Unifying different notions of quantum incompatibility into a strict hierarchy of resource theories of communication
Francesco Buscemi, Kodai Kobayashi, Shintaro Minagawa, Paolo Perinotti, Alessandro Tosini
Abstract
While there is general consensus on the definition of incompatible POVMs, moving up to the level of instruments one finds a much less clear situation, with mathematically different and logically independent definitions of incompatibility. Here we close this gap by introducing the notion of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>p</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mi>b</mml:mi><mml:mi>i</mml:mi><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>t</mml:mi><mml:mi>y</mml:mi></mml:math>, which unifies different notions of POVMs, channels, and instruments incompatibility into one hierarchy of resource theories of communication between separated parties. The resource theories that we obtain are<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>p</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi></mml:math>, in the sense that they contain complete families of free operations and monotones providing necessary and sufficient conditions for the existence of a transformation. Furthermore, our framework is fully<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi><mml:mi>p</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mi>a</mml:mi><mml:mi>l</mml:mi></mml:math>, in the sense that free transformations are characterized explicitly, in terms of local operations aided by causally-constrained directed classical communication, and all monotones possess a game-theoretic interpretation making them experimentally measurable in principle. We are thus able to pinpoint exactly what each notion of incompatibility consists of, in terms of information-theoretic resources.