Question About Qubits and the Representation of the Bloch Sphere

[captainhampto]

I have a general question about extracting information from measurement of a qubit. Theoretically a qubit in a superposition state contains an infinite amount of information, but when measured collapses to a definite state and result. My question is this:

Is there a way to obtain a value from the qubit once measured between a predefined boundary (for example 15 < x < 20) and if so, does this correspond to some restriction on the surface of the Bloch sphere for some geometric interpretation?

If not, I'd appreciate any feedback on the matter. As I am not a physicist by profession, please excuse my ignorance. Thanks for your kindness. Also, if any follow up questions are needed by me for clarification, please do not hesitate to ask I will respond promptly.

 

[Ameno]

The state of a qubit is (in the Bloch ball representation) described by a point in a ball (or on a sphere, if one restricts to pure states), so one might hope that one can store an infinite amount of information in a qubit and retrieve it by measurement(s). But this is not the case. The famous Holevo bound implies, roughly speaking, that one cannot get more than n (classical) bits of information out of n qubits. So one qubit can serve to store at most one bit of information (in a retrievable way). So for this particular task, qubits do not offer an advantage over classical bits.

Statements like "One can save an infinite amount of information in (the state of) a qubit but one cannot retrieve it" does not make much sense, I would say. A quantum state is quite a theoretical concept, and one should not fill it with too much ontological content.

 

[captainhampto]

Right, I understand that saying the qubit contains an infinite amount of information prior to measurement is misleading and perhaps treading too far away from hard science which I certainly do not want to do. I am vaguely familiar (at least with the overall result) of the Holevo bound which restricts the transmission of qubit/bits to an equal amount of information. So, let me rephrase my question in another way.

Let's say I still wanted to restrict the amount of information I obtain from a quantum system. In this case, let's say ( 4 <= x <= 7 ). This quantum system of course must consist of multiple qubits as opposed to just one. My idea is this: If I keep the initial bit static (or append a classical bit of 1) (|1XX>) while allowing the two "X's" to behave as qubits, does the measurement of the quantum system guarantee to yield at least a "4" (|100>) or at most a "7" (|111>)? Seeing as the post-measurement information obtained from a quantum system is classical I don't see an issue with this.

I tried to convey this as clearly as I could, but if something is unclear on my part please let me know, I'd be more than happy to attempt to explain my logic. Thanks again for your help Ameno, I greatly appreciate it.

(Possible restricted information from my proposed system)
|100>
|101>
|110>
|111>

 

[Ameno]

OK, let's do another attempt to approach your issue.

As usual in quantum information theory, we make examples involving two parties, called "Alice" and "Bob". Say that Alice and Bob agree on the following procedure:
First, Alice prepares a three-qubit system in some state. Then, she gives this three-qubit system to Bob. He uses a special measurement device to measure the qubit. This device can output integers between 0 and 7 as measurement results. These numbers are related to post-measurement states as follows (please let me know if you don't know what this means):

0 <--> |000>
1 <--> |001>
2 <--> |010>
3 <--> |011>
4 <--> |100>
5 <--> |101>
6 <--> |110>
7 <--> |111>

In principle, one could indeed build such a measurement device.
If Alices restricts herself to preparing states where the first qubit is in the definite state 1, but the other two qubits are not subject to any constraint, then indeed Bob will measure 4, 5, 6 or 7, but certainly he will not measure 0, 1, 2 or 3.

Unfortunately, in the case of a three-qubit system, we do not have such an elegant visualisation like the Bloch ball for the one-qubit system, so we cannot talk about a "region on the Bloch ball associated to the values 4, 5, 6 and 7". But instead, we have an 8-dimensional state space (or, if we take normalisation into account, we have something like the 7-sphere, i.e. a 7-dimensional sphere). In this 8-dimensional state space, the subset of states for which the first qubit is in the fixed state 1, is a 4-dimensional subspace. So indeed, we have something like a "well-defined subspace associated to a restricted amount of information" (although we have formulated it quite vague here, but one can make this more precise).

But notice: In (quantum) information theory, one has to be aware of a fact which one could roughly state as "information is independent of labelling". Instead of the measurement device described above, one could also build a device which has the following bit <--> post-measurement state associations:

0 <--> |000>
1 <--> |001>
2 <--> |010>
5 <--> |011>
4 <--> |100>
3 <--> |101>
6 <--> |110>
7 <--> |111>

or something like that. Moreover, if you restrict the first qubit to be in a definite state (e.g. 1), then you can just as well drop it. You could use a two-qubit system to encode like this:

4 <--> |00>
5 <--> |01>
6 <--> |10>
7 <--> |11>

The state space of this system would be the same as the sub-space in the three-qubit state space where the first qubit is fixed.





Does this help you?

 

[captainhampto]

Yes this helps a great deal Ameno, thanks again for your very sound and vivid explanation. I sincerely appreciate the time and effort you spent writing your reply. Have a great day and thanks again for the help.