Gleason-Busch theorem for sequential measurements

Two-time state formalism for quantum eavesdropping

“…We prefer the prepare-and-measure scheme for reasons of conceptual simplicity (e.g. it does not require ancillary qubits to be invoked) as well as noting that it allows us to make a link back to the quantum reconstructions work [3] which inspired it, detailing the principles upon which security relies.…”

“…This approach contrasts with quantum mechanics as it is often presented, as a predictive theory in which a state is modified by a sequence of measurements. In a recent paper [3] we developed a framework, closely linked to that of two-time states, more suitable to the task at hand. In that work, we consider physical processes in which a quantum state is prepared and subsequently measured twice.…”

“…In a recent article [3] inspired by quantum reconstructions [16,17], we were interested in the minimal set of physical principles which make Eq. (1) unique.…”

Conditional probability and interferences in generalized measurements with or without definite causal order

“…In the framework of the general measurement formalism of positive-operator-valued measures (POVM, also called probability operator measures), in 2003, Busch [4] extended Gleason's theorem for any dimension and for imperfect measurements described by positive operators, effects E i , instead of projectors. Recently (2017), in the same context of POVM, Flatt, Barnett and Croke applied the Gleason-Busch theorem to subsequent measurements [5]. Considering the operators E i and F j associated with the measurements i and j, with i before j, Flatt and coworkers proved that P r(i, j) takes the general form…”

Conditional probabilities of measurements, quantum time and the Wigner's friend case

“…In both cases, time plays a special role in the description of a system's evolution. Moreover, the formal description of a system before and after a measurement (via a short interaction) requires considering two distinct Hilbert spaces H in ⊗ H out [13,[16][17][18]. This complexification and the central role of the time coordinate can be eliminated at once by including the measurement of time itself in a more general description of the studied interaction together with a clock system.…”