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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218 views

Benefit of phase shift in quantum computing

I am new to quantum computing. I compare Pauli-X gate and Pauli-Y gate as equivalent to NOT gate in classical computers. Though I am not very sure when to use Pauli-X and Pauli-Y gates as the result ...
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1answer
206 views

Does composition of two single qubit rotations yield a single rotation around a unit vector?

$ \newcommand{\coefcos}[0]{c_1 c_2 - s_1 s_2 \hat{n}_1 \cdot \hat{n}_2} \newcommand{\coefsin}[0]{s_1 c_2 \hat{n}_1 + c_1 s_2 \hat{n}_2 - s_1 s_2 \hat{n}_2 \times \hat{n}_1}$This question relates to ...
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1answer
80 views

What is the "phase" of a state in terms of the Bloch sphere?

I am pretty new to Quantum Computing and am not exactly sure what the phase is. Could you please explain in terms of the Bloch sphere and point out how to mathematically calculate and represent the ...
3
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2answers
112 views

when we specify quantum states in $\mathbb C^2$, why do we only have $2$ basis states?

I am just starting to get up to speed with quantum computing via the Quiskit learning path: online tutorial Here they explain the Dirac notation and use it to describe quantum states as elements in $\...
2
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1answer
43 views

$N(\frac{1}{2},2)=3$ for vectors in a Hilbert Space

Came across This question regarding the maximum number of almost orthogonal vectors one can embed in a Hilbert space. They state that $N(\frac{1}{2},2)=3$, and that explicit construction of the ...
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2answers
615 views

How are eigenvectors and eigenvalues expressed in the Bloch sphere?

I'm relatively new to the subject of quantum computing, and I recently came across the idea of eigenvalues and eigenvectors. I believe I understand the relationship between the two, where eigenvalues ...
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0answers
70 views

Reduced Density Matrix Equation of Motion to describe an Ellipse

Given a pure two qubit state $|\psi_{AB}\rangle$. If we trace out system $B$, the remaining density matrix $\rho_A = Tr_B|\psi_{AB}\rangle\langle\psi_{AB}|$, can be represented as a point lying ...
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0answers
75 views

Why an element of SU(2) acts as a rotation for Majorana representation of states?

I know that for a given spin-j quantum state, say $\vert\psi\rangle = (\psi_0 , \psi_1 , \cdots , \psi_{2j})$, we can construct a polynomial as follows $ w(z) = \sum_{k = 0}^{2j} (-1)^k \psi_k \sqrt{\...
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1answer
69 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
161 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
205 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
119 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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1answer
81 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
77 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
172 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
229 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
78 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
246 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
222 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
95 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
30 views

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
1k 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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524 views

How many classical bits are needed to represent a qubit?

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
250 views

How to get the Bloch sphere angles given an arbitrary qubit?

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
154 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
188 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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1answer
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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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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
253 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
134 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
926 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
165 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
135 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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2answers
110 views

Do quantum gates rotate a qubit around the Bloch sphere, or do quantum gates rotate the Bloch sphere around a qubit?

Do quantum gates rotate a qubit around the Bloch sphere? Or do quantum gates rotate the Bloch sphere around a qubit? "The simplest quantum gates are the Paulis: X, Y, and Z. Their action is to ...
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1answer
144 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
72 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
437 views

Does the trace distance have a geometric 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 $\sigma$...
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2answers
208 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
228 views

Do pure qudit 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
118 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
1k 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
470 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)|...
3
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1answer
316 views

What is the rotation matrix corresponding to a point on the Bloch sphere?

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
421 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
65 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 ...
4
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1answer
593 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
350 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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4answers
5k 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 ...
3
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2answers
194 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
833 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?