The unavoidable information flow to environment in quantum measurements

Abstract

One of the basic lessons of quantum theory is that one cannot obtain information on an unknown quantum state without disturbing it. Hence, by performing a certain measurement, we limit the other possible measurements that can be effectively implemented on the original input state. It has been recently shown that one can implement sequentially any device, either channel or observable, which is compatible with the first measurement [8]. In this work we prove that this can be done, apart from some special cases, only when the succeeding device is implemented on a larger system than just the input system. This means that some part of the still available quantum information has been flown to the environment and cannot be gathered by accessing the input system only. We characterize the size of the post-measurement system by determining the class of measurements for the observable in question that allow the subsequent realization of any measurement process compatible with the said observable. We also study the class of measurements that allow the subsequent realization of any observable jointly measurable with the first one and show that these two classes coincide when the first observable is extreme. arXiv:1710.08681v1 [quant-ph]

Noise–Disturbance Relation and the Galois Connection of Quantum Measurements

“…, which is equivalent to B A by combining (2) and (5). For general Λ ∈ C, however, τ c (Λ) is not a principal ideal.…”

“…The qualitative noise-disturbance relation, presented in [2] and further developed in [3,4,5], characterizes the compatible channels for any given observable: the set of compatible channels is a principal ideal, generated by the so-called least disturbing channel of that observable. We would like to point out that the work that led to [2] started when Paul recommended two of the authors, not known to each other before, to meet for a scientific interaction.…”

Post-processing of quantum instruments

“…A special kind of post-processing, called relabeling, is one where all the elements of the stochastic post-processing matrix are either 0 or 1. Following [20], this can be formalized by the existence of a function f ∶ Ω A → Ω B such that ν xy = δ f (x),y , where δ x,x ′ is the Kronceker delta, so that…”

Entanglement protection via periodic environment resetting in continuous-time quantum-dynamical processes

“…As an example of a post-processing we consider the case of relabeling where all of the elements of a post-processing matrix are either 0 or 1. Formally, following [97], we say that an observable A ∈ O(Ω, S) is a refinement of an observable B ∈ O(Λ, S) if there exists a function f : Ω → Λ such that B y = x∈f −1 (y) A x for all y ∈ Λ. Relabeling thus consists of not only literal bijective relabeling of outcomes but also merging of different outcomes.…”