Talk:Bloch sphere

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Thanks for correcting this. CSTAR 13:53, 8 Jul 2004 (UTC)

level?[edit]

what means "2-level" quantum mechanical system? Does it mean 2 dimensional Hilbert space? -Lethe | Talk 00:01, Jan 29, 2005 (UTC)

Yeah. For instance two energy levels for a Hamiltonian (assuming multiplicity 1 etc).CSTAR 00:09, 29 Jan 2005 (UTC)

OK. I am familiar with the term "energy level". But what are the "levels" in the spin Hilbert space of a spin-1/2 particle? L_z levels? This terminology looks strange to me. -Lethe | Talk 03:54, Jan 31, 2005 (UTC)

It's terminology used by quantum information theorists for some reason.CSTAR 03:57, 31 Jan 2005 (UTC)

Is it really all that strange? After all, in QIT one doesn't care about the physical representation of the qubit. Just as bits have two "levels", so do qubits. — Preceding unsigned comment added by 207.171.180.101 (talk) 22:40, 7 February 2007 (UTC)[reply]

why discussed so often?[edit]

Does anyone know what the _significance_ of the Bloch sphere is? Why does it come up so often in discussions, given how simple it is?

Thank-you in advance to anyone who answers!

-)

--70.27.140.234 22:07, 2 May 2005 (UTC)[reply]

What the _significance_ of the Bloch sphere

I'm not sure I can really answer that question. However, one can point out that it does provide a clear geometrical picture for what the set of superpositions of two states looks like; similar questions about superpositions of more than two states can be also asked and answered. The answers in the general case are a little more complicated, however, involving quotient spaces of compact Lie groups.--CSTAR 05:12, 3 May 2005 (UTC)[reply]

-- dmack

_significance"

Simply representing qubits as a superposition of base states requires two complex (therefore four real) parameters, when these are actually encoding redundant information - since this is a pure state and therefore on the unit sphere, and since global phase is irrelevant, we can represent the same state in half the number of parameters. This is why the Bloch sphere is useful, it is a 3d visualisation of the distinct state space. Also, as it has its own quirks, learning about it prompts useful thought and discussion. —Preceding unsigned comment added by 163.1.167.234 (talk) 21:18, 14 March 2010 (UTC)[reply]

Answer to: why discussed so often?[edit]

Hi, although the quantum description SU(2) appears to have more degrees of freedom, than the classical description SO(3), the length of the vector is fixed to one and the mixed state (called "pure" in the page) is in fact equivalent to the classical description under that restriction.

The Pauli matrices was the first link between a classical theory of a magnetic nuclear moment and a quantum theory using a two level system. The gap between both theory lead to consequences extremely hard to admit even today, therefore one has to use all possible tools to illustrate those equations. The Bloch vector model, the graphical equivalent of the Pauli model is a nice visual tool.

Alain Michaud 07:37, 11 June 2006 (UTC)[reply]

6 entangled states[edit]

The article Many worlds interpretation says that this article will explain me why the space of entangled states of two block spheres is 6 dimensional. If it's here, I don't see it, and I would like to know why. I was expecting it to be 4 dimensional. -lethe ^{talk +} 03:17, 6 February 2006 (UTC)[reply]

I wrote that; its the dimension given by the manifold of two qubits.

Corollary. The real dimension of the pure state space of an m qubit quantum register is 2^(m+1) − 2.

Plugging in m = 2 gives 6. Is this wrong? Please tell me. I will even do penance.--CSTAR 03:22, 6 February 2006 (UTC)[reply]

Oh, I'm not saying I see a mistake. I just don't get it. I didn't read carefully the generalization section and its theorem, I just read the first section, expecting to see the statement for 2 dimensional spaces there. [...reading...]. OK, having read and think about the general statement, I'm still a little surprised at its assertion. I expect that for projective space, dim PV = dim V – 1 for any vector space V, so there's something I'm not getting. Am I wrong that the set of pure states in H is just PH? -lethe ^{talk +} 04:11, 6 February 2006 (UTC)[reply]

Youre right about the state space being projective space. However, a two qubit Hilbert space has complex dimension 4 = 2 × 2, real dimension 8. Doesn't this given the real dimension of the state space to be 6 as claimed? The usual caveat I give is that I could be wrong, misguided or insane.--CSTAR 04:26, 6 February 2006 (UTC)[reply]

Ack. Real dimension. Right, so if dim_C PV = dim_C V – 1, then dim_R PV = dim_R V – 2. OK, everything's clear now. I'm an idiot. -lethe ^{talk +} 04:31, 6 February 2006 (UTC)[reply]

Well I still could be misguided or insane. At least I'm not wrong. For this.--CSTAR 04:34, 6 February 2006 (UTC)[reply]

Agreed. Thanks for the clarification. So lemme just make sure I get what's going on with this theorem: this quotient group G acts transitively on PH and so there is an isomorphism between the group and the space of pure states. Don't we need the group action to also be faithful for that to hold? I think this action is faithful, so the result is OK. -lethe ^{talk +} 04:42, 6 February 2006 (UTC)[reply]

The group action is not going to be faithful, I don't think. It has to be transitive. Usual caveat:Wrong, misguided or insane.--CSTAR 04:45, 6 February 2006 (UTC).[reply]

BTW the isomorphism is not between the group and projective space, but between a coset space of the group and projective space. Projective space is thus a symmetric space.--CSTAR 06:14, 6 February 2006 (UTC)[reply]

It's a symmetric space, that's right! that means the action is faithful and transitive! This is exactly my point! -lethe ^{talk +} 08:35, 6 February 2006 (UTC)[reply]

Errr... actually it means free and transitive. I think I probably meant to say free all along. -lethe ^{talk +} 08:45, 6 February 2006 (UTC)[reply]

Relation to Felix Bloch?[edit]

Is Bloch sphere related to the NMR physicist Felix Bloch? Thanks for any information or reference. --KasugaHuang 09:06, 3 March 2006 (UTC)[reply]

Answer:[edit]

Yes the rotation of the Bloch vector on the Bloch sphere is nothing but the simple description of a NMR experiment. It was not until the 40th that the NMR experiment were done on solids (Purcel and xx), following the measurements of Rabi on atomic beams (1937).

In the 30th, the situation was like this: The simplest system would have two levels, and the quantum mechanics at best would describe the probability that the system jumps from one state to the other. On the other hand the spectroscopists would measure 'common sense' physical properties like magnetic moment, etc...

There were many people that were involved in the evolution from concept of shroedinger equation, which is the simplest wave equation, to the concept of "probability", the concept of "spin", the correspondence principle, the NMR experiments, etc... Among others, they were: Otto Stern who saw the first "doublet", Isidore Rabi, who induced a transition, Zeeman and Landé who made early spectroscopic measurements, Ulenbeck and Goudsmith who had the weird idea of an half integer cinetic moment, Pauli who gave a model for it, Bloch, Thomas, Dirac, etc...

During all those years, the gap between the classical and quantum theory remained and even today most of the literature for the general (non scientific) public is devoted to explain this apparent paradox.

Nevertheless, the complete 'exact' theoritical description was given in the 50 with the work of von Neuman who introduced the concept density operator. This was the birth of what is now called "measurement theory": Measurements made on a large ensemble of identical particles show the "statistical" picture, while measurements made on an isolated particle reveal the "probabilistic" nature.

(Also, it is worth mentioning the book of Hietler in 1935 which was very advanced for its time!)

Alain Michaud 06:55, 11 June 2006 (UTC)[reply]

Layout of the page[edit]

Hi,

I think this page goes too fast! In the first paragraph the "qubit" is introduced, while there is a section on the density operator at the end of the page.

I did not change anything, as I do not want to destroy someone else's work. Here would my approach:

Section 1- Low level introduction: Something that a High-school student could read: The bloch sphere realy is a visual aid for illustrating the link between the R^3 space to the U^2 space. The first one is easy to see while the second one is more "abstract" as the only "aspect" we can observe is the projection along one or two axis!

Section 2 - History: This model was introduced after the early spectroscopy measurements as link between classical (3D real) and quantum (2D complex) physics. This was only the tip of a gigantic iceberg!

Section 3 - A complete mathematical description requires the introduction of the density operator (introduced in the 50 by von Neuman) and also of the concept of spin (the simplest 2 level system model), etc...

Section 4 - A modern application is the use of the same formalism but on single particle or small systems, which leads to completely unexpected situations like entangled stated, etc... (All these phenomenons are a consequence of a "mismatch" between the R^3 ans U^2 spaces, which can be visualized by the bloch sphere)

As you see there is a lot here, but at least the first few paragraph should contain just the right information. very simple, but exact and not misleading.

Thank you for reading

Alain Michaud 05:53, 11 June 2006 (UTC)[reply]

"Natural"?[edit]

Doing a disambig run on Natural, and I'm not enough of a mathematician to know if the word "natural" used in the "Generalization" section of this article should go to Nature or Natural transformation - or, indeed, somewhere else. Could someone please help? Thanks. Tevildo 04:05, 16 December 2006 (UTC)[reply]

Only pure states?[edit]

Any reason this article doesn't mention the ability for the inside of the Bloch sphere to represent mixed states? — Preceding unsigned comment added by 207.171.180.101 (talk) 22:46, 7 February 2007 (UTC)[reply]

I think this is the great weakness of this article: there seems to be a misconception of the significance of the Bloch sphere. the real significance is that the interior of the ball gives one all the mixed states and decomposition of a mixed state can be done through a barycentric calculus. The non-uniqueness of that decomposition is the key to many of the quantum paradoxes. See the book by Beltrametti and Cassinelli (the Logic of Quantum Mechanics) for details ---- Eluard — Preceding unsigned comment added by Eluard (talk • contribs) 03:41, 12 August 2009 (UTC)[reply]

---I agree this should be fixed, and one must mention that it is actually a ball if one wishes to use it to describe an arbitrary density matrix. This fact is elaborated in Neilson and Chauang, the foremost textbook in quantum computing. Danski14^(talk) 06:13, 20 February 2012 (UTC)[reply]

Lack of any actual states on the Bloch sphere figure?[edit]

Slightly confused by the inclusion of a picture of a sphere that doesn't actually show any states on it. In essense it then is just a sphere with an xyz axis, which is a pity since the whole beauty of the bloch sphere is that it allows a visualisation of the different possible states of a single qubit.

Would it be possible to update the picture. I guess i could have a bash if that isn't stepping on anyone's feet.

I would imagine that something like the figure in this paper would be good:

http://arxiv.org/pdf/0803.1554

--Wideofthemark (talk) 12:06, 15 January 2009 (UTC)[reply]

The picture includes a parametrization of states as described in the article in the body of the article. Why isn't that enough?--CSTAR (talk) 15:53, 15 January 2009 (UTC)[reply]

Because it doesn't explicitly label the poles corresponding to |0> and |1>. -KR 13:11, 2 June 2009 (UTC) —Preceding unsigned comment added by 67.194.1.176 (talk)

Reference[edit]

Consider adding the following reference:

Geometry of a Qubit by Maris Ozols.

In this essay I discuss various ways to represent a qubit state. Maybe this reference is relevant also for the article on Qubit.

— Preceding unsigned comment added by Marozols (talk • contribs) 22:32, 22 February 2009 (UTC)[reply]

Grammar problem?[edit]

2nd para, 2nd sentence: "The space of pure states of a quantum system is given by the one-dimensional subspaces the Hilbert space (the "points" of projective Hilbert space)." Should there be an "of" after subspaces? Or am I even unable to read sentences on QP? Forton (talk) 18:24, 16 November 2009 (UTC)[reply]

image problem[edit]

The diagram is awful, not only is it ugly it also lacks vital information (what different points correspond to) making it unhelpful. If you have time and inclination, please replace it! —Preceding unsigned comment added by 163.1.167.234 (talk) 21:19, 14 March 2010 (UTC) Another problem: the two antipodes are labelled "z" and "-z", which is of course incorrect. The labels should be simply |0> and |1> or some such thing. — Preceding unsigned comment added by 2601:740:8200:E10:3CE3:EC91:30D1:1466 (talk) 22:09, 28 December 2017 (UTC)[reply]

Occurrence of U(n) (and half-angles)[edit]

The presence of $U(n)$ and half-angles is never explained. There is the attempt,

"To prove this fact, note that there is a natural group action of

U(n)

on the set of states of

H n

".

The blue link "natural" leads to abstract nonsense. Better might be to say simply that $U(n)$ is the (unique) group that preserves the standard Hermitian form (inner product) on $H n$ , see classical group. Before I possibly do anything, I'd like some feedback. YohanN7 (talk) 16:35, 13 August 2014 (UTC)[reply]

the Bloch CPU[edit]

The Bloch CPU is a shpere that has many transistors on it's surface. It acts like a probabilistic CPU when noise inputs inserted, and many Bloch spheres usually work together as qubits do. It is zillion times slower than an actual quantum computer, also exports rounded results, but it may work well if many Bloch-sheres combine their states and if avegaged many results to provide a mean. It can be used as an input for a real quantum CPU, when even the question we pose is complex enough and probabilistic. Also it can be used as a filter for the output of a quantum-CPU, when we require some extra rotations [usually fixed ones] — Preceding unsigned comment added by 2.84.219.136 (talk) 22:18, 15 May 2015 (UTC)[reply]

Unless there is a reliable source establishing some connection between the topic of this section and the Bloch sphere, beyond the name itself, I suggest that the section be removed from this talk page as being off topic. Vaughan Pratt (talk) 03:06, 21 October 2017 (UTC)[reply]

Consistent variable naming[edit]

The diagram uses theta and lower case phi but the definition uses theta and phi. Suggest to change one or the other to keep them consistent. — Preceding unsigned comment added by 124.19.49.38 (talk) 22:28, 20 February 2018 (UTC)[reply]

Severe problem with image[edit]

The diagram is not only incorrect, it points the reader into a severly wrong direction by labeling the down spin direction as -z. This is complete rubbish as the consequence, algebraically, would be -|0> = |1>. Please tidy this up, it points beginners into an absolutely wrong direction. — Preceding unsigned comment added by 217.95.166.162 (talk) 16:51, 19 January 2018 (UTC)[reply]

You misunderstand what the image represents. — Aldaron • T/C 23:14, 19 January 2018 (UTC)[reply]

Nope.

Maybe my comment was too short or you did not bother to read it, the choice is yours. I am not suffering from the common misconception which is explained in the stackexchange article. The issue I meant is the following: The image is labeled ẑ=|0> at the top and -ẑ=|1> at the bottom. Multiplying -ẑ=|1> with (-1) we obtain ẑ=-|1>. Together with ẑ=|0> and transitivity of equality we obtain the completely absurd result |0> = -|1>

The author of the drawing probably meant to indicate that |0> corresponds to a "spin-up" state when measuring in z direction and |1> to a "spin-down" in z direction. In many textbooks the former often is written as |z+> and the latter as |z->. Also an index notation is quite common. Sometimes the "hat" notation is also used to designate the Bloch-vector and ẑ could mean the Bloch vector in real 3-space in z direction. Then, the 3-space Bloch vector ẑ would in fact correspond to the complex ray formed by |0> and -ẑ would correspond to |1>. However, if this were intended, then the equality sign is wrong. But the two labelings, as they exist on the sphere, lead to the erroneous conclusion |0> = -|1>

There are, of course, many more aspects in this article, which go wrong.

The notation of a "hat" as in ẑ is used and never properly explained; it is not really a standard notation which can be assumed.
Qubit space is correctly introduced as ℂℙ1 ("a state is a ray"), but in the definition this concept is no longer used and the article falls back to "a state is a vector and only relative phases matter". Later, yet another concept seems to be used, "a state is a vector of norm = 1". While all this is, of course, not completely wrong if the reader already knows the concepts of the Bloch sphere, it confuses the reader who consults this article, to get a better understanding. One should start with the ray definition (which is the only one which really is fully correct) and then mention how the other interpretations follow.
${\vec {\sigma }}$ is an abuse of notation, which suggests that it is a vector in 3-space, which it is not. It is a sloppy notation where every entry in the vector is a 2x2 matrix. (This can be saved from formal criticism using product tensor notation, but this is not an article for the specialist but an entry in an encyclopedia. Writing this as $a_{1}\sigma _{1}+a_{2}\sigma _{2}+a_{3}\sigma _{3}$ would circumvent this problem)
The above issue pops up again in the "explanation" of the theorem in the pure state section. In the sentence "Any $g$ of U(n) that leaves $|\psi \rangle$ invariant must have $|\psi \rangle$ as an eigenvector" the notation $|\psi \rangle$ first is used as a ray and then as a vector in one and the same sentence. The definition of the isotropy group goes wrong along similar lines, since the article is consistent in its use of ket-vectors (are they vectors or states, i.e. rays)

But, well, yes, of course, I simply misunderstood the diagram :-). Killing the messenger is also a way of "quality" "control", and it is very effective. Actually, it is the reason why I no longer contribute to Wikipedia articles but only to Wikipedia discussion pages. At least it saves those from misunderstandings who care to read the discussion page.

I will now revert the article to the form which included the warning template, but only once. I am not interested in edit wars and everybody who believes an article should be improved by removing warning templates should just do so.

— Preceding unsigned comment added by 217.95.168.141 (talk) 14:59, 25 January 2018 (UTC)[reply]

I think the

-\mathbf {\hat {z}} \equiv |-z\rangle \equiv |1\rangle

, no? — Aldaron • T/C 20:53, 25 January 2018 (UTC)[reply]

That is exactly the problem with the article. It leads to "I think ... no?" type of questions, which is not what we want from a good encyclopedic article.

Your modification points into the right direction and it could be what the initial author had wanted to do. It may produce more misunderstandings, though. In the original Dirac bra-ket notation, inside of the

|...\rangle

there should be no arithmetic operations but only state labels, whereas in a common notation of the Hermitian inner product (which often uses the same symbols), such operations are permitted, as in

\langle \phi +\psi |\alpha \rangle =\langle \phi |\alpha \rangle +\langle \psi |\alpha \rangle

. I have seen this to produce issues as well. My personal suggestion for fixing this would be

|z_{+}\rangle =|0\rangle

and

|z_{-}\rangle =|1\rangle

. Or, even better, just to use

|z_{+}\rangle

and

|z_{-}\rangle

and to drop the notation using

|0\rangle

and

|1\rangle

. The latter two, that's just another way of writing the same thing, although quite common in quantum computer literature.

— Preceding unsigned comment added by 217.95.168.141 (talk) 22:47, 25 January 2018 (UTC)[reply]

If those are indeed your concerns then you should raise them (concisely) in the image page in Commons. — Aldaron • T/C 17:53, 26 January 2018 (UTC)[reply]

For now, I've substituted a version of the image that doesn't have the issue you're concerned with. — Aldaron • T/C 14:33, 28 January 2018 (UTC)[reply]

Looks cool. Thank you. Particularly this aspect is much better now !! — Preceding unsigned comment added by 217.95.173.244 (talk) 18:35, 29 January 2018 (UTC)[reply]

Image wrong wrong wrong[edit]

It looks like somebody takes consistent fun in repeatedly replacing the image by an image which is wrong. The wrong image is labeled ẑ=|0> at the top and -ẑ=|1> at the bottom. Multiplying -ẑ=|1> with (-1) we obtain ẑ=-|1>. Together with ẑ=|0> and transitivity of equality we obtain the completely absurd result |0> = -|1> Please do Wikipedia a favor and stop replacing the corrected image by an image with this absurd annotation. — Preceding unsigned comment added by 87.163.198.107 (talk) 22:06, 30 September 2018 (UTC)[reply]

If you want to help, remove that absurd picture in all the other national variants of this article. — Preceding unsigned comment added by 87.163.198.107 (talk) 22:08, 30 September 2018 (UTC)[reply]

This confused me too and deserves an explanation, but the diagram is not wrong. The block sphere does not live in the same space as the |0>, |1> vector space because of the use of theta/2 instead of \theta in the first equation under Definition. "states which are orthogonal are not orthogonal vectors on the Bloch sphere. Thus for instance, |0⟩ and |1⟩ are orthogonal, but are represented on the Bloch sphere as θ = 0 and θ = pi" See this article by Dave Bacon. Pulu (talk) 04:17, 2 May 2019 (UTC)[reply]

Sigh. Well - NOW it is correct, but it WAS wrong before you came here :-) — Preceding unsigned comment added by 217.95.163.145 (talk) 22:48, 5 December 2019 (UTC)[reply]

Definition of isotropy group[edit]

The article states:

For any state

|\psi \rangle

, the isotropy group of

|\psi \rangle

, (defined as the set of elements

g

of U(n) such that

g|\psi \rangle =|\psi \rangle

) is isomorphic to the product group

\operatorname {U} (n-1)\times \operatorname {U} (1).

Actually, what is meant is the set of elements $g$ such that $g|\psi \rangle$ is some complex number (of absolute value 1) times $|\psi \rangle .$ Could we make this clear? As it stands, the definition of this isotropy group is wrong and would be isomorphic only to $\operatorname {U} (n-1)$ . Eric Kvaalen (talk) 07:11, 2 April 2019 (UTC)[reply]

Quantum logic gates[edit]

As the article has some kind of a role as an introductory article for those wanting to develop concrete intuitions about qubits, would it benefit from examining a couple of examples of distinctively quantum logic gates? — Charles Stewart (talk) 07:31, 9 May 2022 (UTC)[reply]