The quasiprobability behind the out-of-time-ordered correlator

TL;DR

This work unifies the out-of-time-ordered correlator (OTOC) with the Kirkwood-Dirac quasiprobability by introducing the behind-OTOC quasiprobability tilde A_rho, which encodes richer information than the OTOC itself. It develops three parallel tracks: (i) experimental pathways to infer tilde A_rho and its coarse-grained version using weak measurements and projection tricks, (ii) numerical and analytical studies of tilde A_rho inIsing and Brownian-circuit models, showing nonclassical features and clear scrambling signatures, and (iii) a theoretical framework that generalizes KD quasiprobabilities, Bayes-type retrodiction, and their relation to TOC/OTOC formalisms. The results reveal that tilde A_rho often exhibits negativity and nonreality, persists under averaging, and that scrambling breaks certain symmetries in the quasiprobability landscape. These findings expand quasiprobability theory, provide tools for near-term scrambling experiments, and offer a foundation for deeper connections between quantum chaos, thermodynamics-like relations, and information-theoretic measures.

Abstract

Two topics, evolving rapidly in separate fields, were combined recently: The out-of-time-ordered correlator (OTOC) signals quantum-information scrambling in many-body systems. The Kirkwood-Dirac (KD) quasiprobability represents operators in quantum optics. The OTOC has been shown to equal a moment of a summed quasiprobability. That quasiprobability, we argue, is an extension of the KD distribution. We explore the quasiprobability's structure from experimental, numerical, and theoretical perspectives. First, we simplify and analyze the weak-measurement and interference protocols for measuring the OTOC and its quasiprobability. We decrease, exponentially in system size, the number of trials required to infer the OTOC from weak measurements. We also construct a circuit for implementing the weak-measurement scheme. Next, we calculate the quasiprobability (after coarse-graining) numerically and analytically: We simulate a transverse-field Ising model first. Then, we calculate the quasiprobability averaged over random circuits, which model chaotic dynamics. The quasiprobability, we find, distinguishes chaotic from integrable regimes. We observe nonclassical behaviors: The quasiprobability typically has negative components. It becomes nonreal in some regimes. The onset of scrambling breaks a symmetry that bifurcates the quasiprobability, as in classical-chaos pitchforks. Finally, we present mathematical properties. The quasiprobability obeys a Bayes-type theorem, for example, that exponentially decreases the memory required to calculate weak values, in certain cases. A time-ordered correlator analogous to the OTOC, insensitive to quantum-information scrambling, depends on a quasiprobability closer to a classical probability. This work not only illuminates the OTOC's underpinnings, but also generalizes quasiprobability theory and motivates immediate-future weak-measurement challenges.

Figures (20)

• Figure 1: Spin-chain example: A spin chain exemplifies the quantum many-body systems characterized by the out-of-time-ordered correlator (OTOC). We illustrate with a one-dimensional chain of $N$ spin-$\frac{1}{2}$ degrees of freedom. The vertical red bars mark the sites. The dotted red arrows illustrate how spins can point in arbitrary directions. The OTOC is defined in terms of local unitary or Hermitian operators $\mathcal{W}$ and $V$. Example operators include single-qubit Paulis $\sigma^x$ and $\sigma^z$ that act nontrivially on opposite sides of the chain.
• Figure 2: Quantum processes described by the probability amplitudes $A_\rho$ in the out-of-time-ordered correlator (OTOC): These figures, and parts of this caption, appear in YungerHalpern_17_Jarzynski. The OTOC quasiprobability $\tilde{A}_{\rho}$ results from summing products $A_\rho^*( . ) A_\rho( . )$. Each $A_\rho( . )$ denotes a probability amplitude [Eq. \ref{['eq:Amp']}], so each product resembles a probability. But the amplitudes' arguments differ---the amplitudes correspond to different quantum processes---because the OTOC operators $\mathcal{W} (t)$ and $V$ fail to commute, typically. Figure \ref{['fig:Protocoll_Trial1']} illustrates the process described by the $A_\rho( . )$; and Fig. \ref{['fig:Protocoll_Trial2']}, the process described by the $A_\rho^*( . )$. Time, as measured by a laboratory clock, increases from left to right. Each process begins with the preparation of the state $\rho = \sum_j p_j \lvert j \rangle\!\langle j \rvert$ and a measurement of the state's eigenbasis. Three evolutions ($U$, $U^\dag$, and $U$) then alternate with three measurements of observables ($\tilde{ \mathcal{W} }$, $\tilde{V}$, and $\tilde{ \mathcal{W} }$). Figures \ref{['fig:Protocoll_Trial1']} and \ref{['fig:Protocoll_Trial2']} are used to define $\tilde{A}_{\rho}$, rather than showing protocols for measuring $\tilde{A}_{\rho}$.
• Figure 3: Quantum circuit for inferring the coarse-grained OTOC quasiprobability $\tilde{ \mathscr{A} }_{\rho}$ from weak measurements: We consider a system of $N$ qubits prepared in a state $\rho$. The local operators $\mathcal{W} = \sigma^{ \mathcal{W} } \otimes \mathbbm{1}^{ \otimes (N - 1) }$ and $V = \mathbbm{1}^{ \otimes (N - 1) } \otimes \sigma^{ V }$ manifest as one-qubit Paulis. Weak measurements can be used to infer the coarse-grained quasiprobability $\tilde{ \mathscr{A} }_{\rho}$. Combining values of $\tilde{ \mathscr{A} }_{\rho}$ yields the OTOC $F(t)$. Figure \ref{['fig:D_subcircuit']} depicts a subcircuit used to implement a weak measurement of $n = \mathcal{W}$ or $V$. An ancilla is prepared in a fiducial state $\lvert 0 \rangle$. A unitary $R_{ n }^\dag$ rotates the qubit's $\sigma^{ n }$ eigenbasis into its $\sigma^z$ eigenbasis. $R_y ( \pm \phi )$ rotates the ancilla's state counterclockwise about the $y$-axis through a small angle $\pm \phi$, controlled by the system's $\sigma^z$. The angle's smallness guarantees the measurement's weakness. $R_{ n }$ rotates the system's $\sigma^z$ eigenbasis back into the $\sigma^{ n }$ eigenbasis. The ancilla's $\sigma^z$ is measured strongly. The outcome, $+$ or $-$, dictates which partial-projection operator $D_{\pm}^n$ updates the state. Figure \ref{['fig:Full_circuit']} shows the circuit used to measure $\tilde{ \mathscr{A} }_{\rho}$. Three weak measurements, interspersed with three time evolutions ($U$, $U^\dag$, and $U$), precede a strong measurement. Suppose that the initial state, $\rho$, commutes with $\mathcal{W}$ or $V$, e.g., $\rho = \mathbbm{1} / d$. Figure \ref{['fig:Full_circuit']} requires only two weak measurements.
• Figure 4: Real (upper curve) and imaginary (lower curve) parts of $F(t)$ as a function of time. $T=\infty$ thermal state. Nonintegrable parameters, $N=10$, $\mathcal{W}=\sigma_1^z$, $V=\sigma_N^z$.
• Figure 5: Real part of $\tilde{ \mathscr{A} }_{\rho}$ as a function of time. $T=\infty$ thermal state. Nonintegrable parameters, $N=10$, $\mathcal{W}=\sigma_1^z$, $V=\sigma_N^z$. There are many degeneracies. The upper curves include $0000$ and $1010$, while the top of the lower pitchfork includes $1110$ and the bottom of the lower pitchfork includes $0001$.

Theorems & Definitions (15)

• Definition 1: $\mathscr{K}$-extended Kirkwood-Dirac quasiprobability
• Example 1: Real $\tilde{A}_{\rho}$ #1: $t = 0$, shared eigenbasis, arbitrary $\rho$
• Example 2: Real $\tilde{A}_{\rho}$ #2: $t = 0$, nonshared eigenbases, $\rho = \mathbbm{1} / d$
• Example 3: Nonreal $\tilde{A}_{\rho}$: $t = 0$, nonshared eigenbases, $\rho$ nondiagonal relative to both
• Theorem 1: Retrodiction about composite observables
• proof
• Theorem 2: Weak-value space saver
• proof
• Corollary 1: Retrodictive application of $\tilde{A}_{\rho}$
• Theorem 3