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Probing Asymmetry in Spatial-temporal Correlations in Quantum Causal Inference

Meng Yu, Zhenghao Liu, Zhikuan Zhao, Peng Yin, Yi‐Tao Wang, Wei Liu, Zhipeng Li, Yuan-Ze Yang, Zhaoan Wang, Jin‐Shi Xu, Shang Yu, Jian‐Shun Tang, Chuan‐Feng Li, Guang‐Can Guo

Published Mar 15, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.21203/rs.3.rs-311195/v1

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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.
``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.
The traditional formalism of nonrelativistic quantum theory allows the state of a quantum system to extend across space, but only restricts it to a single instant in time, leading to distinction between theoretical treatments of spatial and temporal quantum correlations. Here we unify the geometrical description of two-point quantum correlations in space-time. Our study presents the geometry of correlations between two sequential Pauli measurements on a single qubit undergoing an arbitrary quantum channel evolution together with two-qubit spatial correlations under a common framework. We establish a symmetric structure between quantum correlations in space and time. This symmetry is broken in the presence of nonunital channels, which further reveals a set of temporal correlations that are indistinguishable from correlations found in bipartite entangled states.
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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.
In ordinary, non-relativistic, quantum physics, time enters only as a parameter and not as an observable: a state of a physical system is specified at a given time and then evolved according to the prescribed dynamics. While the state can, and usually does, extend across all space, it is only defined at one instant of time. Here we ask what would happen if we defined the notion of the quantum density matrix for multiple spatial and temporal measurements. We introduce the concept of a pseudo-density matrix (PDM) which treats space and time indiscriminately. This matrix in general fails to be positive for measurement events which do not occur simultaneously, motivating us to define a measure of causality that discriminates between spatial and temporal correlations. Important properties of this measure, such as monotonicity under local operations, are proved. Two qubit NMR experiments are presented that illustrate how a temporal pseudo-density matrix approaches a genuinely allowed density matrix as the amount of decoherence is increased between two consecutive measurements.
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.