Spatial Incompatibility Witnesses for Quantum Temporal Correlations

Abstract

We introduce a witness-based framework for certifying quantum temporal correlations via the pseudo-density matrix (PDM) formalism, which is a spatiotemporal generalization of the density matrix. We define spatial incompatibility (SI) as the minimum distance between a PDM and valid density matrices. For trace-norm distance, we show that this reduces to the PDM's negativity, enabling the construction of experimentally accessible SI witnesses. We derive a tight bound on SI for quantum channels and analyze the respective roles of state and channel coherence in witnessing SI. Our approach, unlike the LG framework, exploits measurements that generate coherence through state disturbance. We further show that channels satisfying the LG inequality for incoherent states can still exhibit detectable SI, demonstrating that measurement disturbance enhances the certification of temporal correlations.

Figures (2)

• Figure 1: Setup. (a) Measuring the two-time correlator $\langle \{\sigma_i, \sigma_j \} \rangle$. (b) Defining the measure of spatial incompatibility (SI) $\mathcal{T}_p(R)$ as the minimal distance between $R$ and the set of density matrices (DMs). The SI witness determines a hyperplane tangent to this set. The wider set of pseudo-density matrices (PDMs) is non-convex.
• Figure 2: Hierarchy of Incoherent Operations. The set NCGD contains both CI and DI operations as its subsets, with OI belonging to DI. Operations inside OI cannot be detected by either SI or the LG inequality smirne2018coherence. In contrast, operations outside OI but still within NCGD remain undetectable by the LG inequality yet can be detected by SI witness, demonstrating the broader capability of our approach in certifying temporal correlations.

Theorems & Definitions (20)

• Definition 1: Spatial incompatibility
• Definition 2: SI witness
• Proposition 1: Existence of SI witness
• Proposition 2: Maximal qubit SI degree
• Lemma 1: Spatially compatible pairings of channels and incoherent input states
• Proposition 3: OI channels and any fixed incoherent state spatially compatible
• Proposition 4: SI for coherent input states and classical channels
• Definition 3
• Proposition 5: Spatial statistics cannot violate LG
• Proposition 6: Cases where LG is respected but SI witness exists