We’ve been reading a great deal about the latest successes of AI. A range of open mathematical problems have been solved by it. The hotly debated question is whether mathematicians themselves would become obsolete in the near future. Presumably followed by physicists and other scientists.

Photo by Igor Omilaev on Unsplash

But no one I talked to is aware of the exact details behind the said achievements. How much was AI prompted by humans? Did it really invent new mathematical techniques, or did it just apply an existing method (originally established by humans) to something that is axiomatically perfectly well defined? In other words, how much “creativity” did AI exhibit (if any at all)?

Needless to say, I have no answer to any of these questions, but I can tell you about my own experience with AI. And you can draw your own conclusions.

I tried to get AI to solve a problem that came from my own PhD work. It is to find a closed form for the amount of entanglement in an arbitrary state of two qubits. Entanglement is the phenomenon Einstein called a “spooky action at a distance”, but it turns out that it’s real, and that it is a key resource for quantum computation. To quantify this most elusive of quantum phenomena, we must turn to the concept of relative entropy of entanglement, the method I pioneered during my doctorate.

Think of entanglement not as an all-or-nothing feature, but as a measure of how far a quantum state departs from classicality. The relative entropy of entanglement defines this precisely: it is the distance—measured via the quantum relative entropy—between a given entangled state and the nearest state that is completely separable, or classical. In essence, it tells us how much extra information is packed into the quantum correlations that cannot be explained by any local classical description. It answers the fundamental question: How surprised would a classical physicist be by this state?

What makes this geometric approach so profound is its universality. It satisfies all the necessary criteria for a good entanglement measure, remaining non-increasing under local operations and classical communication (LOCC). It connects the abstract mathematics of Hilbert spaces directly to thermodynamics and information theory, reminding us that entanglement is the ultimate resource in our quest to understand the universe. And, needless to say, the more entanglement a state possesses, the better it is at processing quantum information.

However, it turns out that, in general, this amount of entanglement is difficult to calculate. We can write a computer program to find the closest classical state for every entangled state, but doing this analytically is hard. This is why we still lack a closed formula for the relative entropy of entanglement. Imagine trying to find the exact footprint of a specific person from a messy, chaotic stampede. That is what is known as a forward problem: finding the “closest classical state” for a messy quantum state. It is too mathematically complex to solve universally.

About a decade ago, physicists Miranowicz and Ishizaka flipped the script with their inverse formula. Instead of looking for the footprint from the crowd, they started with a pristine classical footprint and engineered a precise map of every possible quantum state that could have left it behind. They elegantly solved the puzzle backwards, proving that while finding the destination is hard, tracing the origin path is perfectly clear. If one is given the separable state on the edge of the set of separable states, then we can find a line of entangled states for which this separable state is the closest. But this method does not work in the forward direction, which is what I was attempting to solve during my PhD.

The forward problem for entanglement soon became one of the best-known open problems in the field of quantum information. It has not been solved for the last 30 years or so. Which is why I have been so excited by the recent mathematical advances of AI.

But, despite processing petabytes of data, artificial intelligence cannot find the general closed-form for the two-qubit relative entropy of entanglement either. I tried it 6 months ago and then again a couple of weeks ago. Each time, I let it run for days with prompts. This was very disappointing for me to see.

I have an argument for why AI cannot do it. The reason for the failure is due to the fact that the barrier is fundamentally geometric, not computational. AI excels at finding patterns and interpolating functions, but it cannot bypass a structural mismatch in mathematical language.

Computing this metric requires finding a unique closest separable state along a boundary in a high-dimensional quantum landscape. The boundary of this separable state space is not smooth; it is a complex, intricately stitched patchwork of algebraic surfaces. When an AI or a neural network attempts to find a universal formula, it runs into the fact that no single, continuous algebraic expression can describe this jagged boundary. A machine learning model can easily approximate the value for any specific state numerically, but it cannot synthesize an elegant, one-size-fits-all equation because the equation itself does not exist out there in the database in standard closed form.

AI operates within the laws of logic and mathematics; it cannot create an analytical shortcut where topology dictates a dead end. Just as AI cannot express a transcendental number as a simple fraction, it cannot force a deeply fragmented, piecewise geometric boundary into a unified algebraic formula.

The bottom line is that finding a closed form for the relative entropy of entanglement for even two qubits requires human creativity, a leap of imagination. It requires us to find an isomorphism (a likeness) between two mathematical structures. And this is the kind of pattern recognition that humans excel at. So, despite the AI setback, I am confident that the problem will be solved (for two qubits, if not for general bi-partite states). But my bets are on a human mathematician or a theoretical physicist.