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arXiv:2305.15649 · Advanced Quantum Technologies

The Spatiotemporal Doubled‐Density Operator: A Unified Framework for Analyzing Spatial and Temporal Quantum Processes

Zhian Jia, Dagomir Kaszlikowski

Published Aug 4, 2024• 7 citations
arXivHTMLPDF

Abstract

Abstract The measurement statistics for spatial and temporal quantum processes are produced through distinct mechanisms. Measurements that are space‐like separated exhibit non‐signaling behavior. However, time‐like separated measurements can only result in one‐way non‐signaling, as the past is independent of the future, but the opposite is not true. This work presents the doubled‐density operator as a comprehensive framework for studying quantum processes in space‐time. It effectively captures all the physical information of the process, with the measurement and Born rule showing uniformity for both spatial and temporal cases. It is demonstrated that the equal‐time density operator can be derived by performing a partial trace operation on the doubled‐density operator. Furthermore, the temporality of the quantum process can be detected by conducting a partial trace operation on either the left or right half of the doubled‐density operator.

DOI
10.1002/qute.202400102

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Abstract The measurement statistics for spatial and temporal quantum processes are produced through distinct mechanisms. Measurements that are space‐like separated exhibit non‐signaling behavior. However, time‐like separated measurements can only result in one‐way non‐signaling, as the past is independent of the future, but the opposite is not true. This work presents the doubled‐density operator as a comprehensive framework for studying quantum processes in space‐time. It effectively captures all the physical information of the process, with the measurement and Born rule showing uniformity for both spatial and temporal cases. It is demonstrated that the equal‐time density operator can be derived by performing a partial trace operation on the doubled‐density operator. Furthermore, the temporality of the quantum process can be detected by conducting a partial trace operation on either the left or right half of the doubled‐density operator.
Contrary to general relativity, quantum theory treats space and time in fundamentally different ways. In particular, while joint probabilities associated with spacelike separated measurements are defined in terms of the Born rule, joint probabilities associated with measurements performed in sequence are defined in terms of the state-update rule. In this work, we show that one obtains a more unified perspective of space and time in quantum theory by embracing a quasiprobabilistic description of sequential measurements. More precisely, we show that there exists a unique \emph{pseudo}-density operator encoding canonical quasiprobabilities associated with sequential measurements in precisely the same manner that a density operator encodes joint probabilities associated with spacelike separated measurements, thus providing a natural extension of the Born rule into the temporal domain. As an application, we show how such a spatiotemporal Born rule combined in conjunction with a quantum Bayes' rule yields an operational notion of time-reversal symmetry for sequential measurements on an \emph{open} quantum system.
Temporal quantum states generalize the multipartite density operator formalism to the time domain, enabling a unified treatment of quantum systems with both timelike and spacelike correlations. Despite a growing body of temporal state formalisms, their precise operational relationships and conceptual distinctions remain unclear. In this work, we resolve this issue by extending the Kirkwood-Dirac (KD) quasiprobability distribution to arbitrary multi-time quantum processes and, more broadly, to general spatiotemporal settings. We define left, right, and doubled temporal KD quasiprobabilities, together with their real components, which we identify as temporal Margenau-Hill (MH) quasiprobabilities. All of these quantities are experimentally accessible through interferometric measurement schemes. By characterizing their nonclassical features, we show that the generalized KD framework provides a unified operational foundation for a wide class of temporal state approaches and can be directly implemented via temporal or spatiotemporal Bloch tomography.
Quantum correlations often defy an explanation in terms of fundamental notions of classical physics, such as causality, locality, and realism. While the mathematical theory underpinning quantum correlations between spacelike separated systems has been well-established since the 1930s, the mathematical theory for correlations between non-spacelike separated systems is much less developed. In this work, we develop the theory of what we refer to as "local-density operators", which we view as joint states for possibly non-spacelike separated quantum systems. Local-density operators are unit trace operators whose marginals are genuine density operators, which we show not only subsumes the notion of density operator, but also several extensions of the notion of density operator into the spatiotemporal domain, such as pseudo-density operators and quantum states over time. More importantly, we prove a result which establishes a one-to-one correspondence between local-density operators and what we refer to as "Dirac measures", which are complex-valued measures on the space of separable projectors associated with two quantum systems. In the case that one of the systems is the trivial quantum system with a one-dimensional Hilbert space, our result recovers the fundamental result known as Gleason's Theorem, which implies that the Born rule from quantum theory is the only way in which one may assign probabilities to the outcomes of measurements performed on quantum systems in a non-contextual manner. As such, our results establish a direct generalization of Gleason's Theorem to measurements performed on possibly non-spacelike separated systems, thus extending the mathematical theory of quantum correlations across space to quantum correlations across space and time.
Measurements on a single quantum system at different times reveal rich nonclassical correlations similar to those observed in spatially separated multipartite systems. Here we introduce a theory framework that unifies the description of temporal, spatial, and spatiotemporal resources for quantum correlations. We identify and experimentally demonstrate simple cases where an exact mapping between the domains is possible. We then identify correlation resources in arbitrary situations, where not all spatial quantum states correspond to a process and not all temporal measurements have a spatial analog. These results provide a starting point for the systematic exploration of multipoint temporal correlations as a powerful resource for quantum information processing.
Spatial and temporal correlations in quantum noise challenge the local Markovian models commonly used in quantum information processing. We develop a unified operational framework for quantifying temporal, spatial, and total spatiotemporal correlations in general quantum processes. Using process tensors, we define these quantities through optimal distinguishability from Markovian, spatially local, and fully uncorrelated processes. Optimization over admissible probing combs ensures monotonicity under every superprocess that preserves the corresponding free set, while Choi-state functionals and restricted probes provide accessible lower bounds. We establish hierarchy and interpolation relations among the correlation measures and derive dimension-dependent upper bounds on temporal correlations transmitted by quantum memories, with tighter bounds for classical memory, together with universal ceilings imposed by the system dimension. These bounds turn certified lower estimates into witnesses of minimum memory dimension, nonclassicality under a memory-dimension constraint, and inconsistencies in the assumed process model. We illustrate the results with effective superconducting-qubit models featuring ZZZZZZ and XYXYXY interactions, exhibiting saturation of classical-memory, quantum-memory, and system-dimension ceilings. These results pave the way toward practical approaches to characterizing and addressing spatiotemporally correlated errors in quantum devices.
The notion of a spatiotemporal quantum state extends the conventional concept of a spatial quantum state to the spatiotemporal domain. Such states are represented by unit-trace operators that encode correlations among quantum events distributed across space and time. In this work, we use the spatiotemporal Kirkwood-Dirac phase space to provide a unified characterization of spatiotemporal quantum states. Spatiotemporal states obtained from Kirkwood-Dirac distributions are generally non-Hermitian and nonnormal; whereas those constructed from Margenau-Hill distributions are Hermitian. We provide a unification via (quasi)probabilistic mixture of Kirkwwod-Dirac spatiotemporal states which encompasses almost all existing formulations of spatiotemporal quantum states. We derive recursive expressions for spatiotemporal states and elucidate the relation between Kirkwood-Dirac nonclassicality and the temporality of spatiotemporal states. We further extend the construction to many-fold correlation functions and establish its connection with out-of-time-ordered correlators (OTOCs). We also develop a more general unifying framework based on (quasi)probabilistic mixture of sss-parametrized spatiotemporal states and establish their Petz time reversal and application in studying Kubo-Martin-Schwinger (KMS) condition in two-time setting. Finally, we apply this framework to characterize quantum entanglement in spacetime and analyze spatiotemporal entanglement using several complementary entropy measures.