A theory of quantum measurement was introduced some time ago that was based on the notion of the so-called separation status. This separation status had a spatial, local character, so that the theory worked only in special cases. Nevertheless, it enabled a description of state reduction process that was specific in where, when and under which objective conditions the process occurs and that preserved the unitary transformation symmetry of quantum mechanics. Now, in an accompanying paper (arxiv:1411.5524), a completely general mathematical definition of the status is given and analyzed. The present paper reformulates the theory of state reduction accordingly. A general mathematical form of the process is postulated and illustrated by examples of Stern-Gerlach experiment and of screening.
The reduction of the state vector is rightly held as one of the most puzzling aspects of quantum theory. Before adding a few more pages to the inflatory literature on the subject, I wish to voice a somewhat sobering remark. Namely, there is a striking contrast between the amount of theoretical discussions about the nature and significance of this notion, and its practical import. Indeed, the so-called “axiom” of reduction of the state vector has a rather limited use in the effective working of quantum theory. There are very few situations where we really need to assert that, after a measurement, the state vector has been reduced to the eigenvector of the physical property corresponding to the eigenvalue resulting from the measurement (“observed”). Most of the time, in effect, the system under investigation is forgotten or discarded after the measurement - or even destroyed by it (photons). We rarely make successive observations of the same system, so that we do not usually need to know what the state of a system is after a measurement. One might argue that the reduction of the state vector is operationally used mainly to describe the preparation of a system before acting upon it, that is, to ensure that we are indeed dealing with spin up electrons or velocity selected protons such as we wish to experiment about. Anyway, the statement of reduction is to be considered more as a kind of consistency condition to be required at the beginning or end of a theoretical analysis than as an effective tool of this analysis - such as, for instance, the statement of unitary evolution, which leads to the concrete construction of the evolution operator, hence the S-matrix, cross-sections, etc. In other words, the practical status of the state vector reduction, even if it is considered as an axiom of quantum theory, is different enough from that of the constructive axioms of the theory.
The question as to whether quantum mechanics can be applied to a macroscopic system, which has recently been raised through discussions of the problem of how to interpret the so-called “reduction of a wave packet”, is discussed. It is shown by a detailed analysis of the destruction of interference which brings about the reduction, that a supplementary assumption on the preciseness of measurements must at least be introduced in order to interpret the reduction process as an objective process, and it does not, however, necessarily imply inapplicability of quantum mechanics to a macroscopic system.
The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs in this regime: an explicit 45-copy rank-two input state
A, published as a 1,472-byte witness, with certified
Ic>0 at the exact rational
p=16239/250000=0.064956 (per-Pauli convention; total error rate
3p=0.194868) -- beyond both the best value printed in the Agarwal et al. paper (
0.064657) and the strongest state in the same authors' public repository (
≈0.064911), both numerical. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in
p on
[0,1/4], so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of
A and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than
1/25000 -- to our knowledge the first proven ordering of the positivity boundaries of two explicit competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.
The paper reviews and discusses four ideas scattered in previous papers of the author. First, objective properties of quantum systems are not associated with observables but are defined by preparations. Second, measurable results of classical theories are not sharp. All such results can be obtained as high-entropy limits of quantum mechanics. Third, a careful study of exchange symmetry requires a modification of the theory of measurement. Fourth, screens and detectors are closely associated with state reduction. A new, rather improved understanding of quantum mechanics is achieved without any deep changes of the theory itself. In the framework of a model approach to the philosophy of science, the RCU interpretation is developed step by step by adding specific supplementary hypotheses to the Minimum Interpretation of quantum mechanics.