Tests of Linearity

A test of quantum linearity asks whether a specified evolution departs from linear quantum dynamics. Linearity of state-vector evolution and affine dependence of operational evolution on density operators are related, distinct requirements: a state-dependent global phase can be nonlinear on vectors while leaving the density operator unchanged, and affine quantum channels can describe dissipative evolution. The test needs an explicit alternative and a measurable deviation. Nonlinear effective equations in optics or many-body physics are not automatically violations of fundamental quantum mechanics.

A state-dependent precession frequency

As an illustrative pure-state modification, let a normalized qubit evolve with

$$H[\psi]=\frac{\hbar}{2}\left(\omega_0+\kappa \langle Z\rangle_\psi\right)Z.$$

This Hermitian state-dependent generator preserves the norm. Its $Z$ populations remain constant, while transverse components precess with frequency $\omega_0+\kappa z$, where $z=\langle Z\rangle_\psi$. A Ramsey experiment can prepare several values of $z$ and search for a population-dependent phase after a controlled time $t$.

For two preparations $z_1,z_2$ with detectable transverse coherence, the model predicts

$$\Delta\phi=\kappa(z_1-z_2)t.$$

If the uncertainty in the residual differential phase is $\delta\phi$, the simple sensitivity scale is $|\kappa|\lesssim\delta\phi/(|z_1-z_2|t)$. This is a model-dependent estimate, not a confidence interval without a likelihood and nuisance treatment.

Bollinger and collaborators used hyperfine spectroscopy of trapped beryllium ions to test a Weinberg-type nonlinear theory in 1989. The important experimental idea is controlled variation of populations, with conventional frequency shifts constrained independently.

Conventional effects and effective nonlinearities

AC Stark shifts, mean fields, density-dependent interactions, state-preparation errors, and magnetic drifts can all create preparation-dependent frequencies. An experiment must determine whether the proposed signal exceeds their allowed contribution.

The Gross–Pitaevskii equation is nonlinear in a condensate order parameter because it approximates interacting many-body dynamics. It is not a universal evolution law for arbitrary isolated quantum states. Similarly, a nonlinear classical optical response does not replace the linear Hilbert-space evolution of the fully modeled field and material.

An alleged fundamental modification must specify how entangled systems, mixtures, measurements, and different ensemble decompositions are handled. A pure-state differential equation alone is incomplete for these questions.

Mixture consistency and signaling

Ordinary channels obey

$$\mathcal E\!\left(\sum_jp_j\rho_j\right) =\sum_jp_j\mathcal E(\rho_j).$$

Without this property, two preparations with the same initial density operator might evolve to observably different states. Combined with the usual remote steering and collapse rules, such ensemble dependence can permit signaling. This is a consistency constraint on the extension, not a proof that every conceivable nonlinear framework is already experimentally excluded.

The lesson is to test the whole alternative, including its rules for composite systems. See Tests of No-Signaling for the distinct operational marginal test.

Exercise

Prepare an equal mixture of pure Bloch vectors $\boldsymbol r$ and $-\boldsymbol r$, where $\boldsymbol r=(\sin\theta,0,\cos\theta)$, and set $\omega_0=0$. Under the illustrative state-dependent pure-state evolution, compute the average Bloch vector.

Solution

The two vectors rotate around $z$ with opposite angular velocities $\pm\kappa\cos\theta$. Their average is $\bigl(0,\sin\theta\sin(\kappa t\cos\theta),0\bigr)$. It is generally nonzero even though the initial mixture is $I/2$. An equal mixture of $Z$ eigenstates remains $I/2$. Thus naive ensemble-by-ensemble use of this pure-state rule depends on the decomposition; a complete theory must address that ambiguity.

References

• Bollinger, J. J., Heinzen, D. J., Itano, W. M., Gilbert, S. L., and Wineland, D. J., “Test of the linearity of quantum mechanics by rf spectroscopy of the $^9$Be$^+$ ground state,” Physical Review Letters 63, 1031–1034 (1989), doi:10.1103/PhysRevLett.63.1031 — population-dependent spectroscopy.
• Weinberg, S., “Testing quantum mechanics,” Annals of Physics 194, 336–386 (1989), doi:10.1016/0003-4916(89)90276-5 — a concrete nonlinear extension.