What are the applications of Renyi entropy

Quantum entanglement made visible

If two particles are entangled quantum mechanically, they coordinate their behavior “spookily”, even over great distances, more closely than is possible according to the laws of classical physics. The state of the individual particles is completely indeterminate. Although entanglement is already being used in quantum information processing and quantum cryptography, there has not yet been a simple method of measuring it. Researchers working with Markus Greiner at Harvard have now made the entanglement of atoms in a light grid directly visible and quantified using a microscope. You published your results in the journal "Nature".

They placed four rubidium atoms in a one-dimensional light grid so that the particles sat individually in adjacent potential wells of the grid. If the atoms were excited to glow, their spatial distribution in the lattice could be observed with a microscope. Sufficiently high potential mountains between the troughs ensured that the atoms were stuck and could not get into neighboring troughs. Their quantum state corresponded to that of an electrical insulator, the electrons of which are also immobile.

The isolated atoms behaved independently of one another and were therefore not entangled. The researchers checked this by making an identical copy of the four atoms sitting in potential wells. Then they made these two four-particle systems interfere with each other. To do this, they made it possible for each atom to tunnel from its trough into the corresponding trough in the other system with a probability of 50 percent. Due to the paired interference of the isolated atoms, which were bosons, there were ultimately only an even number of atoms in each well, but never an odd number, as a look through the microscope confirmed.

Then the researchers separated the two systems again and returned them to their original state. They lowered the potential mountains so far that the atoms within their system could tunnel from one through to the other. This created a kind of superfluid. This brought the four atoms into an entangled quantum state that was clearly defined and “pure”, while the state of the individual atoms was completely indeterminate and “mixed”. With the so-called Rényi entropy one has a measure for the indeterminacy of the respective state. Accordingly, this entropy for the individual atoms is greater than that for the entire system - which would be impossible in classical physics.

The two four-particle systems were then brought into interference again by allowing the atoms to tunnel between them. But since the particles were now in an entangled state, the interference could lead to an odd number of atoms getting into a through. For example, if a well contained exactly one atom, it was in an indeterminate, mixed state and therefore had to be entangled with the other atoms. If the through contained two atoms, these were in a pure state and not entangled with the other atoms.

In this way, after looking through the microscope, it was possible to tell which atoms of the four-particle system were entangled with one another. The Rényi entropy (second order) could be determined from the observed particle numbers. By interfering with three or more identical copies of the system, one could also determine Rényi entropies of a higher order and thereby obtain more and more detailed information about the entangled state. To do this, it is not necessary to reconstruct the condition using complex condition tomography. The method should also work for a larger number of atoms. In this way, for example, insights into the entanglement behavior of quantum many-body systems in the vicinity of quantum phase transitions could be obtained.