Gaussian Atemporality Robustness

• Gaussian atemporality robustness is a measure that quantifies the minimal additional Gaussian noise required to render non-temporal quantum correlations explainable by completely positive Gaussian channels.
• It is efficiently computable via explicit closed-form expressions that capture directional asymmetries and indicate an intrinsic arrow of time in continuous-variable quantum systems.
• The measure refines the hierarchy of quantum correlations by distinguishing spatially sourced nonclassical effects that cannot be reproduced by standard temporal dynamics, extending beyond mere entanglement.

Gaussian atemporality robustness is a concept in continuous-variable quantum information that quantifies the non-equivalence of certain Gaussian correlations to any sequence of temporal (causal) operations, thus revealing quantum correlations that cannot be explained by any direct-causal (Gaussian channel) mechanism and instead demand the assumption of a spatially distributed common cause. This robustness is both efficiently computable and operationally meaningful as the minimal extra Gaussian noise required to make atemporal quantum correlations compatible with a valid temporal (channel-based) description (Song et al., 15 Aug 2025). The measure extends previous notions of quantum nonlocality, capturing nonclassical correlations beyond entanglement, and can signal the presence of an intrinsic arrow of time in Gaussian systems.

1. Definition and Conceptual Basis

Gaussian atemporality characterizes situations where the covariance matrix of quadrature measurements from bipartite quantum systems is unphysical under any possible temporal sequence of operations—that is, no Gaussian channel G : A → B exists such that the measured covariances arise from a single system evolving in time via G. In these cases, the observed statistics force the postulation of a spatial (common-cause) quantum origin rather than any time-ordered dynamical evolution.

Gaussian atemporality robustness, f(V), is defined as the minimal amount of (typically isotropic, thermal) Gaussian noise that must be added to one mode (see below) such that the full covariance matrix becomes compatible with a completely positive (CP) Gaussian channel explanation. This measure detects the degree to which quantum correlations are irreducibly atemporal.

2. Mathematical Structure of Atemporality

A bipartite Gaussian quantum system with quadratures (q_A, p_A) for Alice and (q_B, p_B) for Bob is fully characterized by its covariance matrix: $$V_{AB} = \begin{pmatrix} V_A & C \ C^{\top} & V_B \end{pmatrix}$$ where V_A and V_B are the local 2×2 covariance matrices of each mode, and C is the 2×2 cross-correlation matrix.

To describe a possible temporal realization, it is necessary that there exists a Gaussian channel from A → B with linear map T and noise matrix N such that: $V_B = T V_A T^{\top} + N$

$C = V_A T^{\top}$

and crucially, the CP condition for the channel: $N + i I - i T T^{\top} \geq 0$ If no such T and N exist—with physical, positive semidefinite noise matrix N—then the correlations are atemporal.

Atemporality robustness for the “forward” direction (A→B) is defined by the minimal μ ≥ 0 such that

$N + μ I + i I - i T T^{\top} \geq 0$

This is operationally interpreted as the minimal noise level to regularize the channel.

For the class of covariance matrices considered in (Song et al., 15 Aug 2025), an explicit closed-form expression is provided: $$f_{→}(V_{AB}) = \max\left\{0,\, |ω| - \sqrt{\det N}\right\}$$ with

$$ω = 1 - \frac{\det C}{\det V_A},\qquad \det N = \frac{\det V_{AB}}{\det V_A}$$

Similar logic yields f_{←}(V_{AB}) for the reverse direction.

3. Operational Significance and Computability

The operational meaning of f(V) is as a feasible, efficiently computable test for atemporality: a statistically significant, nonzero value certifies that the observed statistics cannot arise from any temporal Gaussian channel, even with moderate small perturbations. The measure quantifies the “distance in noise” from the set of temporal (CP-channel explainable) Gaussian correlations.

Moreover, the atemporality robustness can differ for the forward and reverse temporal directions: given the same bipartite covariance, it is possible for f_{→}(V_{AB}) ≠ f_{←}(V_{AB}). This asymmetry provides a rigorous way to define and quantify an intrinsic arrow of time in the correlations.

4. Examples and Thresholds

The paper provides concrete classes of states illustrating the relevance and subtlety of Gaussian atemporality robustness:

• Two-mode squeezed thermal states: Given local thermal variances v ≥ 1 and squeezing parameter r, the covariance is atemporal only when r exceeds a threshold r_{atemp} (distinct, and strictly larger, than the entanglement threshold r_{ent}). Thus, there exist entangled states that do not exhibit atemporality, clarifying that Gaussian atemporality measures quantum correlations beyond just separability violation.
• Beam splitter with asymmetric single-mode squeezing: For local squeezing parameters u, v, the atemporality region is determined by inequalities involving their quadratic functions. States can be entangled (via PPT criterion) yet still have zero atemporality robustness.
• Lossy channels: If a state with variances (v_1, v_2) is subjected to loss with transmissivity η, the “forward” temporal pseudo-channel to the receiver is always CP. However, the “reverse” pseudo-channel may not be; in this case, the atemporality robustness is nonzero only if the rescaled variances v_1^{(η)} v_2^{(η)} < 1. This establishes that some space–time Gaussian correlations possess a true temporal irreversibility.

5. Relationship to Entanglement and Quantum Signature

Gaussian atemporality robustness is strictly stronger than entanglement. All atemporality-robust states are entangled, but the converse is false: there are entangled states that remain compatible with a temporal description (f(V) = 0). Only when the logarithmic negativity exceeds a critical value (state is sufficiently strongly entangled) does atemporality robustness become positive. Thus, f(V) distinguishes a hierarchy of quantum correlations:

• Separable (classical or merely quantum)
• Entangled (negative partial transpose, PPT violation)
• Atemporal (not temporally simulatable by any valid Gaussian channel)

6. Practical and Foundational Implications

The availability of a simple, analytic formula for Gaussian atemporality robustness offers a pragmatic tool for experimental quantum physics, particularly in continuous-variable optics, where covariance matrices are experimentally accessible.

Practically, by comparing measured covariance matrices and adding calibrated noise, one can operationally certify not just the presence of quantum entanglement, but also whether that entanglement is fundamentally atemporal—i.e., irreducible to any dynamical causal explanation. This has ramifications for quantum communication, secure protocols, and the emerging paper of quantum causal inference.

On the foundational level, Gaussian atemporality robustness joins a broader class of measures probing the distinction between spatial (common-cause) and temporal (direct-cause) origins of quantum correlations. The explicit demonstration of quantum correlations with an intrinsic arrow of time, and the ability to sharply distinguish them from standard (possibly time-symmetric) entanglement, provides novel insight into the structure of quantum correlations and the constraints imposed by causality in quantum mechanics.

7. Connections and Future Directions

This work refines the theoretical apparatus for distinguishing spatiotemporal structures in quantum statistics, directly connecting to modern concepts of pseudo-channels and pseudo-density operators. Open questions remain about the generalization to multimode settings, non-Gaussian regimes, and operational protocols leveraging atemporality. Further exploration of the interplay between atemporality robustness and resource theories of quantum causal structures is likely to deepen the understanding of nonclassicality in continuous-variable systems and its practical exploitation.

---

In summary, Gaussian atemporality robustness provides a rigorous, efficient, and operationally meaningful measure that separates quantum correlations demanding a common cause from those reproducible by temporal Gaussian processes, quantifies the arrow of time in such correlations, and identifies a strict class of quantum phenomena beyond standard entanglement (Song et al., 15 Aug 2025).