We show the relation between the Heisenberg averaging of regularized 2-point out-of-time ordered correlation function and the 2-point spectral form factor in bosonic quantum mechanics. The generalization to all even-point is also discussed. We also do the direct extension from the bosonic quantum mechanics to the noninteracting scalar field theory. Finally, we find that the coherent state and large-[Formula: see text] approaches are useful in the late-time study. We find that the computation of the coherent state can be simplified by the Heisenberg averaging. Therefore, this provides a simplified way to probe the late-time quantum chaos through a coherent state. The large-[Formula: see text] result is also comparable to the [Formula: see text] numerical result in the large-[Formula: see text] quantum mechanics. This can justify that large-[Formula: see text] technique in bosonic quantum mechanics can probe the late time, not the early time. Because the quantitative behavior of large-[Formula: see text] can be captured from the [Formula: see text] numerical result, the realization in experiments should be possible.
We show the relation between the Heisenberg averaging of regularized 2-point out-of-time ordered correlation function and the 2-point spectral form factor in bosonic quantum mechanics. The generalization to all even-point is also discussed. We also do the direct extension from the bosonic quantum mechanics to the noninteracting scalar field theory. Finally, we find that the coherent state and large-[Formula: see text] approaches are useful in the late-time study. We find that the computation of the coherent state can be simplified by the Heisenberg averaging. Therefore, this provides a simplified way to probe the late-time quantum chaos through a coherent state. The large-[Formula: see text] result is also comparable to the [Formula: see text] numerical result in the large-[Formula: see text] quantum mechanics. This can justify that large-[Formula: see text] technique in bosonic quantum mechanics can probe the late time, not the early time. Because the quantitative behavior of large-[Formula: see text] can be captured from the [Formula: see text] numerical result, the realization in experiments should be possible.
We prove that in bosonic quantum mechanics the two-point spectral form factor can be obtained as an average of the two-point out-of-time ordered correlation function, with the average taken over the Heisenberg group. In quantum field theory, there is an analogous result with the average taken over the tensor product of many copies of the Heisenberg group, one copy for each field mode. The resulting formula is expressed as a path integral over two fields, providing a promising approach to the computation of the spectral form factor. We develop the formula that we have obtained using a coherent state description from the JC model and also in the context of the large-N limit of CFT and Yang-Mills theory from the large-N matrix quantum mechanics.
We establish a rigorous connection between quantum coherence and quantum chaos by employing coherence measures originating from the resource theory framework as a diagnostic tool for quantum chaos.We quantify this connection at two different levels: quantum states and quantum channels.At the level of states, we show how several well-studied quantifiers of chaos are, in fact, quantum coherence measures in disguise (or closely related to them).We further this connection for all quantum coherence measures by using tools from majorization theory.Then we numerically study the coherence of chaotic-versus-integrable eigenstates and find excellent agreement with random matrix theory in the bulk of the spectrum.At the level of channels, we show that the coherence-generating power (CGP)-a measure of how much coherence a dynamical process generates on average-emerges as a subpart of the out-of-time-ordered correlator (OTOC), a measure of information scrambling in many-body systems.Via numerical simulations of the (nonintegrable) transverse-field Ising model, we show that the OTOC and CGP capture quantum recurrences in quantitatively the same way.Moreover, using random matrix theory, we analytically characterize the OTOC-CGP connection for the Haar and Gaussian ensembles.In closing, we remark on how our coherence-based signatures of chaos relate to other diagnostics, namely, the Loschmidt echo, OTOC, and the Spectral Form Factor.
The spectral form factor (SFF), which is the Fourier transform of the two-point spectral correlation, is a fundamental diagnostics of quantum chaos. Nevertheless, its experimental detection in quantum many-body systems was less unexplored until recently. Here, we establish a relationship between the SFF with the quantum Fisher information (QFI) by means of quantum sensing protocols. Furthermore, when the eigenvalues and eigenstates of the disordered Hamiltonian are statistically independent, we prove that QFI is typical both in short and long timescales while numerics suggest that it is also typical at intermediate timescales, which ensures self-averaging of QFI. The self-averaging property renders the experimental measurement of QFI with our protocol less noisy in contrast with the direct measurement of SFF which is known to be non-self-averaging. Our results reveal an intriguing connection between quantum noise and the signatures of quantum chaos.
A bstract We develop a finite-dimensional formulation of the recently introduced notion of “timelike entanglement”, defined in terms of two-point functions between operators supported on different Cauchy slices. Using a local orthonormal operator basis, we recast this construction in terms of a generalized response tensor . Building on this, we introduce a generalized spacetime density kernel (GSDK) corresponding to higher point correlation functions, including time-ordered as well as out-of-time-ordered correlators. Motivated by the structure of Haar probe averaging, we contract the (2 N )-leg GSDK with cyclic permutation operators and thereby define a kernel form factor . This quantity is computable even when Haar averaging is not feasible, and it reduces to the (2 N )-th moment of the spectral form factor, evaluated at an N -enhanced effective temperature. The correlation functions of the GSDK operators also yield the SFF, with an effective (1/ N )-reduction of the physical time-scales. The GSDK places both early time scrambling diagnostics and late time spectral statistics on a similar footing and clarifies how higher-point correlators and non-trivial time ordering capture fine-grained dynamical information of a quantum system.
In this work, the term “quantum chaos” refers to spectral correlations similar to those found in the random matrix theory. Quantum chaos can be diagnosed through the analysis of level statistics using, e.g., the spectral form factor, which detects both short- and long-range level correlations. The spectral form factor corresponds to the Fourier transform of the two-point spectral correlation function and exhibits a typical slope-dip-ramp-plateau structure (aka correlation hole) when the system is chaotic. We discuss how this structure could be detected through the quench dynamics of two physical quantities accessible to experimental many-body quantum systems: the survival probability and the spin autocorrelation function. The survival probability is equivalent to the spectral form factor with an additional filter. When the system is small, the dip of the correlation hole reaches sufficiently large values at times which are short enough to be detected with current experimental platforms. As the system is pushed away from chaos, the correlation hole disappears, signaling integrability or localization. We also provide a relatively shallow circuit with which the correlation hole could be detected with commercially available quantum computers.
We study symmetry-resolved Krylov complexity in finite-dimensional chaotic quantum many-body systems. When both the Hamiltonian and the initial operator commute with a conserved charge, the operator dynamics decomposes into independent symmetry sectors, each with its own Krylov chain. We show that, after saturation, the unresolved Krylov complexity is additive over symmetry sectors. In the absence of additional Liouvillian degeneracies, the late-time contribution of a sector with Hilbert-space dimension dq is controlled by dq(dq−1), leading to a dimension-weighted equipartition that approaches the simple large-sector scaling dq2/∑q′dq′2. This late-time rule differs from the early-time weighted-average discussed in the literature and is governed instead by the dimensions of the accessible operator spaces. We support the analytic prediction with numerical studies of the real and complex SYK models, a chaotic bosonic spin model, and the mixed-field Ising chain. Our results show that resolving exact symmetries is essential for interpreting the saturation value of Krylov complexity as a diagnostic of chaotic operator growth.