Bloch Sphere Demystified

This article aims to provide a simplified and more intuitive explanation of the Bloch sphere assuming that the reader has a basic understanding of quantum computing such as what is a qubit, what are basis states and a basic understanding of high school geometry. The Bloch sphere at its core the projective space of all complex numbers $\mathbb{C}^2$ which is also the true state space of a single qubit. This is well explained in this video by Gabriele Carcassi¹. I’ll try to explain it in a more intuitive way in this blog post.

I. Defining the Bloch Sphere

According to wikipedia², we can define the Bloch sphere as follows:

Definition 1 (Bloch Sphere). In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system* (or a qubit), named after the physicist Felix Bloch.

Note: *A two-level quantum mechanical system is simply a system that when observed is found to exist in only one of two distinct states. For example, electron spin is two-level quantum mechanical. Whenever you attempt to measure the spin, you will find it either spin up or spin down.

While this definition is technically accurate, it may not be very intuitive for someone new to the concept. Let’s break it down and define some of our terms:

1. Two-level quantum mechanical system: A two-level quantum mechanical system is simply a system that can exist in two distinct states such as spin the up and spin down states on an electron. Generally, these systems are used to create qubits in quantum computing and their states are labeled as $\ket{0}$ and $\ket{1}$.
2. Pure state: A pure state is the most basic form of quantum state that can be fully described as a linear combination of basis states. For example, the states ‘$\alpha\ket{0} + \beta \ket{1}$’ is a pure state as it is written purely as a superposition of the basis states.

Armed with these definitions, we can now understand what the Bloch Sphere is trying to represent. The Bloch sphere is a way to visualize the state of a single qubit (a two-level quantum system). However, before we can delve deeper into the true nature of the Bloch Sphere, we need to understand the concept of geometric representation.

II. Geometric Representations

In really simple terms, a geometric representation translates abstract information into a spatial, visual format, making it easier to understand relationships, patterns, and structures. For example, we can represent real numbers on a number line, which is a one-dimensional geometric space. Similarly, we can represent complex numbers on a two-dimensional plane called the complex plane.

Fig 1. Geometric Representations Complex Numbers $(5+5i)$ and $(3-4i)$

“Geometric representations translate abstract information into a spatial, visual format, making it easier to understand relationships, patterns, and structures.”

As the figure shows, complex numbers can be geometrically represented on a 2D plane.

\[z = x + iy \in \mathbb{C} \forall x,y \in \mathbb{R}\]

In the example, the key idea is that every single unique assignment of $(x,y)$ corresponds to a unique point on the 2D plane. More formally, we can define a geometric representation as follows:

Definition 2 (Geometric Representation). A geometric representation is mapping from some abstract data space $\mathcal{D}$ to a mathematical space $\mathcal{S}$.

Where a mathematical space is a set with some added structure such as a metric or topology³. A vector space is an example of a mathematical space where the elements of the set are vectors and the added structure is vector addition and scalar multiplication.

Why do we care? Geometric representations allow us to visualize and analyze complex data in a more intuitive way. They are especially useful in the analysis of transformations and/or functions that map data from one space to another. Single Qubit gates for instance are often represented as rotations on the Bloch sphere.

Geometric Space of the Bloch Sphere: As stated earlier, the Bloch sphere is a specific type of ‘space’ known as a ‘Projective’ space of all complex numbers $\mathbb{C}^2$.

III. Projective Spaces

A projective space aims to capture the idea of direction without magnitude. To better understand this idea let’s define a new coordinate system in terms of two variables, $\theta$ and $\lambda$, where $\theta$ is the angle between the vector and the $y$-axis and $\lambda$ is a scaling factor.

Fig 2. Vectors as a scaled unit vector

As we can see in the picture above, we can represent any vector in the 2D plane $\ket{v} \in \mathbb{R}^2$ in terms of a point on the unit circle defined by the angle $\theta$ and a scaling factor $\lambda$. Formally, the semicircle is termed the real projective line and is denoted as $\mathbb{RP}^1$.

\[\ket{v} = \lambda(\cos\theta, \sin\theta) \;\; \forall \;\; \theta \in [0,\pi], \lambda \in \mathbb{R}\]

This idea allows us to define the notion of a projective ray -

Definition 3 (Projective Ray). A projective ray is a line that starts at the origin, passes through some point in the 2D plane, and extends infinitely in that direction. It can be defined by the angle it makes with an axis. All points on the same ray are considered equivalent in projective space. They can be defined using the equation $\ket{r} = [cos\theta, \sin\theta]^T$ where $\theta$ is the angle the ray makes with the $y$-axis.

Any point in the 2D plane can be represented as a point on the unit circle that lies intersects with a given ray and a scaling factor $\lambda$ - $\ket{v} = \lambda \ket{r}$. The significance of a ray is that all points on the same ray are considered equivalent in projective space or more simply, we don’t care about the magnitude of a vector in projective space.

Real Projective Line

The interactive plot below shows how we create the ‘real projective line’ from the 2D plane. Try moving changing the angle of the ‘rotating vector’ and see how it affects the ‘projective shadow’ as well as the point on the real projective line or the ‘Projected Vector’.

Fig 3. Interactive Real Projective Space

Move the angle of the rotating vector and see how it affects the projected vector on the real projective line.

In this plot the coordinates on the real projective line $(p,\theta)$ represent the intercept point of $y=1$ and the angle of the rotating vector respectively. Here are some key takeaways from the plot to better understand the concept of projective spaces:

1. The scaling factor $\lambda \in {-1,1}$ represents if we are reflecting the vector across the origin or not, i.e. when $\lambda = -1$ all points from the 3rd quadrant are reflected to the 1st quadrant and all points from the 4th quadrant are reflected to the 2nd quadrant.
2. As the vector rotates, the project shadow traces its path on the $x$-axis. However, the path is traced for semicircle occupying the 1st and 2nd quadrants only, therefore, when the angle is greater than $\frac{\pi}{2}$, $\lambda$ becomes negative and the angles start from $-\frac{\pi}{2}$ to $0$, which is the other side of the $x$-axis.
3. As the angles go past $\frac{\pi}{2}$, the Projected vector on the real projective line continues to move in the same direction, this is because the points for $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ are equivalent in projective space. Therefore, while they are opposite ends of the line for the projected shadow, they are the same point on the real projective line.

We can use these observations to conclude that all points in $\mathbb{R}^2$ that are on the same ray from the origin (i.e. have the same angle $\theta$) map to the same point on the real projective line. Additionally, since the scaling factor $\lambda$ can be either positive or negative, we can represent points in all four quadrants of $\mathbb{R}^2$ using just the angles from $0$ to $\pi$ or half the unit circle.

IV. Quantum States and Projective Spaces

Let us think about an arbitary vector ($\ket{v}$) in 2D Complex Hilbert Space ($\mathbb{C}^2$). We can also represent this vector in terms of its basis vectors $\ket{0}$ and $\ket{1}$ as follows:

\[\ket{v} = x\ket{0} + y \ket{1} \;\; \forall \;\; x,y \in \mathbb{C}\]

We can apply the same logic as before and every possible $\ket{v}$ in terms of a point on the unit circle in $\mathbb{C}^2$ and a scaling factor $\lambda$ as follows:

\[\ket{v} = \lambda(\alpha\ket{0} + \beta \ket{1}) \;\; \forall \;\; \alpha,\beta,\lambda \in \mathbb{C} \;\; and \;\; |\alpha|^2 + |\beta|^2 = 1\]

Additionally, any complex number can be represented in terms of its polar coordinates $re^{i\phi}$ where $r\in\mathbb{R}$ and $\phi \in [0,\pi]$. While not immediately obvious, this polar representation is just an extension of the projective space idea we discussed earlier. Therefore,

\[\ket{v} = r\cdot e^{i\phi}(\alpha\ket{0} + \beta \ket{1}) = r \cdot \ket{\Phi}\]

where, $r\in\mathbb{R}$, $\alpha,\beta \in \mathbb{C}$ and $\phi \in [0,\pi]$. What we have defined here is a point on a unit circle $\ket{\Phi} \in \mathbb{C}^2$ and a phase factor $e^{i\phi}$ of unit norm. As you might’ve noticed that this is exactly the same as the general equation of a valid quantum state. We can use this information to define a quantum state geometrically as follows:

Definition 4 (Quantum State). A valid quantum state represents a ray from the origin to any point in the complex Hilbert space $\mathbb{C}^2$. They can be mathematically represented as $\ket{\Phi} = e^{\phi}(\alpha\ket{0} + \beta \ket{1})$ where $\alpha,\beta \in \mathbb{C}$ and $|\alpha|^2 + |\beta|^2 = 1$.

Valid vs Measurable States - A Geometric Perspective

If we define a valid qubit state as a ray, what can be say about the difference between a valid state and a measurable state? A measurable state is simply a valid state which does not have a global phase factor. Therefore, Every measurable state is a valid state, but every valid state is not a measurable state. This is due to the magnitude operator $|.|^2$ which removes the global phase factor when we measure a quantum state.

As we saw earlier, all points in the 1st and 2nd quadrants differ from points in the 3rd and 4th quadrants by a global phase factor of $e^{i\pi} = -1$. Therefore, we can say that The set of all measurable states forms the projective space of $\mathbb{C}^2$.

Definition 5 (Measurable Quantum State). The projective vector of a valid quantum state $\ket{\Psi}$ is its measurable state($\ket{\psi}$). It can be mathematically represented as $\ket{\psi} = \alpha\ket{0} + \beta \ket{1})$ where $\alpha,\beta \in \mathbb{C}$ and $|\alpha|^2 + |\beta|^2 = 1$ and $\alpha,\beta$ do not share a global phase factor.

Put another way - The projective space of $\mathbb{C}^2$ forms the set of all measurable quantum state. As we read further, what we will find is that the Bloch Sphere is the Project Space of $\mathbb{C}^2$ ($\mathbb{CP}^1$).

V. Bloch Sphere as $\mathbb{CP}^1$

The Bloch Sphere is a unit sphere where the computational basis ($\ket{0}$ and $\ket{1}$) are represented on the poles of the $z$-axis, the sign basis ($\ket{+}$ and $\ket{-}$) are represented on the poles of the $x$-axis and the imaginary basis ($\ket{+i}$ and $\ket{-i}$) are represented on the poles of the $y$-axis. This is shown in figure 4 below:

Fig 4. Bloch Sphere with all 3 axes

In order to better understand the Bloch Sphere we need to first understand the projective space of $\mathbb{C}^2$ which is denoted as $\mathbb{CP}^1$. A good first step is understanding the structure of $\mathbb{C}^2$. As you might’ve learned in high school, any complex number $c \in \mathbb{C}$ can be represented as a point in a 2D plane. Therefore, $\mathbb{C}^2$ can be represented as a 4D real space where each complex dimension is represented as a 2D plane. This makes it very hard to visualize.

Projective Space of Measurable States

While we cannot visualize the full 4D space of single qubit states, we can leverage the global phase property of measurable states to create a 3D space for all measurable states up to a global phase factor. The interactive plot below plots this 3D space where the imaginary axis of $\ket{0}$ is removed. Try moving the unit sphere to see how the $\ket{0}, \ket{1}, \ket{+}, \ket{-}, \ket{+i}, \ket{-i}$ states all relate to each other.

Compressed C2 Space

Note: While not plotted for clarity, the $\ket{-i}$ and $\ket{-}$ also have an angle of $\pi/4$ between them.

Fig 5. Interactive C2 Space without the $i\ket{0}$ axis.

Scrollwheel might not work if cursor is on the figure.

What this animation allows us to see is how in the compressed $\mathbb{C}^2$ space, the axes of the Bloch Sphere exist relative to each other. The most important is to notice how the $\ket{+},\ket{-}$ basis and $\ket{+i},\ket{-i}$ basis are at an angle of $\frac{\pi}{4}$ to each other while being on perpendicular planes. To better understand how this space creates the Bloch Sphere lets look at those planes in isolation. The real projective line for each of the individual planes can be made using the same approach as explained in Section III.

$\ket{+},\ket{-}$-Basis Plane

The $X-Z$ plane which contains the $\ket{+},\ket{-}$ basis is formed by the real axes of the $\ket{0}$ and $\ket{1}$ basis states. The resultant projective line can be termed $Re(\mathbb{C})\mathbb{P}$ or Real Component Projective Line. Here we consider the 1st and 4th quadrants as our primary space. This is completely valid since the choice of which half of the unit is entirely arbitrary. From there we can define the states $\ket{+}$ and $\ket{-}$ as the $\ket{0}$ vector rotated by $\pm \frac{\pi}{4}$ respectively. We can easily verify this to be the case by examining their state vectors. This is shown in the Figure below:

Fig 6. Projective Space of the Real Axes of a single qubit

What figure 6 shows is that on the projective line -

1. The $\ket{0}$ and $i\ket{1}$ states occupy the poles of the same axis - we can label this the $z$-axis.
2. The $\ket{+}$ and $\ket{-}$ states occupy the poles of the axis perpendicular to the $z$-axis - we can label this the $x$-axis.

$\ket{i},\ket{-i}$-Basis Plane

Now, we can repeat the same process to create the $Y-Z$-plane which consists of the imaginary axis for $\ket{1}$, which is $i\ket{1}$, and $ket{0}$. We call this ($Im(\mathbb{C}^2)$) or Imaginary Component Projective Line. In this case, we can define the $\ket{+i}$ and $\ket{-i}$ states as the $\ket{0}$ vector rotated by $\pm \frac{\pi}{4}$ respectively in the imaginary plane. This is shown in the figure below:

Fig 7. Projective Space of the Imaginary Axes of a single qubit

What figure 7 shows is that on the projective line -

1. The $\ket{0}$ and $\ket{1}$ states occupy the poles of the same axis - we can label this the $z$-axis.
2. The $\ket{i}$ and $\ket{-i}$ states occupy the poles of the axis perpendicular to the $z$-axis - we can label this the $y$-axis.

Note: The terms for both projective lines are defined for clarity and are not standard terms.

Combining Real and Imaginary Axes

As we have established any two points $p, \lambda p$ which differ only by a single scalar multiple $\lambda \in \mathbb{C}$ are equivalent in projective space. Therefore, the states $\ket{1},-\ket{1},i\ket{1}, -i\ket{1}$ are all equivalent in projective space. This further evidenced by the fact that all of those 4 states are collinear to the $\ket{0}$ state in the projective lines of the $X-Z$ and $Y-Z$ planes respectively. However, this is not the case for the $\ket{+},\ket{-}$ and $\ket{+i},\ket{-i}$ states.

To understand why, let’s look at the equation of each of these states:

\[\begin{align*} \ket{+} &= \frac{1}{\sqrt{2}}(\ket{0} + \ket{1}) \quad \quad &\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})\\ \ket{i} &= \frac{1}{\sqrt{2}}(\ket{0} + i\ket{1}) \quad \quad &\ket{-i} = \frac{1}{\sqrt{2}}(\ket{0} - i\ket{1}) \end{align*}\]

As we can see, due the relative phase difference between the states $\ket{+}$ and $\ket{i}$, we cannot transform one to the other by multiplying a scalar to the entire state. Multiplying $i$ to $\ket{+}$ for example would also affect the coefficient of $\ket{0}$ creating $\frac{1}{\sqrt{2}}(i\ket{0} + i\ket{1})$ which is not the same as $\ket{i}$. Therefore, while the axes formed by the states $\ket{+},\ket{-}$ and $\ket{+i},\ket{-i}$ are both perpendicular to the $\ket{0},\ket{1}$ axis, they are not the same. Additionally, we know that the angle between the $\ket{+}$ - $\ket{i}$ states and $\ket{-}$ - $\ket{-i}$ states respectively is $\frac{\pi}{4}$ as shown in figure 5. This can be verified by calculating the inner product between the two states. Due to the doubling of angles between state vectors, this implies that the planes formed by these axes are also perpendicular to each other. We can also arrive at this conclusion by looking at the fact that the 2 planes that the projective lines were formed from are also perpendicular to each other.

Note: While this is not a rigorous proof, it serves as a good intuitive method to understand the relationship between these basis states

Compressed C2 Space to Bloch Sphere

Now that we understand how the Real and Imaginary component projective lines and how they relate to each other, we can combine them to form the Bloch Sphere. The animation below shows how the compressed $\mathbb{C}^2$ space transforms to the Bloch Sphere.

Fig 10. Animated plot showing the transformation of compressed $\mathbb{C}^2$ space to the Bloch Sphere.

Use the slider to see different timesteps to better understand how the axes transform.

As we can see the distinct states $\ket{1},-\ket{1},i\ket{1}, -i\ket{1}$ converge onto the same point, while the $\ket{+},\ket{-}$ and $\ket{+i},\ket{-i}$ states remain distinct and form axes.

VI. Conclusion

In this article we explored the Bloch Sphere from a geometric perspective. What we see is that following the rules of what defines a ‘measurable’ state geometrically naturally leads us to the Bloch Sphere representation. Therefore, what the Bloch sphere really represents is the space of all measurably distinct quantum states, or the set of states that can be distinguished from each other through measurement in any given valid basis.

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References

1. https://www.youtube.com/watch?v=KEzmw6cKOlU ↩︎

2. https://en.wikipedia.org/wiki/bloch_sphere ↩︎

3. https://en.wikipedia.org/wiki/Space_(mathematics) ↩︎