Visualizing Dephasing and Amplitude Damping on the Bloch Sphere

Introduction

In our previous lesson, we established the operator-sum representation (or Kraus formalism) as the mathematical framework for describing open quantum systems. We saw how this formalism allows us to model the evolution of a quantum state under the influence of environmental noise. We derived the Kraus operators for three fundamental noise models: the bit-flip, phase-flip, and depolarizing channels.

This lesson builds directly on that foundation. We will now analyze two more physically crucial noise channels: dephasing and amplitude damping. Unlike the more abstract Pauli channels, these models directly correspond to common physical processes in real quantum hardware: loss of phase coherence and energy relaxation.

Our goal is to derive the action of these channels on single-qubit states and, crucially, to visualize their geometric effect on the Bloch sphere. Your experience with physical modeling will be valuable here, as we connect the abstract mathematical machinery of quantum channels to tangible physical phenomena and their geometric consequences. By the end of this lesson, you will have a strong intuition for how these primary forms of decoherence corrupt quantum information.

From Density Matrix to Bloch Sphere

Before we dive into the new channels, let's briefly formalize the connection between the density matrix $\rho$ and its geometric representation as a point inside the Bloch sphere. As we touched upon in the last lesson, any single-qubit density matrix can be written in terms of the identity matrix $I$ and the Pauli matrices $\vec{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)$ as:

$$\rho =\frac{1}{2}(I+\vec{P}\cdot \vec{\sigma })$$

Here, $\vec{P}=(P_x,P_y,P_z)$ is the Bloch vector, a real three-dimensional vector. For pure states, $|\vec{P}|=1$, and the vector lies on the surface of the sphere. For mixed states, $|\vec{P}|<1$, and the vector lies inside the sphere. The maximally mixed state $\rho =I/2$ corresponds to the origin, $\vec{P}=(0,0,0)$.

By inverting this relationship, we can express the components of the Bloch vector in terms of the density matrix elements:

$$P_x=\text{Tr}(\rho {\sigma }_x)={\rho }_{01}+{\rho }_{10}$$ $$P_y=\text{Tr}(\rho {\sigma }_y)=i({\rho }_{10}-{\rho }_{01})$$ $$P_z=\text{Tr}(\rho {\sigma }_z)={\rho }_{00}-{\rho }_{11}$$

This duality is powerful: we can analyze the algebraic action of a channel on $\rho$ and then translate it into a geometric transformation of the Bloch vector $\vec{P}$.

The Dephasing (Phase Damping) Channel

The first process we will examine is dephasing. This type of noise describes the loss of quantum coherence between the $|0⟩$ and $|1⟩$ states without any change in the populations (${\rho }_{00}$ and ${\rho }_{11}$). Physically, this can be thought of as the environment "measuring" the qubit in the computational basis, which destroys the superposition.

We will again use John Preskill's lecture notes as our guide.

Action on the Density Matrix

As derived in the notes, the dephasing channel acts on a density matrix $\rho$ as:

$$\mathcal{E}(\rho )=\left(\begin{array}{cc}{\rho }_{00}& (1-p){\rho }_{01}\\ (1-p){\rho }_{10}& {\rho }_{11}\end{array}\right)$$

where $p$ is the probability of a phase-damping event. Notice that the diagonal elements, which represent the probabilities of being in state $|0⟩$ or $|1⟩$, are unchanged. The off-diagonal elements, which represent the coherence between these states, decay by a factor of $(1-p)$.

If this process occurs continuously in time with a rate $\Gamma$, then after a time $t$, the off-diagonal elements are suppressed by a factor of $e^{-\Gamma t}$. The characteristic time for this decay, $T_2=1/\Gamma$, is often called the dephasing time or transverse relaxation time.

Geometric Effect on the Bloch Sphere

Let's see what this means for the Bloch vector $\vec{P}$. Using our mapping from $\rho$ to $\vec{P}$:

• The new $P_z^{\prime }$ is ${\rho }_{00}^{\prime }-{\rho }_{11}^{\prime }={\rho }_{00}-{\rho }_{11}=P_z$. The z-component is unchanged, as expected, since populations are conserved.
• The new $P_x^{\prime }$ is ${\rho }_{01}^{\prime }+{\rho }_{10}^{\prime }=(1-p)({\rho }_{01}+{\rho }_{10})=(1-p)P_x$.
• The new $P_y^{\prime }$ is $i({\rho }_{10}^{\prime }-{\rho }_{01}^{\prime })=i(1-p)({\rho }_{10}-{\rho }_{01})=(1-p)P_y$.

So, the transformation on the Bloch vector is:

$$(P_x,P_y,P_z)⟶((1-p)P_x,(1-p)P_y,P_z)$$

This is a very clear geometric picture: the Bloch sphere is compressed horizontally towards the z-axis. The components of the Bloch vector in the x-y plane, which represent coherence, shrink, while the z-component, representing energy/population, is preserved. A pure state on the surface of the sphere is mapped to a mixed state inside a prolate spheroid, whose poles are still at $(0,0,\pm 1)$.

The Amplitude Damping Channel

Next, we consider a process that involves energy exchange with the environment: amplitude damping. The canonical example is an excited atom spontaneously emitting a photon and decaying to its ground state. For a qubit, this corresponds to the state $|1⟩$ decaying towards the state $|0⟩$. This process involves both energy relaxation and dephasing.

Action on the Density Matrix

From the physical model of spontaneous emission, the Kraus operators are derived as:

$$M_0=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{1-p}\end{array}\right),M_1=\left(\begin{array}{cc}0& \sqrt{p}\\ 0& 0\end{array}\right)$$

where $p$ is the probability of decay over a given time interval. $M_1$ represents the "quantum jump" from $|1⟩$ to $|0⟩$, while $M_0$ describes the evolution if no jump occurs.

Applying the operator-sum formula $\mathcal{E}(\rho )=M_0\rho M_0^{†}+M_1\rho M_1^{†}$ gives the transformed density matrix:

$$\mathcal{E}(\rho )=\left(\begin{array}{cc}{\rho }_{00}+p{\rho }_{11}& \sqrt{1-p}{\rho }_{01}\\ \sqrt{1-p}{\rho }_{10}& (1-p){\rho }_{11}\end{array}\right)$$

Here we see two effects: the population in the excited state, ${\rho }_{11}$, decays, and this lost population is added to the ground state, ${\rho }_{00}$. Simultaneously, the off-diagonal coherence terms also decay, but with a different factor.

In the continuous time limit with decay rate $\Gamma$, the population of the excited state, ${\rho }_{11}(t)$, decays as $e^{-\Gamma t}$. The characteristic time for this process, $T_1=1/\Gamma$, is called the energy relaxation time or longitudinal relaxation time.

Geometric Effect on the Bloch Sphere

Let's derive the transformation on the Bloch vector $\vec{P}$:

• $P_x^{\prime }={\rho }_{01}^{\prime }+{\rho }_{10}^{\prime }=\sqrt{1-p}({\rho }_{01}+{\rho }_{10})=\sqrt{1-p}{P}_x$
• $P_y^{\prime }=i({\rho }_{10}^{\prime }-{\rho }_{01}^{\prime })=i\sqrt{1-p}({\rho }_{10}-{\rho }_{01})=\sqrt{1-p}{P}_y$
• For $P_z^{\prime }$, we have: $$P_z^{\prime }={\rho }_{00}^{\prime }-{\rho }_{11}^{\prime }=({\rho }_{00}+p{\rho }_{11})-(1-p){\rho }_{11}={\rho }_{00}-{\rho }_{11}+2p{\rho }_{11}=P_z+2p{\rho }_{11}$$ Since ${\rho }_{11}=(1-P_z)/2$, we can substitute this in: $$P_z^{\prime }=P_z+2p\frac{1-P_z}{2}=P_z+p(1-P_z)=(1-p)P_z+p$$

The full transformation on the Bloch vector is:

$$(P_x,P_y,P_z)⟶(\sqrt{1-p}{P}_x,\sqrt{1-p}{P}_y,(1-p)P_z+p)$$

The geometric picture is more complex than for dephasing. The mapping is an affine transformation. The Bloch sphere shrinks towards a single point: $(0,0,1)$, which corresponds to the ground state $|0⟩$. This beautifully visualizes the process of energy relaxation.

An important consequence derived from the time evolution is the relationship between the two relaxation times. The off-diagonal elements ${\rho }_{01}$ decay with a factor of $\sqrt{1-p}\approx e^{-\Gamma t/2}$, which implies a dephasing time $T_2=2/\Gamma$. Since the energy relaxation time is $T_1=1/\Gamma$, for the amplitude damping channel we have the fundamental relationship $T_2=2T_1$. This means that energy relaxation is always accompanied by some amount of dephasing.

Test your understanding!

Consider a qubit prepared in the state $|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$.

1. What is its initial Bloch vector ${\vec{P}}_{init}$?
2. The qubit is subjected to a dephasing channel with parameter $p$. What is the new Bloch vector ${\vec{P}}_{dephased}$? Describe the trajectory of the state vector.
3. Instead, the qubit is subjected to an amplitude damping channel with parameter $p$. What is the new Bloch vector ${\vec{P}}_{damped}$? Describe its trajectory.

Show answer

1. The state $|+⟩$ has density matrix $\rho =|+⟩⟨+|=\frac{1}{2}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$. Its Bloch vector is ${\vec{P}}_{init}=(1,0,0)$. It lies on the equator of the Bloch sphere, pointing along the x-axis.

2. Under dephasing, the transformation is $(P_x,P_y,P_z)\to ((1-p)P_x,(1-p)P_y,P_z)$. Applying this to ${\vec{P}}_{init}$:
   ${\vec{P}}_{dephased}=((1-p)\cdot 1,(1-p)\cdot 0,0)=(1-p,0,0)$
   The state vector moves from the surface of the sphere straight along the x-axis towards the center. Its coherence is reduced, but it remains "pointing" along the x-direction.

3. Under amplitude damping, the transformation is $(P_x,P_y,P_z)\to (\sqrt{1-p}{P}_x,\sqrt{1-p}{P}_y,(1-p)P_z+p)$. Applying this to ${\vec{P}}_{init}$:
   ${\vec{P}}_{damped}=(\sqrt{1-p}\cdot 1,\sqrt{1-p}\cdot 0,(1-p)\cdot 0+p)=(\sqrt{1-p},0,p)$
   The state vector moves from $(1,0,0)$ along a curved path in the x-z plane. As $p$ goes from 0 to 1, the path traces an arc from $(1,0,0)$ to the north pole $(0,0,1)$. The qubit loses both coherence (the x-component shrinks) and energy (the z-component increases towards +1).

Conclusion

In this lesson, we have moved beyond the abstract Pauli error models to analyze two physically ubiquitous noise channels and their geometric effects.

Key Takeaways:

• Dephasing (Phase Damping) models the loss of coherence without energy relaxation. Its geometric effect is to squash the Bloch sphere into a prolate spheroid along the z-axis. The transverse components $(P_x,P_y)$ decay, while the longitudinal component $P_z$ is preserved. This process is characterized by the dephasing time $T_2$.

• Amplitude Damping models energy relaxation, such as an excited state decaying to the ground state. Its geometric effect is an affine transformation that shrinks and shifts the entire Bloch sphere towards the ground state $|0⟩$ at the north pole. This process affects all components of the Bloch vector and is characterized by the energy relaxation time $T_1$.

• For amplitude damping, energy relaxation inherently causes dephasing, leading to the relation $T_2=2T_1$. In more general physical systems, additional dephasing mechanisms can be present, leading to the inequality $T_2\le 2T_1$.

Understanding these geometric transformations provides a powerful intuition for the dynamics of decoherence, turning the abstract algebra of Kraus operators into a visual narrative of how quantum information is lost.

Preview of the next lesson:

Having thoroughly examined single-qubit states, gates, and noise, we are now ready to consider multi-qubit systems. The true power of quantum computation arises from the interactions between qubits. In the next lesson, we will introduce the mathematical tool for describing multi-qubit systems—the tensor product—and use it to apply fundamental two-qubit gates like CNOT, CZ, and SWAP.