Density Matrices and Von Neumann Entropy

Hello! Welcome to your first lesson in our deep dive into recent advances in quantum computing.

Given your background in astrophysics and numerical simulations, you're already well-equipped with the mathematical and physical intuition that underpins much of this field. This course aims to build upon that foundation, bridging any gaps you might have in formal quantum information theory and exploring the cutting edge of research.

Today, we'll start with a foundational concept that is essential for describing any realistic quantum system: the density matrix. While introductory quantum mechanics often focuses on "pure" states described by a state vector $|\psi ⟩$, real-world systems are almost always subject to noise and interactions, leading to "mixed" states. The density matrix formalism provides a universal language to describe both.

Our learning outcome for this lesson is to derive the density matrix formalism for mixed quantum states and calculate their von Neumann entropy. We will cover:

• The distinction between pure and mixed quantum states.
• The definition and properties of the density operator, $\rho$.
• How to use the density matrix to calculate measurement outcomes and expectation values.
• The von Neumann entropy, $S(\rho )$, as a measure of a state's uncertainty or "mixedness."

This topic is the bedrock for understanding quantum noise, error correction, and entanglement—all of which are central to modern quantum computing.

1. Pure vs. Mixed States: Why We Need a New Tool

In quantum mechanics, even if we have perfect knowledge of a system's state, represented by a state vector $|\psi ⟩$, measurement outcomes are still probabilistic. This is an inherent feature of the quantum world.

However, what if our knowledge of the state itself is incomplete? For instance, imagine a source that prepares a qubit in state $|{\psi }_1⟩$ with probability $p_1$, or in state $|{\psi }_2⟩$ with probability $p_2$, and so on. This is a statistical ensemble of pure states, and we call the resulting state a mixed state. This kind of uncertainty is classical in nature, stemming from our lack of information. Your experience with statistical mechanics, which deals with macroscopic systems described by probability distributions over microstates, provides a strong parallel here.

To understand this distinction more clearly, please watch the first part of the following video.

2. The Density Operator

The state vector formalism is insufficient for describing mixed states. The correct tool is the density operator, denoted by $\rho$. Let's build it up from pure states and then generalize.

2.1. Density Operator for a Pure State

For a pure state $|\psi ⟩$, the density operator is defined as the outer product of the state vector with itself:

$$\rho =|\psi ⟩⟨\psi |$$

This operator is simply a projector onto the state $|\psi ⟩$.

Using this operator, the expectation value of an observable $A$ can be written using the trace operation:

$$⟨A⟩=⟨\psi |A|\psi ⟩=\text{tr}(|\psi ⟩⟨\psi |A)=\text{tr}(\rho A)$$

This equivalence relies on the cyclic property of the trace, $\text{tr}(XYZ)=\text{tr}(ZXY)$, which you may recall from linear algebra.

2.2. Generalizing to Mixed States

Now, consider our mixed state, which is an ensemble $\{p_k,|{\psi }_k⟩\}$. The density operator for this mixed state is the weighted average of the density operators for each pure state in the ensemble:

$$\rho =\sum _k{p}_k|{\psi }_k⟩⟨{\psi }_k|=\sum _k{p}_k{\rho }_k$$

where ${\rho }_k=|{\psi }_k⟩⟨{\psi }_k|$ is the density operator for the pure state $|{\psi }_k⟩$, and the probabilities sum to one, ${\sum }_k{p}_k=1$.

The great utility of this formalism is that the rules for calculating physical quantities maintain the same form. For example, the expectation value of an observable $A$ for a mixed state is still given by:

$$⟨A⟩=\text{tr}(\rho A)$$

This is because the expectation value for the ensemble is just the average of the individual expectation values: $⟨A⟩={\sum }_k{p}_k⟨{\psi }_k|A|{\psi }_k⟩={\sum }_k{p}_k\text{tr}({\rho }_{k}A)=\text{tr}\left(\left({\sum }_k{p}_k{\rho }_k\right)A\right)=\text{tr}(\rho A)$.

The following video segments provide a detailed derivation of the density operator for mixed states and show how the formulas for normalization, expectation values, and time evolution generalize seamlessly.

3. Properties and Purity

All density operators, whether for pure or mixed states, share three fundamental properties:

1. Hermiticity: $\rho ={\rho }^{†}$.
2. Unit Trace: $\text{tr}(\rho )=1$.
3. Positive Semidefiniteness: $⟨\varphi |\rho |\varphi ⟩\ge 0$ for any state $|\varphi ⟩$. This implies that all eigenvalues of $\rho$ are non-negative.

A crucial question is: given a density matrix $\rho$, how can we tell if it represents a pure or a mixed state? The answer lies in a property called purity, defined as $\text{tr}({\rho }^2)$.

• For a pure state, ${\rho }^2=(|\psi ⟩⟨\psi |)(|\psi ⟩⟨\psi |)=|\psi ⟩⟨\psi |\psi ⟩⟨\psi |=|\psi ⟩⟨\psi |=\rho$. Because $\rho$ is a projector, it is idempotent. This leads to $\text{tr}({\rho }^2)=\text{tr}(\rho )=1$.
• For a mixed state, it can be shown that $\text{tr}({\rho }^2)<1$.

So, the purity provides a simple test:

• $\text{tr}({\rho }^2)=1⟺$ the state is pure.
• $\text{tr}({\rho }^2)<1⟺$ the state is mixed.

The final segment of the video explains this test in detail.

4. Von Neumann Entropy: Quantifying Uncertainty

While purity tells us if a state is mixed, we need a way to quantify how mixed it is. This is the role of the von Neumann entropy, the quantum analogue of the classical Shannon entropy. It is defined as:

$$S(\rho )=-\text{tr}(\rho \log \rho )$$

The logarithm of a matrix is defined via its spectral decomposition. If $\rho$ has eigenvalues $\{{\lambda }_i\}$, it can be diagonalized, and the entropy becomes:

$$S(\rho )=-\sum _i{\lambda }_i\log {\lambda }_i$$

By convention in quantum information, $\log$ is often base 2, giving the entropy in units of bits. In physics contexts, the natural logarithm ($\ln$) is more common. We will use $\ln$ here, consistent with the physics literature.

Let's examine the two extreme cases:

• Pure state: The eigenvalues are $\{1,0,0,\dots \}$. The entropy is $S(\rho )=-(1\ln 1+0\ln 0+\dots )=0$. This makes intuitive sense: a pure state represents a state of complete knowledge, so its uncertainty is zero.
• Maximally mixed state: In a $D$-dimensional Hilbert space, this state is $\rho =\frac{1}{D}I$, where $I$ is the identity matrix. The eigenvalues are all ${\lambda }_i=1/D$. The entropy is $S(\rho )=-{\sum }_{i=1}^D\frac{1}{D}\ln \left(\frac{1}{D}\right)=-D\left(\frac{1}{D}(-\ln D)\right)=\ln D$. This is the maximum possible entropy for a system of this dimension.

The following resources provide a formal definition and explore the properties of von Neumann entropy. The video offers a clear walkthrough, while the excerpt from Preskill's renowned lecture notes gives a concise, formal summary that you may appreciate.

5. The Physical Origin of Mixed States

We've established that mixed states represent a lack of knowledge. This lack of knowledge can arise in two fundamentally different ways.

1. Imperfect Preparation (Classical Ignorance): As discussed, we might simply not know which pure state was prepared in our apparatus. This is a classical statistical mixture.

2. Entanglement (Quantum Ignorance): This is a more profound, purely quantum-mechanical source of mixedness. If a system A is entangled with another system B, then even if the total system AB is in a pure state, system A by itself must be described as a mixed state.

To find the state of subsystem A, we "trace out" the degrees of freedom of subsystem B. This operation is called the partial trace. If the total system is described by ${\rho }_{AB}$, the state of subsystem A is:

$${\rho }_A={\text{tr}}_B({\rho }_{AB})$$

Let's see a concrete example. Consider the Bell state $|{\Phi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$, which is a pure state of a two-qubit system (AB). The global density matrix ${\rho }_{AB}=|{\Phi }^{+}⟩⟨{\Phi }^{+}|$ represents a pure state, so $S({\rho }_{AB})=0$.

However, if we trace out qubit B to find the state of qubit A, we get:

$${\rho }_A={\text{tr}}_B({\rho }_{AB})=\frac{1}{2}|0⟩⟨0|+\frac{1}{2}|1⟩⟨1|=\left(\begin{array}{cc}1/2& 0\\ 0& 1/2\end{array}\right)$$

This is the maximally mixed state for a single qubit! Its entropy is $S({\rho }_A)=\ln 2$. This means that if you only have access to qubit A, your measurement outcomes will be completely random, even though the global state is perfectly known. The information is not located in A or B individually, but is encoded in their correlations.

The following reading provides a clear definition of the partial trace and works through this exact example.

Conclusion

In this lesson, we have established the essential framework for describing general quantum states.

Key Takeaways:

• Quantum systems can be in pure states (full knowledge, described by a state vector $|\psi ⟩$) or mixed states (statistical ensembles of pure states).
• The density operator $\rho$ is a universal tool that can describe any quantum state. It is defined as $\rho ={\sum }_k{p}_k|{\psi }_k⟩⟨{\psi }_k|$ for a mixed state and $\rho =|\psi ⟩⟨\psi |$ for a pure state.
• The purity, $\text{tr}({\rho }^2)$, is a simple test to distinguish pure states ($\text{tr}({\rho }^2)=1$) from mixed states ($\text{tr}({\rho }^2)<1$).
• The von Neumann entropy, $S(\rho )=-\text{tr}(\rho \log \rho )$, quantifies the uncertainty of a quantum state. It is zero for pure states and positive for mixed states, reaching a maximum of $\ln D$ for a maximally mixed state in a $D$-dimensional space.
• A mixed state can arise from classical ignorance about state preparation or, more profoundly, as the state of a subsystem of a larger, entangled pure state.

This formalism is not just a mathematical convenience; it is the natural language for quantum information science, especially when dealing with the effects of noise and the nature of entanglement.

Preview of the Next Lesson:

In our next session, we will focus on single-qubit operations. The learning outcome is to "Analyze single-qubit gates as rotations on the Bloch sphere, identifying the rotation axis and angle for each standard gate." We will see how the Bloch sphere provides a beautiful geometric picture for single-qubit pure states, and I will show you how the density matrices we've studied today allow us to represent mixed states as points inside the sphere, giving a geometric interpretation to purity and mixedness.