Diagnosis of information scrambling from Hamiltonian evolution under decoherence

We apply a quantum teleportation protocol based on the Hayden-Preskill thought experiment to quantify how scrambling a given quantum evolution is. It has an advantage over the direct measurement of out-of-time ordered correlators when used to diagnose the information scrambling in the presence of decoherence effects stemming from a noisy quantum device. We demonstrate the protocol by applying it to two physical systems: Ising spin chain and SU(2) lattice Yang-Mills theory. To this end, we numerically simulate the time evolution of the two theories in the Hamiltonian formalism. The lattice Yang-Mills theory is implemented with a suitable truncation of Hilbert space on the basis of the Kogut-Susskind formalism. On a two-leg ladder geometry and with the lowest nontrivial spin representations, it can be mapped to a spin chain, which we call the Yang-Mills-Ising model and is also directly applicable to future digital quantum simulations. We find that the Yang-Mills-Ising model shows the signal of information scrambling at late times.

Figure 1

The configuration of input and output Hilbert spaces of evolution operator $U_{\text{Ising}}$. The states on ${\mathcal{H}}_A$ and ${\mathcal{H}}_D$ are respectively represented by red and blue wires. The rest of wires in input and output wires are the states on ${\mathcal{H}}_B$ and ${\mathcal{H}}_C$, respectively.

Figure 2

The ideal Hayden-Preskill recovery protocol with $U=U_{\text{Ising}}$. (a) The projection probability (2.9). (b) The fidelity (2.10). The parameters on each panel are $N≔\log d$, $N_A≔\log d_A$, $N_D≔\log d_D$, and $dt≔t/M$. The green, red, and blue points correspond to $(h,m)=(0,0)$ [classical], (1.0,0) [critical], and $(-1.05,0.5)$ [chaotic], respectively. The black line represents the $P_{\mathrm{EPR}}$ (2.13) and $F_{\mathrm{EPR}}$ (2.14) computed with $U$ given by a Haar random unitary operator.

Figure 3

The Hayden-Preskill recovery protocol with $U=U_{\text{Ising}}$ in the presence of decoherence (3.3). (a) The projection probability (2.18). (b) The fidelity (2.15). The parameters on each panel are $N≔\log d$, $N_A≔\log d_A$, $N_D≔\log d_D$, $dt≔t/M$, and $p$ is the error probability (3.3). The green, red, and blue points correspond to $(h,m)=(0,0)$ [classical], (1.0,0) [critical], and $(-1.05,0.5)$ [chaotic], respectively. The black line represents the $P_{\mathrm{EPR}}$ (2.13) and $F_{\mathrm{EPR}}$ (2.14) computed with $U$ given by a Haar random unitary operator without decoherence effects.

Figure 4

Two-leg ladder model of $SU(2)$ Yang-Mills theory.

Figure 5

The configuration of input and output Hilbert spaces of evolution operator $U_{\mathrm{YM}}$. The states on ${\mathcal{H}}_A$ and ${\mathcal{H}}_D$ are respectively represented by red and blue wires. The rest of wires in input and output wires are the states on ${\mathcal{H}}_B$ and ${\mathcal{H}}_C$, respectively. $A$ and $D$ are kept away from the boundaries to reduce possible boundary effects.

Figure 6

The ideal Hayden-Preskill recovery protocol with $U=U_{\mathrm{YM}}$. (a) The projection probability (2.9). (b) The recovery fidelity (2.10). The parameters on each panel are $N≔\log d$, $N_A≔\log d_A$, $N_D≔\log d_D$, and $dt≔t/M$. The green, red, and blue points correspond to $K=0$, $K=0.5$, and $K=2$, respectively. The black line represents the $P_{\mathrm{EPR}}$ (2.13) and $F_{\mathrm{EPR}}$ (2.14) computed with $U$ given by a Haar random unitary operator.

Figure 7

The Hayden-Preskill recovery protocol with $U=U_{\mathrm{YM}}$ in the absence of decoherence. The projection probability (a) and the recovery fidelity (b) against the rescaled time $Kt$. The lower panels show the $K$-dependence of the late-time values of $P_{\mathrm{EPR}}$ (c) and $F_{\mathrm{EPR}}$ (d). The blue data are the values of $P_{\mathrm{EPR}}$ ($F_{\mathrm{EPR}}$) at $t=1000$, and the red data are obtained by averaging $P_{\mathrm{EPR}}$ ($F_{\mathrm{EPR}}$) over the time duration $100\le t\le 1000$. The solid and dotted black lines represent those by the Haar random evolution and classical Ising Hamiltonian (YM-Ising model at $K=0$).

Figure 8

The Hayden-Preskill recovery protocol with $U=U_{\mathrm{YM}}$ in the presence of decoherence (3.3). (a) The projection probability (2.18). (b) The recovery fidelity (2.15). The parameters on each panel are $N≔\log d$, $N_A≔\log d_A$, $N_D≔\log d_D$, $dt≔t/M$, and $p$ is the error probability (3.3). The green, red, and blue points correspond to $K=0$, $K=0.5$, and $K=2$, respectively. The black line represents the $P_{\mathrm{EPR}}$ (2.13) and $F_{\mathrm{EPR}}$ (2.14) computed with $U$ given by a Haar random unitary operator without decoherence effects.

Figure 9

The Hayden-Preskill recovery protocol for the Ising spin chain in the absence of decoherence. (a) The projection probability. (b) The recovery fidelity. The data points are obtaind by the computation with the Suzuki-Trotter decomposition applided to the evolution operator, while the curves are obtaind by the exact Hamiltonian evolution.

Figure 10

The Hayden-Preskill recovery protocol for the YM-Ising model in the absence of decoherence. (a) The projection probability. (b) The recovery fidelity. The data points are obtained by the computation with the Suzuki-Trotter decomposition applied to the evolution operator, while the curves are obtained by the exact Hamiltonian evolution.

Figure 11

The Hayden-Preskill recovery protocol for the Ising spin chain in the absence of decoherence. The projection probability (a), and the recovery fidelity (b) are computed with system sizes $N=6$, 7, 8.

Figure 12

The Hayden-Preskill recovery protocol for the YM-Ising model with $K=2$ in the absence of decoherence. The projection probability (a), and the recovery fidelity (b) are computed with system sizes $N=6$, 7, 8.

Figure 13

The Hayden-Preskill recovery protocol for the YM-Ising model in the absence of decoherence. The projection probability (a), and the recovery fidelity (b) against the rescaled time $Kt$.

Figure 14

Label of a physical state of $SU(2)$ Yang-Mills theory in a two-leg ladder geometry.