Complementary Observables in Quantum Mechanics

Abstract

To the memory of Paul Busch, our friend and colleague One may view the world with the p-eye and one may view it with the q-eye but if one opens both eyes simultaneously then one gets crazy.Wolfgang Pauli in a letter to Werner Heisenberg, 19 October 1926. We hope to have demonstrated that one can safely open a pair of complementary 'eyes' simultaneously.He who does so may even 'see more' than with one eye only. The means of observation being part of the physical world, Nature Herself protects him from seeing too much and at the same time protects Herself from being questioned too closely: quantum reality, as it emerges under physical observation, is intrinsically unsharp. It can be forced to assume sharp contours -real properties -by performing repeatable measurements. But sometimes unsharp measurements will be both, less invasive and more informative. Paul Busch et co in the Epilogue of [1].Abstract. We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity -some of which new -in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position-momentum, position-energy, number-phase, as well as periodic observables relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Paul's work on operational formulation of quantum measurements and central to his philosophy of unsharp reality.

An attempt to understand relational quantum mechanics

“…Neither does the so-called unsharp (phase space) localization help in this respect, see, for instance, [26,27].…”

“…We call this hypothesis a local collapse, a thing which happens to S only with respect to S ′ . 27 Being an ad hoc solution, it remains unexplained which of the possible values of E gets actualized and, in particular, how does that happen, unless S ′ has a capacity to recognize that Z has now taken the value Z(X k ), say, and thus concludes, by strong correlation, the value of E is E(X k ). 28 Remark 8.…”

“…26 If E is sharp, the orthogonality requirement is redundant. 27 Dorato [16] calls this process a "primitive, mutual manifestation of dispositional properties". Dorato also writes [ibid.…”

Classical Systems in Quantum Mechanics

“…Many existing theories of quantum measurements which have appeared up to the present day analyze systematically possible results of various measurements (of corresponding observables) as well as their mutual relations like their mutual consistency or 'complementarity', see e.g. [83,62,63,174]. These theories, called by their authors "operational", are purely phenomenological, built on the formal structure of quantum kinematics and usually manifested no interest in the description of specific dynamics of the considered processes.…”

Jordan products of quantum channels and their compatibility

“…We first introduce the different settings for the compatibility of states, measurements, and channels that are considered in this paper. (The interested reader is referred to the reviews in references [1][2][3][4] for further discussions on these topics.) This overview is followed by a summary of our main results.…”