Scientists study the spread of information in the interacting boson system

Illustration of an effective light cone. Here, we describe the interacting bosons in terms of the Bose-Hubbard-type Hamiltonian (1). Let's start by considering the speed at which the boson particles move towards distant regions, as shown in Figure 2. The light cone transmitted by the boson particles was shown to be almost linear before logarithmic correction (indicated by a blue hatched line), as shown in Result 1. Conversely, if we consider the propagation of complete information (see also Figure 3), the speed may be much faster than particle transmission. The effective light cone is shown to be polynomial, and the exponent is equal to the spatial dimension D (denoted by the orange hatched line), where the mathematical form of the Lieb-Robinson boundary is given in Result 2. We can explicitly construct a protocol to implement the light cone using dynamics with a transient Bose-Hubbard-type Hamiltonian. Image source: Nature Communications (2024). DOI： 10.1038/s41467-024-46501-7

A new study by Japanese scientists explores the propagation of quantum information in boson systems such as Bose-Einstein condensates (BEC)-interacting bosons, revealing the potential for accelerated transmission unlike previously thought.

Quantum many-body systems, such as interacting boson systems, are used in various branches of physics and are therefore fundamentally important. The propagation of information in quantum many-body systems is controlled by Lieb-Robinson constraints. This quantifies the speed at which information or changes propagate through a quantum system.

When you make a change to one part of the system, the Lieb-Robinson boundary describes how fast the change affects the rest of the system. In practice, this means that the effects of the initial change will spread outward from its origin, affecting the immediate vicinity of the system.

However, the Lieb-Robinson of the interactive boson system has long been a challenge.

Researchers led by Dr. Tomotaka Kuwahara, head of the RIKEN Hakubi team at the RIKEN Center for Quantum Computing, addressed this challenge in their new Nature Communications study.

Dr. Kuwabara explained the importance of their work to Phys.org, emphasizing the importance of understanding quantum systems that contain elementary particles such as bosons and fermions.

"The boson system has no energy limit in principle, which makes Lieb-Robinson in the boson system very challenging," he said.

The bondage of Lieb Robinson

As mentioned earlier, the Lieb-Robinson boundary provides a quantitative limit for the correlation or the speed at which the effect propagates between spatially separated regions of a quantum system.

This means that propagation cannot be immediately ubiquitous, but is limited to effective light cones. Inspired by Einstein's theory of relativity, the light cone represents all points in space and time that can be reached by the light signal emitted by an event. This forms a double cone: one for the past and one for the future.

The same applies to information propagation in quantum many-body systems, i.e., systems with more than two quantum particles.

"The Lieb-Robinson boundary sets a universal speed limit for the speed at which information travels through these systems," Dr. Kuwahara explains.

According to the Lieb-Robinson boundary, the spread of information is finite and exponentially decays with distance or time. The details of attenuation depend on the individual system and the interactions that may occur within the system.

The Lieb-Robinson boundary, formulated by Elliott Lieb and Derek Robinson in 1972, applies only to non-relativistic systems, that is, information travels at a speed well below the speed of light.

Bose-Hubbard model

The interacting boson system is made up of many bosons, such as photons. These systems, while common, also present many challenges, such as the long-range interaction between bosons and infinite energies, which makes it difficult to develop simulation and theoretical models.

However, since the discovery of the BEC, models such as the Bose-Hubbard model have been developed to study the boson system. The Bose-Hubbard model is a theoretical framework for understanding the behavior of bosons when confined to a lattice structure, such as atoms in a crystal.

The model takes into account two main factors. The first is the jump of the boson from one lattice site to another, represented by the runout parameter. This is followed by the field interaction parameter, which represents the repulsive force between bosons when they occupy the same place. As more bosons occupy the same position, this interaction energy also increases.

These factors include the interaction between bosons, which is why the researchers chose the Bose-Hubbard model to study the Lieb-Robinson boundary in the interacting boson system.

upper limit

The researchers chose to study the Lieb-Robinson binding of a D-dimensional lattice (interactive boson system) controlled by a Bose-Hubbard model. They found three results of this system.

Outcome 1

This result solves the interaction of bosons within the lattice. The researchers found that the speed at which the boson is transported is finite, even in systems with long-range interactions. This rate, while limited, will at most grow with the logarithm of time, which is relatively slow.

This discovery provides important insights into the dynamics of the boson system, setting an upper bound for its velocity.

Outcome 2

This result focuses on the propagation of system operators over time. Operators are basically variables of a system, just like momentum. As these operators propagate, they deviate from the ideal evolution, resulting in the accumulation of errors.

This false propagation determines the speed at which information travels through the system. For example, if the error is large, it indicates that the information is propagating slower or more constrained because the approximation deviates significantly from the ideal evolution of the system.

Similarly, if the error is small, the information spreads quickly. This is consistent with the Lieb-Robinson boundary, indicating that there is an upper limit to error propagation.

Although there is an upper limit for error propagation, the interaction between bosons induces clustering in a specific region. These regions are characterized by higher boson concentrations, which help accelerate the spread of information along certain lattice paths or directions.

This phenomenon is consistent with the Lieb-Robinson boundary. However, this acceleration is bounded and has a polynomial growth depending on the dimensionality of the system.

Outcome 3

This result provides a way to model these systems using fundamental quantum gates, such as CNOT. The researchers have provided an upper bound on the number of fundamental quantum gates needed to effectively model the temporal evolution of interacting boson systems.

Comparison with the Fermi subsystem

The Fermi subsystem has a limited speed limit on the speed at which information can be propagated. Prior to this work, scientists had the same assumptions about the boson system, which was incorrect.

"The light cone diffuses faster and is nonlinear, i.e., accelerates over time. Specifically, if you're looking at a three-dimensional space, the distance "information" can travel grows with the square of time. So, in this sense, bosons can send information faster than fermions, especially over time," Dr. Kuwabara explained.

It depends on the number of bosons that can occupy the same state at the same time. Essentially, the more bosons are added, the faster the information will travel.

"However, since bosons can only move at a finite speed, it takes some time for many of them to come together, resulting in a limited speed at which information can travel. Over time, as more and more bosons cooperate, the speed at which they send information increases," Dr. Kuwabara said.

This work opens a new window for exploring the interactive boson systems of information propagation.

"I expect that the algorithm will be used to simulate condensed matter physics, which could lead to the discovery of new quantum phases. It should also prove useful in simulating quantum thermalization, helping to solve fundamental questions about how closed quantum systems enter a steady state over time," Dr. Kuwahara concluded.

More information: Tomotaka Kuwahara et al., Effective light cones and numerical quantum simulations of interacting bosons, Nature Communications (2024). DOI： 10.1038/s41467-024-46501-7.

期刊信息： Nature Communications