For the last three decades, quantum information theory has been exploiting the non-classical features of quantum theory to conduct otherwise impossible communication tasks, such as secret key distribution, quantum teleportation and dense coding. However, almost no attention has been paid to mechanical scenarios. It so happens that, thanks to new levitation techniques, nowadays we have an unprecedented experimental control over massive nanoparticles. Could these new experimental setups be hiding a surprise?

The goal of this workshop is to integrate the small community of researchers who are interested in the operational differences between quantum and classical mechanical systems. The talks will cover topics like times of arrival, quantum backflow, Tsirelson's harmonic oscillator paradox, operational theories in phase space, and the mathematics of Wigner functions.

The workshop will take place at the Institute of Quantum Optics and Quantum Information (IQOQI) Vienna. Boltzmanngasse, 3, 1090. A list of the talks and abstracts can be found here.

Invited speakers: Chris Fewster, Eric A. Galapon, Arseni Goussev, Zaaw Lin Htoo, Harkan Kirk-Karakaya, Matthias Kleinmann, Clemens Mawby, James Millen, Martin Plávala, David Trillo and Reinhard F. Werner.

Workshop program:

Times | Monday, 11th | Tuesday, 12th | Wednesday, 13th |

9:00-10:00 | Zaw Lin Htoo | Reinhard F. Werner | Discussions day |

10:00-11:00 | Matthias Kleinmann | Chris Fewster | |

11:00-11:15 | COFFEE/TEA | COFFEE/TEA | |

11:15-12:15 | Martin Plávala | Clemens Mawby | |

12:15-14:00 | LUNCH | LUNCH | |

14:00-15:00 | David Trillo | James Millen | |

15:00-16:00 | Arseni Goussev | Eric A. Galapon | |

16:00-16:15 | Coffee/Tea | Coffee/Tea | |

16:15-17:15 | Harkan Kirk-Karakaya | Uros Delic |

Monday, 11^{th}

**Detecting Nonclassicality in Precessions: Beyond Harmonic Systems**

Speaker: Zaw Lin Htoo

Abstract: Since a harmonic oscillator undergoes the same uniform precession in both classical and quantum theory, one would not expect to find quantumness in the time evolution of a quantum harmonic oscillator. Yet, a protocol introduced by Tsirelson showed the contrary: by measuring the position of a harmonic oscillator at a randomly-chosen time every round, and calculating the score as the probability that the position was found to be positive, suitable quantum states can attain scores larger than the maximum possible classical score [1]. Therefore, nonclassicality can be detected by demonstrating a violation of the classical bound, under the sole assumption that the system is a harmonic oscillator with angular frequency ω. However, it can be desirable to relax this assumption to some extent—for example, if the oscillator suffers from dissipation, or if ω is only known to be within some range of frequencies, or if the system is anharmonic. Partly based on a recent work [2], I will cover how the protocol can be adjusted to accommodate these scenarios. It will be seen that a quantum mechanical advantage is still present in many of the above cases, thereby demonstrating that the protocol is robust to some changes to its primary assumption.

[1] Tsirelson, B. (2006). How often is the coordinate of a harmonic oscillator positive? arXiv:quant-ph/0611147; Zaw, L.H., Aw, C.C., Lasmar, Z., & Scarani, V. (2022). Detecting quantumness in uniform precessions. Phys. Rev. A 106, 032222.

[2] Zaw, L.H. & Scarani, V. (2022). Dynamics-based quantumness certification of continuous variables with generic time-independent Hamiltonians. arXiv:2212.06017

**Operational theories in phase-space**

Speaker: Matthias Kleinmann

Abstract: Quantum theory has been established to be a special case of the generalized probabilistic theories, which encompass all theories featuring preparation procedures and measurements. Intriguingly, those theories include instances where the correlations reach beyond anything possible according to quantum theory or classical probability theory. However, in this framework, there is no constructive way to suggest an experiment that could achieve such correlations and hence test the validity of quantum theory in that respect. We address this defect by a phase-space description of operational theories allowing us to describe experiments on common systems, such as the harmonic oscillator or the hydrogen atom. Accordingly, we can make predictions on the spectrum, position density, or the time-evolution for theories that are neither classical nor quantum mechanical. I will give an introduction to this framework and show how it adds new insights into quantum mechanics and how new experiments can test the structure of any theory of microscopic systems.

**Black-box approach and maximal violations of Tsirelson inequalities**

Speaker: Martin Plávala

Abstract: Tsirelson inequalities were developed by Boris Tsirelson as a simple test of non-classicality of states of the quantum harmonic oscillator; they were recently generalized to spins and anharmonic systems as well as used to construct robust entanglement witnesses. All of these inequalities are formulated as inequalities between measurable physical quantities, or, equivalently, as inequalities between the respective operators, but a unified framework that would connect all of the applications is so far missing. In the first half of the talk we address this by presenting a device-independent theory of Tsirelson inequalities, which treats the measurement devices as black boxes with some set of inputs and outputs. Using this approach we derive new inequalities that are violated by the quantum harmonic oscillator, but we also prove that the original inequality introduced by Boris Tsirelson is unique and tight for the considered scenario. In the second half of the talk we discuss maximal violations of Tsirelson inequalities for the harmonic oscillator by quantum theory as well as operational theories in phase space.

**Quantum supremacy in mechanical tasks: projectiles, rockets and quantum backflow**

Speaker: David Trillo

Abstract: We consider a scenario where a non-relativistic one-dimensional quantum particle is prepared in some bounded region of space and left to propagate freely. After a certain amount of time, we check if the particle has reached some distant target region. We find that there exist "ultrafast" ("ultraslow") quantum states, whose probability of arrival is greater (smaller) than that of any classical system prepared in the same region with the same momentum distribution. We prove that the maximum possible difference between quantum and optimal classical arrival probabilities for projectiles, as well as for self-propelling particles or rockets, is limited by the Bracken-Melloy constant c_{bm}, introduced in 1969 to characterize the maximum expression of the phenomenon known as quantum backflow. This mathematical correspondence extends to other examples of mechanical effects with a quantum advantage, whose study we advance by deriving the first rigorous upper bound c_{bm}\leq 0.0725. We also prove that the hard limit given by can be overcome in a variant of the original projectile scenario: if the classical particle is required to possess, not just the same momentum distribution as the quantum particle, but also the same position distribution, then the difference between arrival probabilities can reach 0.1228.

**Quantum backflow for rotational motion**

Speaker: A. Goussev

Abstract: Quantum backflow is a counterintuitive phenomenon in which the probability density of a free particle moves opposite to its momentum. This effect, known for its delicate nature, has not yet been observed experimentally. In my talk, I will provide a brief introduction to quantum backflow and present some recent findings. The talk will mainly focus on the manifestation of quantum backflow in rotational motion.

**A new approach to the calculation of the backflow constant**

Speaker: Harkan Kirk-Karakaya

Abstract: In 1D quantum mechanics, quantum backflow is the phenomenon whereby a state consisting purely of positive momenta can exhibit an increase in probability of being measured in the region x < 0. Of particular interest is the maximum amount this probability can increase, the backflow constant, and the associated states that exhibit this maximum backflow. In this talk, I will discuss a new method for answering these questions by taking matrix elements of the Bracken-Melloy operator with respect to a sequence of non-orthonormal $L^2(\mathbb{R}^+)$ vectors. I will present progress towards obtaining a more accurate estimate of the backflow constant.

Tuesday, 12^{th}

**Wigner Functions revisited**

Speaker: R. Werner

Abstract: I will show how Wigner functions answer questions about the possible time dependence of expectation values, and also present a generalization of Wigner functions, a unique tempered distribution that contains as marginals the quantum distributions of arbitrary linear combinations of n operators.

**Classical and Quantum Energy Inequalities**

Speaker: Chris Fewster

Abstract: In many models of classical matter, the energy density is everywhere non-negative according to all observers; that is, the Weak Energy Condition is satisfied. A variety of classical energy conditions are utilised in classical general relativity, for example as hypotheses for the singularity theorems. However, it is known that no quantum field theory can satisfy any of the classical energy conditions, opening a possible door to quantum (dis?)advantage. It turns out that in many cases QFTs obey weaker Quantum Energy Inequalities (QEIs) which can place limits on the extent to which the classical bounds can be violated. In this talk I will review the subject of QEIs and their applications. If time allows, I will draw connections between this subject and that of quantum backflow.

**Identifying quantum behaviour in continuous variable systems using Leggett-Garg tests for macrorealism**

Speaker: Clemens Mawby

Abstract: Macrorealism is the worldview that certain quantities may take definite values at all times irrespective of past or future measurements and may be experimentally falsified via violation of the Leggett-Garg (LG) inequalities. We put this worldview to the test for systems described by a continuous variable x by seeking LG violations for measurements of a dichotomic variable Q = sign(x) in a quantum harmonic oscillator, for initial states consisting of energy eigenstates, superpositions thereof, and coherent states. To obtain a physical picture of the LG violations for coherent states, we exploit the continuous nature of the underlying position variable and analyze the relevant quantum-mechanical currents and Bohm trajectories, thereby noting that diffraction in time is the physically relevant mechanism. We briefly discuss possible connections with quantum backflow and the Tsirelson harmonic oscillator paradox.

C. Mawby and J. J. Halliwell, Phys. Rev. A 105, 022221 (2022); Phys. Rev. A 107, 032216 (2023).

J. J. Halliwell, H. Beck, B. K. B. Lee, and S. O'Brien, Phys. Rev. A 99, 012124 (2019).

**Bath Engineering for Levitated Mechanical Systems**

Speaker: James Millen

Abstract: Levitated particles have gained interest as a mechanical system due to the fact that they are physically decoupled from their environment. This yields the potential for quantum control in room temperature environments, and the subsequent free evolution of quantum states of motion. In this talk, I will consider the ability to use optical and electrical fields to engineer artificial (non-thermal) environments for levitated mechanical systems. This will cover cooling of mechanical motion, chaotic non-linear dynamics and the study of thermodynamic processes.

**The need to update quantum measurement theory **

Speaker: Eric A. Galapon

Abstract: Our investigations on quantum time of arrival operators show that the spectral properties of a time operator can acquire unambiguous interpretation independent of the postulates of standard quantum measurement theory. In particular, the spectral properties may be tied with the dynamics of the system, such as the eigenfunctions of the confined quantum time of arrival operators unitarily arriving at some point at their respective eigenvalues. This calls for a certain modification of the standard quantum measurement theory, which makes no exceptions on the interpretation of the spectral properties of observables. How the modified quantum theory of measurement will eventually look like is not yet clear. But we foresee that the modified theory will require classification of quantum observables into, which we can provisionally call, temporal and non-temporal observables, with the former falling into the standard quantum measurement theory and the latter falling into the more general measurement theory that accommodates time as a dynamical observable. But how can this classification be made? We suspect that temporal observables comprise the Hilbert space solutions to the time–energy canonical commutation relation for a given Hamiltonian. The most surprising aspect of the theory of time of arrival operators is that it connects the quantum time of arrival problem and the problem of the appearance of a particle in quantum mechanics. In standard quantum mechanics, the appearance of a particle arises out of the collapse of the wave function brought about by a position measurement, and the collapse occurs not before but at the moment the particle appears on our detectors. However, the dynamics of the confined time of arrival operators suggests that the appearance of a particle arises as a combination of the collapse of the initial wave function into one of the eigenfunctions of the time of arrival operator, followed by the unitary Schrodinger evolution of the eigen-function. This implies that particles, at least within the quantum arrival setting, do not arise out of position measurements but out of time of arrival measurements and that the collapse of the wave function on the appearance of a particle is not fundamental but decomposable into a series of casually separated processes. This invites us to reconsider our beliefs on quantum measurement theory.

E.A. Galapon, “Post-Pauli’s Theorem Emerging Perspective on Time in Quantum Mechanics,” Lect. Notes Phys. 789, 25–63 (2009) DOI 10.1007/978-3-642-03174-8 3.

**Collective mechanical effects in many-particle levitated experiments**

Speaker: Uros Delic

Abstract: In my talk, I will introduce a novel many-particle quantum system: an optical trap array of dielectric nanoparticles interacting via highly tunable and non-reciprocal light-induced dipole-dipole forces. I will show how we realize quantum control over the motion of a single particle and program arbitrary non-Hermitian dynamics in a particle array. I will present a recent observation of parity-time symmetry breaking in our system, which results in a collective mechanical lasing phase. Finally, I will present the path toward exploring multipartite entanglement in a macroscopic system.