Composition laws in quantum theory
Quantum theory predicts music that cannot compose. We classify when it does by treating measurement as a test of compositional consistency across temporal boundaries.
Quantum transition probabilities—the likelihood of one outcome following another—do not multiply the way classical probabilities do. Checking a quantum process step by step can change its predicted endpoint statistics. We classify exactly when stepwise and endpoint statistics agree, introducing the Born–Chapman–Kolmogorov current to measure their disagreement.
arXiv:2609.11402 · Paper (PDF) · Musical demonstrations · GitHub project page
The problem. Intermediate measurements erase interference, so probabilities inferred step by step can differ from the final outcome distribution.
The question. When can a sequence of quantum operations be measured at every intermediate step without disturbing the final outcome distribution? How badly can this fail?
The result. Universal composition for a composition-closed family occurs exactly when every operation is a phase-dressed permutation—the quantum counterpart of a deterministic reversible process. Prescribed sequences can be richer: they may compose from their initial boundary yet fail when the unchanged remaining operations begin from a fresh preparation at an internal boundary. We quantify this failure and prove sharp bounds. Separately, promised output-probability prediction for general quantum circuits is PromiseBQP-complete.
The musical consequence. Unitary operations are notes and temporal boundaries are cues. A classical output map assigns sounds to recorded outcomes. Readout timing gives the rhythm, and Born-kernel composition determines which rhythms preserve the endpoint distribution. The stepwise test can hold from the entrance cue and fail after fresh preparation at an internal cue.
For a unitary operation U, the Born kernel Q(U) records its transition probabilities in a fixed measurement basis:
Q(U)yx = |⟨y|U|x⟩|².
𝒥𝒞(V,U) := Q(VU) − Q(V)Q(U).
The current is the signed difference between the coherent endpoint law and the stepwise-checked law. A positive entry is probability added to a transition by interference; a negative entry is probability removed. Every column sums to zero: the current moves probability but does not create it.
Its normalized Frobenius norm is
0 ≤ ν𝒞(V,U) := 1/√(d−1) ‖𝒥𝒞(V,U)‖F ≤ 1.
Here 𝒞 labels the fixed measurement basis and d is its number of outcomes. The value 0 means exact composition; 1 is the largest possible failure, and the bound is attained.
Let S record whether an intermediate check was made, X the prepared input, and Y the endpoint. Then the same consistency test can be read four ways:
diagram commutes ⇔ 𝒥𝒞(V,U) = 0 ⇔ ν𝒞(V,U) = 0 ⇔ 𝓘(S;Y | X) = 0.
For the paper’s first three qubit operations, with mixing parameter λ = 1/2, the six contiguous passages and their Born-composition tests are:
Prefix consistency
Boundary stability
Here Q(U₁) = Q(U₂) = Q(U₃) = K(1/2), the one-note law with probability 3/4 of retaining an outcome and 1/4 of changing it. The paradox. The score passes every fully stepwise test from cue 0, but the unchanged suffix U₃∘U₂ fails when freshly begun at cue 1.
Each row applies U₁, then U₂, then U₃. A chosen beat duration Δ gives the time between adjacent operation boundaries. The block lengths specify successive sound holds; a dot marks every readout, including the final one. The initial sound is specified separately from the prepared quantum state.
Admissibility means preserving the coherent endpoint distribution for every context-basis input, and hence every classical mixture of those inputs. The physically possible schedule (1,2), with readouts after U₁ and then U₃, fails this condition. Removing the readout after U₂ from (1,1,1) gives (1,2); removing the remaining intermediate readout restores admissibility as (3).
Hear these three rhythms in the paper’s two-qubit realization →
The mathematical claim
This paper gives quantum music a mathematical theory of composition: an exact law, a boundary paradox, a sharp capacity limit, and a quantum-complete prediction problem. The score is where the composition law becomes audible.
Theorem 2.1
For a finite-dimensional composition-closed family, Q(VU) = Q(V)Q(U) for every allowed pair if and only if every operation is a phase-dressed permutation. At the probability level, the only universally compositional quantum processes are reversible deterministic state machines.
Theorem 4.2
Genuinely mixing qubit sequences exist at arbitrary finite length—and as an explicit infinite sequence—that agree with stepwise statistics at every prefix from the opening boundary, yet fail when the unchanged continuation begins from a fresh context-basis preparation at an internal boundary.
Theorem 4.3
In dimension d, a boundary-stable sequence contains at most d full-support steps. The bound is attained in every prime dimension: a cue-stable d-outcome score can contain exactly d fully mixing notes.
Theorem 5.2
Given a polynomial-size quantum circuit, deciding whether a designated output probability lies above an upper input threshold or below a lower one, separated by an inverse-polynomial gap, is PromiseBQP-complete under classical polynomial-time reductions. The circuit size grows with the input; the theorem applies equally to musical labels and ordinary machine output symbols. An efficient classical randomized solver for all promised instances would imply BQP = BPP.
Theorem D.4
The normalized magnitude of the Born-composition defect satisfies 0 ≤ ν𝒞(V,U) ≤ 1 in every finite dimension. Inverse complex-Hadamard pairs attain the maximum.
Taken together, the results establish a theorem-driven theory of quantum composition: its exact classical limit, its irreducibly quantum failure, its finite-dimensional capacity, and its computational frontier.
A score prescribes unitary operations; its readout schedule determines the sound holds. Every enabled readout starts the assigned sound anew, even when the pitch repeats. Sounds are classical, but musical rules need not be.
The paper’s musical example
A qutrit chooses among (3), (2,1), and (1,1,1): a three-beat hold, a two-beat hold followed by one beat, or three one-beat holds. Each schedule applies the same U₁, U₂, U₃ and preserves the endpoint distribution.
Swipe across the score to see all five passes.
Loading the realization…
Filled note: Δ · open note: 2Δ · dotted open note: 3Δ. Each readout starts its assigned sound anew.
The selector outcomes 1, 2, 0, 1, 2 choose five complete passes. Each data pass begins in |00⟩, while the preceding sound is held until the next readout. All three schedules have endpoint probabilities (9,3,3,1)/16 for outcomes 00, 01, 10, 11. Changing the selection probabilities changes the rhythms while preserving this endpoint law.
The final readout at 15Δ produces C₄, held until 16Δ, followed by silence until 17Δ. This is the paper’s displayed realization, seed 1766.
A continuing E₂–F₂ pulse
An infinite qubit score with a driving, repeating pulse. The player’s λ = −1/2 variant changes pitch with probability 3/4; the paper’s λ = 1/2 realization favors repeated pitches. Both preserve the endpoint distribution of every fully checked prefix from their entrance cue.
Explore a two-interval passage
A measured selector enables or skips the intermediate readout. An enabled readout starts its assigned pitch; a skipped readout holds the sounding key. Watch the circuit and piano score unfold together, and choose new takes in this separate two-interval example.
An eight-readout video of the two-interval example, showing the selector, sound readouts, and earlier paths. The five-pass realization above follows the current paper’s timing.
Jacob Biamonte. arXiv:2609.11402 [quant-ph] (2026).
Preprint · First version: .