Quantum Concept Music Framework

• Quantum Concept Music (QCM) is a framework that models musical entities as quantum states, incorporating superposition, entanglement, and probabilistic measurement.
• It employs Hilbert space representations and unitary transformations to redefine musical evolution, enabling non-classical compositional and performance practices.
• QCM offers innovative applications in algorithmic composition and interactive performance via diagrammatic formalisms like ZX-calculus to capture relational musical structures.

Quantum Concept Music (QCM) designates a rigorous framework and emerging field wherein musical composition, performance, and perception are reconceptualized within a quantum-theoretical paradigm. Instead of treating music as a deterministic progression of discrete, classically-defined tones and gestures, QCM models musical objects as quantum states—embedding non-classical phenomena such as coherent superposition, entanglement, probabilistic measurement, and contextual evolution via unitary transformations. By mapping the mathematics of quantum mechanics (Hilbert spaces, state vectors, gates, measurement, and diagrammatic formalisms such as ZX-calculus) onto both the ontology of musical materials and the modalities of their presentation, QCM fundamentally extends both the compositional and experiential space of music beyond classical boundaries (Putz et al., 2015, Abdyssagin et al., 6 Oct 2025).

1. Mathematical and Physical Foundations

The foundational gesture of QCM is to represent musical entities as vectors in a Hilbert space, with the set of possible classical notes (e.g., the white notes of a C major scale) forming an orthonormal basis $$B = \{\lvert c \rangle, \lvert d\rangle, ..., \lvert b\rangle\}$$ in $\mathbb{C}^7$ (for the seven-tone example) (Putz et al., 2015). Each note—or musical event—is modeled as a quantum state, either as a basis vector or, crucially, as a superposition:

$$\lvert \psi \rangle = a_c\lvert c \rangle + a_d\lvert d \rangle + ... + a_b\lvert b \rangle, \quad \text{with} \quad \sum_{k} |a_k|^2 = 1$$

Alternatively, field-mode models treat each tone as a quantum mode with associated occupation numbers. In the fermionic restriction, each note can be “on” ($|1\rangle$) or “off” ($|0\rangle$), with general states given by

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,\quad |\alpha|^2 + |\beta|^2 = 1$$

Unitary transformations (e.g., the Hadamard gate $H = \frac{1}{\sqrt{2}} \left[\begin{smaLLMatrix}1 & 1\1 & -1\end{smaLLMatrix}\right]$) and more complex gate structures facilitate evolution and interference patterns within this space. These mechanisms allow for the construction of non-classical musical experiences, such as coherent superpositions and entangled states (Putz et al., 2015, Putz et al., 2021, Abdyssagin et al., 6 Oct 2025).

2. Superposition, Measurement, and Quantum Listening

A central operation in QCM is the coherent superposition of classically incompatible musical events—a direct analog to quantum superposition. For example, a 50:50 mixture of the presence and absence of a note $n$ is represented as

$$|\psi_{50:50}\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$$

Conceptually, this “gray tone” neither is fully “on” nor “off” until measured: upon a “musical measurement” (i.e., heard by a listener), the superposed state collapses probabilistically to either $|0\rangle$ (silent) or $|1\rangle$ (sounding), with outcome statistics governed by the quantum amplitudes (Putz et al., 2015, Putz et al., 2021).

QCM embraces quantum parallelism in perception: repeated exposures (across listeners or auditions) to the same quantum musical state can yield divergent classical musical experiences. This non-determinism is not stochastic noise but a reflection of quantum measurement: the musical work is, in effect, a many-worlds entity until collapsed (Putz et al., 2015).

3. Entanglement and Correlated Musical Structures

Extending beyond superposition, QCM exploits entanglement—states in which two or more musical entities become inseparably correlated, such that the measurement of one instantaneously constrains the possible outcomes of the other. A musically-entangled Bell state between two tones is furnished as:

$$|\Psi^{-}\rangle = \frac{1}{\sqrt{2}}(|0_e\rangle|1_a\rangle - |1_e\rangle|0_a\rangle)$$

Measurement of the $e$-tone immediately fixes the state (and thus the audible result) of the $a$-tone, and vice versa. This enables “quantum musical parallelism,” in which ensemble members or audience subsets experience distinct but interdependent musical realities—a resource unavailable in classical music (Putz et al., 2015, Putz et al., 2021).

The necessary and sufficient condition for such entanglement is $x_1x_4 \neq x_2x_3$ for a state $$|\psi\rangle = x_1|0_e\rangle|0_a\rangle + x_2|0_e\rangle|1_a\rangle + x_3|1_e\rangle|0_a\rangle + x_4|1_e\rangle|1_a\rangle$$ (Putz et al., 2015).

4. Formalisms for Quantum Music: Diagrammatics and CQM

QCM formalizes music not just in symbolic or notational terms but through categorical quantum mechanics (CQM) and the diagrammatic ZX-calculus, as presented in Quantum Picturalism (QPict) (Abdyssagin et al., 6 Oct 2025). Musical objects/functions—notes, chords, phrases, ensemble relationships—become processes or morphisms in a compact-closed category. Their interactions are encoded diagrammatically, using constructs such as spiders, wires, and measurement “gates.”

This formalism captures relational and processual aspects of music that escape linear notation: for instance, two players as a Bell pair, their performance “collapse” upon mutual measurement, and the propagation of information/influence throughout an interaction network. Musically, this allows scores that specify interactive, entanglement-inspired performance practice—covering both composed and improvisational contexts.

Diagrammatic notation directly supports algorithmic manipulation and automation. For example:

$\begin{tikzpicture}[scale=1.0, baseline=(current bounding box.center)] \node (A) at (0,0) [circle, draw, fill=blue!20] {%%%%16%%%%}; \node (B) at (2,0) [circle, draw, fill=blue!20] {%%%%17%%%%}; \draw[->, thick] (A) -- node[above] {%%%%18%%%%} (B); \end{tikzpicture}$

Here, circles denote musical “states,” and the measurement arrow ($\mathcal{M}$) represents a measurement-induced transition and enforced relational dependency (Abdyssagin et al., 6 Oct 2025).

5. Differentiation from Classical Musical Models

QCM marks a paradigmatic break from classical music. In classical frameworks, musical materials are sequenced, deterministic, and objective: a note or phrase is either played or not, with no ambiguity, and the realized work is uniquely defined for all listeners (Putz et al., 2015). By contrast, in QCM:

• Musical states are fundamentally probabilistic and may exist in arbitrary superpositions before “collapse.”
• Entanglement allows for correlated structures transcending local description.
• Evolution is governed by unitary transformations (e.g., Hadamard, permutation, general quantum gates), which generate non-classical evolution and interference patterns.
• Performance and perception themselves become transformative, as measurement, not mere reproduction, is an integral part of the musical process.
• The category-theoretic and diagrammatic approach under ZX-calculus or QPic transcends the limitations of one-dimensional, timeline-based scoring, making explicit relational and interactive musical structures (Abdyssagin et al., 6 Oct 2025).

6. Applications and Implications

QCM is relevant for a number of domains:

• Compositional innovation: new modes of structuring and varying musical materials by exploiting quantum-theoretic mechanisms.
• Algorithmic and AI-based music: the inherently algebraic and process-friendly nature of QCM notation is conducive to computer-assisted composition, simulation, and even machine learning-based creativity (Abdyssagin et al., 6 Oct 2025).
• Performance practice: enables scores and directions where performers interact dynamically via entangled protocols, responding in real time to “measured” outcomes.
• Automatable systems: the algebraic structure is amenable to manipulation in computer-algebra and automated reasoning tools, facilitating a new generation of generative/interactive electronic music systems.
• Interactivity and genre-spanning composition: QCM is style-agnostic, supporting classical, avant-garde, experimental, and popular idioms that can benefit from relational, non-linear structures.

7. Conceptual Summary and Outlook

Quantum Concept Music (QCM), grounded in quantum formalism and categorical diagrammatics, is fundamentally an exploration of how musical meaning, organization, and interaction can be transformed by encoding genuinely non-classical behavior at the core of compositional, performative, and notational practice. QCM generalizes and extends the possibilities of the musical artform by importing not just the surface features of quantum mechanics (randomness, noise) but its deep structural features—superposition, entanglement, informational flow, and non-local relationality—into the heart of music’s syntax and semantics (Putz et al., 2015, Abdyssagin et al., 6 Oct 2025).

This framework allows the creation of scores and systems where the music realized by different listeners or performers becomes a contextual instantiation of a deeper quantum musical state—an object only fully addressed in the language of quantum information theory, process categories, and the operational formalism of measurement, evolution, and interaction.