Talk:Bloch sphere
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[edit] Layout of the page
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)
Thanks for correcting this. CSTAR 13:53, 8 Jul 2004 (UTC)
[edit] level?
what means "2-level" quantum mechanical system? Does it mean 2 dimensional Hilbert space? -Lethe | Talk 00:01, Jan 29, 2005 (UTC)
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- Yeah. For instance two energy levels for a Hamiltonian (assuming multiplicity 1 etc).CSTAR 00:09, 29 Jan 2005 (UTC)
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- It's terminology used by quantum information theorists for some reason.CSTAR 03:57, 31 Jan 2005 (UTC)
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- 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.
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[edit] why discussed so often?
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!
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--70.27.140.234 22:07, 2 May 2005 (UTC)
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- 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)
[edit] Answer to: why discussed so often?
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)
[edit] 6 entangled states
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)
- 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)
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)
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- 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)
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)
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- 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).
- 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)
[edit] Relation to Felix Bloch?
Is Bloch sphere related to the NMR physicist Felix Bloch? Thanks for any information or reference. --KasugaHuang 09:06, 3 March 2006 (UTC)
[edit] Answer:
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)
[edit] "Natural"?
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)
[edit] Only pure states?
Any reason this article doesn't mention the ability for the inside of the Bloch sphere to represent mixed states?
