4-dimensional lattice models for the quantum range

Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4..

Un-Weyl-ing the Clifford Hierarchy

The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.

Simulating a Quantum Harmonic Oscillator by Introducing it to a Bosonic System

The goal of this paper is to associate a Quantum Harmonic Oscillator to a Bosonic System, try to simulate it in IBMQ-Experience (at 8192 shots) and further study it. We associated the concept of Pauli Matrix equivalent to Bosonic Particles and used it to calculate the Unitary Operators which helped us to theoretically visualize each Quantum states and further simulate our system.

Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System

Here, we derive the Pauli Matrix Equivalent for Spin-1 particles (mainly Z-Boson and W-Boson). Pauli Matrices are generally associated with Spin- 1/2 particles and it is used for determining the properties of many Spin- 1/2 particles. But in our case, we try to expand its domain and attempt to implement it for calculating the Unitary Operators of the Harmonic oscillator involving the Spin-1 system and study it.

Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model

The purpose of this paper is to study the exceptional eigenvalues of the asymmetric quantum Rabi models (AQRMs), specifically, to determine the degeneracy of their eigenstates. Here, the Hamiltonian $H_{\textrm{Rabi}}^{\varepsilon }$ of the AQRM is defined by adding the fluctuation term $\varepsilon \sigma _x$, with $\sigma _x$ being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry. The spectrum of $H_{\textrm{Rabi}}^{\varepsilon }$ contains a set of exceptional eigenvalues, considered to be remains of the eigenvalues of the uncoupled bosonic mode, which are further classified in two types: Juddian, associated with polynomial eigensolutions, and non-Juddian exceptional. We explicitly describe the constraint relations for allowing the model to have exceptional eigenvalues. By studying these relations we obtain the proof of the conjecture on constraint polynomials previously proposed by the third author. In fact we prove that the spectrum of the AQRM possesses degeneracies if and only if the parameter $\varepsilon $ is a halfinteger. Moreover, we show that non-Juddian exceptional eigenvalues do not contribute any degeneracy and we characterize exceptional eigenvalues by representations of $\mathfrak{s}\mathfrak{l}_2$. Upon these results, we draw the whole picture of the spectrum of the AQRM. Furthermore, generating functions of constraint polynomials from the viewpoint of confluent Heun equations are also discussed.

A Self-Stabilizing Phase Decoder for Quantum Key Distribution

Self-stabilization quantum key distribution (QKD) systems are often based on the Faraday magneto-optic effect such as “plug and play” QKD systems and Faraday–Michelson QKD systems. In this article, we propose a new anti-quantum-channel disturbance decoder for QKD without magneto-optic devices, which can be a benefit for the photonic integration and applications in magnetic environments. The decoder is based on a quarter-wave plate reflector–Michelson (Q–M) interferometer, with which the QKD system can be free of polarization disturbance caused by quantum channel and optical devices in the system. The theoretical analysis indicates that the Q–M interferometer is immune to polarization-induced signal fading, where the operator of the Q–M interferometer corresponding to Pauli Matrix σ2 makes it satisfy the anti-disturbance condition naturally. A Q–M interferometer based time-bin phase encoding QKD setup is demonstrated, and the experimental results show that the QKD setup works stably with a low quantum bit error rate about 1.3% for 10 h over 60.6 km standard telecommunication optical fiber.

Robustness of quantum correlation for accelerating two atoms coupled with electromagnetic field

In this paper, we investigate the behaviors of quantum correlation quantified by trace norm measurement-induced nonlocality (TMIN) for two uniformly accelerated atoms coupled with electromagnetic vacuum fluctuation in the Minkowski vacuum. We firstly discuss the solving process of master equation that governs the system evolution based on the Pauli matrix presentation for two-qubit state. Similar to [Y. Q. Yang et al., Phys. Rev. A 94, 032337 (2016)], we analyze the degradation, creation, revival and enhancement of quantum correlation for different initial states and polarizations of two atoms, and the influence of interatomic separation and acceleration on quantum correlation. Compared with the entanglement dynamics discussed in [Y. Q. Yang et al., Phys. Rev. A 94, 032337 (2016)], it is found that quantum correlation exhibits better robustness than entanglement. This may be helpful for quantum information processing.

Veldkamp spaces: From (Dynkin) diagrams to (Pauli) groups

Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type [Formula: see text], [Formula: see text], it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG[Formula: see text]. Proper labeling of the vertices of the diagram (for [Formula: see text]) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG[Formula: see text]. The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space [Formula: see text] of the PG[Formula: see text] whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin–Peres magic square. In the case of [Formula: see text], a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG[Formula: see text], the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG[Formula: see text] are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin–Peres square. Moreover, save for [Formula: see text], each Veldkamp space is also endowed with some exceptional point(s). Interestingly, two such points in the [Formula: see text] case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix [Formula: see text].