Experimental verification of the area law of mutual information in a quantum field simulator

This study experimentally verifies the area law of quantum mutual information in gapped one-dimensional quantum field theories using an ultra-cold atom simulator, overcoming the challenges of measuring von Neumann entropy in spatially extended subsystems.

The Gist

Imagine you have a giant, invisible blanket made of quantum particles. If you look at a tiny patch of this blanket, how much information does that patch share with the rest of the blanket? This is the big question the scientists in this paper asked.

In the world of quantum physics, there's a famous rule called the "Area Law." Think of it like this: If you have a room full of people talking, the amount of gossip a small group of people shares with the rest of the room usually depends on how many people are standing at the edge of that group (the surface area), not on how many people are sitting inside the group (the volume).

For a long time, physicists knew this rule should exist in theory, but proving it in a real, messy quantum system was incredibly hard. It's like trying to count every single grain of sand on a beach to understand the shape of the shore.

The Experiment: A Quantum "Twin" Setup

The team at TU Wien in Vienna built a special laboratory setup to test this. Here is how they did it, using a simple analogy:

1. The Twins: They took a cloud of ultra-cold atoms (specifically Rubidium) and split it into two identical "twins" sitting side-by-side in a double-well trap. Think of these as two synchronized swimmers who are holding hands.
2. The Snap: They let these twins interact and cool down together until they reached a calm, balanced state (thermal equilibrium). Then, in a split second, they "snapped" the connection between them. The two clouds were now free to swim on their own.
3. The Snapshot: As the clouds moved apart, the scientists took hundreds of "photos" (measurements) of how the waves in the clouds interfered with each other. Because the clouds were quantum objects, these photos showed them the hidden "phase" (the timing of the waves) of the atoms.

The Detective Work: Reconstructing the Invisible

The scientists couldn't see the atoms directly, but they could see the ripples they made. By taking thousands of these photos at different moments in time, they used a mathematical trick called tomography (similar to a CT scan for a human body) to reconstruct the entire "state" of the system.

They built a giant map (called a covariance matrix) that described how every part of the cloud was connected to every other part. Once they had this map, they could calculate the Mutual Information—a fancy term for "how much two pieces of the cloud know about each other."

The Big Discovery: The Area Law is Real

When they looked at the data, they found exactly what the theory predicted:

• The Volume Law (The "Noise"): When they measured the total "disorder" (entropy) of a chunk of the cloud, it grew as the chunk got bigger. This is like a noisy party: the more people you have in a room, the louder it gets. This part followed the "Volume Law."
• The Area Law (The "Secret"): However, when they measured how much information one chunk shared with the rest of the cloud, the amount of shared information stopped growing once the chunk got big enough. It hit a "plateau."

The Analogy: Imagine a long line of people passing a secret note.

• If you ask, "How much noise is in this line?" the answer gets bigger the longer the line is (Volume Law).
• But if you ask, "How much does the first 10 people know about the last 10 people?" the answer is almost zero if they are far apart. If they are close, they know a lot. But once you look at a big block of people in the middle, the amount of information they share with the outside world depends only on the two people at the very edges of the block, not the hundreds of people inside. That is the Area Law.

What They Found About Distance and Heat

The team also tested two other things:

1. Distance: They moved two separate chunks of the cloud further apart. As expected, the "shared information" dropped off quickly, like a radio signal fading as you walk away from the tower. They measured exactly how fast it faded, which matched the theoretical "correlation length" (how far the quantum connection reaches).
2. Temperature: They checked if heating up the cloud changed the rules. They found that while the total noise increased with heat, the fundamental rule about shared information (the Area Law) held true.

Why This Matters (According to the Paper)

The paper states that this is a crucial step forward. Before this, scientists could only guess that the Area Law existed in these complex quantum fields. Now, they have experimentally verified it.

They also noted that while they successfully measured the "shared information," they couldn't yet measure "entanglement" (a deeper, stranger quantum connection) because their system was still a little too "warm" and their cameras a little too blurry to see the tiniest details. But this experiment proved the foundation is solid, paving the way for future experiments to probe even deeper into the secrets of quantum fields.

In short: They built a quantum simulator, took a "CT scan" of it, and proved that in the quantum world, information is shared mostly across boundaries, not through the bulk, just like the famous Area Law predicts.

Technical Summary

Technical Summary: Experimental Verification of the Area Law of Mutual Information in a Quantum Field Simulator

Problem Statement
The scaling of entropies and mutual information is a fundamental property of quantum many-body systems, with profound implications for condensed matter physics, quantum field theory, and gravity. While theoretical frameworks predict that the von Neumann (vN) entropy of ground states in gapped systems follows an "area law" (scaling with surface area rather than volume), and that thermal states of such systems exhibit an area law for mutual information (MI), experimental verification has been elusive. Measuring vN entropy is notoriously difficult because it requires complete knowledge of the system's density matrix, necessitating full state tomography. While techniques exist to measure purity (and thus second-order Rényi entropy) or vN entropy in specific lattice setups, measuring vN entropy for spatially extended subsystems in continuous quantum field theories, and verifying the predicted area law scaling of MI, has remained a significant challenge.

Methodology
The authors utilize an ultra-cold atom simulator to model a one-dimensional (1D) quantum field theory. The experimental setup involves a pair of tunnelling-coupled quasi-1D Bose gases ($^{87}\text{Rb}$) confined in a double-well potential on an atom chip.

1. State Preparation: The system is prepared in a thermal equilibrium state described by a massive Klein-Gordon (KG) Hamiltonian. This is achieved by cooling atoms in a strongly coupled double-well potential. The relative phase $\phi(z)$ between the two condensates and the relative density fluctuations $\delta\rho(z)$ serve as the canonical conjugate fields.
2. Quench and Evolution: To probe the system, the inter-well tunnelling rate $J$ is rapidly quenched to zero. The system then evolves independently under a Tomonaga-Luttinger liquid (TLL) Hamiltonian. This evolution rotates the initial density correlations into phase correlations and vice versa.
3. Tomography: The researchers employ a quantum tomography protocol to reconstruct the full covariance matrix $\Gamma$ of the initial state. By measuring the spatially resolved relative phase at various evolution times after the quench, they fit the time-dependent phase-phase correlations to extract the initial phase-phase ($Q$), density-density ($P$), and phase-density ($R$) correlation matrices.
4. Gaussianity Verification: The authors verify that the initial state is Gaussian by measuring the normalized connected fourth-order correlation function, confirming that higher-order correlations are negligible. This allows the full state to be described entirely by the covariance matrix.
5. Entropy Calculation: Using the reconstructed covariance matrix, the symplectic spectrum is calculated. The vN entropy for any subsystem $A$ is derived from the symplectic eigenvalues $\gamma_n$ using the formula:
   $$S(\Gamma_A) = \sum_{n=1}^{N} \left[ \left(\gamma_n + \frac{1}{2}\right) \ln \left(\gamma_n + \frac{1}{2}\right) - \left(\gamma_n - \frac{1}{2}\right) \ln \left(\gamma_n - \frac{1}{2}\right) \right]$$
   The mutual information $I(A:B)$ is then calculated as $S_A + S_B - S_{A \cup B}$.

Key Contributions and Results

• Experimental Verification of the Area Law for MI: The study provides the first experimental verification of the area law for quantum mutual information in a continuous quantum field simulator. The results show that while the vN entropy of a subsystem scales linearly with its size (volume law, as expected for thermal states), the mutual information between a subsystem and its complement saturates to a constant value in the bulk of the system, independent of the subsystem size.
• Dependence on Separation and Temperature: The authors investigate the dependence of MI on the distance between two disjoint subsystems. They observe an exponential decay of MI with separation, extracting a decay length ($l_{fit} \approx 5.1 \, \mu\text{m}$) that agrees with the theoretically calculated correlation length ($l_C \approx 6.8 \, \mu\text{m}$). Furthermore, they demonstrate that while vN entropy depends linearly on temperature, the mutual information reaches a finite asymptotic value determined by classical correlations, regardless of the ultraviolet (UV) cut-off, though the asymptotic value is reduced by finite mode resolution.
• Robustness of the Method: The reconstruction of the covariance matrix is performed without assuming the Gaussianity of the state a priori; Gaussianity is confirmed post-hoc. The method relies only on the assumption that the post-quench dynamics follow the TLL Hamiltonian.

Significance
The paper claims this work is a "crucial step" toward employing ultra-cold atom simulators to probe entanglement in quantum field theories. By successfully measuring the vN entropy of spatially extended subsystems and verifying the area law of mutual information, the authors demonstrate a pathway to accessing quantum information measures in continuous systems that were previously inaccessible.

The authors modestly note that while the reconstruction of the full covariance matrix enables the calculation of various entanglement measures (such as logarithmic negativity), the detection of genuine entanglement in their current experiments is limited by finite temperature (which keeps mode occupation numbers high) and finite optical resolution (which introduces a soft UV cut-off). However, the successful verification of the area law for MI in thermal states establishes the validity of the experimental platform and methodology for future investigations into entanglement in interacting many-body quantum systems.