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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3answers
513 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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2answers
912 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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3answers
288 views

Can I find the axis of rotation for any single-qubit gate?

Suppose I have an arbitrary qiskit $U_3$ gate: $U_3(\theta,\phi,\lambda)$. Is there a way I can find which axis the gate is rotating around? In other words, given any real numbers $\theta,\phi,\lambda$...
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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 +\...
21
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5answers
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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 ...
9
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2answers
2k views

What is the relation between these two forms of a single-qubit unitary operation?

I want to understand the relation between the following two ways of deriving a (unitary) matrix that corresponds to the action of a gate on a single qubit: 1) HERE, in IBM's tutorial, they ...
15
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4answers
2k views

Why are half angles used in the Bloch sphere representation of qubits?

Suppose we have a single qubit with state $| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$. We know that $|\alpha|^2 + |\beta|^2 = 1$, so we can write $| \alpha | = \cos(\theta)$, $| \beta | ...
10
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2answers
517 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 \\\...
3
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1answer
818 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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2answers
187 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
150 views

How is a two qubit mixed state represented in the form of Bloch vector?

How is a two qubit mixed state represented in the form of Bloch 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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2answers
1k 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 ...
4
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1answer
247 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(\...
6
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2answers
356 views

How to prove that Z operator rotates points on Bloch sphere about Z axis through 180°?

My idea was to apply $Z$ operator 𝐭𝐰𝐢𝐜𝐞, which leads us back to the point where we started from, and also show that after applying the $Z$ operator just 𝐨𝐧𝐜𝐞 we are not at the same point ...
4
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1answer
535 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 ...
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 ...
3
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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
241 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 ...
4
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1answer
56 views

Why did I get two solutions to solve for the parameters of this $U_3$ gate? (I only expected one of them)

I have the following complex vector in $\mathbb{C}^2$: Vec= [[ 0.89741876+0.j] [-0.33540402+0.28660724j]] I'm trying to implement a $U_3$ gate to prepare this ...
3
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3answers
347 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 ...
2
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1answer
87 views

Why can any density operator be written this way? (quantum tomography)

From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product: \begin{aligned} \rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
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2answers
109 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 ...
2
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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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0answers
67 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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1answer
191 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
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 ...
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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 ...
-3
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1answer
132 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 $$ \left| \varphi \right>=\frac{1+i}{\sqrt{3}} \left| 0 \right> + {\sqrt{\frac{1}{3}}} \left| 1\right> $$...