I Qubit two-state quantum system

[cianfa72]

TL;DR Summary

About the Hilbert space representing the state space of two-state quantum system

Very simple question: for a two-state quantum system (qubit), is its state space a two-dimensional Hilbert space over the complex field $\mathbb C$ isomorphic to $\mathbb C^2$ or is it $\mathbb C^2$ itself ?

 

[jambaugh]

We are often a bit sloppy with the notation $\mathbb{C}^n$ could mean the n-dimensional complex vector space with no specific inner product or to the Hilbert space with a given Hermitian inner product (since they are all equivalent). Similarly with $\mathbb{R}^n$. The author should give some context to distinguish their use of the notation.

At a more nuanced level, we often speak of algebraic objects like SO(3) or $\mathbb{C}^2$ as if there is a singular one (as a categorical object). Other times we mean a specific presentation and other examples are merely isomorphic. In the end its not too big an issue so far as I've seen. I've not yet run across a situation where the ambiguity led to an actualizable error.

 

[cianfa72]

We are often a bit sloppy with the notation $\mathbb{C}^n$ could mean the n-dimensional complex vector space with no specific inner product or to the Hilbert space with a given Hermitian inner product (since they are all equivalent).

You mean $\mathbb C^n$ could mean just the "naked" set of complex number pairs without any further structure. Or the Hilbert space one gets taking a specific Hermitian inner product within it (the standard one).

However I believe it is better to understand the qubit's state space as an abstract 2-dimensional Hilbert state space over the field $\mathbb C$ isomorphic to $\mathbb C^2$ endowed with its standard Hermitian inner product structure that turns it into a 2-dimensional Hilbert space over $\mathbb C$.

 

[PeterDonis]

However I believe it is better to understand the qubit's state space as an abstract 2-dimensional Hilbert state space over the field $\mathbb C$

It's not just "better to understand"--this is the qubit's state space.

isomorphic to $\mathbb C^2$ endowed with its standard Hermitian inner product structure that turns it into a 2-dimensional Hilbert space over $\mathbb C$.

Not just "isomorphic to"--this is the 2-dimensional Hilbert space over the field $\mathbb{C}$.

 

[cianfa72]

It's not just "better to understand"--this is the qubit's state space.

Ok, you draw a distinction between the qubit's state space and the 2-dimensional Hilbert space over $\mathbb C$.

Not just "isomorphic to"--this is the 2-dimensional Hilbert space over the field $\mathbb{C}$.

I'm not sure that the 2-dimensional Hilbert space over $\mathbb C$ is exactly the set $\mathbb C^2$ endowed with the standard Hermitian inner product structure. This is, let me say, just the "prototype" of it.

 

[PeterDonis]

Ok, you draw a distinction between the qubit's state space and the 2-dimensional Hilbert space over $\mathbb C$.

Um, no, I'm saying they're the same.

I'm not sure that the 2-dimensional Hilbert space over $\mathbb C$ is exactly the set $\mathbb C^2$ endowed with the standard Hermitian inner product structure. This is, let me say, just the "prototype" of it.

I have no idea what you mean by this.

 

[cianfa72]

Um, no, I'm saying they're the same.

Ok.

I have no idea what you mean by this.

I mean the following: what is for instance the Euclidean space of dimension 3 ? Is it the set $\mathbb R^3$ as affine space endowed with the Euclidean inner product structure, or is it any abstract set of "points" with an affine structure and an Euclidean inner product?

 

[PeterDonis]

I mean the following: what is for instance the Euclidean space of dimension 3 ? Is it the set $\mathbb R^3$ as affine space endowed with the Euclidean inner product structure

Um, yes? What else would it be?

or is it any abstract set of "points" with an affine structure and an Euclidean inner product?

What does "any abstract set of points" even mean?

I think you're getting yourself tangled up over words instead of looking at the concepts.

 

[cianfa72]

What does "any abstract set of points" even mean?

In this specific context it is the set of qubit's quantum states. We assign it a vector space structure over $\mathbb C$ and an Hermitian inner product turning it into a 2-dimensional Hilbert space over $\mathbb C$ (let's call it $\mathbb H^2$). The latter is then isomorphic to $\mathbb C^2$ with its standard vector space structure over $\mathbb C$ and the standard Hermitian inner product (the isomorphism isn't canonical since requires to pick a basis in both).

By the way, from a physical viewpoint, what is relevant are actually the elements of the projective space $\mathbf P(\mathbb H^2)$ represented by the Riemann or the Bloch sphere.

Although we had a thread some time ago about it, I ask: consider an "instance" of Bloch sphere, do points on it represent superposition of qubit's spin about a single given axis in (physical) 3D space ?

 

[PeterDonis]

In this specific context it is the set of qubit's quantum states.

Ok, but that's not an "abstract set of points." It already has a particular structure.

We assign it a vector space structure over $\mathbb C$ and an Hermitian inner product turning it into a 2-dimensional Hilbert space over $\mathbb C$ (let's call it $\mathbb H^2$).

No, we don't "assign" this--we discover that the set of a qubit's quantum states has this structure by looking at how qubits behave physically in experiments. (With one caveat, which you give later on--see below.)

The latter is then isomorphic to $\mathbb C^2$ with its standard vector space structure over $\mathbb C$ and the standard Hermitian inner product (the isomorphism isn't canonical since requires to pick a basis in both).

Um, no, it isn't "isomorphic"--it's the same thing. I don't understand why you think these are somehow two different mathematical objects that are "isomorphic". They're just one mathematical object.

from a physical viewpoint, what is relevant are actually the elements of the projective space $\mathbf P(\mathbb H^2)$ represented by the Riemann or the Bloch sphere.

Yes, that's correct.

 

[PeterDonis]

onsider an "instance" of Bloch sphere

What does this mean?

do points on it represent superposition of qubit's spin about a single given axis in (physical) 3D space ?

What do you think?

 

[cianfa72]

Um, no, it isn't "isomorphic"--it's the same thing. I don't understand why you think these are somehow two different mathematical objects that are "isomorphic". They're just one mathematical object.

I'd say they are actually two "instances" of the same mathematical object.

What do you think?

Yes, fixed an axis in the physical world and a spin measurement apparatus aligned along it, the Bloch sphere North and South poles represent "spin-up" and "spin-down" along that axis respectively. Every other point on the sphere represents a superposition of the above two states, and not the spin along a different axis in space.

 

[PeterDonis]

I'd say they are actually two "instances" of the same mathematical object.

You keep throwing these words around without explaining what they mean.

Yes, fixed an axis in the physical world and a spin measurement apparatus aligned along it, the Bloch sphere North and South poles represent "spin-up" and "spin-down" along that axis respectively.

If you choose to interpret it that way, yes.

Every other point on the sphere represents a superposition of the above two states

Yes.

and not the spin along a different axis in space.

No. Every point on the Bloch sphere represents an eigenstate of qubit spin along some axis.

In other words, a qubit's spin about some axis is also a superposition of spin states about some other axis. They're not two different things only one of which is true. They're two different ways of viewing the same quantum state.

 

[cianfa72]

No. Every point on the Bloch sphere represents an eigenstate of qubit spin along some axis.

In other words, a qubit's spin about some axis is also a superposition of spin states about some other axis. They're not two different things only one of which is true. They're two different ways of viewing the same quantum state.

Ah ok, so a pair of antipodal points on the Bloch sphere represent the "spin-up" and "spin-down" states respectively along some spatial axis (along which a spin measurement apparatus alike Stern-Gerlach might be aligned).

 

[PeterDonis]

a pair of antipodal points on the Bloch sphere represent the "spin-up" and "spin-down" states respectively along some spatial axis

Yes.

 

[jbergman]

All 2 dimensional complex vector spaces are isomorphic so I am not sure that the question is important.

 

[cianfa72]

All 2 dimensional complex vector spaces are isomorphic so I am not sure that the question is important.

Exactly, although they are isomorphic that doesn't mean they are actually the same.

 

[jbergman]

Exactly, although they are isomorphic that doesn't mean they are actually the same.

Right but isomorphism is sufficient. We don't really care about sameness. I would disagree also with @PeterDonis and say they are only isomorphic and *not* the same, but when talking about these vector spaces most don't even make these distinctions.

 

[PeterDonis]

All 2 dimensional complex vector spaces

Can you give examples of more than one?

 

[jbergman]

Can you give examples of more than one?

Complex polynomials of degree 2 or less with complex coefficients is isomorphic to $\mathbb C^2$ as a vector space.

 

[PeterDonis]

Complex polynomials of degree 2 or less with complex coefficients is isomorphic to $\mathbb C^2$ as a vector space.

Is it? It takes three complex constants to define such a polynomial, not two.

 

[PeterDonis]

isomorphism is sufficient. We don't really care about sameness.

I would put this a bit differently. I would say that the abstract space involved is the space of 2-tuples of complex numbers, with an appropriate vector space structure, and there is only one such abstract space, mathematically speaking; but we can interpret this space in different ways. But this might be more a matter of how we describe the math in ordinary language than anything else.

 

[jbergman]

Is it? It takes three complex constants to define such a polynomial, not two.

You are right. Good catch. Ok, the polynomials of degree 1.

 

[cianfa72]

I would say that the abstract space involved is the space of 2-tuples of complex numbers, with an appropriate vector space structure, and there is only one such abstract space, mathematically speaking; but we can interpret this space in different ways.

No, I don't think so. The abstract space involved is a set of "abstract points" with the appropriate vector space structure over the field $\mathbb C$. Mathematicians don't know what "points" are (they admit don't know what they're talking about), what is relevant is the assigned structure alone. $\mathbb C^2$ is just an "instance" or "realization/incarnation" of the abstract vector space of dimension 2 over the field $\mathbb C$.

 

[jbergman]

I would put this a bit differently. I would say that the abstract space involved is the space of 2-tuples of complex numbers, with an appropriate vector space structure, and there is only one such abstract space, mathematically speaking; but we can interpret this space in different ways. But this might be more a matter of how we describe the math in ordinary language than anything else.

If we are being pedantic a vector space is just a set with the additional structure that satisfies the axioms of a vector space.

So we just need an abelian group structure on that set and a distributive multiplication of a field.

The polynomials of max degree 1 with complex coefficients satisfy these axioms.

The isomorphism is a linear invertible map from the polynomials to $\mathbb C^n$ or the 2-tuple of complex numbers as you describe it. There are actually multiple possible isomorphisms here as vector spaces.

But again these distinctions aren't very important for physics or even much math.

 

[PeterDonis]

No, I don't think so.

And yet you have given no difference between this...

a set of "abstract points" with the appropriate vector space structure over the field $\mathbb C$.

...and this:

an "instance" or "realization/incarnation" of the abstract vector space of dimension 2 over the field $\mathbb C$.

Unless and until you can state some actual difference between an "instance" or "realization" of an "abstract space" and the "abstract space" itself, you have no basis for claiming that they are different.

 

[PeterDonis]

There are actually multiple possible isomorphisms here as vector spaces.

Can you describe two distinct ones?

 

[cianfa72]

Can you describe two distinct ones?

Just pick two different basis pairs (i.e. not the same pair), the isomorphism depends on the basis you pick on both vector spaces.

 

[jbergman]

Can you describe two distinct ones?

$f(ax+b)=(a,b)$ and $g(ax+b)=(b,a)$ for example. When we think of polynomials there isn't really a natural sense in which coefficient should map to the 1st coordinate or the 2nd. Of course these choices don't matter. We could even choose something like $h(ax+b) = (3a,b)$. Now if we have an orientation and hermitian inner product on both vector spaces you are more constrained.

 

[PeterDonis]

Just pick two different basis pairs (i.e. not the same pair), the isomorphism depends on the basis you pick on both vector spaces.

No, isomorphism between two vector spaces is not basis dependent.