Foundations of quantum mechanics · Theory
Bargmann-invariant witness of the graviton commutator
arXiv:2610.35253v1 ↗PDF ↗Submitted 26 October 2026
- Question
- Experiments proposed to show that gravity is quantum look for gravitationally induced entanglement or decoherence, but Newtonian entangling phases can be mimicked by assigning a classical field to each branch, and decoherence can be mimicked by classical noise. Is there an interference observable that no branchwise-definite classical gravitational field can reproduce?
- Result
- Yes. For matter coupled linearly to linearized gravity, each branch coherence factorizes exactly into a conservative phase, the characteristic functional of the graviton state, and a phase from the field commutator. For two interferometers, the phase \(\mathfrak B\) of a four-cycle Bargmann invariant cancels everything except the commutator: it is independent of the graviton state and of all Newtonian interactions, and it vanishes for every branchwise-definite classical model. It is bounded by which-path decoherence, \(\mathfrak B^2\le16\,\Gamma_X\Gamma_Y\).
The paper in detail
Each matter branch \(a\) sources a classical stress tensor \(T_a\). Because the coupling is linear, the evolution in branch \(a\) is \(U_a=e^{iS_a}e^{ih[J_a]}\), and tracing out the gravitons gives \[M_{ab}=e^{i(S_a-S_b)}\,e^{\frac i2(T_a|G_{\rm PJ}|T_b)}\,\chi[T_a-T_b].\] The first factor contains the Newtonian potential and all conservative post-Newtonian interactions, the last is the characteristic functional of the graviton state, and the middle one comes from the commutator (Pauli–Jordan) kernel, which is state independent and fixed by on-shell gravitons.
The comparison class contains every model in which each branch acquires a deterministic phase plus the phase of one classical random field acting on all branches; it covers semiclassical gravity, quantum-controlled classical fields and Diósi–Penrose-type noise. If the graviton state has a nonnegative Wigner functional (vacuum, thermal, coherent, squeezed, inflationary relic states) and the commutator terms between branch differences vanish, all coherences are reproducible in this class. For a single interferometer they always vanish, so its decoherence can always be explained classically.
With two interferometers \(A\) and \(B\) with difference sources \(X\) and \(Y\), the phase \(\mathfrak B=\arg(\rho_{00,10}\rho_{10,11}\rho_{11,01}\rho_{01,00})\) cancels the conservative phases and turns the \(\chi\) factors into a nonnegative product, leaving \(\mathfrak B=(X|G_{\rm PJ}|Y)\); every classical-class model gives zero. Operationally, \(\mathfrak B\) is the difference between the fringe shift that \(B\) induces on \(A\) and vice versa: quantum fields give one-way, causal influence, which a branchwise-definite model can match only by allowing signalling. An uncertainty relation for displacement operators bounds the witness by the visibilities.
For slow sources the witness reduces to the cross term of the Burke–Thorne radiation-reaction potential at 2.5 post-Newtonian order. In a Gaussian trajectory model the ratio to its budget reaches 0.906 for parallel axes at a delay \(\delta\simeq0.62\), so gravitational radiation nearly saturates the bound, but the absolute size is about \(3\times10^{-60}\). In a trapped-ion analog, qubits play the branches and a shared motional mode the field; open displacements \(\alpha_X,\alpha_Y\) in orthogonal quadratures give \(\mathfrak B=2\operatorname{Im}(\alpha_Y^*\alpha_X)\) for any motional state, about 0.5 for \(|\alpha|\simeq0.5\).
Transferable methods
A cycle of coherences that cancels everything classical
Products of coherences around a closed cycle of branches are invariant under local rephasings, so branch-dependent phases cancel, and noise factors pair up as moduli. Whatever phase survives must come from non-commuting contributions, here the field commutator.
Around \((00)\to(10)\to(11)\to(01)\), the terms \(S_a-S_b\) telescope to zero. The noise factors give \(\chi[-X]\chi[-Y]\chi[X]\chi[Y]=|\chi[X]\chi[Y]|^2\ge0\) because \(\chi[-T]=\chi[T]^*\). The commutator phases add to \(\mathfrak B_{\rm Q}=(X|G_{\rm PJ}|Y)=2\operatorname{Im}\langle\alpha_Y,\alpha_X\rangle\).
In the classical class, \(M^{\mathscr C}_{ab}=e^{i(s_a-s_b)}\Phi_P[T_a-T_b]\), so the same cycle gives \(\arg|\Phi_P[X]\Phi_P[Y]|^2=0\) for every distribution \(P\) and every choice of phases \(s_a\).
Editorial check: simulating two qubits that displace a shared oscillator mode by \(\alpha_X=0.5\) and then \(\alpha_Y=0.5i\), the four-cycle phase is \(-0.500\) for both the vacuum and a thermal state with mean occupation 0.5, matching \(2\operatorname{Im}(\alpha_Y^*\alpha_X)\) in magnitude, while a classical Gaussian random-field model gives \(0.000\). The budget \(|\sin(\mathfrak B/2)|V_{X+Y}\le\sqrt{(1-V_X^2)(1-V_Y^2)}\) holds in both cases (0.193 against 0.221 for the vacuum).
How to reuse it
To separate a genuinely quantum effect from classical phases and noise in any multi-branch interference experiment, look for a rephasing-invariant cycle of coherences. Branchwise phases cancel, noise contributes only moduli, and the surviving phase isolates commutator or geometric-phase physics.
Bounding a commutator signal by decoherence
The commutator of two displacement operators is a phase times another displacement, so the uncertainty relation for unitary operators turns the Bargmann phase into a quantity bounded by the visibilities that measure which-path information.
For unitaries, \(\Delta U\,\Delta V\ge\tfrac12|\langle[U,V]\rangle|\) with \(\Delta U^2=1-|\langle U\rangle|^2\). With \(U=e^{ih[J_X]}\), \(V=e^{\pm ih[J_Y]}\), \(|\langle[U,V]\rangle|=2|\sin(\mathfrak B/2)|\,|\chi[X\pm Y]|\), which gives \[|\sin(\mathfrak B/2)|\,V_{X\pm Y}\le\sqrt{(1-V_X^2)(1-V_Y^2)},\] and in the weak-coupling regime \(\mathfrak B^2\le16\Gamma_X\Gamma_Y\).
Editorial check: with \(I(\delta)=\sqrt{\pi/2}(\delta^5-10\delta^3+15\delta)e^{-\delta^2/2}\), the parallel-axis budget ratio \(|I|/8\) peaks at 0.906 for \(\delta=0.617\), and the identical-interferometer ratio \(|\mathfrak B|/\Gamma_{\rm vac}=|I|_{\max}/2=3.62\), as stated.
How to reuse it
Wherever a coherent effect is mediated by a field that also records which-path information, an uncertainty relation of this form bounds the effect by the decoherence it causes. It is a quick feasibility test: a predicted signal larger than the budget signals an error in the model.
Context & limitations
Why it matters. It identifies precisely which interferometric quantity carries the quantum nature of the field, explains why decoherence alone cannot certify it, and ties the signal to a decoherence budget. The same algebra gives a trapped-ion analog where the witness is of order one.
Limitations. For gravity the effect is about \(10^{-60}\) radians for the most optimistic proposed parameters, requiring about \(10^{119}\) repetitions. A nonzero witness excludes branchwise-definite models but not hybrid classical–quantum dynamics, which reproduce the same phase and differ only in their noise. The analysis assumes linear coupling, linearized gravity and additive sources.