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.
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.
I present a derivation of quantum probabilities (Born rule) from a minimal set of spacetime composition axioms. By treating boundary states on spacetime slices as fundamental and imposing linearity under sequential composition, tensor factorization for spacelike-separated regions, and unitary invariance, I show that the quadratic probability measure arises uniquely. My derivation interprets quantum amplitudes as intrinsic features of spacetime: probabilities naturally emerge from how amplitudes on boundaries glue together, without introducing additional operators or postulates. This submission establishes priority for a specific, elegant argument linking the Born rule to the symmetries of spacelike factorization and foliation invariance, providing a first-principles foundation for quantum probabilities from spacetime considerations.
We show that, given an evolving quantum system and the quasiprobability distribution generated by the spatiotemporal generalization of the Born rule in pseudo density-matrices (PDMs), this distribution deviates from the sequential measurements probability distribution, given by the Lüders von-Neumann distribution, if and only if the non-signaling in time (NSIT) is violated; equivalently, if and only if the macroscopic realism (MR) is violated. Furthermore, we propose a definition of temporal entanglement according to the structure of the PDMs that is analogous to the definition of spatial entanglement in density matrices, showing that temporal entanglement is necessary for the violation of temporal Bell inequalities and the violation of MR. We employ our results to study the relationship between the negativity of the PDM, temporal entanglement, violation of temporal Bell inequalities, and MR.
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.
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.
While quantum correlations between two spacelike-separated systems are fully encoded by the bipartite density operator associated with the joint system, there does not exist an analogous operator representing general quantum correlations across space and time. This is in stark contrast to the case of classical random variables, which make no distinction between spacelike and timelike correlations. Despite this, we show that spatiotemporal correlations between light-touch observables (i.e., observables whose eigenvalues are all equal in magnitude) admit a unique operator representation for arbitrary timelike-separated quantum systems. A special case of our result reproduces generalized Pauli observables and pseudo-density matrices, which have, up until now, only been defined for multi-qubit systems. In the case of qutrit systems, we use our results to illustrate an intriguing connection between light-touch observables and symmetric, informationally complete, positive operator-valued measures (SIC-POVMs).