On measures of distance between quantum observables and a new joint measurement uncertainty relation

          @misc{ pirsa_PIRSA:06100016,
            doi = {10.48660/06100016},
            url = {https://pirsa.org/06100016},
            author = {Busch, Paul},
            keywords = {Quantum Foundations},
            language = {en},
            title = {On measures of distance between quantum observables and a new joint measurement uncertainty relation},
            publisher = {Perimeter Institute},
            year = {2006},
            month = {oct},
            note = {PIRSA:06100016 see, \url{https://pirsa.org}}
          }
          

Abstract

I will discuss various different ways of quantifying the differences between two quantum observables (POVMs). Each of these approaches gives rise to a notion of approximately measuring one observable by means of measuring some other observable. This will be illustrated in the case of position and momentum by studying the question which POVMs on phase space can reasonably be said to represent a joint approximate determination of these observables. A new, universally valid trade-off relation for the associated inaccuracies will be rigorously formulated. I will sketch the proof which is an adaptation of some interesting techniques and properties of covariant phase space observables used recently by R Werner in a related project. Recommended reading (optional): quant-ph/0405184 (R Werner), quant-ph/0609185 (PB et al), and also for further background information quant-ph/0309091 (M Hall), quant-ph/0310070 (M Ozawa), quant-ph/9803051 (DM Appleby).