Quasiprobability behind the out-of-time-ordered correlator

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 was shown to equal a moment of a summed quasiprobability [Yunger Halpern, Phys. Rev. A 95, 012120 (2017)]. 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 Yunger Halpern's 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. We define an extended KD quasiprobability that generalizes the KD distribution. 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.

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 Ref. [37]. The OTOC quasiprobability ${\tilde{A}}_{\rho }$ results from summing products $A_{\rho }^{*}(...)A_{\rho }(...)$. Each $A_{\rho }(...)$ denotes a probability amplitude [Eq. (32)], so each product resembles a probability. But the amplitudes' arguments differ (the amplitudes correspond to different quantum processes) because the OTOC operators $\mathcal{W}\left(t\right)$ and $V$ fail to commute, typically. Panel (a) Illustrates the process described by the $A_{\rho }(...)$; Panel (b), 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 $\rho ={\sum }_j{p}_j|j\rangle \langle j|$ and a measurement of the state's eigenbasis. Three evolutions ($U$, $U^{†}$, and $U$) then alternate with three measurements of observables ($\tilde{\mathcal{W}}$, $\tilde{V}$, and $\tilde{\mathcal{W}}$). Panels (a) and (b) 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 ${\tilde{\mathcal{A}}}_{\rho }$ from weak measurements. We consider a system of $N$ qubits prepared in a state $\rho$. The local operators $\mathcal{W}={\sigma }^{\mathcal{W}}\otimes {\mathbb{1}}^{\otimes (N-1)}$ and $V={\mathbb{1}}^{\otimes (N-1)}\otimes {\sigma }^V$ manifest as one-qubit Paulis. Weak measurements can be used to infer the coarse-grained quasiprobability ${\tilde{\mathcal{A}}}_{\rho }$. Combining values of ${\tilde{\mathcal{A}}}_{\rho }$ yields the OTOC $F\left(t\right)$. Panel (a) depicts a subcircuit used to implement a weak measurement of $n=\mathcal{W}$ or $V$. An ancilla is prepared in a fiducial state $|0\rangle$. A unitary $R_n^{†}$ rotates the qubit's ${\sigma }^n$ eigenbasis into its ${\sigma }^z$ eigenbasis. $R_y(\pm \varphi )$ rotates the ancilla's state counterclockwise about the $y$ axis through a small angle $\pm \varphi$, 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. (b) Shows the circuit used to measure ${\tilde{\mathcal{A}}}_{\rho }$. Three weak measurements, interspersed with three time evolutions ($U$, $U^{†}$, and $U$), precede a strong measurement. Suppose that the initial state $\rho$ commutes with $\mathcal{W}$ or $V$, e.g., $\rho =\mathbb{1}/d$. Panel (b) requires only two weak measurements.

Figure 4

Real (upper curve) and imaginary (lower curve) parts of $F\left(t\right)$ as a function of time. The imaginary curve is nearly indistinguishable from the $x$ axis. $T=\infty$ thermal state. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$.

Figure 5

Real part of ${\tilde{\mathcal{A}}}_{\rho }$ 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.

Figure 6

Imaginary part of ${\tilde{\mathcal{A}}}_{\rho }$ as a function of time. $T=\infty$ thermal state. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$. To within machine precision, $\text{Im}\left({\tilde{A}}_{\rho }\right)$ vanishes for all values of the arguments.

Figure 7

Real (upper curve) and imaginary (lower curve) parts of $F\left(t\right)$ as a function of time. $T=1$ thermal state. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$.

Figure 8

Real part of ${\tilde{\mathcal{A}}}_{\rho }$ as a function of time. $T=1$ thermal state. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$. The upper curve includes 0000, the middle curve includes 0011, and the lower cluster of curves includes 0100.

Figure 9

Imaginary part of ${\tilde{\mathcal{A}}}_{\rho }$ as a function of time. $T=1$ thermal state. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$. The various curves display similar looking oscillations as a function of time; some of the curves are, roughly, interrelated by a factor of $-1$, e.g., 1111 and 1110.

Figure 10

Real (upper curve) and imaginary (lower curve) parts of $F\left(t\right)$ as a function of time. The imaginary part is nearly indistinguishable from the $x$ axis. $T=\infty$ thermal state. Integrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$.

Figure 11

Real part of ${\tilde{\mathcal{A}}}_{\rho }$ as a function of time. $T=\infty$ thermal state. Integrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$. The highest curve includes 0000. Of the curves near the $x$ axis, the upper curve includes 1110, and the lower includes 0001.

Figure 12

Real (upper curve) and imaginary (lower curve) parts of $F\left(t\right)$ as a function of time. Random pure state. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$.

Figure 13

Real part of ${\tilde{\mathcal{A}}}_{\rho }$ as a function of time. Random pure state. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$. The upper cluster includes 0000 and 0011. The upper prong of the lower pitchfork includes 1110; the lower prong includes 1001.

Figure 14

Imaginary part of ${\tilde{\mathcal{A}}}_{\rho }$ as a function of time. Random pure state. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$. A familiar pattern of oscillations is visible, with some curves being more or less related by a factor of $-1$, for example, 0011 and 1000.

Figure 15

Real (upper curve) and imaginary (lower curve) parts of $F\left(t\right)$ as a function of time. Product ${|+x\rangle }^{\otimes N}$ of $N$ copies of the $+1{\sigma }^x$ eigenstate. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$.

Figure 16

Real part of ${\tilde{\mathcal{A}}}_{\rho }$ as a function of time. Product ${|+x\rangle }^{\otimes N}$ of $N$ copies of the $+1{\sigma }^x$ eigenstate. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$. The top of the upper cluster includes 0101, while the bottom of the upper cluster includes 0011. The top of the lower pitchfork includes 1110, while the bottom of the lower pitchfork includes 1001.

Figure 17

Imaginary part of ${\tilde{\mathcal{A}}}_{\rho }$ as a function of time. Product ${|+x\rangle }^{\otimes N}$ of $N$ copies of the $+1{\sigma }^x$ eigenstate. Nonintegrable parameters, $N=10$, $\mathcal{W}={\sigma }_1^z$, $V={\sigma }_N^z$. Similar physics to other depictions of the imaginary part.

Figure 18

Quantum processes described by the probability amplitudes $A_{\rho }^{\mathrm{TOC}}$ in the time-ordered correlator (TOC) $F_{\mathrm{TOC}}\left(t\right)$: $F_{\mathrm{TOC}}\left(t\right)$, like $F\left(t\right)$, equals a moment of a summed quasiprobability (Theorem t16). The quasiprobability ${\tilde{A}}_{\rho }^{\mathrm{TOC}}$ equals a sum of multiplied probability amplitudes $A_{\rho }^{\mathrm{TOC}}$ [Eq. (143)]. Each product contains two factors: $A_{\rho }^{\mathrm{TOC}}(j;v_1,{\lambda }_{v_1};w_1,{\alpha }_{w_1})$ denotes the probability amplitude associated with the “forward” process in (a). The system $S$ is prepared in a state $\rho$. The $\rho$ eigenbasis $\left\{|j\rangle \langle j|\right\}$ is measured, yielding outcome $j$. $\tilde{V}$ is measured, yielding outcome $(v_1,{\lambda }_{v_1})$. $S$ is evolved forward in time under the unitary $U$. $\tilde{\mathcal{W}}$ is measured, yielding outcome $(w_1,{\alpha }_{w_1})$. Along the abscissa runs the time measured by a laboratory clock. Along the ordinate runs the $t$ in $U:=e^{-iHt}$. The second factor in each ${\tilde{A}}_{\rho }^{\mathrm{TOC}}$ product is $A_{\rho }^{\mathrm{TOC}}{(j;v_2,{\lambda }_{v_2};w_1,{\alpha }_{w_1})}^{*}$. This factor relates to the process in (b). The operations are those in (a). The processes' initial measurements yield the same outcome. So do the final measurements. The middle outcomes might differ. Complex conjugating $A_{\rho }^{\mathrm{TOC}}$ yields the probability amplitude associated with the reverse process. Panels (a) and (b) depict no time reversals. Each analogous OTOC figure [Figs. 2 and 2] depicts two.

Figure 19

$\overline{\mathcal{K}}$-fold out-of-time-ordered correlator (OTOC): the conventional OTOC [Eq. (28)] encodes just three time reversals. The $\overline{\mathcal{K}}$-fold OTOC $F^{\left(\overline{\mathcal{K}}\right)}\left(t\right)$ encodes $2\overline{\mathcal{K}}-1=\mathcal{K}=3,5,...$ time reversals. The time measured by a laboratory clock runs along the abscissa. The ordinate represents the time parameter $t$, which may be inverted in experiments. The orange, leftmost dot represents the state preparation $\rho$. Each green dot represents a $\mathcal{W}\left(t\right)$ or a $V$. Each purple line represents a unitary time evolution. The diagram, scanned from left to right, represents $F^{\left(\overline{\mathcal{K}}\right)}\left(t\right)$, scanned from left to right.

Figure 20

Three interrelated ideas: relationships among the out-of-time-ordered correlator, quasiprobabilities, and Schwinger-Keldysh path integrals were articulated recently.