In standard quantum mechanics, the state vector of a pre-selected system provides the probability distribution for the outcomes of ideal measurements and plays a central role in the computation of the average values of physical observables. However, when the information about the pre-selected system is incomplete or less than maximal, the state vector formalism becomes inadequate. In such cases, the density matrix formalism is employed as a more comprehensive framework, capable of encoding all measurement statistics within a single mathematical construct. The density matrix, representing a statistical mixture of state vectors, contains the minimal set of parameters required to calculate the expectation values of observables. Algebraically, it is a Hermitian, positive semi-definite, and normalized operator acting on a Hilbert space. Despite its versatility, there are cases where even the density matrix formalism falls short, particularly in scenarios involving unconventional causal relationships. Examples include single systems defined by both past and future conditions [1], measurements too weak to collapse superpositions [2], quantum entanglement between temporally separated systems [3], and processes corresponding to superpositions of distinct pasts and futures [4]. Our research agenda in quantum-causal relativity seeks to address these limitations by identifying and characterizing alternative mathematical structures that can efficiently encode the measurement statistics of physical systems whose states defy representation within the conventional framework of quantum mechanics.
[1] See two-state vector formalism;
[2] See weak measurements and weak values;
[3] See quantum states over time and pseudo-density matrix/operator formalism;
[4] See indefinite causal orders and quantum SWITCH.
To reconcile quantum mechanics and general relativity without prioritizing one as fundamentally superior, it is essential to retain the distinct and radically non-classical features of both theories. This suggests that a potential theory of quantum gravity must include superpositions of causal structures. Contemporary experiments have demonstrated the simulation of indefinite causal structures in photonic setups, which have been shown to enhance the performance of emerging technologies like quantum computing. However, to harness this resource effectively, it first needs to be identified. Current indefinite causal order witnesses described in the literature can only detect such structures after the process is complete.
To address this challenge, we recently integrated three distinct operational frameworks [U.1]. The first framework is the quantum SWITCH formalism developed by Giulio Chiribella, which enables the simulation of indefinite causal orders. The second is the two-state vector formalism introduced by Yakir Aharonov and colleagues to describe pre- and post-selected systems. The third is the pseudo-density operator formalism proposed by Vlatko Vedral, which facilitates the description of composite states of temporally separated systems.
Combining these three formalisms, we introduced the notion of a single-time pseudo-state to encompass previous works and approaches. we found that weak measurements are sufficient to investigate the relation between the state at a given time and the uncertainty in the order of events before and after it. We also proposed experimental realizations of witnessing superpositions of causal orders by weak measurements. Now, we explore the theoretical foundation of this representation through quasi-probabilities, such as Wigner and Kirkwood-Dirac distributions.
