Measuring unconventional causal structures in monitored dynamics

TL;DR

This work tackles how causal structure in monitored quantum dynamics can deviate from classical intuition, introducing XEQCI as an experimentally accessible measure of quantum causal influence that remains robust to post-selection overhead. By combining a general framework with a concrete experimental protocol, it demonstrates, through Clifford-circuit simulations and analytically tractable models, that the arrow of time tends to flow from low to high entropy and that causal influence can propagate in inverted or nonlocal patterns depending on initial/final states and monitoring. The study uncovers universal aspects of QCI near measurement-induced phase transitions and reveals richer causal geometries in dual-unitary and Brownian models, illustrating the deep interplay between entropy, scrambling, and causal structure. These insights pave the way for practical measurements on current quantum hardware and offer new perspectives on information flow, causality, and emergent spacetime in quantum many-body systems.

Abstract

Causality underpins all logical reasoning. However, the causal structure in quantum processes can be far from intuitive, often differing from its classical counterpart in relativity, which is defined by the light cone. In particular, in systems with measurement and post-selection, causal influence can occur between spacelike separated regions. In this work, we study the causal structure and emergent "arrow of time" in monitored quantum dynamics, particularly their dependence on initial and final states. We propose a new measure, the cross-entropy quantum causal influence, to quantify the extent of causal influence, whose simulation demonstrates exotic causal structures, such as inverted light cones. This quantity can be measured in current quantum computing platforms. Additionally, we provide an analytical understanding of the relation between time arrow and entropy by studying two types of models that are analytically tractable: a quantum Brownian evolution model and a dual-unitary circuit model.

Figures (8)

• Figure 1: (a) In a standard brickwork circuit evolution, the conventional light cone is formed by the spacetime coordinates that a local unitary perturbation $U_A$ can influence. Here, each square represents a two-qubit gate, each triangle represents a single-qubit pure state, and the bending curve at the final time denotes that the circuit is contracted with an identical (but Hermitian-conjugated) backward evolution circuit. (b) For the same brickwork circuit, if the initial state is maximally mixed and the final state is post-selected, the region that $U_A$ can influence forms a light cone propagating backward in time. (c) Schematic of the central idea underlying our proposed cross-entropy measure of QCI. A physical system subjected to a local unitary perturbation $U_A$ is compared with an otherwise identical simulated copy in which $U_A$ is not applied. We perform measurement in another region $B$. The QCI from $A$ to $B$ is quantified by how "different" the measurement outcome ("$b$") distributions are.
• Figure 2: QCI landscapes and light cone inversion. In each labeled figure, the color is the value of $-\log\overline{\chi}$, where $\chi$ is the circuit-averaged cross-entropy measure of QCI (see Eq. \ref{['eq:chibar']}).
• Figure 3: (a-c) Illustration of possible paths through which $A$ can influence $B$, and how they are impacted by mid-circuit measurements. (d-f) The QCI landscape in hybrid circuits with different measurement rates. (a) and (d): small-$p$ phase, where the QCI only spreads an $O(1)$ distance. (b) and (e): close to the critical point $p_c$, where the QCI spreads a maximal distance. (c) and (f): large-$p$ phase, where the spreading is suppressed to an $O(1)$ range again.
• Figure 4: Correlation time of XEQCI and universality. Here, $\tau$ is the correlation time and $T$ is the total time of the simulation. By choosing $p_c = 0.154$ and $\nu = 1.33$, data for different total time $T$ collapse onto one universal curve.
• Figure 5: Causal influence in BGUE toy model in \ref{['ssec:brownian']}. We fix the whole system being $20$ qubits ($d_\text{tot}=2^{20}$), total evolution time being $T=30$ and $U_A,O_B$ acting on single qubit ($d_A=d_B=2$). The final state is a product of pure state on each qubit and the initial state is a product mixed state on each qubit with local purity set to a generic value $0.504$. We fix the location of $U_A$ insertion at $t_U=T/2$, and draw the function of $\overline{\text{CI}}$ as a function of $t_O\in[0,T]$. The blue curve is when $U_A,O_B$ acting on the same qubit (so QCI has discontinuity when $t_O$ cross $t_U$) and orange curve is when $U_A,O_B$ acting on different qubits (QCI is continuous when $t_O$ cross $t_U$).