## Majorana representation for hermitian operators

2019/10/17 In 1932, Ettore Majorana published a famous paper in which he showed how to visualize quantum states of any quantum system with finite-dimensional Hilbert space by a configuration of stars on the unit-sphere - the stellar representation was born. This was 6 years before he disappeared mysteriously on a boat trip from Palermo to Naples, where he had become a professor of theoretical phyiscs just a year before. But his stellar representation lived on, owing to its beauty and invariance properties. E.g. when the physical system is a spin or angular momentum and you rotate the lab and then look at the state, its stellar configuration has rigidly rotated just in the same way as the lab, even though they exist in completely differen spaces, of course. For the most classical states, all the stars coincide. The more "quantum" the state, the more the stars are spread out. We have used the stellar representation earlier to visualize for example the "queens of quantum", the most spin-states with the most "quantumness" possible.

Now we have generalized the stellar representation to mixed quantum states, i.e. states with a certain classical uncertainty, as for example in thermal states, and in fact even to any hermitian operator on a finite dimensional Hilbert space! We show how important operations such as taking the partial trace of the density matrix, or products of operators can be expressed in terms of the polynomials that determine the stars of the representation, compare representations of Schrödinger cat-states with their decohered mixtures, and show the connection to the tensor-representation of spin-states, as well as the Husimi- and P-functions, i.e. quasiprobability distributions familiar mostly from quantum optics that also describe the state of the system. Read more and discover some nice examples of stellar representations in this preprint...

## More sensitive quantum sensors with reinforcement learning

2019/08/22 At least since Alpha Go made headlines by beating a human world champion in the board game of Go, machine learning, and in particular reinforcement learning has been in the limelight of public and scientific interest. How we can profit from machine learning in all disciplines of science is a major long-term research topic, one that is highly relevant for society, and also part of the scientific program that won the University of Tübingen again its place in the nobel class of "Excellence Universities". In this new preprint we show that** reinforcement learning can strongly improve the sensitivity of quantum sensors**. These are devices with which one can measure e.g. magnetic fields already now extremely precisely - the current world record is below a femto Tesla per square root of Hertz (roughly hundred thousand million times smaller than the magnetic field of Earth at its surface) - and this by using a tiny little gas cell with about 10 billion atoms! Such sensors are already used for monitoring more precisely heart and brain activities than was possible before. Recently, we had found that these sensor can be improved substantially by "kicking" them periodically with laser pulses, rendering them chaotic. Now our analysis shows that even more sensitivity can be gained by adjusting the laser pulses via re-inforcement learning. In addition, the sensor becomes more robust against unavoidable decoherence: the machine learns by itself how to optimally shoot the laser pulses to maximize the sensitivity, and can even adapt to changing environments. We were stunned!

## Trade-off relations for operation entropy of complementary quantum channels

2019/08/09 Quantum channels map the quantum mechanical state of a system to another one. Non-unitary maps arise from interactions with an environment, which leads typically to decoherence and the destruction of quantum mechanical interference effects. Less studied is what happens to the environment in such a case. The map that propagates the system´s initial state to the environment´s final state is called the "complementary channel". In the case of a quantum communication channel you can think of it as propagating quantum information from the system to an evesdropper. The loss of quantum information measured by the entropy of the channel is zero if the system evolves unitarily, but can become large in the case of strong interaction. It turns out that there is a trade-off relation between between the entropies of the channel and the complementary channel: The sum of the entropies of the original channel and its complementary channel is bounded from below. Hence, if the original system evolves unitarily, its channel entropy is zero, whereas the entropy of the complementary channel is maximal, meaning that an eavesdropper gets no information, and vice versa. We prove the exact lower bound in the case of two qubits, find the channels that saturate it, and conjecture the bound for higher dimensional systems based on extensive numerical evidence. Apart from the mathematical insight into quantum information processing, this work also led to graphs of stunning beauty...

## Time-evolution of nonlinear optomechanical systems

2019/08/02 Non-linear systems are notoriously difficult to solve. However, there are exceptions. In this new preprint we describe how we can exactly solve the dynamics of an optomechanical system consisting of a cavity mode and a movable mirror, or any other mechanical oscillator such as a trapped micro-bead, to which the light couples through its light pressure. The solution is based on realizing that the operators in the Hamiltonian form a closed Lie algebra, which allows one decouple it into several parts. This works in principle even for arbitrary time-dependent parameters, with the caveat that the resulting differential equations that need to be solved may only be solvable numerically and might be unstable. Nevertheless, this method marks real progress compared to previous analyses of such a systems that approximated the dynamics by assuming that Gaussian initial states remain approximately Gaussian. This is something we can check now via our exact solutions, and we also investigate the interplay between non-Gaussianity and squeezing. This is a milestone towards rigorously investigating the usefulness of this system for quantum metrology, in particular the measurement of extremely weak forces.

## Outcoupling from a Bose-Einstein condensate in the strong-field limit

2019/06/12 Have you ever seen a laser that emmits not light, but cold, coherently propagating atoms? Have a look at our new preprint that investigates such matter waves both theoretically and experimentally! The bose laser works essentially by producing a Bose-Einstein condensate of atoms trapped in a harmonic potential, and then cutting a hole into the trap, such that atoms simply fall out. While none of this is new, lead author Caroline Arnold developed full 3D simulations and was able to probe the regime of strong outcoupling. She found that the intensity saturates in that regime - a purely theoretically finding first, that was then confirmed by experiments here in Tübingen.

## Quantum parameter-estimation of frequency and damping of a harmonic-oscillator

2019/05/20 The harmonic oscillator is one of the most important systems in all of physics, be it classical or quantum. Its importance arises from the fact that a.) for small amplitudes most stable systems can be approximated by a set of harmonic oscillators, and b.) that it can be solved exactly. Moreover, it has only two free parameters: its frequency and its damping. How precisely can one determine these in principle? While this question was answered many years ago for an undamped oscillator in this paper, in reality there is always some damping, and taking it into account can change the result a lot. In a new preprint, we solve the problem for Gaussian states, which are a broad class of experimentally relevant states. We predict that with existing carbon nanotube resonators it should be possible to achieve a mass sensitivity of the order of an electron mass per square root of Hz, orders of magnitude lower than the current record, due to the shift of frequency from an adsorbed mass.

## Maximal sensitivity for mixed states

2019/05/15 More often than not, quantum states come with a certain amount of uncertainty about themselves, which means in quantum mechanics jargon that they are "mixed states", rather than pure ones. This uncertainty reduces on average the nice quantum mechanical effects such as quantum interference that we want to use for enhancing e.g. sensitivity of sensors. The question is then, how to optimally prepare an initial mixed state through unitary transformations to obtain maximal sensitivity in a subsequent evolution that encodes the parameter to be measured in the state. This is an old open problem that we resolved completely in recent work. Here is how.

## Fast entanglement detection

13/2/2019 With the increasing size of existing quantum processors, the question of how to test their functionality as efficiently as possible has become a challenge. In a new preprint, "Optimal measurement strategies for fast entanglement detection" we show that one of the core goodies of quantum information processing, namely quantum entanglement, can be verified much more rapidly than through full state tomography. The approach is based on our work on truncated moment sequences, which deals very naturally with missing data. One set of measurements turns out to be particularly efficient.

## Farday effect and optical activity in the gravitational field of a laser beam

11/12/2018 Another chapter in our exploration of the gravitational interaction of light with light: Our preprint "Rotation of polarization in the gravitational field of a laser beam - Faraday effect and optical activity" is now finally out. It turns out that the gravitational field of a circularly polarized laser beam leads to the rotation of the polarization of a probe beam in its vicinity - a gravitational effect that is clearly beyound the scope of Newtonian gravity, but that can be precisely calculated using General Relativity. The rotation of polarization mixes effects reminiscent of the Faraday effect and optical activity in media which have been known for a long time in optics. In the latter, the effect is undone when the probe-beam propagates back, in the former, it is doubled. Unfortunately, the magnitude of the effect is very very small: of the order of the power of the laser beam divided by the Planck power, where the latter is the Planck energy divided by the Planck time. Optimistic estimates for the current most powerful lasers let us expect rotation angles of order 10^{-32 }- and that includes already an amplification using an optical cavity. Clearly not an effect that will be measured in the lab any time soon. Yet, the effect is of substantial fundamental interest, as it demonstrates gravitational spin-spin coupling in the well-defined and tested frame-work of General Relativity, while the effect is also predicted by certain quantum gravity theory candidates.