Data tables, dimension witnesses, and QKD

          @misc{ pirsa_PIRSA:11050045,
            doi = {10.48660/11050045},
            url = {https://pirsa.org/11050045},
            author = {Brunner, Nicolas},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Data tables, dimension witnesses, and QKD},
            publisher = {Perimeter Institute},
            year = {2011},
            month = {may},
            note = {PIRSA:11050045 see, \url{https://pirsa.org}}
          }
          

Abstract

We address the problem of testing the dimensionality of classical and quantum systems in a ‘black-box’ scenario. Imagine two uncharacterized devices. The first one allows an experimentalist to prepare a physical system in various ways. The second one allows the experimentalist to perform some measurement on the system. After collecting enough statistics, the experimentalist obtains a ‘data table’, featuring the probability distribution of the measurement outcomes for each choice of preparation (of the system) and of measurement. Here, we develop a general formalism to assess the minimal dimensionality of classical and quantum systems necessary to reproduce a given data table. To illustrate these ideas, we provide simple examples of classical and quantum ‘dimension witnesses’. In general quantum systems are more economical than classical ones in terms of dimensionality, in the sense that there exist data tables obtainable from quantum systems of dimension d which can only be generated from classical systems of dimension strictly greater than d. By drawing connections to communication complexity one can find data tables for which this classical/quantum separation is dramatic. Finally, these ideas can also be used to demonstrate security of one-way QKD in a semi-device-independent scenario, in which devices are uncharacterized, but only assumed to produce quantum systems of a given dimension.