Questions tagged [bloch-sphere]

For questions related to the Bloch sphere. In quantum mechanics, 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. (Wikipedia)

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52 views

Is $e^{i\beta} R_Z(-2\beta)$ equivalent to $U_1(2\beta)$?

As far as I know the single qubit gate $$ e^{i\beta\sigma_z} = \begin{bmatrix} e^{i\beta} & 0 \\ 0 & e^{-i\beta} \end{bmatrix} = e^{i\beta} \begin{bmatrix} 1 & 0 \\ 0 & e^{-i2\beta} \...
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2answers
46 views

Does normalizing a mixed state give a pure state?

According to bloch sphere interpretation, any point on the surface of the sphere corresponds to a pure state and any point inside the sphere corresponds to a mixed state. Suppose you have a point ...
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1answer
28 views

What is the intuition of the outer product of two states?

If it is possible give me an intuition both with vectors on the plane and on the Bloch-Sphere.
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1answer
83 views

Where is $i|0\rangle$ located on the bloch sphere?

I understand linear transformations on the plane but cannot understand the Bloch sphere. How can a three dimensional sphere be generated by two linearly dependent vecotrs (the basis states 0 and 1)?
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31 views

Cannot interpret transformations on the bloch sphere as matrix multiplications

I understand that X,Y and Z gates are rotations around the axes with the respective letters, but I cannot understand how can Y gate multiply the amplitude of 0 with unreal number and have it landing ...
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1answer
47 views

What kind of transformation does the Y-gate do on the bloch sphere?

I'm going through "Quantum Computation & Quantum Information" by Michael A. Nielsen and Isaac L. Chuang, and as a high school student with no previous knowledge, I cannot understand some things ...
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1answer
49 views

Effect of Pauli X gate on minus state using bloch sphere

As I understood, the X gate flips the state around : $X(|0\rangle) = |1\rangle$. It can also be visualized with a $\pi$ rotation around the $x$ axis in the Bloch sphere. I have no problem with that. ...
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1answer
108 views

Represent a pure state in terms of 2 antipodal points on the Bloch sphere

I recently had an assignment where the question is based on the assumption that we can write any pure state qubit $|\phi \rangle$ as: $$|\phi \rangle = \gamma |\psi\rangle + \delta |\psi^\perp ...
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1answer
42 views

Find local state and compute Bloch coordinates, like Quirk

In a multi-qubit system I can find the amplitudes of a state and compute probabilities, $\theta$, and $\varphi$. This falls out from simulation with simple numpy arrays. For example, after application ...
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1answer
54 views

$\lambda$ parameter for U3 gate in qiskit Bloch sphere visualisation

It is easy to see how $\theta$ (rotation from the positive z-axis) and $\varphi$ (rotation from the positive z-axis) affect the initial state of the qubit when looking at the Bloch sphere but I have ...
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1answer
40 views

Bloch sphere, where are magnitude and phase of a qbit?

Where are magnitude and phase of a qbit on Bloch sphere? Phase is angle φ. What do you mean by magnitude? Amplitudes? They are given by angle θ - amplitude of |0⟩ is cos(θ/2) and amplitude of |1⟩ ...
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1answer
40 views

How is a qubit represented on a bloch sphere?

A quantum state can be represented as linear combination of 2 states: In Chuang and Nielsens book, it states that because the squared amplitudes sum to 1: That the combination can be rewritten as: ...
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0answers
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Deriving Bloch vector $dr$ from master equation

I am trying to derive the Bloch vector $dr$ for a measurement of a observable in any arbitrary direction $\theta$. For context this is the setup and derivation I have for continuous measurement along ...
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2answers
285 views

How should I understand the change of qubit's basis as a rotation?

I have a little difficulty with understanding. How do I properly visualize the change of qubit's basis as a rotation? Let's say that we have classical basis vectors, $|0\rangle$ and $|1\rangle$. Now, ...
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2answers
75 views

Information content of qubits

I have two question concerning information content of qubit. Question 1: How many classical bits are needed to represent a qubit: A qubit can be represented by a vector $q = \begin{pmatrix}\alpha \\\...
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1answer
86 views

How to get Bloch sphere angles given arbitrary qbit as linear combination of basis vectors?

I understand that given a pure state $ |\psi\rangle$, we can express it in terms of two angles $\theta$ and $\varphi$ such that $|\psi\rangle = \cos(\theta/2)|0\rangle + \mathrm{e}^{i\varphi}\sin(\...
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0answers
55 views

Affine Map of the Bloch sphere

I am referring to Equation (8.89) to (8.92) in Chapter 8 of "Quantum Computing and Information 10th Anniversary Edition" by Nielsen and Chuang. This section deals with the geometric picture of single ...
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2answers
106 views

Can a qubit be in the inside of the Bloch Sphere?

Can qubit be inside Bloch Sphere, i.e. its length is less than 1? If yes, how we represent that state since we have only parameter for angles and not the length (norm) of the vector?
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131 views

Find coordinates $\theta$ and $\phi$ on the Bloch sphere of a given qubit state [duplicate]

In last time there is a lot of questions how to find $\theta$ and $\phi$ for particular state on Bloch sphere. I think that it would be useful to solve one example to stop stream of very similar ...
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52 views

Simplify the system to find the $\theta$, $\phi$ on bloch sphere [duplicate]

I have problem to simplify the quantum system states on bloch sphere to get $\theta$, $\phi$ values, My question is not duplicated! $$ \left| \varphi \right>=\frac{1+i}{\sqrt{3}} \left| 0 \right>...
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2answers
125 views

What is the probability of a single qubit state lying over the surface of Bloch sphere?

I want to compute the POVM $E_{(\theta, \phi)}$ of the measure which gives the probability of a qubit state lying over the surface of Bloch sphere, with angles $\theta, \phi$. How can I handle this? ...
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1answer
102 views

what are the angles 𝜃, and ϕ values of the following quantum state? [duplicate]

I need to find the coordinate 𝜃 and ϕ values of the quantum state on the bloch sphere $$ \left| \varphi \right>=\frac{1+i}{\sqrt{3}} \left| 0 \right> + {\sqrt{\frac{1}{3}}} \left| 1\right> $$...
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2answers
225 views

How can I find the $\theta$ and $\phi$ values of a qubit on the Bloch sphere?

If I have the following state: $$ \left| \varphi \right>=\frac{1}{\sqrt{2}}\left(\left(\frac{1+i}{\sqrt{2}} \right)\left| 0 \right> + \left| 1\right>\right) $$ How can I find the $\theta$ ...
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4answers
120 views

Can you use Rz to flip from $|+\rangle$ to $|-\rangle$?

Here's the Rz matrix: $$ Rz(\theta) = \begin{bmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{bmatrix} $$ As I understand it, Rz rotates around the Z axis on the Bloch sphere. Since $|+\...
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1answer
104 views

What is the corresponding code for finding the state of a qubit on the Bloch sphere?

To find the state of a qubit on the Bloch sphere we use the following formula: \begin{equation} |\psi\rangle=\mathrm{cos}\frac{\theta}{2}|0\rangle+\mathrm{e}^{i\phi}\mathrm{sin}\frac{\theta}{2}|1\...
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1answer
88 views

Bloch sphere and quantum operations [closed]

Evidently the Bloch Sphere is used as a graphical representation for any single qubit system, although what does it mean at an intuitive level? Moreover, the manipulation of qubits still seems unclear,...
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1answer
53 views

Which angle is estimated by the phase estimation algorithm?

Is it $\theta$ or $\varphi$ as usually depicted on the Bloch sphere? In other words, is it the angle projected on the $xy$-plane or is it the one on a plane that intersects the $z$-axis of the Bloch ...
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1answer
200 views

Does the trace distance have a geometrical interpretation?

Consider the trace distance between two quantum states $\rho,\sigma$, defined via $$D(\rho,\sigma)=\frac12\operatorname{Tr}|\rho-\sigma|,$$ where $|A|\equiv\sqrt{A^\dagger A}$. When $\rho$ and $\...
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2answers
74 views

Probability of equal outcomes measuring a Bell state in the directions $\vec{n}_1,\vec{n}_2$

Take the Bell state $$\frac{1}{\sqrt{2}}\left(|00\rangle +|11\rangle\right) $$ and measure the first qubit with respect to some axis $\vec{n}_1$ on the Bloch sphere, i.e. measure the observable $$\vec{...
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1answer
162 views

Do pure states lie on a hypersphere in the Bloch representation?

It is known that every state $\rho$ of a $d$-level system (or if you prefer, qudits living in a $d$-dimensional Hilbert space) can be mapped into elements of $\mathbb R^{d^2-1}$ through the mapping ...
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1answer
79 views

Can a mixed state be represented on a Bloch sphere?

I have a hard time getting the exact difference between concepts of superposed and mixed states. Is it possible to represent the second one on the Bloch sphere, to show the main difference?
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2answers
390 views

How to prove that antipodal points on the Bloch sphere are orthogonal?

I started by assuming two antipodal states $$ |(\theta,\psi)\rangle = \cos\dfrac{\theta}{2}|0\rangle + \sin\dfrac{\theta}{2}e^{i\psi}|1\rangle\\ |(\theta+\pi,\psi+\pi)\rangle= \cos\dfrac{\theta+\pi}{2}...
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1answer
159 views

How are measurements on $Z$ and $X$ axes interpreted in the Bloch sphere?

I'm having trouble understanding how the measurement on $z$ and $x$ axes can be interpreted in terms of the Bloch sphere representation. . I know that the state can be written as $$∣𝜓⟩=\cos(𝜃/2)|...
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1answer
150 views

Bloch Sphere - Rotation Matrix

A qubit is given in the following form: $\left|\psi\right\rangle = \cos\left(\dfrac{\theta}{2}\right)\left|0\right\rangle + e^{i\phi}\sin\left(\dfrac{\theta}{2}\right)\left|1\right\rangle$. Let's ...
4
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2answers
336 views

Maximum number of “almost orthogonal” vectors one can embed in Hilbert space

In a Hilbert space of dimension $d$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is small but finite compared ...
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0answers
50 views

How to add states on the Bloch sphere from a master equation?

Can you please help me to find an answer to this question: I am using qutip for the study of quantum systems. suppose I have calculated the solution of the master equation using the ...
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1answer
403 views

Plotting Bloch sphere in QuTiP

Is there anyone who reproduced the Bloch sphere given in the paper QuTiP: An open-source Python framework for the dynamics of open quantum systems by J. R. Johansson, P. D. Nation, Franco Nori? I am ...
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3answers
220 views

How are orthogonal sets of pure states arranged in state space?

It is well known that the state of a (pure) qubit can be described as a point on a two-dimensional sphere, the so-called Bloch sphere. The mapping $\lvert\psi\rangle\mapsto \boldsymbol r_\psi$ that ...
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3answers
1k views

Is there any online Bloch sphere simulator?

While writing this answer I realized it would be really helpful if I could show the OP a video or .gif of how qubit states in Bloch spheres transform under certain unitary operations. I googled up a ...
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2answers
112 views

Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?

Consider an arbitrary qudit state $\rho$ over $d$ modes. Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation: $$\rho=\frac{1}{d}\left(\mathbb I +\...
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1answer
424 views

What is the meaning of writing a state in its Bloch representation?

What is the meaning of writing a state $|\psi\rangle$ in its Bloch representation. Would I be correct in saying it's just writing out its Bloch vector?
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1answer
302 views

Drawing tangent vectors to the Bloch sphere with qutip

I need to plot drawings of qubit dynamics on the Bloch sphere. I know QuTip allows to do such drawings but I specifically need to represent evolution velocities on the Bloch sphere so I need to draw ...
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3answers
249 views

Purity of mixed states as a function of radial distance from origin of Bloch ball

@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
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1answer
291 views

Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere

The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia: Quantum ...
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1answer
147 views

What utility is provided by the Bloch sphere visualization?

The Bloch sphere is a mainstay of introductions to quantum computing, but what utility does it actually provide? It can't usefully represent multiple qbits because of entanglement and requires a weird ...
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1answer
562 views

What's a vector in the Bloch sphere representation?

The Bloch sphere isn't so intuitive for me. But I am not sure how you are supposed to manipulate it using vectors and matrices. How do you actually represent a vector on it? Is it $(\cos(a) , e^i\...
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0answers
31 views

Spin precession using a laser

According to Christopher Monroe: "Modular Ion Trap Quantum Networks: Going Big", the hyperfine states of the valence electron in the Yb+ is used as a qubit. I know that we can change the spin ...
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2answers
628 views

Why is an entangled qubit shown at the origin of a Bloch sphere?

I'm unclear why the Bloch sphere representation of a maximally entangled qubit shows the state of the bit as being at the origin of the sphere. For example, this illustration shows the effect of ...
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2answers
783 views

How to obtain Y rotation with only X and Z rotations gates?

Let's say you have a system with which you can perform arbitrary rotations around the X and Z axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the Y axis? ...
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5answers
3k views

Can the Bloch sphere be generalized to two qubits?

The Bloch sphere is a nice visualization of single qubit states. Mathematically, it can be generalized to any number of qubits by means of a high-dimensional hypersphere. But such things are not easy ...