Detecting Initial System-Environment Correlations from a Single Observable

Abstract

We address the problem of detecting initial system--environment correlations when the environment is not directly accessible. Most existing approaches rely on full state tomography or multiple system preparations, which can be experimentally demanding. We show that, for a known interaction, it can be sufficient to monitor a single expectation value of the system. Focusing on a qubit interacting with an environment via isotropic Heisenberg exchange, we derive exact bounds on the signal $z(t)=\langleσ_z^S\rangle(t)$ that hold for all factorized initial states. These bounds define a \emph{factorized envelope}: if an observed trajectory exits this envelope at any time, initial system--environment correlations are certified. From a reduced-dynamics perspective, the envelope admits a clear operational interpretation as the admissible region generated by the standard product assignment (embedding) map, which serves as a null model for uncorrelated preparations. Envelope violations therefore rule out the entire product-assignment class using only a single calibrated observable. We illustrate the method using three families of correlated initial states and observe clear envelope violations, including cases in which the reduced system state is maximally mixed. We further show that the same single-observable logic extends to an exactly solvable pure-dephasing spin--boson model with an infinite environment, where factorized initial states generate a simple coherence envelope whose violation certifies initial correlations. Overall, our results demonstrate that single-axis measurements, combined with a one-time calibration of $ρ_S(0)$, can certify initial system--environment correlations without tomography or environment access.

Figures (5)

• Figure 1: Illustration of the factorized-envelope witness. The shaded region represents the factorized envelope $\mathcal{R}(t)=[z_{\min}(t),z_{\max}(t)]$, containing all trajectories $z(t)=\langle\sigma_z^S\rangle(t)$ obtainable from factorized initial states $\rho_{SE}(0)=\rho_S(0)\otimes\rho_E(0)$. The blue curve shows a representative factorized trajectory lying entirely inside the envelope, while the orange curve corresponds to a correlated initial state that exits the envelope, thereby certifying initial system–environment correlations.
• Figure 2: (Color online) Geometric picture of exchange-driven Bloch-vector motion. (a) Factorized preparation: the system Bloch vector $\vec{s}_u(t)$ evolves in the plane spanned by the initial system vector $\vec{s}$ and the environment vector $\vec{e}$, tracing a partial-swap arc with rotation axis $\vec{s}\times\vec{e}$. (b) Correlated preparation: initial correlations generate an additional component along $\vec{s}\times\vec{e}$, so the system trajectory $\vec{s}_c(t)$ leaves the $\vec{s}$-$\vec{e}$ plane. This out-of-plane motion is incompatible with any factorized initial state with the same calibrated $\rho_S(0)$.
• Figure 3: (Color online) Example 1. The black curve shows the correlated trajectory $z^{(\mathrm{corr})}(t)=-\sin(4Jt)$ for the maximally entangled initial state. The blue shaded region is the factorized envelope $z_{\min/\max}(t)=\pm\sin^2(2Jt)$ for all product states with $\rho_S(0)=\mathbb{I}/2$. At $t^*=3\pi/(8J)$ (red marker), the correlated value lies outside the admissible region, certifying initial correlations using only $z(t)$.
• Figure 4: (Color online) Example 2. The blue shaded region shows the factorized envelope $z_{\min/\max}(t)=\cos^2(2Jt)\,p \pm \sin^2(2Jt)$ for product initial states with the calibrated $\rho_S(0)$. The black curves show the correlated trajectories $z^{(\mathrm{corr})}(t)=p\cos(4Jt)+(p-1)\sin(4Jt)$. Panels correspond to different values of $p$. For $p<1/3$, the correlated trajectory leaves the envelope (red marker), certifying initial system-environment correlations using only $z(t)$.
• Figure 5: (Color online) Example 3. The blue shaded region shows the factorized envelope $z_{\min/\max}(t)=\pm\sin^2(2Jt)$ for product initial states with the calibrated $\rho_S(0)=\mathbb{I}/2$. The black curves show the correlated trajectories $z^{(\mathrm{corr})}(t)=(p-1)\sin(4Jt)$ for different values of $p$. For smaller $p$ (stronger entangled component), the trajectories clearly leave the envelope, certifying initial $S$-$E$ correlations even though the reduced state is maximally mixed at $t=0$.

Theorems & Definitions (8)

• Proposition 1
• proof
• Lemma 2
• proof
• Corollary 3
• proof
• Theorem 4: Minimal-observable witness
• proof