Neither Contextuality nor Nonlocality Admits Catalysts

Abstract

We show that the resource theory of contextuality does not admit catalysts. As a corollary, we observe that the same holds for non-locality. This adds a further example to the list of "anomalies of entanglement", showing that non-locality and entanglement behave differently as resources. We also show that catalysis remains impossible even if instead of classical randomness we allow some more powerful behaviors to be used freely in the free transformations of the resource theory. Introduction. Contextuality [1] and non-locality [2, 3] play a prominent role in a wide variety of applications of quantum mechanics, with non-locality being used for example in quantum key-distribution [4], certified randomness [5] and randomness expansion [6].Similarly, contextuality powers quantum computation in some computational models [7][8][9][10][11][12] and even increases expressive power in quantum machine learning [13]. Consequently, it is vital to understand how non-locality and contextuality behave as resources.In this Letter, we show that neither contextuality nor non-locality admit catalysts: that is, there are no correlations that can be used to enable an otherwise impossible conversion between correlations and still recovered afterwards. To state this slightly more precisely, let us write d, e, f . . . for various correlations (whether classical or not), d ⊗ e for having independent instances of d and e, and d e (read as "d simulates e") for the existence of a conversion d → e. Then our results state that, in suitably formalized resource theories of contextuality and non-locality, whenever d ⊗ e d ⊗ f , then e f already. This gives a strong indication that contextuality (and non-locality) are resources that get spent when you use them: there is no way of using a correlation d to achieve a task you could not do otherwise while keeping d intact. As entanglement theory famously allows for catalysts [14],this can be seen as yet another "anomaly of non-locality" [15] and thus further testament to the fact that non-locality and entanglement are different resources. [16] We do this by working in precisely defined resource theories of contextuality and nonlocality. These are not strictly speaking quantum resource theories [17], but resource theories in a more general sense [18, 19], as we allow resources such as PR-boxes [20] that are not quantum realizable. The kinds of conversions between correlations we have in mind capture the intuitive idea of using one system to simulate another one, and have been studied in earlier literature [21][22][23][24][25][26][27][28][29]. These roughly correspond to the local operations and shared randomness-paradigm ((LOSR)) or to wirings and prior-to-input classical communication