Quantum Harmonic Oscillator and Wigner Functions

Hello! Welcome to the next lesson in our exploration of recent advances in quantum error correction.

In our last session, we delved into autonomous QEC, where we saw how engineered dissipation could passively protect a quantum state. The primary example we studied, the squeezed cat code, belongs to a class of continuous-variable (CV) codes. These codes don't use discrete two-level qubits but instead encode information in the continuous degrees of freedom of a quantum harmonic oscillator, like a mode of an electromagnetic field in a microwave cavity.

To properly understand and analyze these powerful codes, such as the Schrödinger cat and GKP codes we'll be studying next, we must first master the mathematical and conceptual language of CV quantum systems. This lesson is dedicated to building that foundation.

Today, we will focus on the learning outcome: Apply quantum harmonic oscillator operators to express continuous-variable quantum states and calculate their Wigner functions. We'll revisit the quantum harmonic oscillator, introduce its key states, and then learn how to use the Wigner function—a powerful tool for visualizing and analyzing these states in phase space.

1. The Language of Continuous Variables: Harmonic Oscillator Operators

Your background in physics means you are well-acquainted with the quantum harmonic oscillator (QHO). It serves as the bedrock for describing bosonic fields, such as light, and is therefore the fundamental model for continuous-variable quantum information.

Let's begin with a quick review of the key operators.

As the text reminds us, the system is described by the position ($\hat{q}$) and momentum ($\hat{p}$) operators, which satisfy the canonical commutation relation $[\hat{q},\hat{p}]=i$ (setting $\hslash =1$).

It is often more convenient to work with their linear combinations, the annihilation and creation operators:

These operators satisfy the commutation relation $[\hat{a},{\hat{a}}^{†}]=1$. Using these, we can define the fundamental states of the oscillator:

• Fock States (or Number States) $|n⟩$: These are the energy eigenstates of the QHO Hamiltonian $\hat{H}=\omega ({\hat{a}}^{†}\hat{a}+1/2)$. They are eigenstates of the number operator $\hat{n}={\hat{a}}^{†}\hat{a}$ with integer eigenvalues $n$. A state $|n⟩$ contains exactly $n$ quanta of energy (e.g., photons). They are generated by successive applications of the creation operator on the vacuum state: $|n⟩=\frac{({\hat{a}}^{†}{)}^n}{\sqrt{n!}}|0⟩$.

• Coherent States $|\alpha ⟩$: These are the eigenstates of the annihilation operator: $\hat{a}|\alpha ⟩=\alpha |\alpha ⟩$. The eigenvalue $\alpha$ is a complex number, whose real and imaginary parts correspond to the quadratures of the oscillator. A coherent state can be generated by applying the displacement operator $\hat{D}(\alpha )=e^{\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}}$ to the vacuum: $|\alpha ⟩=\hat{D}(\alpha )|0⟩$. They are considered the "most classical" of quantum states.

2. The Wigner Function: Visualizing States in Phase Space

While state vectors are a complete description, they are abstract and live in an infinite-dimensional Hilbert space. For CV systems, it's often more insightful to use a phase-space representation. The most common of these is the Wigner function, which maps a quantum state (represented by its density matrix $\hat{\rho }$) to a real-valued function $W(q,p)$ on a 2D phase space.

The following resource provides a formal definition and explores its properties.

As defined in the notes (Eq. 52), for a pure state $|\psi ⟩$ with wavefunction $\psi (q)=⟨q|\psi ⟩$, the Wigner function is:

(Note: we've set $\hslash =1$ and made a change of variables $\xi =2y$ from the text for a common form of the definition).

The Wigner function has several crucial properties:

1. It is a quasi-probability distribution. Integrating it over one variable gives the probability distribution of the other. For instance, $\int dpW(q,p)=|\psi (q){|}^2$. However, it is not a true probability distribution because...
2. It can be negative. Regions of negativity in the Wigner function are a definitive signature of the non-classical nature of a quantum state. As stated in the Mari-Veitch theorem, any quantum computation that involves only states and operations with positive Wigner functions can be efficiently simulated on a classical computer. Therefore, Wigner negativity is a necessary resource for a quantum advantage.

3. Calculating Wigner Functions for Key States

Now let's apply this formalism to calculate the Wigner functions for the states we introduced. This will build our intuition for what these functions look like and what they represent. The Polkovnikov notes are an excellent guide for these calculations.

The Vacuum State $|0⟩$

The vacuum state is the ground state of the QHO. Its wavefunction is a Gaussian. As shown in the notes (LINK, Eq. 58), its Wigner function is also a 2D Gaussian centered at the origin of phase space.

Coherent States $|\alpha ⟩$

A coherent state $|\alpha ⟩$ is a vacuum state that has been displaced in phase space by the complex amount $\alpha$. Its Wigner function is simply the Gaussian blob of the vacuum, shifted so its center is at $\alpha$. Like the vacuum, its Wigner function is positive everywhere. This is why coherent states are considered "classical-like."

For convenience, we often describe phase space not with $(q,p)$ but with the complex amplitude $a=(\hat{q}+i\hat{p})/\sqrt{2}$. The Polkovnikov notes show this calculation in Section B (LINK, Eq. 76), resulting in:

Fock States $|n⟩$

Fock states, for $n>0$, are fundamentally non-classical. Their Wigner functions must exhibit negativity. The calculation is more involved, leading to an expression involving Laguerre polynomials.

The result is (Eq. 81):

The Laguerre polynomials $L_n(x)$ oscillate, which, combined with the overall factor of $e^{-2|a{|}^2}$, creates a function with concentric rings of alternating positive and negative values. For $n=1$, this results in a positive outer ring and a negative "dip" at the center, as seen in the figure above.

4. A Glimpse of Continuous-Variable Codes

With these tools in hand, we can now appreciate the structure of the CV error correction codes we will be building. The figure below shows the Wigner functions for a gallery of important non-Gaussian states.

You can now interpret these images:

• (b) Cat code: This is a superposition of two coherent states with opposite phases, $|\psi ⟩\propto |\alpha ⟩+|-\alpha ⟩$. You can see the two positive Gaussian blobs corresponding to $|\alpha ⟩$ and $|-\alpha ⟩$, and crucially, the striped interference pattern between them, which is a region of rapidly oscillating Wigner negativity.
• (h) GKP code: This is a much more exotic state, forming a grid of positive Gaussian "spikes" embedded in a sea of negativity. This structure is what gives it the power to correct for small displacement errors in phase space.

We are now equipped to dissect these states in detail.

Conclusion

In this lesson, we established the fundamental framework for describing continuous-variable quantum systems. We have moved from the abstract Hilbert space picture to the intuitive phase-space picture provided by the Wigner function.

Key Takeaways:

• CV quantum systems are modeled by the quantum harmonic oscillator, described by operators $\hat{q},\hat{p}$ or, more conveniently, $\hat{a},{\hat{a}}^{†}$.
• The Wigner function maps a quantum state onto a real-valued quasi-probability distribution in phase space.
• Wigner negativity is a hallmark of non-classicality and is a necessary ingredient for quantum computational advantage.
• Simple states have characteristic Wigner functions: Gaussians for the vacuum and coherent states, and oscillating, negative-valued functions for Fock states.
• Complex CV code states, like cat and GKP codes, exhibit intricate structures of Wigner negativity that are key to their error-correcting properties.

Preview of the next lesson:
We will immediately put this new formalism to use. In our next lesson, we will dive into the construction and properties of Schrödinger cat codes. We will analyze how their structure, as a superposition of coherent states, allows them to correct for the dominant error in many physical systems: single photon loss.