The Signal
Vardhan and Wang Link Higher-Point OTOCs to a New Chaos Measure
Shreya Vardhan and Jinzhao Wang introduce free mutual information as a measure of quantum chaos and relate it to higher-point out-of-time-ordered correlators. Their theoretical analysis finds that the measure can distinguish some cases of o
Quantum.world Team · Written with AI · fact-checked ·
PRX Quantum published a new measure of quantum chaos on October 26: free mutual information, or FMI, a quantity Shreya Vardhan and Jinzhao Wang propose to track how operators spread through a quantum system. Their paper links FMI to higher-point out-of-time-ordered correlators, offering a way to combine information that the familiar four-point correlator can miss. The work is theoretical; it reports no new laboratory experiment.
A measure of spreading beyond physical space
In a many-body system, an operator that begins acting on a few degrees of freedom can spread across more of the system as it evolves. The standard four-point out-of-time-ordered correlator, or OTOC, is commonly used to track that growth. But the authors distinguish physical-space spreading from a finer question: how broadly an evolved operator ranges across the space of all possible operators, which they describe as doubly exponential in the number of degrees of freedom.
FMI quantifies the volume occupied in that abstract operator space. It compares a coarse-grained set of operators compatible with the evolved operator and a reference operator against the full set of possibilities. As that volume fraction grows, FMI falls; a small late-time value indicates that the evolved operator has become less constrained by its reference. The authors adapt the idea from free probability, a mathematical framework for studying relationships between noncommuting quantities.
Higher-point correlators add up
For observables with the spectrum of Pauli matrices, Vardhan and Wang derive an explicit formula for FMI using the eigenvalues of the product of the time-evolved observable and its reference. They also show that, under the paper’s stated conditions, FMI can be expressed as a weighted sum of higher-point OTOCs. Those terms collectively give an information-theoretic interpretation to correlators beyond the standard four-point version.
The connection offers a route to estimating FMI from correlator measurements: finite partial sums provide increasing lower estimates, with the estimate improving as more higher-point terms are included. That is not the same as a ready-made experimental protocol. The paper presents the relation as useful alongside recent progress in measuring higher-point OTOCs, but leaves further work on practical evaluation and on the measure’s behavior in additional settings.
Where the four-point test misses a distinction
The authors analyze chaotic models including random unitary circuits and chaotic spin chains, where FMI decreases over time even as individual higher-point OTOCs can oscillate. They compare those results with non-chaotic examples, including Clifford and permutation–phase–Clifford unitaries and the transverse-field Ising model. In those cases, the standard four-point OTOC can decay—signaling spreading through physical degrees of freedom—while FMI remains infinite, pointing to a lack of spreading through operator space.
“The FMI is thus a sharper diagnostic of chaos than the standard four-point OTOC.”
Shreya Vardhan and Jinzhao Wang, paper authors
Symmetry still complicates the diagnosis
The paper does not claim FMI cleanly separates every integrable system from every chaotic one. Its discussion notes that an integrable Heisenberg model can resemble chaotic models under FMI for some choices of observables, a behavior the authors connect to symmetry. They identify a symmetry-aware version of the measure as a subject for further work, alongside extending the analysis beyond the maximally mixed reference state used in the paper.