#### What are the displayed shapes?

The quantum superposition states in the outputs of Quantum Collapses are visualized using the Wigner quasi-probability distribution, a widely used tool in quantum optics. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. It has applications in statistical mechanics, quantum chemistry, quantum optics.

This function generalizes for a quantum field the classical concept of probability distribution in phase space. It contains the same information as the wavefunction or the density operator (a generalization of quantum states for open quantum systems). In our project, we considered quantum states for a single mode electromagnetic field. Hence each shape in the output describes a photonic quantum superposition, corresponding to a field suspended between different number of photons.

Why not displaying directly wavefunctions? According to quantum theory, wavefunctions are complex functions (have a real and imaginary part), hence it is rather tricky to display both in a same image. The usually displayed square modulus describes the probability of finding the particle at a given position. However, it does not provide full information about the quantum state. On the contrary, the Wigner function is real and contains all the information on the quantum state. This reflects on the complexity and beauty of the displayed shapes.

#### Time evolution

The shapes in the initial image are quantum superposition states assigned during the process of minting. These initial states animate according to the rules of quantum theory. The time evolution is calculated in real time numerically solving the time-dependent Schrödinger equation. In physics, the generator of time evolution is the Hamiltonian, corresponding to the energy of the system. All the outputs evolve according to the same Hamiltonian, which has been randomly generated, imposing some constraint.

Quantum Collapses will also undergo some evolutionary steps. At some point, all the outputs will display new animations on monthly basis. New real-time evolutions correspond to new, randomly generated Hamiltonians.

#### Controlling photons in a box and exploring the quantum to cassical boundary

It is interesting to know that Wigner functions for a photonic system have been reconstructed experimentally in pioneristic experiments by Serge Haroche and coworkers. This group was able to engineer photonic quantum superpositions, by controlling the interaction of photons in a box with individual atoms, and to probe the obtained states reconstructing the Wigner functions. They were able to demonstrate the creation of cat-like superpositions and to test how they get affected by interactions with the environment. These experiments opened the way to second generation quantum technologies and Serge Haroche was awarded the 2012 Nobel Prize for Physics jointly with David J. Wineland.

#### Dirac background

Paul Dirac is one of the most significant physicists in the hystory of physics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger. Dirac also developed the so-called Dirac notation, widely used to denote quantum states and to calculate expectation values.The notation uses angle brakets and vertical bars to constructs “bras” and “kets” representing quantum states. As a tribute to him, a fraction of the outputs of Quantum Collapses displays a background of scattered angle brakets and bars.

#### CODE TECHNICAL

Quantum Collapses is written in Javascript, it doesn’t depend on any external graphical library. This means that the entire code is stored permanently on the blockchain. A small library of math functions was written from scratch and included in the code. We use math and matrix operations to calculate how the quantum states evolve in real time, according to the laws of quantum theory. The code generates a series of matrices, representing 3D surfaces made of 4096 points.

The data, interpreted as grayscale images, is then passed to the graphic card for further processing. This step is fundamental to achieve high quality images in real time. Graphic cards are the most powerful number crunching component on your computer! They are the first choice for massive parallelized tasks: 3D rendering, artificial intelligence, matrix operations, crypto mining, scientific simulations. In such use cases they can be thousands of time faster than the CPU.

We are utilizing Javascript WebGL2 API technology to run a fragment shader. Shaders are little programs that runs on the GPU, written in a specific programming language, we are using GLSL.

The shader applies bicubic spatial interpolation to the images, increasing the resolution and image quality. It then applies linear temporal interpolation, increasing the framerate. Then all the quantum wigners are composed in a single image, a color palette is applied based on the values of each pixel, and lastly a series of effects are applied in real time to generate a unique look.

We also created a minimal drawing library to draws lines directly to texture, used to generate a unique Dirac background pattern for each output with this trait. Collision detection is used to place the quantum particles on the canvas. To detect sounds from the microphone, the code analyzes frequencies using the Fast Fourier Transform, offered by Javascript APIs.