Structures in the Set of Quantum Observables: Coexistence, Unsharpness, Approximation Paul Busch∗ In the abstract of their seminal paper on quantum logic [1], Garrett Birkhoff and John von Neumann refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom- mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be illustrated for pairs of simple qubit observables. This contribution is based on collaborative work published in [2]–[5]. References [1] Birkhoff, G. and von Neumann, J. (1936), The Logic of Quantum Mechanics. Annals of Mathemat- ics 37, 823–843. [2] Busch, P., Heinonen, T. and Lahti, P. (2007), Heisenberg’s Uncertainty Principle. Physics Reports 452, 155–176. Also at arXiv:quant-ph/0609185v3. [3] Busch, P. and Pearson, D. (2007), Universal joint-measurement uncertainty relation for error bars. J. Math. Phys. 48, 082103/1–10. Also at arXiv:math-ph/0612074v2. [4] Busch, P. and Heinonen, T. (2008), Approximate joint measurements of qubit observables. Quantum Inf. Comp. (in press). Also at arXiv:0706.1415v2 [quant-ph]. [5] Busch, P. and Schmidt, H.-J. (2008), Coexistence of Qubit Effects. Available at arXiv:0802.416v2 [quant-ph]. ∗Department of Mathematics, University of York, UK; e-mail: [email protected] 1.