# Time, Randomness, and Correlations in a Quantum Model

How do you know if something is random? If you were a substitute teacher that only taught on Wednesdays, you might interpret a dip in attendance as a random fluctuation. If you taught that same class every single day, however, the dip might signal the tail end of a local flu epidemic that caused even more students to miss class on Monday and Tuesday. Most “random” events are not as random as they appear.

If you monitor a system, like a class, in short enough intervals, you’ll see that many seemingly random events are actually caused by environmental factors. The influence of these factors may last for a time, but they eventually disappear. This means that your interpretation of what is happening in a system may depend on how often you collect data. Not only is this true in day-to-day life, it’s also true on the microscopic level.

In the school scenario, a teacher can combine attendance numbers with anecdotes from students and news reports to get the big picture of what’s going on. However, when scientists are studying microscopic systems that have never been studied before, doing experiments that have never been done before, the question of how often to take data becomes even more important.

A team of scientists from Brazil and Australia recently published a journal article in the American Physical Society’s Physical Review A that sheds light on this question. They studied a quantum mechanical system and, in doing so, uncovered a mathematical relationship between the length of time an environmental correlation lasts (e.g., how long the flu epidemic lasts), and how often you need to collect data in order to see that correlation.

This might seem kind of unnecessary. After all, why wouldn’t you just collect data as often as possible?

When you do a quantum mechanical experiment, environmental noise—fluctuations in the data that come from the experimental setup—can have short-term effects that obscure the actual data. By collecting data at an interval that is much longer than these short-term effects, you can sometimes get rid of this noise. However, if you choose an interval that is too large, you could miss important changes in the system you are studying. On the flip side, if your goal is to see whether an event is truly random, the data collection interval must be short enough to show any environmental correlations.

In addition, scientists need to consider how quickly their equipment can collect and process data, as well as the related time and expense. Optimizing the interval time, therefore, depends on the goals of the experiment, the equipment, and the details of the physical system.

“Many times in physics, we make approximations [and say] that for them to hold, a given parameter must be much larger than another, but we do not specify what exactly ‘much larger’ means,” says Nadja Bernardes, a researcher at Universidade Federal de Minas Gerais in Brazil and the lead author of the paper. That’s because we don’t always know what “much larger” means.

When you need to take data at an interval that is “much larger” than the correlation time of the environment, should it be ten times as large? One hundred times? One thousand times? The answer depends on the details of the system. In practice, these correlations can be really hard to separate from those caused by other sources.

In previous work, this research team developed what’s called a “collisional model” for studying how environmental correlations impact the behavior of a qubit over time. A qubit is the quantum version of a bit. A qubit is kind of like a bird that can fly all over the globe when nobody is watching, but when you look at it, it only shows up either in the north or south pole—like a coin toss, it only has two possible outcomes.

Imagine collecting data on a coin toss experiment where the outcome of each toss depends on the outcome of past tosses and is shaped by the environment. The team’s collisional model represents this kind of situation in the quantum realm. With their model, it’s possible to investigate environmental correlations independently from other sources.

This new research builds on that work. The team members, who are also good friends, delved deeper and deeper into this subject over the course of two years, analyzing how the interval at which you take data impacts the results of the experiment. In this way, they were able to determine the minimum time interval that you need between measurements if you want to completely overlook the effect of environmental correlations.

In doing so, the team identified what “much larger” means in the form of a clear mathematical relation between interval time and environmental correlation time. If you know the correlation time, this relation tells you the smallest time interval at which you should collect data in order to eliminate changes caused by noise in the environment. The researchers were surprised to find that in some cases, a “much larger” interval time can be just twice the correlation time! Their work shows that this mathematical relation holds true for different types of correlations and environments.

This approach can be used to investigate many different scenarios, and can help researchers better understand the impact of the environment on their data. “Our results are important in both directions,” says Bernardes, “they hint on the required observation frequency if one wants to spot memory effects and better describe the dynamics of a microscopic system or wash them out and generate true randomness, an essential feature in many modern applications such as cryptography.”

Kendra Redmond