Gaussian Atemporality: When Gaussian Quantum Correlations Imply Common Cause

TL;DR

The paper addresses how Gaussian quadrature correlations can reveal whether two systems share a common cause or arise from temporal evolution. By formalizing space-time covariance matrices and forward/reverse Gaussian channels, it defines Gaussian atemporality and derives a closed-form robustness measure that quantifies how much Gaussian noise can be added before temporal explanations become viable. It shows that atemporality can exist beyond entanglement and that a time-arrow can emerge in certain spatiotemporal Gaussian correlations, including asymmetries under time reversal. The results enable a principled classification of Gaussian quantum correlations by their compatibility with causal mechanisms and offer a tractable framework for studying multi-time Gaussian processes and related operational tasks.

Abstract

Conventionally, covariances do not distinguish between spatial and temporal correlations. The same covariance matrix could equally describe temporal correlations between observations of the same system at two different times or correlations made on two spatially separated systems that arose from some common cause. Here, we demonstrate Gaussian quantum correlations that are `atemporal', such that the covariances governing their quadrature measurements are unphysical without postulating some common cause. We introduce Gaussian atemporality robustness as a measure of atemporality, illustrating its efficient computability and operational meaning as the maximum noise which can be added without removing this uniquely quantum phenomenon. We illustrate that (i) specific spatiotemporal Gaussian correlations possess an intrinsic arrow of time, such that Gaussian atemporality robustness is zero in one temporal direction and not the other and (ii) that it measures quantum correlations beyond entanglement.

Figures (5)

• Figure 1: Spatio-temporal causal distribution mechanisms. The blue-shaded regions in the figure show the measurement stations of the parties involved, Alice and Bob. The respective parties perform Gaussian measurements, i.e., homodyne or heterodyne measurements. For simplicity of representation, a) and b) show homodyne measurements in each station, and parties are subject to any allocation for these station; Party 1 could be either Alice or Bob. Likewise, Party 2 could be Bob or Alice. This allows the above diagrammatic descriptions not to be confined by a specific causal order of Alice's and Bob's measurement stations. Consequently, the causal mechanisms can be classified into three types: Fig. a) shows the spatially distributed mechanism where parties 1 and 2 perform the measurement simultaneously on a two-mode squeezed vacuum (TMSV) state. Party 1 can change the measurement basis by imparting an additional phase through the piezoelectric transducer (PZT). Fig. b) shows the temporally distributed mechanism where Party 1 performs continuous variable quantum non-demolition measurements andersen2002nondemolotionbuchler2001nondemolition on the input state and shares the residual state with Party 2. The AM is the amplitude modulator, which is a part of the quantum non-demolition measurement. Fig. c) shows the general spatiaotemporally distributed mechanism that is practically implementable Note_process. The "Meas" represent the Gaussian measurements. The state preparation box represents Gaussian operations that are generally used to prepare states, namely the displacement operation $\mathrm{\hat{D}}(\alpha)$, rotation $\mathrm{\hat{R}}(\theta)$, and squeezing operation $\mathrm{\hat{S}}(r)$, and it depends on the measurement basis and the outcome. The displacement operation and rotation operation, along with the beamsplitting operation $\mathrm{\hat{BS}}(t)$, constitute the elements of the arbitrary two-mode unitary operation between the ancilla and one arm of the TMSV state.
• Figure 2: Entanglement and atemporality for asymmetric states passing through a balanced beam splitter. The red-shaded region represents the set of all unphysical states. The regions where $u<1$ or $v<1$ represent the entangled states. States in the blue region are entangled and Gaussian-atemporal, whereas the states in the green region are entangled but not Gaussian-atemporal. The states in the remaining white region are separable and not Gaussian-atemporal.
• Figure 3: Entanglement and atemporality for randomly generated Gaussian states. Consider randomly generated two-mode squeezed states. We use the logarithmic negativity vidal02negativityplenio2005logarithmic as a measure of entanglement. (a) Plot of Gaussian-atemporality vs entanglement for $5000$ randomly generated pure state (with a squeezing $0 < r <1$). The top green line is the line for a two-mode squeezed vacuum state (with $0<r<1$ and $v=1$ ). The black dashed line is for a state we get from interfering two squeezed states (with $r=1$) while rotating the other. We also get the same behaviour (orange line) by interfering the two squeezed states on an asymmetric beam splitter. (b) Plot of atemporality vs entanglement for $20000$ randomly generated mixed states with the same level of mixedness. There is a finite value of the logarithmic negativity needed before we can observe any Gaussian-atemporality. The top green line is for a two-mode squeezed thermal state of $v=1.5$. The black dashed line is for a state that we get from interfering two squeezed thermal states while rotating one state relative to the other and the orange line is from interfering two squeezed thermal states at an asymmetric beam splitter.
• Figure 4: Time-reversal asymmetry of atemporality. The states constructed are parameterized by $v_1,v_2$. By construction, these states are not forward Gaussian-atemporal. The region above the blue line represents physical space-time Gaussian states. The region below the red (yellow, green, purple) line represents the ones that have non-zero reverse Gaussian-atemporalites, when $\eta=0.3$ ($\eta=0.5, 0.7, 0.9$, respectively). The region between these lines thus includes physical space-time Gaussian states that are not forward Gaussian-atemporal, but are reverse Gaussian-atemporal.
• Figure 5: A map of states with respect to $v,r$. States in the yellow region are separable and not Gaussian-atemporal. States in the light pink region are entangled and not Gaussian-atemporal. The states in the rest (purple) are entangled and Gaussian-atemporal.

Theorems & Definitions (14)

• Definition 1: Space-time covariance matrix in standard form
• Lemma 0
• Definition 2: Forward Gaussian Pseudo-Channels
• Theorem 1
• proof
• Definition 3: Forward Atemporality Robustness
• Theorem 2
• Lemma 3
• proof
• Theorem 3