Unifying theory of quantum state estimation using past and future information

Quantum state estimation for continuously monitored dynamical systems involves assigning a quantum state to an individual system at some time, conditioned on the results of continuous observations. The quality of the estimation depends on how much observed information is used and on how optimality is defined for the estimate. In this work, we consider problems of quantum state estimation where some of the measurement records are not available, but where the available records come from both before (past) and after (future) the estimation time, enabling better estimates than is possible using the past information alone. Past-future information for quantum systems has been used in various ways in the literature, in particular, the quantum state smoothing, the most-likely path, and the two-state vector and related formalisms. To unify these seemingly unrelated approaches, we propose a framework for partially-observed quantum system with continuous monitoring, wherein the first two existing formalisms can be accommodated, with some generalization. The unifying framework is based on state estimation with expected cost minimization, where the cost can be defined either in the space of the unknown record or in the space of the unknown true state. Moreover, we connect all three existing approaches conceptually by defining five new cost functions, and thus new types of estimators, which bridge the gaps between them. We illustrate the applicability of our method by calculating all seven estimators we consider for the example of a driven two-level system dissipatively coupled to bosonic baths. Our theory also allows connections to classical state estimation, which create further conceptual links between our quantum state estimators.Comment: 48 pages, 9 figure

Nonlocality activation in a photonic quantum network

Bell nonlocality refers to correlations between two distant, entangled particles that challenge classical notions of local causality. Beyond its foundational significance, nonlocality is crucial for device-independent technologies like quantum key distribution and randomness generation. Nonlocality quickly deteriorates in the presence of noise, and restoring nonlocal correlations requires additional resources. These often come in the form of many instances of the input state and joint measurements, incurring a significant resource overhead. Here, we experimentally demonstrate that single copies of Bell-local states, incapable of violating any standard Bell inequality, can give rise to nonlocality after being embedded into a quantum network of multiple parties. We subject the initial entangled state to a quantum channel that broadcasts part of the state to two independent receivers and certify the nonlocality in the resulting network by violating a tailored Bell-like inequality. We obtain these results without making any assumptions about the prepared states, the quantum channel, or the validity of quantum theory. Our findings have fundamental implications for nonlocality and enable the practical use of nonlocal correlations in real-world applications, even in scenarios dominated by noise.Comment: Main text and Supplementary Information. Comments welcom