How Can We Describe a Quantum State Using Simple Probabilities?

In 2024, to celebrate the International Year of Quantum Science and Technology, researchers from the Lebedev Institute and UNAM published a study that reshapes how we talk about quantum states. Instead of relying solely on wave functions, they showed that ordinary probability distributions—similar to those we encounter daily—can fully describe a quantum system. These so‑called tomographic distributions capture all the information of a quantum state without the negative values typical of the Wigner function.

The team derived tomographic forms for particles moving in various potentials: the free case, finite and infinite wells, and even the Morse potential, which models molecular vibrations. The results illustrate how the probability of finding a particle with a given position and momentum evolves over time, yet always remains consistent with quantum mechanics.

Another highlight is the extension of the Peres‑Horodecki separability criterion. This well‑known test for entanglement has been adapted to momentum probability distributions, allowing researchers to check whether two asymmetric systems are truly entangled. In practice, one can determine if two particles are “linked” just by looking at their motion probabilities.

These findings open the door to new experimental techniques where measuring classical probabilities could replace more complex procedures. The future of quantum physics may thus become more accessible, even for those without advanced mathematical training.

The tomographic representation relies on the Radon transform, a mathematical technique used in medical tomography to reconstruct images from projections. In quantum mechanics, the projection variable is a linear combination of position \(x\) and momentum \(p\), defined as \(X_\theta = x\cos\theta + p\sin\theta\). By measuring the probability distribution \(w(X_\theta,\theta)\) for various angles \(\theta\), one can reconstruct the Wigner function via the inverse Radon transform. This approach eliminates the need to handle negative values, making visualization more intuitive.

For potential wells, the exact Schrödinger solutions provide eigenstates \(\psi_n(x)\). Applying the Radon transform to \(|\psi_n(x)|^2\) yields tomographic distributions \(w_n(X_\theta,\theta)\), which exhibit sharper peaks for low‑energy states and greater spreading for excited states. In the Morse potential, the exponential nature of the wavefunction leads to asymmetric tomograms, reflecting the anharmonic character of the system.

Extending the Peres‑Horodecki criterion to momentum probabilities is based on constructing a covariance matrix \(\Gamma\) of the moments \(\langle p_i p_j \rangle\). If the partially transposed (PPT) matrix of \(\Gamma\) is not positive semidefinite, the state is entangled. This method is especially useful for real asymmetric states, where momentum correlations can reveal hidden entanglement. Moreover, momentum tomography allows experimental verification of the criterion without resorting to spin or polarization measurements, broadening the possibilities for testing entanglement in ultracold atoms or waveguide photons.

Finally, tomographic representation bridges theory and experiment: probability distributions measured directly in the lab can be fed into tomographic calculations to reconstruct the full quantum state, providing a powerful tool for quantum diagnostics and the development of more robust quantum technologies.