Covariant Measures of Non-Markovianity in Curved Spacetime

TL;DR

This work addresses the challenge of quantifying quantum memory in curved spacetime where no global time foliation exists. It introduces a covariant process-tensor framework built from causal diamonds along timelike worldlines and derives memory kernels from Hadamard two-point functions $W(x,x')$, enabling a foliation-free, operational measure of non-Markovianity, $N(Υ)$, defined as the distance to the set of CP-divisible (Markovian) combs. The approach reveals that acceleration and spacetime curvature generate long-range temporal correlations and horizon-induced memory tails, with effects that can be revealed by multi-time or multi-probe protocols and even exhibit superactivation. The results establish a coordinate-invariant quantification of quantum memory in relativistic settings and highlight curvature and horizons as resources or contaminants for relativistic quantum information processing near black holes or in accelerated frames.

Abstract

Standard measures of quantum non-Markovianity are usually defined in terms of dynamical maps on a preferred time foliation and therefore do not extend straightforwardly to curved spacetimes, where no global time coordinate exists and causal structure is primary. We develop a covariant framework for open quantum dynamics along arbitrary timelike worldlines by building multi-time quantum processes (process tensors) from overlapping causal diamonds. For an Unruh--DeWitt detector weakly coupled to a scalar field in a Hadamard state, we define a foliation-independent measure of non-Markovianity as the operational distance between the physical process tensor and the convex set of Markovian (CP-divisible) processes. Numerical benchmarks in $(1{+}1)$ dimensions compare inertial motion, uniform acceleration, and static and infalling trajectories in Schwarzschild spacetime. Inertial trajectories are found to be almost Markovian, whereas acceleration and curvature generate pronounced long-range temporal correlations and strong non-Markovian behaviour. In Rindler spacetime, acceleration produces horizon-induced memory tails. In Schwarzschild spacetime, near-horizon field correlations cause both static and freely falling observers to experience enhanced memory, which can remain hidden in single-step diagnostics but becomes evident in multi-time protocols and can even be superactivated by combining different time steps. Our results provide, to our knowledge, the first coordinate-independent, operational quantification of quantum memory in relativistic settings. They identify spacetime curvature, horizons, and acceleration as controllable ingredients that can either degrade or be harnessed as resources in relativistic quantum information tasks, including communication and metrology with accelerated detectors and near black holes.

Figures (5)

• Figure 1: (Color online) Schematic illustration of causal diamonds and interventions along a geodesic worldline. (a) In flat (Minkowski) spacetime, a sequence of interventions $A_k$ is performed along a worldline parameterized by the affine parameter $\lambda$. Each segment $\gamma_k = [\lambda_{k-1},\lambda_k]$ generates a causal diamond (outlined diamonds). For the second intervention $A_2$, the light–green and light–orange cross–hatched wedges depict its causal future $J^{+}(A_2)$ and causal past $J^{-}(A_2)$, respectively; their overlap defines the shared causal volume with neighboring segments, which supports field correlations quantified by the Wightman function $W(x,x')$ and hence the memory kernels $\eta_{i-j}$. Green (red) arrows indicate future–directed (past–directed) null rays. (b) In curved spacetime (e.g., near an event horizon), the worldline, causal diamonds, and light cones are distorted, as suggested by the deformed background grid. Gravitational focusing and shear reduce and skew the overlap between $J^{+}$ and $J^{-}$ of neighboring interventions, leading to a smaller and asymmetric shared causal volume.
• Figure 2: Choi representation of process tensors: (a) Markovian comb, factorizing into a chain of CPTP maps $\phi_{k}$ with only nearest-neighbor causal links (blue arrows), implying memoryless dynamics. (b) Non-Markovian comb, including cross-time correlations (red dashed arrows) that capture environment-mediated memory effects, such as those from Wightman functions in curved spacetimes.
• Figure 3: Heatmaps of the real part of the memory kernel $\eta$-matrix across the four benchmark geometries: (a) Minkowski inertial, (b) Rindler ($a=1.0$), (c) Schwarzschild static ($r=6.0M$), and (d) Schwarzschild infalling (from $r_0=8.0M$). The off-diagonal elements ($\eta_{i-j}$, $i \neq j$) signify non-Markovian memory. These elements are negligible for the inertial case but are significantly enhanced by acceleration (Rindler) and spacetime curvature (Schwarzschild), indicating strong history-dependent dynamics.
• Figure 4: Line cuts and parameter scans of the memory kernel $\eta$‑matrix. (a) Median magnitude $(\langle|\eta_{i-j}|\rangle)$ as a function of slot separation for four worldlines: Minkowski inertial (blue), Rindler accelerated (orange), Schwarzschild static (green), and Schwarzschild infall (red). (b–d) Most negative off‑diagonal entries as functions of initial radius $r_0$ (infall), acceleration a (Rindler), and radial position r (static). Together these plots demonstrate how acceleration and curvature modulate long‑range anticorrelations and memory backflow.
• Figure 5: (Color online) Operational bounds and SDP-based estimates of covariant non-Markovianity across the four benchmark worldlines (Minkowski inertial, Minkowski Rindler, Schwarzschild static, Schwarzschild infall). (a) SDP-derived lower-bound proxy for $N(\Upsilon)$, given by the summed absolute prediction error $\sum_{\text{testers}}|\Delta|$, as a function of the number of slots $n$, showing a clear enhancement of non-Markovianity for accelerated and infall trajectories relative to the inertial case. (b) Stitched two-slot activation $\Delta_{\text{act}}$ versus $n$, obtained from the excess negativity of a nonlocal two-slot block over its constituent single-slot blocks Watrous2009FawziRennerCMP2015, which reproduces the same geometry ordering and provides a complementary witness of operational memory activation. (c) Geometry-resolved scatter plots of the SDP proxy for $N(\Upsilon)$ versus the certified lower bound from intermediate Choi negativities, together with linear fits and $R^2$ values, demonstrating that the Choi-based bound tracks the full operational distance with nearly linear behavior in each geometry. (d) Corresponding scatter of the SDP proxy versus the normalized kernel tail mass $S(\Delta=1)$, indicating that processes with heavier long-range tails in $\eta_{i-j}$ systematically display larger non-Markovianity while preserving the hierarchy observed in panels (a) and (b).

Theorems & Definitions (3)

• Proposition 1: Reparametrization and diffeomorphism covariance
• Proposition 2: LB via intermediate Choi
• proof : Proof sketch