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arXiv:2306.09336 · Physical Review Letters

Causal Classification of Spatiotemporal Quantum Correlations

Min-Jeong Song, Varun Narasimhachar, Bartosz Regula, Thomas J. Elliott, Mile Gu

Published Sep 11, 2024• 14 citations
arXivHTMLPDF

Abstract

From correlations in measurement outcomes alone, can two otherwise isolated parties establish whether such correlations are atemporal? That is, can they rule out that they have been given the same system at two different times? Classical statistics says no, yet quantum theory disagrees. Here, we introduce the necessary and sufficient conditions by which such quantum correlations can be identified as atemporal. We demonstrate the asymmetry of atemporality under time reversal and reveal it to be a measure of spatial quantum correlation distinct from entanglement. Our results indicate that certain quantum correlations possess an intrinsic arrow of time and enable classification of general quantum correlations across space-time based on their (in)compatibility with various underlying causal structures.

DOI
10.1103/physrevlett.133.110202

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From correlations in measurement outcomes alone, can two otherwise isolated parties establish whether such correlations are atemporal? That is, can they rule out that they have been given the same system at two different times? Classical statistics says no, yet quantum theory disagrees. Here, we introduce the necessary and sufficient conditions by which such quantum correlations can be identified as atemporal. We demonstrate the asymmetry of atemporality under time reversal and reveal it to be a measure of spatial quantum correlation distinct from entanglement. Our results indicate that certain quantum correlations possess an intrinsic arrow of time and enable classification of general quantum correlations across space-time based on their (in)compatibility with various underlying causal structures.
``Why'' questions are ubiquitous in our everyday lives. This innate curiosity has led to monumental scientific methodologies to find the ``because'' -- the key to unlocking ``why''. In search of the true causes behind complex phenomena, we typically employ statistical inference along with appropriate interventions, observing how systems react to various types of interventions. For such sophisticated causal inference, it is crucial to distinguish between the direct causal relations and the indirect causal relations, mediated by a common cause. These causal relationships form two primary causal structures: (i) a temporal structure, in which systems are in the relation of a direct cause, and (ii) a spatial structure, in which systems are in the relation of a common cause. In order to infer causal structures, what types of interventions are necessary? Recent research has revealed quantum advantages in this question; when the systems in question exhibit non-classical characteristics, effective causal inference is possible even without performing interventions actively, in a way that would be impossible classically. Motivated by these pioneering studies, this thesis aims to characterize quantum correlations across space-time in the absence of active interventions by causal classification, which is based on their compatible causal structures. The causal classification of quantum correlations involves examining the incompatibility of observed correlations with spatial and temporal structures. While certifying incompatibility with spatial structures is known to be relatively straightforward, establishing an effective approach for certifying incompatibility with temporal structures has remained challenging. Here, we refer to as atemporality this incompatibility with temporal structures, and introduce an efficient means to certify atemporality, contributing to a complete set of tools for inferring underlying causal structures. This also reveals various properties of atemporality: (1) asymmetry under time reversal, and (2) entanglement does not necessarily imply atemporality, although strong entanglement does imply atemporality. Atemporality thus opens a new aspect of quantum correlations across space-time, distinct from other known spatial quantum correlations such as entanglement. This insight has the potential to spur foundational interest, akin to that seen in the study of spatial quantum correlations.
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.
Abstract Quantum correlations in space-time encapsulate the most defining aspects of quantum physics. The dual of the spatial and temporal perspectives are bind with a one-to-one correspondence between bipartite quantum states and quantum channels. Consequently, causal relations between quantum events can sometimes be inferred solely from correlation statistics, apparently contradicting the classical \textit{credo}, `correlation does not imply causation'[1-6]. However, since the spatial-temporal duality does not imply a full symmetry of measurement statistics between the two domains[7], the extent to which correlation alone identifies quantum causality ponders inquiry vital for both fundamental and practical interests. Here, demonstrating a unified geometrical representation of spatial-temporal quantum correlation, we show that certain non-unital channels create temporal correlation without spatial analogue and break the spatial-temporal symmetry. By implementing such channels in a photonic architecture, we observe this asymmetry and classify quantum correlations using a distance criterion, thus bringing empirical insight into causal inference in quantum mechanics.
The statistics of local measurements of joint quantum systems can sometimes be used to distinguish the spatiotemporal structure in which they were measured. We first prove that every bipartite separable density matrix is temporally compatible with direct causal influence for arbitrary finite-dimensional quantum systems and measurements of a tomographically complete class of observables, which includes all Pauli observables in the case of multi-qubit systems. Equivalently, if a bipartite density matrix is not temporally compatible with direct causal influence, then it must be entangled. We also provide an operational meaning for the two temporal evolutions consistent with such correlations in terms of generalized dephasing channels and pretty good measurements. The two temporal evolutions are Bayesian inverses of each other, which is different from them being Petz recovery maps of each other. Finally, we prove necessary and sufficient conditions for an arbitrary bipartite quantum state to be temporally compatible, thereby providing a temporal analogue of the positive partial transpose criterion valid for quantum systems of any dimension.
We give a causal inference scheme using quantum observations alone for a case with both temporal and spatial correlations: a bipartite quantum system with measurements at two times. The protocol determines compatibility with five causal structures distinguished by the direction of causal influence and whether there are initial correlations. We derive and exploit a closed-form expression for the spacetime pseudo-density matrix (PDM) for many times and qubits. This PDM can be determined by light-touch coarse-grained measurements alone. We prove that if there is no signalling between two subsystems, the reduced state of the PDM cannot have negativity, regardless of initial spatial correlations. In addition, the protocol exploits the time asymmetry of the PDM to determine the temporal order. The protocol succeeds for a state with coherence undergoing a fully decohering channel. Thus coherence in the channel is not necessary for the quantum advantage of causal inference from observations alone.
We consider how to tell the time-ordering associated with measurement data from quantum experiments at two times and any number of qubits. We define an arrow of time inference problem. We consider conditions on the initial and final states that are symmetric or asymmetric under time reversal. We represent the spatiotemporal measurement data via the pseudodensity matrix space-time state. There is a forward process which is completely positive and trace-preserving (CPTP) and a reverse process which is obtained via an alternative recovery map based on inverting unitary dilations. For asymmetric conditions, the protocol determines whether the data are consistent with the unitary dilation recovery map or the CPTP map. For symmetric conditions, the recovery map yields a valid CPTP map and the experiment may take place in either direction. We also discuss adapting the approach to the Leifer-Spekkens or Process matrix space-time states.