocol achieves quantum multi-party fair exchange of secret information through the utilization of three-particle GHZ states, Pauli matrix, a semi-trusted third party, and polarization state of a single photon. By leveraging the unique physical properties of the GHZ state, this protocol achieves fair exchange of secret information with experimental results that align with theoretical derivation, thus demonstrating its feasibility. Additionally, it provides superdense coding which enhances transmission efficiency while also being resistant to false signal attacks, interception retransmission attacks, and entanglement attacks, ultimately improving security. Furthermore, compared with the classic fair exchange protocol, which requires a trusted third party, it only requires a semi-trusted third party, which raises the security from "computational security" to "unconditional security". The implementation of this protocol resolves the risk of users who hand over information first, presenting a novel solution for digital currency trading, information swapping, and multi-party secure computing. [ABSTRACT FROM AUTHOR]

This paper investigates quantum error correction schemes for fully-correlated noise channels on an n-qubit system, where error operators take the form W ⊗ n , with W being an arbitrary 2 × 2 unitary operator. In previous literature, a recursive quantum error correction scheme can be used to protect k qubits using (k + 1) -qubit ancilla. We implement this scheme on 3-qubit and 5-qubit channels using the IBM quantum computers, where we uncover an error in the previous paper related to the decomposition of the encoding/decoding operator into elementary quantum gates. Here, we present a modified encoding/decoding operator that can be efficiently decomposed into (a) standard gates available in the qiskit library and (b) basic gates comprised of single-qubit gates and CNOT gates. Since IBM quantum computers perform relatively better with fewer basic gates, a more efficient decomposition gives more accurate results. Our experiments highlight the importance of an efficient decomposition for the encoding/decoding operators and demonstrate the effectiveness of our proposed schemes in correcting quantum errors. Furthermore, we explore a special type of channel with error operators of the form σ x ⊗ n , σ y ⊗ n and σ z ⊗ n , where σ x , σ y , σ z are the Pauli matrices. For these channels, we implement a hybrid quantum error correction scheme that protects both quantum and classical information using IBM's quantum computers. We conduct experiments for n = 3 , 4 , 5 and show significant improvements compared to recent work. [ABSTRACT FROM AUTHOR]

The author expounds or proves some results connected with Segal's chronometric theory. He gives short proofs of results about linear representation of the group of nondegenerate complex -matrices on the Minkowski space-time and on the universal covering of the Lie group of unitary -matrices, i.e., on the Einstein Universe, as well as about the Cayley transform of Lie algebras of Lie groups of unitary matrices into these groups. In comparison with the structure of conformal infinity for the Minkowski space, the structure of the set of unitary -matrices, which do not admit the Cayley transform, is found. Some problems are suggested. [ABSTRACT FROM AUTHOR]

In the present paper, we develop a theory of thinking based on an attempt to formalize the construction of mental representations as described in psychoanalytic theory. In previous work, we described Freud's and Matte Blanco's structural Unconscious in a formal model in which the properties of unconscious representations are captured by particular sets-infinite singletons-that can be derived in first-order logic language. Here, we afford the issue of the finitization of unconscious representations by assuming that the mind can form an all-purpose modality, originating from abstraction from infinite singletons; in this way, a symmetric prelogical setting for mental representations is formally created, and this is interpreted in a quantum spin model by a modal (necessity) projector. Then, by introducing time, one can describe the links that mental representations can establish with reality, and hence finitize the representations. The modality is so split into finite components, here termed positive, negative and irreal; the splitting of the modality is traced back to the decomposition of the spin observables by means of the Pauli matrices, which can offer a quantum semantics to the method applied. Here, we suggest that the development of the modal approach and its quantum logic implementation can be considered as a proper formalization of some aspect of the psychoanalytic theory of thinking proposed by Bion; namely, we will show that the process of abstraction leading from raw data to preconceptions, and therefore to the definition of the content-container relationship, is adequately captured by our model, and further correspondences can be detected with Bion's theory about links and transformations, implying different ways in which the mind can get in touch with both internal and external reality. [ABSTRACT FROM AUTHOR]

We give a new way to obtain the standard isomorphisms of complex Clifford algebras, known as the tensor product of Pauli matrices, by representing the complex Clifford algebras on the space of complex valued functions defined over a finite subset of the Cantor set. [ABSTRACT FROM AUTHOR]

The transformation law of the Pauli spinors under rotations, linked to 1/2-spin particles, leads to a mathematical problem that can be rephrased and solved in purely linear algebraic terms. We present it as a possible approach to be incorporated in a first course of quantum mechanics when introducing the concept of the spin. This concept was very elusive at early times and had an intricate development commonly overlooked in modern texts. We also take the opportunity to make some historical comments. [ABSTRACT FROM AUTHOR]

In this paper, we outline the research conducted by the first named author and his associates on the axiom-preserving, thus isotopic completion of quantum mechanics into hadronic mechanics according to the historical legacy by A. Einstein, B. Podolsky and N. Rosen that quantum mechanics is not a complete theory and review the ensuing exact representation of the magnetic moment and spin of the Deuteron in its ground state thanks to the isotopic completion of Pauli's matrices with an explicit and concrete content of D. Bohm's hidden variable λ . We then outline the independent studies conducted by the second named author on the representation of the conventional Pauli's matrices via geometric Clifford algebras. We finally show that the combination of the two studies allows a mathematically rigorous, numerically exact and time invariant geometric representation of the magnetic moment, spin and hidden variable of the Deuteron in its ground state. [ABSTRACT FROM AUTHOR]

In this work, an analogue to the Pauli spin matrices is presented and investigated. The proposed Hermitian spin matrices exhibit four symmetries for spin-1/n particles. The spin projection operators are derived, and the electrodynamics for hypothetical spin-1/2 fermions are explored using the proposed spin matrices. The fermionic quantum Heisenberg model is constructed using the proposed spin matrices, and comparative studies against simulation results using the Pauli spin matrices are conducted. Further analysis of the key findings as well as discussions on extending the proposed spin matrix framework to describe hypothetical bosonic systems (spin-1 particles) are provided.

In this study, Pauli-Blocking has been observed in a 0D/2D MoS2 quantum dots/graphene (MoS2-QDs/graphene) hybrid system. For the observation of room temperature Pauli-Blocking in the 0D/2D system, a photodetector device based on n-type MoS2-QDs and CVD grown graphene has been fabricated using a facile and lithography free technique. The current-voltage characteristics of the device have been performed at room temperature. The fabricated device shows a negative response under visible light (λ ∼ 400 to 700 nm) illumination. The dark to photo current ratio of the device shows variation up to two orders of magnitude. This negative response, which results decrease in current under visible light illumination, may be attributed to the Pauli-Blocking due to high absorbance of photon energy in visible light range. Furthermore, it is believed that the present study may provide an insight into understanding the Pauli-Blocking in 0D/2D hybrid system at room temperature. [ABSTRACT FROM AUTHOR]

In this work, we consider the one‐dimensional Dirac operator DQ+Qsu(x)=1iσ2ddx+Q(x)+Qs(x)u(x),$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s}u(x)=\left(\frac{1}{\mathrm{i}}{\sigma}_2\frac{d}{dx}+\mathrm{Q}(x)+{\mathrm{Q}}_s(x)\right)u(x), $$where σ2$$ {\sigma}_2 $$ is Pauli's matrix, Q$$ \mathrm{Q} $$ is a 2×2$$ 2\times 2 $$‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, Qs$$ {\mathrm{Q}}_s $$ is a singular potential consisting of N$$ N $$ delta distributions Qs(x)=∑j=1NAjδx−xj,$$ {\mathrm{Q}}_s(x)=\sum \limits_{j=1}^N{\mathrm{A}}_j\delta \left(x-{x}_j\right), $$Aj$$ {\mathrm{A}}_j $$ (j=1,⋯,N$$ j=1,\cdots, N $$) are 2×2$$ 2\times 2 $$‐matrices representing the strengths of Dirac deltas, and u=u1u2⊤$$ u={\left({u}^1\kern0.5em {u}^2\right)}^{\top } $$ is a two‐spinor. We associate to the operator DQ+Qs$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s} $$ an unbounded in L2ℝ,ℂ2$$ {L}^2\left(\mathbb{R},{\mathbb{C}}^2\right) $$ symmetric operator denoted by DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$, where Ω=xjj=1N$$ \Omega ={\left\{{x}_j\right\}}_{j=1}^N $$ is the support of singular potential Qs$$ {\mathrm{Q}}_s $$. The operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ includes only the regular potential Q$$ \mathrm{Q} $$ together with certain interaction conditions at each point xj∈Ω$$ {x}_j\in \Omega $$. The paper presents a method for determining the discrete spectrum of the operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ for arbitrary potential Q$$ \mathrm{Q} $$ whose entries are given by L∞(ℝ)$$ {L}^{\infty}\left(\mathbb{R}\right) $$‐functions. The eigenvalues λn$$ {\lambda}_n $$ of the operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ are the zeros of a dispersion equation ηΩ(λ)=0$$ {\eta}_{\Omega}\left(\lambda \right)=0 $$, where the characteristic function ηΩ$$ {\eta}_{\Omega} $$ is determined explicitly in terms of power series involving the spectral parameter λ$$ \lambda $$. The construction of the characteristic function ηΩ$$ {\eta}_{\Omega} $$ from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ from the zeros of certain approximate function η˜Ω$$ {\tilde{\eta}}_{\Omega} $$ which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method. [ABSTRACT FROM AUTHOR]

In modern computing environments, speed of execution of any operations are gaining more attentions. The faster processing of the unit operations, opens the room for the applications such as cryptography, data communication etc. Hence, in this work a new software-oriented approach is proposed to use in the environments that requires faster executions. This proposed method speeds up the computations of the logical operations – bitwise XOR. This paper first elaborates on the procedure of how to construct the bitwise XOR vector space efficiently using recursive approach based on Pauli's matrix. The corresponding time complexity for the construction of the bitwise XOR vector space is computed by master theorem and is O (log 2 (n / 2 n)). Later, the paper presents on the implementation of this bitwise XOR vector space constructed to facilitate the XOR – Free operations. For the evidential proof, the proposed bitwise XOR-free operations are implemented on standard key stream generation scheme, SNOW 3G. The average speed up gained by the bitwise XOR-Free operations on the SNOW 3G is 1.36 x and the corresponding throughput attained is 21.85 Mbps. [ABSTRACT FROM AUTHOR]

The dynamical feature for a p‐wave superconductor model in different parameter regions in terms of the appearance of topologically gapless edge modes in reel geometry is investigated. First, the parameter region with gapless edge modes versus a parameter and quasi‐momentum is shown. The parameter diagram can be reflected by the expectations of Pauli matrices in global manners. In another view, the dynamical feature of the excitation behave differently in the parameter regions with topological gapless edge modes and not. And the cusps of dynamical return rate vanish as the parameter pass the topological phase boundary slowly enough. It is found that the dynamics in the parameter region with gapless edge modes behaves differently to that without edge modes and related mostly to the eigenenergy gap between the pre‐and post‐quench eigenstates. The cusps of the dynamical return rate behave robustly against the noise occurs during evolution in the lattice until localization behavior dominates. This work benefits detecting topological edge modes by dynamical manners. [ABSTRACT FROM AUTHOR]

In this paper, we obtain Euler's and De Moivre's formulas for the 4 × 4 matrix representation of Pauli quaternions. Moreover, we provide De Moivre's formula for the light-like Pauli quaternions. Additionally, we give the nth roots of the matrix representation of Pauli quaternions. Moreover, we exemplify some of the results with illustrative examples to support the main formulas. [ABSTRACT FROM AUTHOR]

The evolution of quantum states serves as good fundamental studies in understanding the quantum information systems which finally lead to the research on quantum computation. To carry out such a study, mathematical tools such as the Lie group and their associated Lie algebra is of great importance. In this study, the Lie algebra of su(8) is represented in a tensor product operation between three Pauli matrices. This can be realized by constructing the generalized Gell-Mann matrices and comparing them to the Pauli bases. It is shown that there is a one-to-one correlation of the Gell-Mann matrices with the Pauli basis which resembled the change of coordinates. Together with the commutator relations and the frequency analysis of the structure constant via the algebra, the Lie bracket operation will be highlighted providing insight into relating quantum circuit model with Lie Algebra. These are particularly useful when dealing with three-qubit quantum circuit problems which involve quantum gates that is derived from the SU(8) Lie group. [ABSTRACT FROM AUTHOR]

Unitary k-designs are probabilistic ensembles of unitary matrices whose first k statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical Z 4 -linear Kerdock codes maps to a unitary 2-design, which established a new classical-quantum connection via graph states. In this paper, we construct a Markov process that mixes this Kerdock 2-design with symplectic transvections, and show that this process produces an ϵ -approximate unitary 3-design. We construct a graph whose vertices are Pauli matrices, and two vertices are connected by directed edges if and only if they commute. A unitary ensemble that is transitive on vertices, edges, and non-edges of this Pauli graph is an exact 3-design, and the stationary distribution of our process possesses this property. With respect to the symmetries of Kerdock codes, the Pauli graph has two types of edges; the Kerdock 2-design mixes edges of the same type, and the transvections mix the types. More precisely, on m qubits, the process samples O (log (N 5 / ϵ)) random transvections, where N = 2 m , followed by a random Kerdock 2-design element and a random Pauli matrix. Hence, the simplicity of the protocol might make it attractive for several applications. From a hardware perspective, 2-qubit transvections exactly map to the Mølmer–Sørensen gates that form the native 2-qubit operations for trapped-ion quantum computers. Thus, it might be possible to extend our work to construct an approximate 3-design that only involves such 2-qubit transvections. [ABSTRACT FROM AUTHOR]

In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing a three-dimensional vector is discussed. This method leads to a unified representation for 3D multivectors with Pauli algebra. A significant advantage is that this approach provides a representation independent of the coordinate system, which does not exist in the conventional vector perspective. Additionally, the Pauli matrix representations of the nabla operator in the different coordinate systems are derived and discussed. Finally, an example on the radiation from a dipole is given to illustrate the advantages of the methodology. [ABSTRACT FROM AUTHOR]

We have developed a software library that simulates noisy quantum logic circuits. We represent quantum states by their density matrices in the Pauli basis, and incorporate possible errors in initialization, logic gates, memory and measurement using simple models. Our quantum simulator is implemented as a new backend on IBM's open-source Qiskit platform. In this paper, we provide its description, and illustrate it with some simple examples. [ABSTRACT FROM AUTHOR]

We give an intriguing construction for establishing a sequence of unital, commutative, associative number systems by introducing the Z 2 -graded algebra S L n (R). We show how the algebra S L n (R) arises naturally from the Pauli matrices. Moreover, we investigate eigenvalues of elements in S L n (R) and the matrix algebra M m (S L n (R)). [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices pertaining to finite-level systems. The system interacts with external bosonic fields, and its Hamiltonian and coupling operators depend linearly on the system variables. This results in a Hudson--Parthasarathy quantum stochastic differential equation (QSDE) whose drift and dispersion terms are affine and linear functions of the system variables. The quasilinearity of the QSDE leads to tractable dynamics of mean values and higher-order multipoint moments of the system variables driven by vacuum input fields. This allows for the closed-form computation of the quasi-characteristic function of the invariant quantum state of the system and infinite-horizon asymptotic growth rates for a class of cost functionals. The tractability of the moment dynamics is also used for mean square optimal Luenberger observer design in a measurement-based filtering problem for a quasilinear quantum plant, which leads to a Kalman-like quantum filter. [ABSTRACT FROM AUTHOR]

Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields χ (ϱ) = υ 1 (ϱ) (ϱ) + υ 2 (ϱ) (ϱ) , where υ 1 2 (ϱ) + υ 2 2 (ϱ) = 1 for the spatial quaternionic curve ψ and χ (ϱ) = λ 1 (ϱ) ⊤ (ϱ) + λ 2 (ϱ) η (ϱ) + λ 3 β 2 (ϱ) , where λ 1 2 (ϱ) + λ 2 2 (ϱ) + λ 3 2 (ϱ) = 1 for the quaternionic curve ϕ. Then, we have given the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we have examined in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. So, considering that helices also take place in electron physics, it is thought that this study will create a bridge between physics and geometry. Finally, we have given the examples and draw the figures of curves in the examples. [ABSTRACT FROM AUTHOR]

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p × p unitary matrices given by the diagonal matrix whose entries are the p -th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v , satisfy u p = v p = 1 p (the p × p identity matrix) and the commutation relation u v = ζ v u , where ζ is a primitive p -th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d × d unitary matrices such that u p = v p = 1 d and u v = ζ v u , then there exists a unital completely positive linear map ϕ : M p (C) → M d (C) such that ϕ (u) = u and ϕ (v) = v. We also show, moreover, that any two pairs of p -th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails. There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63]. This standard construction is generalised herein to the case p ≥ 3 , producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g -tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory. [ABSTRACT FROM AUTHOR]

The one-dimensional Dirac dynamical system $ \Sigma $ is $ \begin{align*} & iu_t+i\sigma_{\!_3}\, u_x+Vu = 0, \, \, \, \, x, t>0;\, \, \, u|_{t = 0} = 0, \, \, x>0;\, \, \, \, u_1|_{x = 0} = f, \, \, t>0, \end{align*} $ where $ \sigma_{\!_3} = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix} $ is the Pauli matrix; $ V = \begin{pmatrix}0&p\\ \bar p&0\end{pmatrix} $ with $ p = p(x) $ is a potential; $ u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix} $ is the trajectory in $ \mathscr H = L_2(\mathbb R_+;\mathbb C^2) $ ; $ f\in\mathscr F = L_2([0, \infty);\mathbb C) $ is a boundary control. System $ \Sigma $ is not controllable: the total reachable set $ \mathscr U = {\rm span}_{t>0}\{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $ is not dense in $ \mathscr H $ , but contains a controllable part $ \Sigma_u $. We construct a dynamical system $ \Sigma_a $ , which is controllable in $ L_2(\mathbb R_+;\mathbb C) $ and connected with $ \Sigma_u $ via a unitary transform. The construction is based on geometrical optics relations: trajectories of $ \Sigma_a $ are composed of jump amplitudes that arise as a result of projecting in $ \overline{\mathscr U} $ onto the reachable sets $ \mathscr U^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $. System $ \Sigma_a $ , which we call the amplitude model of the original $ \Sigma $ , has the same input/output correspondence as system $ \Sigma $. As such, $ \Sigma_a $ provides a canonical completely reachable realization of the Dirac system. [ABSTRACT FROM AUTHOR]

In this paper the local distinguishability of generalized Bell states in arbitrary dimension is investigated. We firstly study the decomposition of a basis which consists of d 2 number of generalized Pauli matrices. We discover that this basis is equal to the union of D number of different sets, where D = 2 Ď• (d) âˆ' t ∈ Z d gcd (t , d) = 1 âˆ' i = 2 ⌊ d t ⌋ Ď• (i) + 1 and Ď• is Euler Ď• -function. Then we define the generator of the matrices in this decomposition, and exhibit an algorithm to calculate generators of a given set of matrices. This algorithm shows that generators of a given set can be calculated simply and efficiently. Secondly, we show that a set L of GBSs can be distinguished by one-way LOCC if the cardinality of G L is less than DĎ• (d), where G L is a set of generators of all the elements in difference set of a set L of GBSs. The previous results in [2004 Phys. Rev. Lett. 92 177905; 2019 Phys. Rev. A 99 022307; 2021 Quantum Inf. Process. 20 52] can be covered by our result. Finally, for the uncovered cases in [2021 Quantum Inf. Process. 20 52], we give a new result to partly solve that problem. [ABSTRACT FROM AUTHOR]

Exact Green's function related to a Dirac particle submitted to the combination of Aharonov–Bohm and Coulomb potentials in (2 + 1) coordinate space is analytically calculated via path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger model. The energy spectrum as well as the corresponding normalized wave functions are extracted following this approach. The interesting properties of the spinors are thus deduced after symmetrization. According to the symmetric form for Green's function, it is shown that the non-relativistic limit of the Dirac particle is undertaken with much ease. [ABSTRACT FROM AUTHOR]

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations e σ i in the spin representation (a.k.a. Lorentz rotors) in terms of their generators σ i : ln (e σ 1 e σ 2 ) = tanh − 1 tanh σ 1 + tanh σ 2 + 1 2 [ tanh σ 1 , tanh σ 2 ] 1 + 1 2 { tanh σ 1 , tanh σ 2 } . This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension ≤ 4 , naturally generalizing Rodrigues' formula for rotations in ℝ 3 . In particular, it applies to Lorentz rotors within the framework of Hestenes' spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex 2 × 2 matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3 -velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle. [ABSTRACT FROM AUTHOR]