Klaus Dietz
m-Qubit states are imbedded in $\mathfrak{Cl}_{2^m}$ Clifford algebras. Their probability spectra then depend on $O(2m)$ or $O(2m+1)$ invariants. Parameter domains for $O(2m(+1))-$ vector and tensor configurations, generalizing the notion of a Bloch sphere, are derived.
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The Qutrit Bloch Sphere
本文提出数学模型，用于将三能级系统可视化为三维Bloch球。
The two-qubit pure state is explicitly parameterized by three unit 2-spheres and a phase factor. For separable states, two of the three unit spheres are the Bloch spheres of each qubit. The third sphere parameterizes the degree and phase of concurrence, an entanglement measure. This sphere may be considered a variable complex imaginary unit where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states. We argue that…
A scheme for implementing the discrete-time quantum walk on the Bloch sphere is proposed, which is closely related to the SU(2) group. A spin cluster serves as the walker, whereas its location on the Bloch sphere is described by the spin coherent state. An additional spin that interacts with the spin cluster plays the role of a coin, whose state determines the rotation of the spin cluster. The Wigner function is calculated to visualize the movement of the walker on the Bloch sphere, with which the probability distribution and the standard deviation are also achieved. The quadratic enhancement …
Entanglement on Two Bloch Spheres: Exploring Two-Qubit Stabilizer Group Structure
本研究借助稳定子形式体系，建立两量子比特态的密度矩阵与双布洛赫球图形表示的关联（二者均源于稳定子群结构），并以此探究最大纠缠态的对称性。
We show how the Majorana representation can be used to express the pure states of an N-qubit system as points on the Bloch sphere. We compare this geometrical representation of N-qubit states with an alternative one, proposed recently by the present authors.
We describe and characterize an equal-fidelity surface in Bloch space targeted for a qubit state by means of equal-distance concept. The distance is generalized and defined as the Euclidean distance between extended Bloch vectors for arbitrary dimensional states. The distance is a genuine distance according to the definition and is related to other distances between quantum states and super-fidelity.
Superposition States on Different Axes of the Bloch Sphere for Cost-Effective Circuits Realization on IBM Quantum Computers
提出利用布洛赫球Y轴与IBM原生sqrt(X)门制备叠加态，替代Hadamard门，确保编译后量子电路成本更低。
Three unit spheres were used to represent the two-qubit pure states. The three spheres are named the base sphere, entanglement sphere, and fiber sphere. The base sphere and entanglement sphere represent the reduced density matrix of the base qubit and the non-local entanglement measure, concurrence, while the fiber sphere represents the fiber qubit via a simple rotation under a local single-qubit unitary operation; however, in an entangled bipartite state, the fiber sphere has no information on the reduced density matrix of the fiber qubit. When the bipartite state becomes separable, the base …
So far there is no closed formula for relative entropy of entanglement of arbitrary two-qubit states. In this paper we present a method, which guarantees the derivation of the relative entropy of entanglement for most states that have $z$-directional Bloch vectors. It is shown that the closest separable states for those states also have $z$-directional Bloch vectors though there are few exceptions.
Entanglement is an important concept in quantum information, quantum communication, and quantum computing. We provide a geometrical analysis of entanglement and separability for all the rank-2 quantum mixed states: complete analysis for the bipartite states, and partial analysis for the multipartite states. For each rank-2 mixed state, we define its unique Bloch sphere, that is spanned by the eigenstates of its density matrix. We characterize those Bloch spheres into exactly five classes of entanglement and separability, give examples for each class, and prove that those are the only classes.
Towards Two Bloch Sphere Representation of Pure Two Qubit States and Unitaries
本文提出双 Bloch 球几何表示纯双量子比特态，利用坐标轴相对方向描述最大纠缠态，并给出 CNOT 门图形表示及基于球面面积的纠缠度量。
Bloch Sphere of the Qutrit System
本文利用 SU(3) 群和 Bloch 轨迹研究 qutrit 系统的 Bloch 空间，发现共振条件下 Bloch 球 S⁷ 分裂为 S⁴ 与 S²，并揭示了其非平凡几何结构。
For one qubit systems, we present a short, elementary argument characterizing unital quantum operators in terms of their action on Bloch vectors. We then show how our approach generalizes to multi-qubit systems, obtaining inequalities that govern when a ``diagonal'' superoperator on the Bloch region is a quantum operator. These inequalities are the n-qubit analogue of the Algoet-Fujiwara conditions. Our work is facilitated by an analysis of operator-sum decompositions in which negative summands are allowed.
We have considered a system of two superconducting charge qubits capacitively coupled to a microwave resonator. The dynamics of the Bloch vectors are investigated for different regimes. By means of the Bloch vectors and cross dyadic we quantify the degree of entanglement contained in the generated entangled state. We consider different values of the system parameters to discuss the dynamics of the channel capacity between the qubits. We show that there is an important role played by initial state settings, coupling constant and the mean photon number on generating entangled state with high deg…
For a quantum system governed by a non-Hermitian Hamiltonian, we studied the problem of obtaining an optimum Hamiltonian that generates nonunitary transformations of a given initial state into a certain final state in the smallest time $τ$. The analysis is based on the relationship between the states of the two-dimensional subspace of the Hilbert space spanned by the initial and final states and the points of the two-dimensional complex Bloch sphere.
We find an optimal quantum cloning machine, which clones qubits of arbitrary symmetrical distribution around the Bloch vector with the highest fidelity. The process is referred to as phase-independent cloning in contrast to the standard phase-covariant cloning for which an input qubit state is a priori better known. We assume that the information about the input state is encoded in an arbitrary axisymmetric distribution (phase function) on the Bloch sphere of the cloned qubits. We find analytical expressions describing the optimal cloning transformation and fidelity of the clones. As an illust…
In this paper we present a simple algorithm for representation of statistical data of any origin by complex probability amplitudes. Numerical simulation with Mathematica-6 is performed. The Bloch's sphere is used for visualization of results of numerical simulation. On the one hand, creation of such a quantum-like (QL) representation and its numerical approval is an important step in clarification of extremely complicated interrelation between classical and quantum randomness. On the other hand, it opens new possibilities for application the mathematical formalism of QM in other domains of sci…
The properties of the geometric phases between three quantum states are investigated in a high-dimensional Hilbert space using the Majorana representation of symmetric quantum states. We found that the geometric phases between the three quantum states in an N-state quantum system can be represented by N-1 spherical triangles on the Bloch sphere. The parameter dependence of the geometric phase was analyzed based on this picture. We found that the geometric phase exhibits rich nonlinear behavior in a high-dimensional Hilbert space.
From Majorana to Bloch: two ways to generalize the Bloch sphere representation
该研究对比纯qudit态的马约拉纳表示与广义布洛赫表示，推导两者间的解析关系，还分析其在对称多量子比特态的扩展，为研究两量子比特态的量子关联提供几何视角。
The Geometry of Qubit Decoherence: Linear vs. Nonlinear Dynamics in the Bloch Ball
本文通过线性与非线性方法分析量子比特退相干的几何特性，揭示了布洛赫球中混合态轨迹与SU(2)对称性对开放量子比特动力学的影响。
Correlation constraints and the Bloch geometry of two qubits
本文提出仅依赖局域 Bloch 向量长度差的两体态纯度不等式，紧化 2-qubit 边缘问题，并构建三维 Bloch 模型以可视化态空间及局域—关联性质。
An arbitrarily dense discretisation of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretised representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretised spheres) entanglement. Unlike Meyer's earlier discretisation of the Bloch Sphere, there are no orthonormal t…
In a direct two-level qubit system, when the Rabi frequency is comparable to the resonance frequency, the rotating wave approximation is not appropriate. In this case, the Rabi oscillation is accompanied by another oscillation at twice the frequency of the driving field, the so called Bloch-Siegert oscillation (BSO), which depends on the initial phase of the driving field. This oscillation may restrict the precise rotation of a qubit made of direct two-level system. Here, we show that in case of an effectively two-level lambda system, the BSO is inherently negligible, implying a greater precis…
Understanding the evolution of a multi-qubit quantum system, or elucidating what portion of the Hilbert space is occupied by a quantum dataset becomes increasingly hard with the number of qubits. In this context, the visualisation of sets of multi-qubit pure quantum states on a single image can be helpful. However, the current approaches to visualization of this type only allow the representation of a set of single qubits (not allowing multi-qubit systems) or a just a single multi-qubit system (not suitable if we care about sets of states), sometimes with additional restrictions, on symmetry o…
For many implementations of quantum computing, 1/f and other types of broad-spectrum noise are an important source of decoherence. An important step forward would be the ability to back out the characteristics of this noise from qubit measurements and to see if it leads to new physical effects. For certain types of qubits, the working point of the qubit can be varied. Using a new mathematical method that is suited to treat all working points, we present theoretical results that show how this degree of freedom can be used to extract noise parameters and to predict a new effect: noise-induced lo…
A two-sphere ("Bloch" or "Poincare") is familiar for describing the dynamics of a spin-1/2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various sub-groups so that fewer parameters suffice, we derive Bloch-like rotatio…
In this study for particular states of bipartite quantum system in 2n?2m dimensional Hilbert space state, similar to m or n-qubit density matrices represented in Bloch sphere we call them generalized Bloch sphere states(GBSS), we give an efficient optimization procedure so that analytic evaluation of quantum discord can be performed. Using this optimization procedure, we find an exact analytical formula for the optimum positive operator valued measure (POVM) that maximize the measure of the classical correlation for these states. The presented optimization procedure also is used to show that f…
CALA-$n$: A Quantum Library for Realizing Cost-Effective 2-, 3-, 4-, and 5-bit Gates on IBM Quantum Computers using Bloch Sphere Approach, Clifford+T Gates, and Layouts
本文提出基于布洛赫球和Clifford+T门的量子布局感知方法，构造2至5量子比特的高成本效益n位门，并证明其转译后量子成本始终低于标准n位门，可用于构建IBM量子计算机量子库。
Extending the Bloch sphere model to an N-qubit system
研究将布洛赫球表示扩展到任意N量子比特系统的问题，提出用2^N - 1个球的新几何模型，通过特定旋转门应用建立参数化，证明与标准态矢表示的严格双射，为量子模拟等提供优势。
Quantum teleportation is the transfer of quantum information between two locations by the use of shared entanglement. Current teleportation schemes broadly fall under one of two categories, of either qubit or continuous variables teleportation. Spin coherent states on spin ensembles can be teleported under the continuous variables approximation for states with small deviations from a given polarization. However, for spin coherent states with large deviations, no convenient teleportation protocol exists. Recently, we introduced a teleportation scheme where a spin coherent state lying on the equ…
The Bloch sphere is a familiar and useful geometrical picture of the dynamics of a single spin or two-level system's quantum evolution. The analogous geometrical picture for three-level systems is presented, with several applications. The relevant SU(3) group and su(3) algebra are eight-dimensional objects and are realized in our picture as two four-dimensional manifolds describing the time evolution operator. The first, called the base manifold, is the counterpart of the S^2 Bloch sphere, whereas the second, called the fiber, generalizes the single U(1) phase of a single spin. Now four-dimens…
Bloch sphere analog of qudits using Heisenberg-Weyl Operators
本研究基于Heisenberg-Weyl算符基提出多能级量子系统布洛赫矢量参数化方法，以三能级qutrit为例分析其非实心布洛赫球结构，并将方法推广到qudit，可用于量子基、量子映射及系综评估等研究。
Geometric Bloch Vector Solution to Minimum Error Discriminations of Mixed Qubit States
本研究针对混合量子比特态的最小误差辨识问题，提出基于几何布洛赫矢量的四步结构化方法，引入拉格朗日算子与几何POVM类别，可求解任意先验概率下N个混合量子比特态的所有最优辨识解，含4量子比特态的新结果。
As is well known, when an SU(2) operation acts on a two-level system, its Bloch vector rotates without change of magnitude. Considering a system composed of two two-level systems, it is proven that for a class of nonlocal interactions of the two subsystems including σ_i\otimesσ_j (with i,j \in {x,y,z}) and the Heisenberg interaction, the geometric description of the motion is particularly simple: each of the two Bloch vectors follows an elliptical orbit within the Bloch sphere. The utility of this result is demonstrated in two applications, the first of which bears on quantum control via quant…
The time evolution of a two-level quantum mechanical system can be geometrically described using the Bloch sphere. By mapping the Bloch sphere evolution onto the dynamics of oscillating electric dipoles, we provide a physically intuitive link between classical electromagnetism and the electric dipole transitions of atomic & molecular physics.
Charging a quantum battery from the Bloch sphere
研究通过分析双组分量子电池模型，揭示了充电过程中量子相干性和人口倒置对能量存储和ergotropy的影响，并指出热力学考虑对充电功率和时间的重要性。
Non-Hermitian Generalization of Bloch Sphere in Spacetime Algebra
该研究利用时空代数建立了非厄米两能级量子系统的布洛赫球几何推广，将态空间扩展为未来光锥，分类了哈密顿量动力学类型并研究了PT对称量子力学的异常点拓扑特征。
A Geometrical Design Tool for Building Cost-Effective Layout-Aware n-Bit Quantum Gates Using the Bloch Sphere Approach
本文提出基于布洛赫球的几何设计方法，用于构建低成本的n位量子门，通过可视化几何交集替代传统矩阵乘法，降低量子成本。
We show that if the Rabi frequency is comparable to the Bohr frequency so that the rotating wave approximation is inappropriate, an extra oscillation is present with the Rabi oscillation. We discuss how the sensitivity of the degree of excitation to the phase of the field may pose severe constraints on precise rotations of quantum bits involving low-frequency transitions. We present a scheme for observing this effect in an atomic beam.
Clauser-Horne-Shimony-Holt Bell-inequality Violability with the Full Poincaré-Bloch Sphere
本文研究了使用相同或不同线性和椭圆偏振基态对光子进行Bell和非Bell状态时，CHSH不等式违反的实验验证，发现Bell态在相同基态下违反不等式，而非Bell态则相反。
The spinor representation of spin-1/2 states can equally well be mapped to a single unit quaternion, yielding a new perspective despite the equivalent mathematics. This paper first demonstrates a useable map that allows Bloch-sphere rotations to be represented as quaternionic multiplications, simplifying the form of the dynamical equations. Left-multiplications generally correspond to non-unitary transformations, providing a simpler (essentially classical) analysis of time-reversal. But the quaternion viewpoint also reveals a surprisingly large broken symmetry, as well as a potential way to re…
Spin Hamiltonian with large fourth order terms: Triple well potentials and Bloch sphere visualization
研究了含四阶项的广义自旋Hamiltonian，利用相干态与突变理论构建参数空间，发现大四阶参数下存在三势阱区域，施加振荡磁场可引发多次基态量子相变，并实现了基态在Bloch球上局域化的可视化。
Finding control fields (pulse sequences) that can compensate for the dispersion in the parameters governing the evolution of a quantum system is an important problem in coherent spectroscopy and quantum information processing. The use of composite pulses for compensating for dispersion in system dynamics is widely known and applied. In this paper, we introduce new pulse elements for correcting pulse errors. These design methods are analytical and can be used to prove arbitrarily good robust performance. Furthermore, the time-to-compensation is superior to existing Fourier Synthesis Methods whi…
Composite pulse sequences, which produce arbitrary pre-defined rotations of a two-state system at an angle $θ$ on the Bloch sphere, are presented. The composite sequences can contain arbitrarily many pulses and can compensate experimental errors in the pulse amplitude and duration to any desired order. A special attention is devoted to two classes of $π/2$ sequences --- symmetric and asymmetric --- the phases of which are given by simple formulas in terms of rational multiples of $π$ for any number of constituent pulses. This allows one to construct arbitrarily accurate $π/2$ composite rotatio…
By use of a generalized Bloch vector construction, we study the decoherence of a system composed of two interacting qubits in a general noisy environment. In particular, we investigate the effects of correlations in the noise acting on distinct qubits. Our treatment of the two-qubit system by use of the generalized Bloch vector leads to tractable analytic equations for the dynamics of the 4-level Bloch vector and allows for the application of geometrical concepts from the well known 2-level Bloch sphere. We find that in the presence of correlated or anticorrelated noise, the rate of decoherenc…
We discuss a generalization to 2 qubits of the standard Bloch sphere representation for a single qubit, in the framework of Hopf fibrations of high dimensional spheres by lower dimensional spheres. The single qubit Hilbert space is the 3-dimensional sphere S3. The S2 base space of a suitably oriented S3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the qubit overall phase degree of freedom. For the two qubits case, the Hilbert space is a 7-dimensional sphere S7, which also allows for a Hopf fibration, with S3 fibres and a S4 base. A main striking result is…
We have developed a formalism suitable for calculation of the output spectrum of a detector continuously measuring quantum coherent oscillations in a solid-state qubit, starting from microscopic Bloch equations. The results coincide with that obtained using Bayesian and master equation approaches. The previous results are generalized to the cases of arbitrary detector response and finite detector temperature.
1-qubit quantum states form a space called the three-dimensional Bloch ball. To compute Holevo capacity, Voronoi diagrams in the Bloch ball with respect to the quantum divergence have been used as a powerful tool. These diagrams basically treat mixed quantum states corresponding to points in the interior of the Bloch ball. Due to the existence of logarithm in the quantum divergence, the diagrams are not defined on pure quantum states corresponding to points on the two-dimensional sphere. This paper first defines the Voronoi diagrams for pure quantum states on the Bloch sphere by the Fubini-Stu…
Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-establish…
A Quantum Dynamics Tutorial: Visualising Dynamical Decoupling Sequences on the Bloch Sphere
本教程从实验视角出发，借助布洛赫球可视化动力学去耦序列的量子动力学，阐明相干时间定义，提供开放脚本以助力新研究者掌握该量子控制工具。
Calibrated Helstrom geometry on the Bloch ball via Connes spectral distance
本文通过有限标量-量子比特-标量模型，证明量子比特态的等先验Helstrom迹距离几何可从Connes谱距离恢复，标量参考扇区通过恒等Dirac链接各向同性耦合到量子比特块，使完整Bloch球继承标准弦迹距离几何，标量扇区距离起校准作用。
A Bloch sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch sphere were suggested by [M. Boyer, R. Liss, T. Mor, PRA 95, 032308 (2017)]. Here we construct a Bloch sphere for the Hilbert space spanned by one of the ground states of Kitaev's toric code model and one of its closest product states. We prove that this sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the …
Quantum registers that combine the attractive properties of different types of qubits are useful for many different applications. They also pose a number of challenges, often associated with the large differences in coupling strengths between the different types of qubits. One example is the non-resonant effect that alternating electromagnetic fields have on the transitions of qubits that are not targeted by the specific gate operation. The example being studied here is known as Bloch-Siegert shift. Unless these shifts are accounted for and, if possible, compensated, they can completely destro…
Nonlinear and non-CP gates for Bloch vector amplification
本文研究从布洛赫球中心制备量子比特态的互补方案，提出基于线性和非线性马尔可夫PTP信道的布洛赫矢量放大方法，克服了线性CPTP信道的临界减速问题，并可用于实验演示非CP动力学。
We consider here the predictions of quantum theory and local hidden variables for the correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by a given angle θ. The predictions of local hidden variable models for projective measurements on qubits correspond to binary colourings of the Bloch sphere with antipodal points oppositely coloured. We prove Bell inequalities separating the predictions of all local hidden variable models from the singlet correlations predicted by quantum theory for all θin the range 0 < θ< π/3. We raise and explore t…
We analyze qubit decoherence in the framework of geometric quantum mechanics. In this framework the qubit density operators are represented by probability distributions which are also the Kähler functions on the Bloch sphere. Interestingly, the complete positivity of the quantum evolution is recovered as ellipticity of the second order differential operator (deformed Laplacian) which governs the evolution of the probability distribution.
Aspects of Complexity in Quantum Evolutions on the Bloch Sphere
本文研究了Bloch球上时间最优与次优量子哈密顿演化的复杂性，通过路径长度和曲率等度量评估演化，发现高效演化复杂性更低，但复杂性并非仅由长度决定，适当弯曲的较长轨迹复杂性可能不高于较短轨迹。
Why qubits are exceptional in Gleason's theorem: a Bloch-space perspective
本文通过布洛赫空间视角简化格莱森定理，揭示二维系统中非Born概率规则的存在及高维系统中其不可能性，强调Born规则在高维下的不可避免性及量子比特的特殊性。
A qubit chosen from equatorial or polar great circles on a Bloch sphere can be remotely prepared with an Einstain-Podolsky-Rosen (EPR) state shared and a cbit communication. We generalize this protocal into an arbitrary longitudinal qubit on the Bloch sphere in which the azimuthal angle phi can be an arbitrary value instead of only being zero. The generalized scheme was experimentally realized using liquid-state nuclear magnetic resonance (NMR) techniques. Also, we have experimentally demonstrated remote state measurement (RSM) on an arbitary qubit proposed by Pati.
Quantum manipulation protocols for quantum sensors and quantum computation often require many single qubit rotations. However, the impact of phase noise in the field that performs the qubit rotations is often neglected or treated only for special cases. We present a general framework for calculating the impact of phase noise on the state of a qubit, as described by its equivalent Bloch vector. The analysis applies to any Bloch vector orientation, and any rotation axis azimuthal angle for both a single pulse, and pulse sequences. Experimental examples are presented for several special cases. We…
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