Abstract
The relation between noise and disturbance is investigated within the general framework of Galois connections. Within this framework, we introduce the notion of leak of information, mathematically defined as one of the two closure maps arising from the observable-channel compatibility relation. We provide a physical interpretation for it, and we give a comparison with the analogous closure maps associated with joint measurability and simulability for quantum observables.
Highlights
A fundamental fact about quantum measurements is the following: measurement that does not cause any disturbance cannot give any information on the measured system
The physical interpretation of the maps involved in the Galois connection is not anymore as direct as in the qualitative noise–disturbance relation
We say that τ σ and σ τ are the closure maps associated with the Galois connection (σ, τ )
Summary
A fundamental fact about quantum measurements is the following: measurement that does not cause any disturbance cannot give any information on the measured system. A quantum observable and a quantum channel are called compatible if they are parts of a single measurement device, otherwise they are incompatible. In this language, the noinformation-without-disturbance theorem states that the identity channel is compatible only with coin tossing observables. A natural generalization of the qualititative noise–disturbance relation is to consider the set of all compatible channels for a collection of observables, instead of a single observable. Forming the Galois connection gives immediately two closure maps, one on the set of observables and another one on the set of channels. The physical interpretation of the maps involved in the Galois connection is not anymore as direct as in the qualitative noise–disturbance relation.
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