Questions tagged [bloch-sphere]

A geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), used in quantum mechanics and computing.

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How to write an arbitrary qubit density operator? [duplicate]

Physics noob here: I am reading the Wikipedia on Density Matrices (https://en.wikipedia.org/wiki/Density_matrix), and in the section labeled "Pure and mixed states", it states "An ...
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Directly get tangent vector of Bloch sphere from quantum state (qubit)?

We know that Bloch sphere is a good way to represent a qubit(two energy quantum systems). Now I want to know the tangent vector in Bloch sphere, e.g. for states $\frac{1}{\sqrt{2}}\left( \begin{array}{...
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Are qubits just analog, continuous classical bits?

Topologically, classical bits (cbits) are essentially special cases of qubits restricted to the poles of the Bloch sphere. However, this restriction doesn't seem to be classical per se, but is simply ...
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Finding $\theta$ and $\phi$ when qubit state is $\frac{1}{\sqrt 2}[i ,1]^T$

Because we know the state of a qubit can be described as: $$ |q\rangle=\cos{\frac{\theta}{2}}|0\rangle+e^{i\phi}\sin{\frac{\theta}{2}}|1\rangle\\\ \\ \theta, \phi \in \mathbb{R} $$ How do I find the ...
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How do I visualize a stepwise rotation on the Bloch Sphere using $SO(3)$ and $SU(2)$?

Searching for an implementation (that helps me to understand how $SO(3)$ and $SU(2)$ relate to each other) I came across this interesting question Visual interpretation, on the Bloch sphere, when ...
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Find coefficient for pure and mixed states

Consider a generic $2\times 2$ Hermitian matrix written as $$\rho =\alpha\sigma_0+\beta\hat{\vec n}\cdot\vec\sigma\quad ,$$ where $\hat{\vec n}$ is a unit vector and the coefficients are real numbers. ...
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How to prove that every mixed one-qubit state admits a Bloch-sphere representation? [duplicate]

A mixed state $\rho$ can be written as $$\rho=\frac{1}{2}\left(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z\right)\qquad\left(\vec{r}:=\left(r_x,r_y,r_z\right)^T\in\mathbb{R}^3; ||\vec{r}||\leq 1\right)$$ ...
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Quantum Computer - Rotation Bloch Sphere [closed]

please can anyone help? What gate combination allows moving from the state between |0> and |1> states. In terms of bloch-sphere from the north pole to the south pole as an example. And how can ...
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Understanding the Bloch Sphere Better

Numerous times I have used the Bloch sphere and visualized gates as rotations. For Z and X rotations, it is a pretty good representation. However, today I found that this does not stand for Y/2 gate. ...
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States $-|0\rangle$ and $i|0\rangle$ in Bloch Sphere?

I am new on quantum computing and starting reading a book about it. Going through it, the Bloch sphere was described for two states. My question about is: where are the states $-|0\rangle$ and $i|0\...
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Bloch sphere representation - rewriting a state

If a quantum state can be represented as $$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$ Then Because of $|\alpha|^2+|\beta|^2=1$, we may rewrite Equation (1.1) as $$|\psi\rangle=e^{i\gamma}\left(\...
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Is the Bloch sphere a bad way to visualize a qubit?

When a qubits is in a quantum state it can be measured as $[0\rangle$ or $[1\rangle$. Then why does the Bloch sphere have these two states on antipodal sides of the spheres? If I want to plot the ...
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What is the topology of non-entangled states region for a 2 qubit Bloch hypersphere?

Preamble A two qubit/spin-1/2 system can be represented as $$|\psi\rangle=\alpha|\uparrow\uparrow\rangle+\beta|\uparrow\downarrow\rangle+\gamma|\downarrow\uparrow\rangle+\delta|\downarrow\downarrow\...
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Is there a nice formula to represent POVMs $\Pi$ in a Bloch-vector-like form?

A way to write a quantum state is to use the Bloch vector representation, i.e., $$ \varrho = \frac{1}{2}\left(\mathbb{I}_2 + \boldsymbol{r} \cdot \boldsymbol{\sigma}\right) $$ In general a POVM for a ...
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Rabi Oscillations: $\pi$-Pulse vs a single photon

I am puzzled by the following: Assume an atom as a two-level-system. A $\pi$-Pulse acting on an atom in the ground state promotes this atom in the excited state. This is done by a continuous ...
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What is the physical significance of phase difference in qubit or spin half? [duplicate]

Whenever it is talked about phase difference or lack of it, double slit example is used as the example. But I want to know what is the physical manifestation of phase difference for a qubit (or spin ...
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90 degree rotation about $z$-axis on Bloch sphere

I am a beginner on quantum information and right now I study it from the lectures by Peter Shor. The lectures can be found on edX and can be viewed on YouTube as well. The lecture that I'm struggling ...
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Optimal control for single qubit gate

Suppose we want to implement a single qubit gate $U$ and are only able to apply rotations around axes in the $xy$ plane of the Bloch Sphere, i.e. we can only apply Hamiltonians of the form $$H(t) = \...
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Can 2-qubit systems be represented in the Bloch sphere representation?

I am studying a system with 2 qubits, so I need, for a given state, a Bloch representation for each qubit. I am having difficulties because I get results that do not have sense at all. For example if ...
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Which paper introduced the concept of the "Bloch sphere"?

Everyone loves Bloch sphere, but which paper of Bloch was it introduced? The Wikipedia article on Bloch sphere (here), as of 17/May/2021 links to this paper of Bloch “Nuclear Induction” but it doesn’t ...
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Can pure qubit states be represented in a unit circle in $\mathbb R^2$? How does such representation relate with the Bloch sphere?

In this video they are regarding 2D circles denoting real-valued states of qubit, like Teacher says it can be extended to 3D to Bloch sphere. But Bloch sphere has |0> at the top and |1> at the ...
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Sketching dynamics of Bloch vector for ground state with dynamics induced by Hamiltonian

How can I sketch a Bloch vector for a system that is in the ground state |g> that is induced by Hamiltonian $ H_x = \frac{ω_x}{2}σ_z$? Is there a general method that I can follow for drawing Bloch ...
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A potential well with 3-fold reflection symmetry

When we are talking about Bloch's theorem and also the tight-binding approximation, we can use them to help finding eigenstates of a system. However, I am so confused how to apply it in this case (...
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Trying to visualize the concept of qubits [closed]

Background Complete newbie, never taken a physics course. Question I'm trying to visualize the concept of qubits. Qubits can achieve a mixed state, called a "superposition" where they are ...
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Pole point on the Bloch sphere

If the state of the qubit is a point at a pole on the Bloch sphere, does this mean that the coefficient of the component corresponding to the other pole is zero?
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Vector from $SU(2)$ to $SO(3)$? [closed]

I know how to change the element(a matrix) in $SU(2)$ to the matrix in $SO(3)$, and I found one way to change the state(vector) from $2\times 1$ to $3\times 1$, but I don't know why. The method is ...
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Hamiltonian with identity operator: how to visualize the (time-evolution) rotation?

For the Hadarmard Hamiltonian, $\hat H = (\hat X+\hat Z)/\sqrt 2$, where $\hat X$ and $\hat Z$ are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a ...
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Bloch sphere representations for multi-qubit quantum systems

Just a short mixed quantum state representation question. Given a single qubit density matrix $\rho$, since the Pauli matrices form a basis for 2x2 complex matrices, the Bloch sphere representation ...
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Average over the Bloch sphere

Consider you have a function of a two-level wave quantum state $f(\vert \psi \rangle ) $, with $\vert \psi \rangle = \alpha \vert 0 \rangle + \beta e^{\rm i \phi} \vert 1 \rangle$. With no loss of ...
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Why can we only perform rotations of the Bloch sphere (with unitary matrices), and not reflections?

It's easy to take a quantum state represented on the Bloch sphere and rotate it around an arbitrary ray emanating from the origin. On the other hand, we can never use a unitary matrix to get a ...
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Outcomes of quantum measurements

I'm pretty new to quantum computing, and I'm wondering how I can compute the outcome of a projective measurement of a spin along the +Z axis followed by a projective measurement along the -Z axis. I ...
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Why an element of $SU(2)$ acts as a rotation for the Majorana representation of states?

I recently asked a question in quantum computing stack exchange and as suggested by someone in the comments, I decided to ask my question here as well: I know that for a given spin-j quantum state, ...
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Prove that qubits can be represented on a unit sphere, avoiding the density matrix formalism [duplicate]

The Bloch Sphere is regarded as the most "intuitive" way of explaining a 2-level quantum system in computation and rotations of states described on Bloch sphere provides a really easy ...
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Quantum Computing without Complex Numbers

p.s. I am trying to get a handle on what actual computing operations a quantum computer program does. Any information on that would be appreciated [noting the issue that that might count as a ...
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What is the physical intuition for Bloch Sphere? [duplicate]

I am very confused about how to think about the Bloch Sphere. How can we relate the concept of expectation value to the Bloch sphere? If my state lies in let's say $yz$ plane how can we say that ...
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Consider the following single-qubit state on the Block sphere [closed]

$$\left| \varphi \right>=\frac{1+i}{\sqrt{3}} \left| 0 \right> + {\sqrt{\frac{1}{3}}} \left| 1\right> $$ I need to find the coordinate 𝜃, and ϕ values of the quantum state.
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Time evolution of a qubit starting on the an x-axis eigenstate

Suppose I have starting qubit like the one described on the Bloch sphere below. Also, denote the state $|0\rangle \equiv |+\rangle$. So my starting state would be: $$ |\psi(0)\rangle=\frac{1}{\sqrt{...
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Terminology for the symmetric vector product on the Bloch sphere

I was reading the paper "The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem" by Diederik Aerts, Massimiliano Sassoli de Bianchi. ...
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Is there a geometrical representation of classical bits, like that of the Bloch Sphere for quantum bits?

Consider only a single qubit. We know that it can be accurately described using the so called Bloch Sphere. Is there a similar geometric construction used to describe classical bits?
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Maximum number of "almost orthogonal" vectors one can embed in Hilbert space [closed]

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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How to compute the axis of a rotation expressed as a combination of Pauli matrices? [closed]

An exercise asked to compute the axis and the angle of the rotation $THTH$. $(S.1)$ is easy to understand by using the identity $exp(i\theta A) = \cos(\theta) \mathbb1 + i \sin(\theta) A$ for $A$ s.t....
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Constraints on higher-dimensional Bloch vectors

I'm interested in the constraints on the $(4^n-1)$-dimensional generalized Bloch vector (the Bloch vector for $n$ qubits). To the best of my knowledge, these are not analytically characterized for ...
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How to transform coordinates of two-level system's eigenstates from spherical to momentum space?

In almost all references for some two-level quantum system (for example, this Wikipedia article), arbitrary eigenstates are defined in spherical coordinates, and can be written separately as follows (...
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Can all expectation values be constructed using only tensor products of local Pauli operators?

I understand that the Pauli operators and the identity matrix span the space of complex 2x2 matrices. Let's say you have two qubits and you can perform projective meausurements. So you can measure ...
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Bloch sphere representation of an eigenvector

I'm trying to work through a problem that wants me to determine the Bloch sphere representation of the eigenvectors of $\sigma_{z}$. I'm working in bra-ket notation so these would be $\ v_{+} = |0\...
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How does this term $e^{i\Phi_0}$ get removed in bloch sphere equation?

A qubit can be represented in the form of $$|\psi\rangle=\alpha|0⟩+\beta|1\rangle$$ where $\alpha$ and $\beta$ are complex numbers. Or a complex number can be expressed by $R e^{i\Phi_0}$. so the ...
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How to obtain $Y$ rotation with only $X$ and $Z$ rotation gates on the Bloch sphere?

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 $Y$ ...
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Quantum Mechanics Notation

I'm studying the Bloch Sphere and just wanted to ask what this notation means: $|\psi\rangle = \alpha|1\rangle$ for example I'm just not familiar with the notation in this context if anyone could ...
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Quantum superposition in density matrix formalism

I was thinking about quantum superposition and stumbled into something that made me quite uncomfortable. Consider a qubit with Hamiltonian eigenstates $|0\rangle$ and $|1\rangle$. To each of these ...
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Characterisation of the generalised Bloch space in spherical coordinates

I'm so confused by the following definition in the "Quantum Error Correction" by Lidar and Brun that not even sure how to formulate the question properly. Let $\mathbf n$ denote a unit vector, i.e.,...
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