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Bloch sphere

The Bloch sphere is a geometric representation of the pure quantum states of a two-level system, such as a qubit, where each state corresponds to a point on the surface of a unit sphere in three-dimensional Euclidean space.^[1] Introduced by physicist Felix Bloch in 1946 as part of his foundational work on nuclear magnetic resonance and the dynamics of spin systems, it provides an intuitive visualization for the state space of spin-1/2 particles or qubits.^[2]^[3] Mathematically, any pure qubit state can be expressed in the form |\psi\rangle = \cos(\theta/2) |0\rangle + e^{i\phi} \sin(\theta/2) |1\rangle, where \theta \in [0, \pi] is the polar angle from the positive z-axis and \phi \in [0, 2\pi) is the azimuthal angle in the xy-plane; this parametrization directly maps the state to a point on the sphere's surface, with the north pole (\theta = 0) representing the basis state |0\rangle and the south pole (\theta = \pi) representing |1\rangle.^[4] Orthogonal states lie at antipodal points on the sphere, and the overall phase factor has no physical significance due to its unobservability in measurements.^[1] The sphere's coordinates are tied to the expectation values of the Pauli matrices, forming the Bloch vector \vec{r} = (\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta), which encodes the state's polarization along the x, y, and z directions.^[5] For mixed states, described by density matrices rather than pure state vectors, the representation extends to the interior of the sphere, known as the Bloch ball, where points inside the unit sphere (|\vec{r}| < 1) correspond to statistical mixtures, with the center (\vec{r} = 0) denoting the maximally mixed state \rho = I/2.^[5] In quantum information science, the Bloch sphere is instrumental for visualizing single-qubit operations: unitary transformations, such as quantum gates, manifest as rotations of the Bloch vector around the sphere, enabling the decomposition of arbitrary single-qubit unitaries into rotations about the coordinate axes.^[4] This framework originated in the context of nuclear induction but has become central to quantum computing, quantum optics, and spin dynamics, facilitating the analysis of coherence, entanglement, and measurement outcomes.^[6]

Background

Historical Development

The concept of representing quantum spin states geometrically traces its roots to the early development of quantum mechanics in the 1920s and 1930s, particularly through the work on spin-1/2 systems. Wolfgang Pauli introduced the Pauli matrices in 1927 as a mathematical framework to describe the spin angular momentum of electrons in the hydrogen atom, providing a two-dimensional complex vector space for these states. This representation laid foundational groundwork for visualizing spin precession and transformations, influencing later geometric interpretations of quantum states. Building on Pauli's exclusion principle from 1925 and the proposal of electron spin by George Uhlenbeck and Samuel Goudsmit in the same year, these efforts established the algebraic tools essential for two-level quantum systems. The Bloch sphere emerged directly from Felix Bloch's pioneering work in nuclear magnetic resonance (NMR) in 1946. In his theoretical paper on nuclear induction, Bloch developed phenomenological equations describing the macroscopic magnetization vector of nuclear spins in a magnetic field, which precesses around the field direction much like a classical magnetic moment.^[3] This vector model, confirmed experimentally in water and paraffin samples, provided an intuitive way to track the evolution of spin ensembles under radiofrequency pulses, effectively parameterizing the state of a two-level system with three real coordinates.^[7] Although Bloch did not explicitly depict a spherical geometry, his equations implied a bounded vector space that later crystallized into the unit sphere representation for pure states. This work earned Bloch the 1952 Nobel Prize in Physics, shared with Edward M. Purcell, for their discoveries concerning nuclear magnetic resonance in solids.^[8] The explicit naming and visualization of the "Bloch sphere" as a geometric tool occurred in 1972, when Charles Stroud and colleagues used it to illustrate superradiant effects in two-level atomic systems, crediting the underlying model to Bloch's NMR framework. This period marked a bridge between classical spin dynamics and quantum optics. Following the emergence of quantum information theory in the 1980s—sparked by Richard Feynman's 1982 proposal for simulating quantum systems on computers—the Bloch sphere evolved into a standard pedagogical and analytical tool for representing qubit states in quantum computing. Its adoption facilitated intuitive understanding of quantum gates as rotations on the sphere, solidifying its role in the field. A brief connection to density operators, introduced by John von Neumann in 1927 for mixed states, later generalized the sphere's interior to the Bloch ball.

Physical Interpretation

The Bloch sphere provides a geometric representation for the state of a spin-1/2 particle, such as an electron or a nuclear spin, interacting with an external magnetic field. In this context, the position of the Bloch vector on or within the sphere corresponds to the quantum state of the particle, where the vector's direction indicates the orientation of the spin relative to the field.^[9]^[10] The Bloch vector specifically aligns with the expectation value of the spin angular momentum operator for the particle, capturing the average direction in which the spin is likely to be measured along the three spatial axes. This expectation value, scaled appropriately, points to a location on the unit sphere, allowing visualization of how the spin state evolves under the influence of the magnetic field, such as through precession around the field lines.^[10]^[9] On the Bloch sphere, the north pole conventionally represents the spin-up state along the z-axis (aligned with a positive magnetic field direction), while the south pole denotes the spin-down state (anti-aligned). These poles illustrate the basis states for measurements along the field axis, with intermediate points reflecting superpositions or partial alignments.^[10]^[11]^[12] This quantum description draws a direct analogy to the classical behavior of a magnetic dipole, such as a nuclear spin ensemble in nuclear magnetic resonance (NMR), where the Bloch equations describe the precession of the macroscopic magnetization vector around the applied magnetic field. In NMR, the classical dipole's torque-induced motion mirrors the quantum spin dynamics on the Bloch sphere, providing an intuitive bridge between microscopic quantum effects and observable macroscopic signals.^[9]^[13]

Mathematical Foundations

Formal Definition

The Bloch sphere is the unit sphere in three-dimensional real space that parameterizes the projective Hilbert space \mathbb{CP}^1 for a qubit, representing the geometry of pure quantum states up to a global phase factor.^[14] This identification arises from the fact that the state space of a qubit, which is a two-dimensional complex Hilbert space \mathbb{C}^2, has its pure states forming rays under scalar multiplication by unit-complex numbers, and the Bloch sphere provides a faithful geometric embedding of this projective space.^[14] Pure states on the Bloch sphere are parameterized in spherical coordinates by the general form

|\psi\rangle = \cos\left(\frac{\theta}{2}\right) |0\rangle + e^{i\phi} \sin\left(\frac{\theta}{2}\right) |1\rangle,

where \theta \in [0, \pi] is the polar angle measuring deviation from the north pole (|0\rangle), \phi \in [0, 2\pi) is the azimuthal angle in the equatorial plane, and |0\rangle, |1\rangle form the standard computational basis. The sphere has radius 1, ensuring normalization of the states, with the north pole corresponding to |0\rangle (\theta = 0), the south pole to |1\rangle (\theta = \pi), and the equator to balanced superpositions.^[14] Each point on this surface uniquely represents a ray in \mathbb{C}^2 modulo phase, capturing the essential degrees of freedom for a qubit's pure state.^[14] The Cartesian coordinates of a point on the Bloch sphere are linked to the expectation values of the Pauli matrices, providing a vector representation \mathbf{r} = (\langle \sigma_x \rangle, \langle \sigma_y \rangle, \langle \sigma_z \rangle) with |\mathbf{r}| = 1 for pure states.^[14]

Relation to Pauli Matrices

The Bloch vector \vec{r} = (r_x, r_y, r_z) provides a means to represent the state of a qubit through its components, defined as r_i = \operatorname{Tr}(\rho \sigma_i) for i = x, y, z, where \rho is the density matrix of the qubit and \sigma_i are the Pauli matrices. This trace-based construction captures the expectation values of the Pauli observables, linking the abstract operator formalism to a three-dimensional vector space. The Pauli matrices form a basis for the space of 2×2 Hermitian traceless matrices and are explicitly given by

\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

These matrices satisfy the commutation relations [\sigma_i, \sigma_j] = 2i \epsilon_{ijk} \sigma_k and anticommutation relations \{\sigma_i, \sigma_j\} = 2 \delta_{ij} I, where I is the 2×2 identity matrix and \epsilon_{ijk} is the Levi-Civita symbol. Their Hermitian and unitary properties ensure that the Bloch vector components r_i are real numbers between -1 and 1. For pure states, where \rho = |\psi\rangle\langle\psi| with |\psi\rangle a normalized state vector, the magnitude of the Bloch vector satisfies |\vec{r}| = 1. This unit length condition geometrically corresponds to points on the surface of the Bloch sphere.

Representation of Quantum States

Pure States

Pure states of a qubit represent coherent superpositions of the computational basis states |0\rangle and |1\rangle, and they correspond precisely to points on the surface of the unit Bloch sphere, where the magnitude of the Bloch vector satisfies |\vec{r}| = 1.^[15] This mapping arises from the expectation values of the Pauli operators, ensuring that pure states are fully characterized by their position on the sphere's boundary.^[16] A general pure qubit state can be parameterized as |\psi\rangle = \cos(\theta/2) |0\rangle + e^{i\phi} \sin(\theta/2) |1\rangle, where \theta \in [0, \pi] and \phi \in [0, 2\pi). This parameterization yields the Bloch vector components r_x = \sin\theta \cos\phi, r_y = \sin\theta \sin\phi, and r_z = \cos\theta, which fully specify the state's location on the sphere.^[16] These coordinates reflect the spherical geometry, with \theta determining the polar angle from the north pole (corresponding to |0\rangle) and \phi the azimuthal angle.^[17] The Bloch sphere representation accounts for the physical equivalence of states differing by a global phase factor e^{i\alpha}, as such phases do not affect observables; thus, the sphere embodies the quotient space \mathbb{CP}^1, the projective Hilbert space for a two-level system.^[18] This quotient structure ensures a one-to-one correspondence between distinct pure states (up to phase) and points on the unit sphere.^[18] In contrast, mixed states occupy the interior of the Bloch ball with |\vec{r}| < 1.^[15]

Mixed States

The density operator \rho for a qubit in a mixed state admits a parametrization in terms of the Bloch vector \vec{r} \in \mathbb{R}^3 as

\rho = \frac{1}{2} \left( I + \vec{r} \cdot \vec{\sigma} \right),

where I is the $2 \times 2 identity matrix and \vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z) denotes the vector of Pauli matrices.

\] This expression inherently satisfies the requirements for a valid density operator: $\rho$ is Hermitian, positive semi-definite, and has unit trace $\mathrm{Tr}(\rho) = 1$, provided that $|\vec{r}| \leq 1$.\[

The condition |\vec{r}| \leq 1 ensures positivity, with the bound corresponding to the unit ball in three-dimensional real space, termed the Bloch ball.$$] Points within the interior of the Bloch ball, where |\vec{r}| < 1, represent mixed quantum states that arise as statistical ensembles of pure states.[

Such states capture incomplete knowledge or decoherence effects, lacking the maximal [coherence](/page/Coherence) of pure states. For instance, the [origin](/page/Origin) $\vec{r} = \vec{0}$ depicts the completely mixed state $\rho = \frac{1}{2} I$, which corresponds to a uniform probabilistic mixture over the basis states of any [orthonormal basis](/page/Orthonormal_basis), maximizing [entropy](/page/Entropy) for a [qubit](/page/Qubit).

] The collection of all valid density operators for a qubit constitutes a convex set in the space of $2 \times 2 Hermitian matrices with unit trace.[

Consequently, every mixed state can be decomposed as a [convex combination](/page/Convex_combination) $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$ of pure states, where the probabilities satisfy $\sum_i p_i = 1$ and $p_i \geq 0$ for all $i$; the pure states $|\psi_i\rangle$ lie on the surface of the Bloch sphere.

] Geometrically, this convexity property implies that the Bloch ball is precisely the convex hull of the sphere's surface points, with interior points arising as weighted averages thereof.[

The components of $\vec{r}$ are extracted from $\rho$ using the traces with [Pauli matrices](/page/Pauli_matrices), $r_i = \mathrm{Tr}(\rho \sigma_i)$ for $i = x, y, z$.

]

Visualization Techniques

Stereographic Projection

The stereographic projection provides a method to map pure states on the Bloch sphere onto the complex plane, facilitating alternative visualizations of qubit states. For a pure state parameterized by polar angle θ and azimuthal angle φ on the sphere, the projection is performed from the south pole (corresponding to the state |1⟩), yielding a complex coordinate z = tan(θ/2) e^{iφ}.^[19] This mapping bijectively associates each point on the sphere's surface—excluding the south pole, which is sent to infinity—with a unique point in the extended complex plane, effectively representing the projective space of two-dimensional spinors.^[19] The inverse transformation recovers the spherical coordinates from z as θ = 2 arctan(|z|) and φ = arg(z), directly linking the complex plane coordinates to the spinor components of the quantum state.^[19] This relation underscores the projection's utility in connecting the geometry of the Bloch sphere to the complex structure of qubit wavefunctions, where the ratio of the lower to upper spinor amplitude is given by z.^[20] As a conformal mapping, the stereographic projection preserves local angles, making it particularly advantageous for visualizing geometric properties of quantum states without distorting infinitesimal structures.^[21] Furthermore, it offers insights into the Hopf fibration, revealing the Bloch sphere as a fiber bundle over the complex plane that encodes the topology of SU(2) projective representations.^[22]

Cartesian Coordinates

The Cartesian coordinates on the Bloch sphere provide a direct real-vector representation of qubit states, where the Bloch vector \vec{r} = (u, v, w) corresponds to the components u = r_x, v = r_y, w = r_z, with the constraint u^2 + v^2 + w^2 \leq 1 ensuring the vector lies within or on the unit sphere.^[10] For pure states, equality holds (u^2 + v^2 + w^2 = 1), placing the vector on the surface, while mixed states occupy the interior.^[23] This parameterization allows straightforward geometric interpretation without relying on complex amplitudes. These coordinates are intimately linked to quantum expectation values, specifically u = \langle \sigma_x \rangle, v = \langle \sigma_y \rangle, and w = \langle \sigma_z \rangle, where \sigma_x, \sigma_y, \sigma_z are the Pauli matrices and the expectation values are computed as \langle \sigma_i \rangle = \operatorname{Tr}(\rho \sigma_i) for the density matrix \rho.^[1] This relation enables direct extraction of the Bloch vector from a given wavefunction or density operator by evaluating these traces, providing a practical bridge between abstract quantum states and measurable observables.^[10] In numerical simulations, the Cartesian form facilitates efficient vector algebra operations, such as computing rotations or evolutions as simple vector manipulations on or within the sphere, which is particularly advantageous for modeling single-qubit dynamics in quantum algorithms and error analysis.^[23] For instance, unitary transformations correspond to rotations of the vector \vec{r}, allowing simulations to track state evolution using standard linear algebra tools without full matrix exponentiation.^[1]

Unitary Rotations

Rotations about Principal Axes

In quantum mechanics, rotations about the principal axes of the Bloch sphere are implemented by unitary operators generated by the Pauli matrices \sigma_x, \sigma_y, and \sigma_z. These operators, denoted R_x(\theta), R_y(\theta), and R_z(\theta), respectively, are defined as R_\alpha(\theta) = \exp\left(-i \frac{\theta}{2} \sigma_\alpha\right) for \alpha = x, y, z, where \theta is the rotation angle.^[24]^[25] The explicit matrix forms in the standard computational basis are: [ R_x(\theta) = \begin{pmatrix} \cos(\theta/2) & -i \sin(\theta/2) \ -i \sin(\theta/2) & \cos(\theta/2) \end{pmatrix},

R_y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix},

R_z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \ 0 & e^{i\theta/2} \end{pmatrix}.

These derive from the [exponential map](/page/Exponential_map) in the [Lie algebra](/page/Lie_algebra) su(2), with the [Pauli matrices](/page/Pauli_matrices) serving as the generators.[](https://arxiv.org/pdf/2203.12943.pdf) When applied to a qubit state, these rotations act on the corresponding Bloch vector $\vec{r}$ by rotating it by angle $\theta$ around the respective principal axis (x, y, or z), equivalent to an SO(3) transformation that preserves the vector's length (and thus the radius within the Bloch ball).[](https://arxiv.org/pdf/quant-ph/0701140.pdf) For the z-axis rotation $R_z(\theta)$, this manifests as a relative [phase](/page/Phase) shift between the basis states $|0\rangle$ and $|1\rangle$, rotating the Bloch vector in the xy-plane without altering its polar angle.[](https://arxiv.org/pdf/2203.12943.pdf) Such operations maintain the unitarity of the transformation, ensuring no change in the state's purity for pure states on the sphere's surface.[](https://arxiv.org/pdf/quant-ph/0701140.pdf) ### Rotations about Arbitrary Axes In [quantum mechanics](/page/Quantum_mechanics), rotations of a qubit state on the Bloch sphere about an arbitrary [axis](/page/Axis) are described by the [unitary operator](/page/Unitary_operator) $ U(\vec{n}, \theta) = \exp\left(-i \frac{\theta}{2} \vec{n} \cdot \vec{\sigma}\right) $, where $\vec{n}$ is a [unit vector](/page/Unit_vector) specifying the rotation [axis](/page/Axis), $\theta$ is the rotation angle, and $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$ denotes the vector of [Pauli matrices](/page/Pauli_matrices).[](https://arxiv.org/pdf/2203.12943.pdf) This operator generates a rotation of the Bloch vector by angle $\theta$ around $\vec{n}$, preserving the unit sphere's geometry and corresponding to an element of the special unitary group SU(2).[](https://arxiv.org/pdf/2203.12943.pdf) This formulation generalizes rotations about the principal [axes](/page/Axis) (x, y, z), where $\vec{n}$ aligns with one of the standard basis vectors.[](https://arxiv.org/pdf/2203.12943.pdf) The effect of this unitary on the Bloch vector $\vec{r}$ (representing the expectation values of the Pauli operators for the [qubit](/page/Qubit) state) is a rigid [rotation](/page/Rotation) in [three-dimensional space](/page/Three-dimensional_space). Specifically, the transformed vector $\vec{r}'$ is given by [Rodrigues' rotation formula](/page/Rodrigues'_rotation_formula):

\vec{r}' = \vec{r} \cos\theta + (\vec{n} \times \vec{r}) \sin\theta + \vec{n} (\vec{n} \cdot \vec{r}) (1 - \cos\theta).

[](https://www.phys.hawaii.edu/~yepez/Spring2013/lectures/Lecture1_Qubits_Notes.pdf) This vector equation directly maps the action of $U(\vec{n}, \theta)$ on the state to a [geometric transformation](/page/Geometric_transformation) on the sphere, ensuring that pure states remain on the surface and the rotation is orthogonal.[](https://www.phys.hawaii.edu/~yepez/Spring2013/lectures/Lecture1_Qubits_Notes.pdf) A key feature of these rotations arises from the group structure: SU(2) provides a double cover of the [rotation](/page/Rotation) group SO(3), meaning that a physical [rotation](/page/Rotation) by angle $\theta$ on the Bloch sphere corresponds to a [phase factor](/page/Phase_factor) involving $2\theta$ in the [spinor](/page/Spinor) ([state vector](/page/State_vector)) representation. Consequently, rotations by $\theta$ and $\theta + 2\pi$ yield equivalent Bloch vectors, but the underlying SU(2) elements differ by a sign, reflecting the non-trivial [topology](/page/Topology) of the double cover. This property is essential for understanding the periodicity and [homotopy](/page/Homotopy) of qubit evolutions under arbitrary-axis rotations. ### Derivation of Rotation Generators The time evolution of a single-qubit state under a Hamiltonian $ H = \frac{\omega}{2} \vec{n} \cdot \vec{\sigma} $, where $ \vec{n} $ is a [unit vector](/page/Unit_vector) specifying the rotation axis, $ \omega $ is the Larmor frequency, and $ \vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z) $ denotes the vector of [Pauli matrices](/page/Pauli_matrices), is governed by the unitary operator $ U(t) = \exp\left( -i H t \right) = \exp\left( -i \frac{\omega t}{2} \vec{n} \cdot \vec{\sigma} \right) $, with $ \hbar = 1 $.[](https://arxiv.org/pdf/2112.07100) This Hamiltonian form arises in contexts such as a spin-1/2 particle in a magnetic field aligned along $ \vec{n} $, inducing precessional dynamics.[](https://arxiv.org/pdf/2112.07100) To evaluate the exponential, note that the operator $ \vec{n} \cdot \vec{\sigma} $ satisfies $ (\vec{n} \cdot \vec{\sigma})^2 = I $, the 2×2 [identity matrix](/page/Identity_matrix), due to the algebraic properties of the [Pauli matrices](/page/Pauli_matrices).[](http://web.cecs.pdx.edu/~mperkows/june2007/bloch-sphere.pdf) Consequently, the Taylor series expansion simplifies via [Euler's formula](/page/Euler's_formula) analog for matrices:

\exp\left( -i \theta , \vec{n} \cdot \vec{\sigma} \right) = \cos \theta , I - i \sin \theta , (\vec{n} \cdot \vec{\sigma}),

where $ \theta = \omega t / 2 $.[](http://web.cecs.pdx.edu/~mperkows/june2007/bloch-sphere.pdf) This closed-form expression reveals $ U(t) $ as an element of the [special unitary group](/page/Special_unitary_group) SU(2). The rotational nature of this unitary stems from the [Lie algebra](/page/Lie_algebra) structure of the [Pauli matrices](/page/Pauli_matrices), which obey the commutation relations $ [\sigma_i, \sigma_j] = 2i \epsilon_{ijk} \sigma_k $, where $ \epsilon_{ijk} $ is the [Levi-Civita symbol](/page/Levi-Civita_symbol).[](https://link.aps.org/doi/10.1103/PhysRevA.93.062320) These relations define the su(2) algebra, with the generators $ -i \sigma_k / 2 $ producing infinitesimal transformations in SU(2), the double cover of the rotation group SO(3).[](https://link.aps.org/doi/10.1103/PhysRevA.93.062320) To confirm the action on the Bloch vector $ \vec{r} = \mathrm{Tr}(\rho \vec{\sigma}) $ for a density operator $ \rho $, consider the infinitesimal case where $ \theta \ll 1 $, so $ U \approx I - i \theta \, \vec{n} \cdot \vec{\sigma} $ and $ U^\dagger \approx I + i \theta \, \vec{n} \cdot \vec{\sigma} $. The evolved density operator is $ \rho' = U \rho U^\dagger \approx \rho - i \theta [\vec{n} \cdot \vec{\sigma}, \rho] $. The change in the Bloch vector is then

\delta \vec{r}_l = \mathrm{Tr} \left( -i \theta [\vec{n} \cdot \vec{\sigma}, \rho] \sigma_l \right) = -i \theta \mathrm{Tr} \left( \rho [\vec{n} \cdot \vec{\sigma}, \sigma_l] \right),

for component $ l = x, y, z $. Substituting the commutation relations yields $ [\vec{n} \cdot \vec{\sigma}, \sigma_l] = 2i (\vec{n} \times \vec{e}_l) \cdot \vec{\sigma} $, where $ \vec{e}_l $ is the unit vector along the $ l $-axis, leading to $ \delta \vec{r} = 2 \theta \, \vec{n} \times \vec{r} $.[](https://arxiv.org/pdf/2112.07100) This cross-product form describes an infinitesimal rotation of $ \vec{r} $ by angle $ 2\theta $ around $ \vec{n} $, generating the SO(3) group actions on the Bloch sphere.[](https://link.aps.org/doi/10.1103/PhysRevA.93.062320) ## Applications and Generalizations ### Role in Quantum Computing In quantum computing, the Bloch sphere serves as a fundamental visualization tool for representing and manipulating the states of a single [qubit](/page/Qubit), enabling intuitive understanding of quantum operations without delving into complex wavefunctions. Pure qubit states are depicted as points on the surface of the unit sphere, where the north pole corresponds to the computational basis state |0⟩, the south pole to |1⟩, and equatorial points represent balanced superpositions of these basis states with varying phases.[](https://www.mathworks.com/help/matlab/math/introduction-to-quantum-computing.html) This geometric mapping facilitates the analysis of qubit evolution under unitary transformations, which manifest as rotations of the [state vector](/page/State_vector) around specific axes on the sphere.[](https://quantum.microsoft.com/en-us/insights/education/concepts/single-qubit-gates) Single-qubit quantum gates, essential building blocks of quantum circuits, correspond directly to rotations on the Bloch sphere. The Pauli X gate, often called the NOT gate, executes a 180° (π radian) rotation around the x-axis, inverting the z-component of the state vector—for instance, mapping the north pole (|0⟩) to the south pole (|1⟩) and vice versa.[](https://www.mathworks.com/help/matlab/math/introduction-to-quantum-computing.html) Similarly, the Pauli Y and Z gates perform 180° rotations around the y- and z-axes, respectively, introducing phase flips or bit flips depending on the initial state; for example, the Z gate leaves |0⟩ unchanged but maps |1⟩ to -|1⟩, effectively rotating superpositions in the x-y plane.[](https://quantum.microsoft.com/en-us/insights/education/concepts/single-qubit-gates) The Hadamard gate, which creates superposition from basis states, is realized as a 180° rotation around the axis defined by the unit vector (1, 0, 1)/√2 (the diagonal in the x-z plane). Applying it to |0⟩ shifts the state from the north pole to the equator at φ = 0, yielding the equal superposition (|0⟩ + |1⟩)/√2, while a second application returns the state to |0⟩ due to the rotation's symmetry.[](https://www.mathworks.com/help/matlab/math/introduction-to-quantum-computing.html) These rotational interpretations allow quantum developers to predict gate effects geometrically, aiding in circuit design and debugging. Bloch sphere diagrams are widely employed in quantum circuit simulation tools to track state evolution step-by-step through sequences of [gates](/page/The_Gates), providing a visual [trajectory](/page/Trajectory) of the qubit's [path](/page/Path) on the sphere that complements matrix-based computations. For example, simulating a [circuit](/page/Circuit) with a NOT gate followed by Hadamard illustrates how the initial flip alters the subsequent superposition, helping to verify algorithmic correctness in environments like MATLAB's [Quantum Computing](/page/Quantum_computing) [Toolbox](/page/Toolbox).[](https://www.mathworks.com/help/matlab/math/introduction-to-quantum-computing.html) Such visualizations enhance pedagogical resources and software interfaces, making abstract [quantum dynamics](/page/Quantum_dynamics) more accessible.[](https://quantum.microsoft.com/en-us/insights/education/concepts/single-qubit-gates) Despite its utility, the Bloch sphere is inherently limited to single-qubit systems (though it represents both pure and mixed states), as multi-qubit states, including entangled ones, cannot be represented on a single sphere and instead require higher-dimensional analogs like the generalized Bloch representation or two-sphere visualizations for pairs of qubits.[](https://rasqberry.org/03-quantum-computing-demos/bloch-sphere) ### Extensions to Higher Dimensions The Bloch sphere, which provides a geometric representation for the states of two-level quantum systems (qubits), extends to higher-dimensional Hilbert spaces for multi-level systems called qudits, where the state space becomes significantly more complex. For a single d-level quantum system, the density operator can be parameterized by a generalized Bloch vector in a (d² - 1)-dimensional real Euclidean space, forming a convex body analogous to the Bloch ball but lacking the simple spherical symmetry of the qubit case.[](https://arxiv.org/pdf/0912.3155) This higher-dimensional structure captures the full range of mixed and pure states, with pure states lying on the boundary, which forms a manifold diffeomorphic to the complex projective space CP^{d-1} of real dimension 2(d-1).[](https://arxiv.org/pdf/2012.00587) For qutrits, which are three-level systems (d=3), the generalized Bloch representation resides in an 8-dimensional real space (ℝ⁸), utilizing the eight [Gell-Mann matrices](/page/Gell-Mann_matrices) to expand the [density matrix](/page/Density_matrix). This space corresponds to the [coset](/page/Coset) SU(3)/U(2), where the set of physically admissible states forms a bounded [convex](/page/Convex) region Ω₃ with ball-like [topology](/page/Topology), a proper [subset](/page/Subset) of the 8-dimensional Hilbert-Schmidt ball, bounded by an outer [sphere](/page/Sphere) of [radius](/page/Radius) √2; its boundary includes spherical surfaces corresponding to pure states at maximum distance √2 from the origin. The pure qutrit states occupy the 4-dimensional [complex projective space](/page/Complex_projective_space) CP² embedded within this 8-dimensional volume, adding further geometric richness but complicating direct analogies to the [qubit](/page/Qubit) Bloch sphere.[](https://arxiv.org/pdf/2012.00587) In higher dimensions, such as for n-qubit systems where the effective level count is 2ⁿ, the state space dimensionality scales to 4ⁿ - 1 for mixed states, rendering intuitive geometric interpretations challenging due to the loss of low-dimensional symmetries and visualizability.[](https://arxiv.org/pdf/0912.3155) To address these visualization difficulties, the Majorana stellar representation maps qudit states onto configurations of points (stars) on the familiar 2-sphere, where for a d-level system, the state is depicted by d-1 stars whose positions encode the probability amplitudes, preserving some geometric intuition at the cost of projecting higher-dimensional information.[](https://arxiv.org/abs/1804.06184) This method, originally developed for angular momentum states, facilitates analysis of entanglement and dynamics in qudits despite the underlying high-dimensional complexity.[](https://arxiv.org/abs/1804.06184)

References

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[PDF] bloch-sphere.pdf
The Bloch sphere is a geometric representation of qubit states as points on the surface of a unit sphere. Many operations on single qubits that are commonly ...
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[PDF] Geometrical phases on the Poincaré and Bloch spheres
Apr 1, 2015 · Section III will look at the quantum mechanical Bloch sphere and Berry's phase, and again give the historical contexts in which they were.Missing: origin | Show results with:origin
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[4]
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### Summary of Bloch Sphere Content
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[PDF] Lecture Notes for Ph219/CS219: Quantum Information Chapter 2
This ball is usually called the Bloch sphere (although it is really a ball, not a sphere). The boundary. |~P| = 1 of the ball (which really is a sphere) ...
[6]
Towards Two Bloch Sphere Representation of Pure Two-Qubit ... - NIH
Mathematically, the Bloch sphere representation relies on the use of complex vector spaces to visually depict quantum states. This approach provides a geometric ...
[7]
https://doi.org/10.1103/PhysRev.70.474
[8]
The classical Bloch equations | American Journal of Physics
Oct 1, 2014 · We illustrate the concept of the Bloch sphere and provide an intuitive understanding of coherent control experiments by discussing Rabi ...
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[PDF] The Bloch Sphere A single spin-1/2 state, or "qubit", is represented ...
This is the same direction as the expectation value of the vector of spin-operators < S >. This direction can be represented as a unit vector, pointing to a ...
[10]
[PDF] Thesis - Purdue Engineering
The spin-up and the spin-down states of an electron or a spin-1/2 nucleus ... on a Bloch sphere, as shown in Fig. 2.6. The south pole and the north ...
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[PDF] Appendix I: Bloch Sphere Representation
In the Bloch sphere representation, the state |ψi is depicted as a vector pointing from the origin to a point on the surface of the unit sphere; ... this state is ...
[12]
[PDF] 14 The Classical Description of NMR - CSCI
Solving the Bloch equation for a given field B(t) tells us M(t), that is, how the magnetization evolves with time. This equation is central to MRI and ...
[13]
[PDF] The Bloch Sphere for Topologists - DTIC
Nov 5, 2008 · fibration from S3 to S2 ∼. = CP1. CP. 1 is complex projective 1-space which is formed from C2 −. (0. 0. ) by iden- tifying points in C2 −. (0. 0. ).
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Tensor Representation of Spin States | Phys. Rev. Lett.
Feb 25, 2015 · Pure states correspond to points on a unit sphere, the so-called Bloch sphere, and mixed states fill the inside of the sphere, the “Bloch ball”.Missing: parameterization | Show results with:parameterization
[15]
[PDF] Vector Field Visualization of Single-Qubit State Tomography - Cerfacs
For a given state ãin (θ, φ) on the Bloch sphere, four variants of the tomography circuit. (Figure 2) are executed, one for each (α, β) combination in the ...Missing: parameterization | Show results with:parameterization
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[PDF] arXiv:2009.01877v3 [quant-ph] 28 Mar 2021
Mar 28, 2021 · The initial spin state of the beam was a pure state defined by its Bloch vector s = (sin θ cos φ, sin θ sin φ, cos θ), where θ = 1.91 and φ = ...
[17]
Interplay between geometric and dynamic phases in a single-spin ...
Sep 23, 2020 · The Bloch sphere is an example of a projective Hilbert space: Quantum states with different overall (global) phases are mapped to the same ...
[18]
[PDF] arXiv:1111.4205v2 [quant-ph] 28 Aug 2012
Aug 28, 2012 · the stereographic projection map ξ = tan(θ/2)eiϕ pro- vides the Bloch sphere with standard coordinates. Thus, any physical state can be ...
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[PDF] arXiv:quant-ph/0201014v2 16 Sep 2002
Fig. 1 also illustrates the definition of the stereographic projection, which maps a point q = (x1,x2,x3) on the unit sphere S2 (the Bloch sphere) to a point z ...
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[PDF] arXiv:0808.3089v2 [math-ph] 11 Sep 2008
Sep 11, 2008 · Here is a comparison diagram that shows how the quaternion action and the Bloch coordinate projection fit into the generic scheme (2). From now ...
[21]
[PDF] arXiv:quant-ph/0310053v1 8 Oct 2003
... tan (θ/2)Q), S(Q′) and Vi,j,k(Q′) being respectively the scalar and ... here to c = 2 |C2|. 3.3. Generalized Bloch sphere for the two-qubit case. Let ...
[22]
https://arxiv.org/pdf/quant-ph/0310053
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https://arxiv.org/pdf/2403.01047.pdf
[24]
https://arxiv.org/pdf/2203.12943.pdf
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https://arxiv.org/pdf/2402.18518.pdf
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[PDF] Lecture notes: Qubit representations and rotations
When you learn about the Bloch sphere (discussed below) you will see why the alternate symbols |+i and |−i are used to denote logical states.1.
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[PDF] arXiv:2112.07100v2 [quant-ph] 18 May 2022
May 18, 2022 · The origin of this similarity can be explained as follows. In the quantum mechanical Bloch sphere formalism, the origin represents a maximally ...
[28]
Entangled Bloch spheres: Bloch matrix and two-qubit state space
Jun 20, 2016 · Section VIII applies the formalism to the characterization of maximally entangled, pure states, and generalized isotropic/Werner states.Missing: parameterization | Show results with:parameterization
[29]
Introduction to Quantum Computing - MATLAB & Simulink
### Summary of Bloch Sphere and Quantum Gates from MATLAB Help
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Single-qubit gates - Microsoft Quantum
The Bloch sphere is a 3D sphere that represents the possible states of a qubit, with the north and south poles of the sphere corresponding to the |0⟩ and |1⟩ ...
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Bloch Sphere Demo - RasQberry Two
The Bloch Sphere demo is an interactive visualization tool that helps users understand how quantum gates affect single-qubit states on the Bloch sphere.
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[PDF] arXiv:0912.3155v1 [quant-ph] 16 Dec 2009
Dec 16, 2009 · generalization of the Bloch vector to higher dimensions is to give an intuitive representation of the dynamics of the underlying system. The ...
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[PDF] The shape of higher-dimensional state space: Bloch-ball analog for ...
Jun 22, 2021 · A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum ...
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[1804.06184] Generalized Weyl-Heisenberg algebra, qudit systems ...
Apr 17, 2018 · In the so-called Majorana scheme, the qudit (d-level) states are represented by N points on the Bloch sphere; roughly speaking, it can be said ...