Amplitude mapping:
The amplitude (loudness) of the audio signal directly maps the amplitude of the probability density given by the Husimi Q-function Q(θ,ϕ). This choice reflects the fundamental quantum mechanical interpretation of Q(θ,ϕ) as a quasi-probability distribution in phase space (Husimi 1940) The mapping follows A(t) proportional to Q(θ,ϕ) where A(t) represents the time-dependent amplitude. This proportional relationship ensures that regions of high quantum probability density produce more audible signals, while low-probability regions become perceptually negligible. The logarithmic nature of human loudness perception (Moore 2003) makes this mapping particularly effective for distinguishing significant quantum features.
We propose a sonification procedure which transforms the Husimi function Qₜ(θ,φ) and the bipartite enanglement entropy SvN(t) during quantum evolution into auditory features. In practice, this means that one sound wave is generated for each point of the Husimi function at a given time, with its amplitude, spatial position, pitch, and timbre determined by the mappings described below. This mapping tries to reflect the full distribution of the function in the most intuitive manner.
Timbre representation:
Waveform complexity serves as an auditory representation of entanglement entropy SvN(ρ). We implement a continuous timbre space where:
• Minimal entropy (SvN ≈ 0) produces pure sinusoidal tones, corresponding to separable, unentangled quantum states.
• Maximal entropy generates complex, harmonic-rich waveforms incorporating features such as triangle waves and ring modulation.
• Intermediate entropy levels are represented by an interpolation between pure sinusoidal and complex waveforms, with the proportion of harmonic complexity increasing continuously as the entropy rises.
Ring modulation involves multiplying two sine waves—an input tone and a carrier oscillator—resulting in a signal composed solely of the sum and the difference of the two frequencies. Mathematically, this is expressed as:
sin(2π f₁ t) · sin(2π f₂ t) = ½ [ cos(2π(f₁ − f₂)t) − cos(2π(f₁ + f₂)t) ],
where f₁ and f₂ are the input and carrier frequencies, respectively. This process eliminates the original frequencies and produces a metallic, bell-like timbre that is highly sensitive to frequency variation and phase relationships.
While the von Neumann entropy cannot be directly represented on the Bloch sphere—requiring separate plots to visualize its temporal evolution—our sonification approach provides a way to capture its dynamics in real time through auditory perception. By mapping the increasing entropy to progressive changes in waveform complexity, the method intuitively mirrors the growth of quantum correlations and the rising structural complexity of the quantum state. This auditory encoding enables simultaneous exploration of both the system’s phase-space structure and its entanglement dynamics, offering a multidimensional understanding of quantum behavior.
Stereo spatialization:
We employ the azimuthal angle ϕ to control stereo panning, creating a spatial representation of quantum phase space, where panning is proportional to ϕ, translating the angular coordinate into a left-right audio distribution. This approach mirrors the spatial symmetry of the quantum system, with ϕ = 0 corresponding to the left channel and ϕ = π to the right channel. The linear proportionality maintains an intuitive correspondence between quantum phase space and auditory space. The sonification could be further enhanced by rendering it in a domed ambisonic environment, which would enable a more faithful spatial mapping of the full azimuthal (ϕ) range.
Pitch encoding:
Frequency (pitch) is determined by the polar angle θ relative to a reference initial state. Using f_init = 440Hz as our reference frequency (A4), corresponding to the angle θ_init at which the system is prepared, we implement:
f = f_init − |θ − θ_init| * f_init * spread
This mapping generates pitch deviations that correspond to angular displacements on the Bloch sphere. The modulation factor reflects the distributional spread of the Husimi Q-function across the sphere. When regions of high probability density are more broadly distributed, the resulting frequency shifts are more pronounced. This approach makes pitch sensitive not only to directionality, but also to how widely the state is supported in phase space, offering a perceptual cue for quantum state delocalization (Sakurai and Napolitano 2017).