Intrinsic Quantum Mpemba Effect in Markovian Systems and Quantum Circuits

Abstract

The quantum Mpemba effect (QME) describes the counterintuitive phenomenon in which a system farther from equilibrium reaches steady state faster than one closer to equilibrium. However, ambiguity in defining a suitable distance measure between quantum states has led to varied interpretations across different contexts. Here we propose the intrinsic quantum Mpemba effect (IQME), defined using the trajectory length traced by the quantum state as a more appropriate measure of distance--distinct from previous trajectory-independent metrics. By treating quantum states as points in a Riemannian space defined by statistical distance, the trajectory length emerges as a more natural and accurate characterization of the counterintuitive dynamics, drawing an analogy to the classical Brachistochrone problem. We demonstrate the existence of IQME in Markovian systems and extend its definition to quantum circuits, thereby establishing a unified framework applicable to both open and closed systems. Notably, we observe an IQME in a $U(1)$-symmetric circuit, offering new insights into the rates of quantum thermalization for different initial states. This work deepens our understanding of quantum state evolution and lays the foundation for accurately capturing novel quantum dynamical behaviour.

Figures (3)

• Figure 1: Comparison for intuition, fact, IQME and QME. (a) The leftmost column presents four cases, which can be analogized to particles moving in a classical gravitational field, where one might intuitively expect that a greater trajectory distance leads to a longer travel time. In each case, the longer trajectory is shown in cyan, while $A$ is always geodesically closer to the steady state. However, as in the counterintuitive Brachistochrone problem, these intuitions can be misleading. The occurrence of IQME corresponds precisely to scenarios where intuition fails. (b-e) Quantum state evolution under Eq. (\ref{['eq2']}) with $\alpha=100$; $\gamma’=0.94$ in (b,d,e) and $\gamma’=0.52$ in (c). (b) $L_A = 0.890 < L_B = 1.046$, while $d_A(0) = 0.782 < d_B(0) = 1.046$, corresponding to case (i). (c) $L_A = 1.019 > L_B = 0.781$, while $d_A(0) = 0.663 < d_B(0) = 0.781$, corresponding to case (ii). (d) $L_A = 1.214 > L_B = 0.885$, while $d_A(0) = 0.780 < d_B(0) = 0.885$, corresponding to case (iii). (e) $L_A = 0.658 < L_B = 1.013$, while $d_A(0) = 0.658 < d_B(0) = 0.908$, corresponding to case (iv).
• Figure 2: IQME and QME in Markovian systems. The left column shows $R(t)$, where a crossing signifies the presence of IQME, while the right column shows $d(t)$, where a crossing indicates QME. Insets display $R_A(t)-R_B(t)$ and $d_A(t)-d_B(t)$ at the point of crossing, highlighted by the red curve crossing zero. We label the geodesically closer point as $A$ and depict the longer trajectory in blue in all cases. (a,b) Parameters correspond to case (i) in Fig. \ref{['fig1']}. $(y_A(0), z_A(0))=(0.5, 0.0)$, $(y_B(0), z_B(0))=(0.0, 0.5)$. (c,d) Parameters correspond to case (ii). $(y_A(0), z_A(0))=(-0.95, -0.25)$, $(y_B(0), z_B(0))=(0.0, 0.0)$. (e,f) Parameters correspond to case (iii). $(y_A(0), z_A(0))=(0.9, 0.0)$, $(y_B(0), z_B(0))=(0.0, 0.2)$. (g,h) Parameters correspond to case (iv). $(y_A(0), z_A(0))=(0.0, -0.25)$, $(y_B(0), z_B(0))=(0.5, 0.25)$.
• Figure 3: IQME in a $U(1)$-symmetric quantum circuit. We consider a system of $N=16$ qubits and averaged over $10,000$ trajectories. (a) The quantum circuit comprises layers of $U(1)$-symmetric two-qubits gates arranged in a brick-wall pattern, as depicted by orange rectangles. We focus on the evolution of a single qubit as discussed in the main text. (b) State evolution for tilted Néel initial states with $\theta=0.1\pi$ and $\theta=0.5\pi$, where the color fades along the direction of the evolution. (c) $\overline{R(t)}$ for tilted Néel initial states with $\theta=0.1\pi$ and $\theta=0.5\pi$. (d) $\overline{R(t)}$ for tilted ferromagnetic initial states.