This Tutorial will introduce the mathematical framework for describing

systems of identical particles, and explain the notion of indistinguishability.

We will then focus our attention on dynamical systems of free particles and

formally introduce the concept of many-particle interference. Its impact on

many-particle transition probabilities is computationally challenging to

evaluate, and it becomes rapidly intractable for systems with large numbers of

identical particles. Hence, this Tutorial will build up towards alternative,

# All

The growing field of quantum computing is based on the concept of a q-bit

which is a delicate superposition of 0 and 1, requiring cryogenic temperatures

for its physical realization along with challenging coherent coupling

techniques for entangling them. By contrast, a probabilistic bit or a p-bit is

a robust classical entity that fluctuates between 0 and 1, and can be

implemented at room temperature using present-day technology. Here, we show

that a probabilistic coprocessor built out of room temperature p-bits can be

S-money [Proc. R. Soc. A 475, 20190170 (2019)] schemes define virtual tokens

designed for networks with relativistic or other trusted signalling

constraints. The tokens allow near-instant verification and guarantee

unforgeability without requiring quantum state storage. We present refined two

stage S-money schemes. The first stage, which may involve quantum information

exchange, generates private user token data. In the second stage, which need

only involve classical communications, users determine the valid presentation

Photosynthetic organisms use networks of chromophores to absorb sunlight and

deliver the energy to reaction centres, where charge separation triggers a

cascade of chemical steps to store the energy. We present a detailed model of

the light-harvesting complexes in purple bacteria, including explicit

interaction with sunlight; energy loss through radiative and non-radiative

processes; and dephasing and thermalizing effects of coupling to a vibrational

bath. An important feature of the model is that we capture the effect of slow

We establish the general framework of quantum fluctuation theorems by finding

the symmetry between the forward and backward transitions of any given quantum

channel. The Petz recovery map is adopted as the reverse quantum channel, and

the notion of entropy production in thermodynamics is extended to the quantum

regime. Our result shows that the fluctuation theorems, which are normally

considered for thermodynamic processes, can be a powerful tool to study the

We study bosons in a one-dimensional hard wall box potential. In the case of

contact interaction, the system is exactly solvable by Bethe ansatz, as first

shown by Gaudin in 1971. Although contained in the exact solution, the boundary

energy for this problem is only approximately calculated by Gaudin at the

leading order at weak repulsion. Here we derive an exact integral equation that

enables one to calculate the boundary energy in the thermodynamic limit at an

We propose a quantum classifier, which can classify data under the supervised

learning scheme using a quantum feature space. The input feature vectors are

encoded in a single qu$N$it (a $N$ level quantum system), as opposed to more

commonly used entangled multi-qubit systems. For training we use the much used

quantum variational algorithm -- a hybrid quantum-classical algorithm -- in

which the forward part of the computation is performed on a quantum hardware

We present a method to generate NOON states with three photons by injecting

photons in an array of three waveguides. Conditional measurements project the

wave function in a given (desired) state. In passing, we show how the array of

three waveguides, that effectively reproduces the interaction of three fields,

may be reduced to the interaction of two fields.

The technological refinement of experimental techniques has recently allowed

the generation of two-photon polarization entangled states at low Earth orbit,

which have been subsequently applied to quantum communications. This

achievement paves the way to study the interplay between General Relativity and

Quantum Mechanics in new setups. Here, we study the generation of two-photon

entangled states via large scale Franson and Hugged interferometric arrays in

the presence of a weak gravitational field. We show that for certain

Para positronium composed by an electron-antielectron pair is an unstable

system decaying into two high energetic gamma photons via self annihilation

process, due to the conservation of the charge conjugation parity in

electromagnetically interacting systems. Therefore, the spectrum covering all

fundamental properties of the para-positronium system includes an imaginary

part corresponding to the proper decay time besides the real parts

corresponding to the total annihilation energy and binding energy,