A rigorous theory of quantum state reduction, the state change of the measured system caused by a measurement conditional upon the outcome of measurement, is developed fully within quantum mechanics without leading to the vicious circle relative to the von Neumann chain. For the basis of the theory, the local measurement theorem provides the joint probability distribution for the outcomes of local successive measurements on a noninteracting entangled system without assuming the projection postulate, and the quantum Bayes principle enables us to determine operationally the quantum state from a given information on the outcome of measurement.
A rigorous theory of quantum state reduction, the state change of the measured system caused by a measurement conditional upon the outcome of measurement, is developed fully within quantum mechanics without leading to the vicious circle relative to the von Neumann chain. For the basis of the theory, the local measurement theorem provides the joint probability distribution for the outcomes of local successive measurements on a noninteracting entangled system without assuming the projection postulate, and the quantum Bayes principle enables us to determine operationally the quantum state from a given information on the outcome of measurement.
A rigorous theory of quantum state reduction, the state change of the measured system caused by a measurement conditional upon the outcome of measurement, is developed fully within quantum mechanics without leading to the vicious circle relative to the von Neumann chain. For the basis of the theory, the local measurement theorem provides the joint probability distribution for the outcomes of local successive measurements on a noninteracting entangled system without assuming the projection postulate, and the quantum Bayes principle enables us to determine operationally the quantum state from a given information on the outcome of measurement.
An operational approach to quantum state reduction, the state change of the measured system caused by a measurement of an observable conditional upon the outcome of measurement, is founded without assuming the projection postulate in any stages of the measuring process. Whereas the conventional formula assumes that the probe measurement satisfies the projection postulate, a new formula for determining the state reduction shows that the state reduction does not depend on how the probe observable is measured, or in particular does not depend on whether the probe measurement satisfies the projection postulate or not, contrary to the longstanding attempts in showing how the macroscopic nature of probe detection provokes state reduction. 1.
A state reduct,ion is the stat1e change caused by a lueasurement on a quan- tum system conditional upon the outcome.A rigorous theorv of the state reduction is developed with nlathematiCa1 formalism, I)hysiCa1 interpret.ation,and nιodels .A special em1JhasiS is on the pure state reduction which trans- forms a pure 1) riorStatCe to the pure posterior state for every outcome.Mathe- 1natiCa1 structure of general pure state reductions is discussed and it is 1) roVed that every pure state reduction is decomposed into just two types, called tlle von Neumann-Davies (ND)tyl)e and the Gordon-Louisell (GL) type; a state reduction $\psirightarrow\psi_{x}$ ' is of the ND type if the 1IlaP1in ) gψ ) ↦P(x∣ψ)1/2?l′x ' is linear, where P(x∣ψ′) is the probability density of the outcome x , and of the GL tyl ) e if ψx depends only on the outcome x (independent of the prior state )) .
A persistent focus on the concept of emergence as a core element of the scientific method allows a clean separation, insofar as this is possible, of the physical and philosophical aspects of the problem of outcomes in quantum mechanics. The philosophical part of the problem is to explain why a closed system has definite experimental outcomes. The physical part is to show mathematically that there exists a limit in which the contradiction between unitary Schroedinger dynamics and a reduction process leading to distinct outcomes becomes negligible according to an explicitly stated criterion, and to make this criterion as objective as possible. The physical problem is solved here by redefining the notion of a quantum state and finding a suitable measure for the change of state upon reduction. The appropriate definition of the quantum state is not as a ray or density operator in Hilbert space, but rather as an equivalence class consisting of all density operators in a given subspace, the members of which all describe the same experimental outcome. For systems containing only subsystems that are integrated with their environments, these equivalence classes can be represented mathematically by projection operators, and the resulting formalism is closely related to that used by von Neumann to study the increase of entropy predicted by the second law of thermodynamics. However, nearly isolated subsystems are reduced only indirectly, as a consequence of their interaction with integrated subsystems. The reduced states of isolated subsystems are the same conditional states used in the definition of quantum discord. The key concepts of decoherence theory can all be adapted to fit this definition of a quantum state, resulting in a unified theory capable of resolving, in principle, all aspects of the quantum measurement problem. The theory thus obtained is weakly objective but not strongly objective.
Information obtained by a quantum measurement process performed on a physical system and the entropy change of the measured physical system are considered in detail. It is shown that the condition for the amount of information obtained by the quantum measurement process to be represented by the Shannon mutual entropy is that the intrinsic observable of the measured physical system commutes with the operational observable defined by the quantum measurement process. When some measurement outcome is obtained, the decrease of the Shannon entropy of the measured system is compared with that of the von Neumann entropy. Furthermore, a condition is established under which the amount of information that can be established by the quantum measurement process becomes equal to the decrease of the Shannon entropy of the measured physical system.
Is the quantum state real? Does the reduction of the state of a quantum system by a measurement on another system, entangled with it and distant, constitute a real disturbance? As is well acknowledged, there is a lack of consensus among physicists over such ontological questions concerning quantum mechanics (QM). Here we argue that it is possible, surprisingly, to address such questions on the basis of only operational considerations. More generally, we show that the operational perspective itself provides an intrinsic interpretation of QM. The theory's syntax supplies its semantics, as it were. This interpretational framework, which we call ``signaling ontology'', builds on the simple insight that unconditional quantum state reduction, namely the state change of a measured quantum system due to a nonselective measurement, can be used for signaling locally, and that this signal provides a natural, operational criterion for the reality of the state reduction. This argument is formalized in the framework of generalized probability theories and extended to multipartite systems. Among other consequences of this approach, the two questions posed above at the beginning are testably answered in the affirmative. In particular, a steering inequality is proposed whose violation, together with a local signaling condition, provides an experimental test for the reality of the remote disturbance referenced in the second question. Accordingly, quantum correlations that meet this criterion are nonlocal in an operationally real sense. A further ramification of our work is an elucidation of the contrast between quantum no-signaling and relativistic signal locality.