What is Bloch sphere: Definition and 21 Discussions

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space. The pure states of a quantum system correspond to the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). For a two-dimensional Hilbert space, the space of all such states is the complex projective line




{\displaystyle \mathbb {CP} ^{1}.}
This is the Bloch sphere, also known to mathematicians as the Riemann sphere.
The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors



{\displaystyle |0\rangle }



{\displaystyle |1\rangle }
, respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states. The Bloch sphere may be generalized to an n-level quantum system, but then the visualization is less useful.
For historical reasons, in optics the Bloch sphere is also known as the Poincaré sphere and specifically represents different types of polarizations. Six common polarization types exist and are called Jones vectors. Indeed Henri Poincaré was the first to suggest the use of this kind of geometrical representation at the end of 19th century, as a three-dimensional representation of Stokes parameters.
The natural metric on the Bloch sphere is the Fubini–Study metric. The mapping from the unit 3-sphere in the two-dimensional state space



{\displaystyle \mathbb {C} ^{2}}
to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere.

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  1. P

    A Dipole moment operator and perterbations

    I am reading a PHD thesis online "A controlled quantum system of individual neutral atom" by Stefan Kuhr. In it on pg46, he has a Hamiltonian I am also reading a book by L. Allen "optical resonance and two level atoms" in it on page 34 he starts with a Hamiltonian where the perturbation is...
  2. Haorong Wu

    Can Bloch sphere be used to represent mixed states in quantum computation?

    Well, I have no clues for this problem. Since I can get nothing from the definition of ##\rho##, I tried from the right part. Also, I know that ##\left ( \vec r \cdot \vec \sigma \right ) ^2={r_1}^2 {\sigma _1}^2+{r_2}^2 {\sigma _2}^2+{r_3}^2 {\sigma _3}^2##. Plus, ##\rho## is positive; then...
  3. Haorong Wu

    I How to understand the Bloch sphere in the quantum computation?

    I've read that ##\left | \psi \right > =cos \frac \theta 2 \left | 0 \right > + e^{i \phi} sin \frac \theta 2 \left | 1 \right >##, and the corresponding point in the Bloch sphere is as the fig below shows. I think ##\left | 0 \right >## and ##\left | 1 \right >## are orthonormal vectors...
  4. X

    I Bloch Sphere Change of Basis

    Anyone know how to change a basis of a qubit state of bloch sphere given a general qubit state? There are 3 different basis corresponding to each direction x,y,z where |1> ,|0> is the z basis, |+>, |-> is the x basis and another 2 ket notation for y basis. Given a single state in the x basis...
  5. nomadreid

    I Phase, Bloch sphere versus Feynman path

    I am pretty sure that I would be comparing apples and oranges in this question, but as I usually learn something from the responses telling me in detail that my question is silly, here goes: Does the phase used as a weight in Feynman's path integral formulation (i.e., the quantum action S in...
  6. P

    I What does it mean to span the Bloch sphere?

    If I construct a set of qubit gates, say {G1, G2 ... Gk ... Gn}, that can act on a state |ψ>, what does it mean for the set of states Gk |ψ> to span the Bloch sphere? As an example, take the set {G1, G2, G3, G4} = { I, X π/2 , Y π/2, Xπ } Here, X π/2 denotes a π/2 rotation about the x-axis, Y...
  7. polyChron

    Rotations in Bloch Sphere about an arbitrary axis

    Hey, (I have already asked the question at http://physics.stackexchange.com/questions/244586/bloch-sphere-interpretation-of-rotations, I am not sure this forum's etiquette allows that!) I am trying to understand the following statement. "Suppose a single qubit has a state represented by the...
  8. naima

    B The sticky elactic in the Bloch sphere

    I found a funny model of the qubit written by Aerts in Foundations of quantum physics: a general realistic and operational realistic and operational approach. At the beginning the qubit is at the point P on the Bloch sphere. It will be measured along another direction (two opposite points on...
  9. M

    Rotations in Bloch Sphere, and Free Parameters of a Qubit

    This question is mostly about group theory but I would like to understand it in the context of qubits rotating in a Bloch Sphere. What my understanding of things are right now: In the rotation Lie Group ##SO(3)##, we have three free parameters (##\frac{n(n-1)}{2}##), and this is also why we end...
  10. J

    Rotation operators on Bloch sphere

    Can anyone explain to me why the following operators are rotation operators: \begin{align*}R_x(\theta) &= e^{-i\theta X/2}=\cos(\frac{\theta}{2})I-i\sin(\frac{\theta}{2})X= \left(\!\begin{array}{cc}\cos(\frac{\theta}{2}) & -i\sin(\frac{\theta}{2}) \\ -i\sin(\frac{\theta}{2})&...
  11. B

    Bloch sphere and mixed stats

    Homework Statement What is reduced density matrix ##\rho_A## and the Bloch vector representation for a state that is 50% ##|0 \rangle## and 50% ##\frac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle)##Homework Equations The Attempt at a Solution [/B] I haven't seen many (any?) examples of this so I'm...
  12. V

    Conditional rotation in the bloch sphere with a 2-qubit system

    Homework Statement The problem is as follows. I have two spins, m_S and m_I. The first spin can either be \uparrow or \downarrow , and the second spin can be -1, 0 or 1. Now, I envision the situation as the first spin being on the bloch sphere, with up up to and down at the bottom. What I...
  13. K

    Bloch Sphere and the 3-sphere

    I'm having a bit of a brain fart here, so hopefully someone can help. Consider a closed, two-level quantum system. We know we can describe pure states as \alpha |0\rangle + \beta |1 \rangle for some orthonormal basis |0\rangle, |1 \rangle . The normalization conditions means we can...
  14. C

    Question About Qubits and the Representation of the Bloch Sphere

    I have a general question about extracting information from measurement of a qubit. Theoretically a qubit in a superposition state contains an infinite amount of information, but when measured collapses to a definite state and result. My question is this: Is there a way to obtain a value from...
  15. C

    Mapping Points on the Bloch Sphere

    Hey guys, I'm attempting to map some discrete points on the surface of the Bloch sphere: For instance, the full spectrum of ranges for variable theta is 0 < theta < pi. However, my goal is to limit that range from some theta_1 < theta < theta_2. I was going to use a spherical harmonic...
  16. F

    How Does Qubit Orientation on the Bloch Sphere Indicate Quantum States?

    There is something that I don't quite understand in relation to the Bloch Sphere representation of qubits. I've read that any vector on the sphere is a superposition of two basic states, like spin up and spin down, denoted by |1> and |0>. So does this mean that if the vector is at z=0...
  17. Talisman

    Spinors and the Bloch sphere

    So I tried learning about spinors yesterday, and got myself confused. Hopefully someone can tell me if I'm barking up the right tree... The way they were introduced was by exhibiting a homomorphism from C^3 to C^2 by using the dot product: (x1, y1, z1) . (x2, y2, z2) = x1*x2 + y1*y2 +...
  18. G

    Bloch sphere model for many spins?

    The Bloch sphere helps understanding the mathematical results for a one-spin state. One could think of the state as a spin pointing in direction \hat{n}. Then the probability for measureing the spin in the direction \hat{m} is simply P=|<\hat{m}|\hat{n}>|^2=\frac{1+\hat{n}\cdot\hat{m}}{2} and...
  19. M

    Understanding the Bloch Sphere Representation for Quantum States

    Hi could someone please explain the what the bloch sphere representation of a quantum state is useful for? thanks Mark
  20. maverick280857

    Simple question regarding the Bloch Sphere

    I am reading Quantum Computation and Quantum Information by Nelson and Chuang myself and came across the Bloch Sphere representation of a quibit on page 15 (equation 1.4) as: |\psi> = \cos\frac{\theta}{2} |0> + e^{i\psi}\sin\frac{\theta}{2} |1> I have two questions: 1. What is the motivation...
  21. J

    Quantum computing - bloch sphere coordinates

    I have one critical problem with quantum computing. When I have one quantum state on bloch sphere, I do some transformation on that state for example PauliX transformation. As far as I know, we can present quantum state as |y> = cos (theta/2) |0> + e^(i*phi) sin (theta/2) |0> So if we...