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Journal of Physics Conference Series

Pairwise coexistence of effects versus coexistence

Werner Stulpe

Published Oct 1, 2020• 0 citations
Publisher

Abstract

Abstract The concept of coexistence of quantum mechanical effects is reviewed. We distinguish between coexistence and pairwise coexistence and give an example showing the non-equivalence the two concepts. A theorem on some closures of sets of pairwise coexistent effects is proved. Some proofs of the well-known fact that any set of pairwise coexistent (mutually compatible) sharp effects is coexistent are considered. In particular, a proof is presented that is based on the statement that every countably complete Boolean lattice of sharp effects is closed in some topology.

DOI
10.1088/1742-6596/1638/1/012004

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Abstract The concept of coexistence of quantum mechanical effects is reviewed. We distinguish between coexistence and pairwise coexistence and give an example showing the non-equivalence the two concepts. A theorem on some closures of sets of pairwise coexistent effects is proved. Some proofs of the well-known fact that any set of pairwise coexistent (mutually compatible) sharp effects is coexistent are considered. In particular, a proof is presented that is based on the statement that every countably complete Boolean lattice of sharp effects is closed in some topology.
The question of quantifying the sharpness (or unsharpness) of a quantum mechanical effect is investigated. Apart from sharpness, another property, bias, is found to be relevant for the joint measurability or coexistence of two effects. Measures of bias will be defined and examples given. Dedication The impossibility of measuring jointly certain pairs of observables is an intriguing non-classical feature of quantum theory that Pekka Lahti identified as a candidate for a rigorous formulation of the principle of complementarity. While he was investigating this fundamental no-go statement in the early 1980s, he learned from Peter Mittelstaedt that one of his students was aiming to prove the positive possibility of approximate joint measurements of complementary quantities such as position and momentum. Pekka joined our group as an Alexander von Humboldt Fellow, and together we found that a reconciliation between complementarity and (approximate) joint measurability is possible on the basis of the generalized representation of observables as positive operator measures (POMs). Since then we have pursued together our aspirations of understanding quantum mechanics and understanding Nature. I have benefited much from Pekka’s intellectual rigor and have been privileged ever since to enjoy his warm humanity. It is a great pleasure to present this paper to Pekka as a token of thanks and friendship on the occasion of his 60th birthday, with all good wishes for many happy recurrences and productive years to come. 1
This work is a study on the symmetry groups of quantum mechanics accompanied by its applications on the coexistence of qubit effects. We surveyed among some of the mappings known as preservers on the set of quantum effects for their role in conserving specific structures that are important in quantum mechanics. Since the preserves in this work mainly target the quantum effects, we provided principals of its corresponding algebra known as effect algebra. Then we advanced, to study the relations between some of these preservers (here we mean automorphisms) by channels operate on qubit effects. The channels are in convex form and we investigate the role of their convex parameters in conserving or breaking the coexistence of two qubit effects. The preserver that we are interested in is a bijective map which preserves the order and coexistence of effects in both directions. Of course, the coexistence of qubit effects can also be studied in the language of coexistence and compatibility of observables. Then our task is to study the coexistence of observables when there are only two effects in their outcome sets. This task has been studied for observables with arbitrary number of effects in their outcome sets [48]. The bounds they have found for the channels convex parameters are independent of that number. However, when the case narrows for limited number outcomes no analytic solution is available, and the study should be carried numerically. Here, we considered observables each having only two nontrivial effects in their out come sets, and asked for the demanding conditions that break their compatibility partially or completely.
The set of effects is not a lattice with respect to its natural order. Projection operators do have the greatest lower bounds (and the least upper bounds) in that set, but there are also other (incomparable) effects which share this property. However, the coexistence, the commutativity, and the regularity of a pair of effects are not sufficient for the existence of their infima and suprema. The structure of the range of an observable (as a normalized POV measure) can vary from that of a commutative Boolean to a noncommutative non-Boolean subset of effects.
In a previous paper (https://dx.doi.org/10.2139/ssrn.3648127 ), I have demonstrated an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction if the case for “n = 3” is granted as proved only arithmetically (a fact a long time ago). Now the same case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1968): the eventual validity of Fermat’s equation would correspond to an admissible disjunctive division of a qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen –Specker theorem. The proof involves set theory, but only within the case “n=3” and implicitly, within any next levels of “n”. Thus the application of set theory and arithmetic can remain disjunctively divided as well: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided. However, the mathematical structure relevant to the proof cannot be restricted to Peano arithmetic but to a generalized structure called “Hilbert arithmetic” and discussed in detail in another previous paper (https://dx.doi.org/10.2139/ssrn.3656179 ). Furthermore, Fermat’s last theorem in virtue of that kind of proof can be considered as another expression of Gleason’s theorem in quantum mechanics (or vice versa): the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information: the availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.
Abstract The notion of coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables. This leads us to a reconsideration of the connection between joint measurability and noncommutativity of observables and of the statistical and individual aspects of quantum measurements.