Incompatibility robustness of quantum measurements: a unified framework

Abstract

In quantum mechanics performing a measurement is an invasive process which generally disturbs the system. Due to this phenomenon, there exist incompatible quantum measurements, i.e. measurements that cannot be simultaneously performed on a single copy of the system. It is then natural to ask what the most incompatible quantum measurements are. To answer this question, several measures have been proposed to quantify how incompatible a set of measurements is, however their properties are not well-understood. In this work, we develop a general framework that encompasses all the commonly used measures of incompatibility based on robustness to noise. Moreover, we propose several conditions that a measure of incompatibility should satisfy, and investigate whether the existing measures comply with them. We find that some of the widely used measures do not fulfil these basic requirements. We also show that when looking for the most incompatible pairs of measurements, we obtain different answers depending on the exact measure. For one of the measures, we analytically prove that projective measurements onto two mutually unbiased bases are among the most incompatible pairs in every dimension. However, for some of the remaining measures we find that some peculiar measurements turn out to be even more incompatible.performed simultaneously by performing the parent measurement. If such a parent measurement does not exist, we say that the measurements are incompatible (or not jointly measurable). We remark here that other notions of compatibility, such as commutativity, non-disturbance and coexistence, are also used in the literature [1,3]; let us for completeness briefly explain how they are related. Commutativity of a measurement pair implies nondisturbance, which in turn implies joint measurability, which then implies coexistence. Moreover, it is known that none of the converse implications hold in general, therefore these notions are strictly distinct [4]. In this work we focus solely on the notion of joint measurability, because the existence (or not) of a parent measurement has a clear operational meaning. Therefore, throughout the present paper we use the terms '(in) compatibility' and '(non-)joint measurability' interchangeably. It is important to notice that whenever two measurements are compatible, they cannot be used to produce quantum advantage in tasks like Bell nonlocality [5] or Einstein-Podolsky-Rosen steering [6,7]. Moreover, it was recently shown that joint measurability is equivalent to a specific notion of classicality, namely, preparation non-contextuality [8,9]. Hence, one may think of compatible measurements as 'classical' , and incompatible measurements as a resource for the above tasks. Therefore, it is of fundamental importance to characterise and understand the structure of incompatible measurements.What is particularly important is to go beyond the dichotomy of compatible and incompatible measurements, and quantify to what extent a pair of measurements is incompatible. A natural framework for this qua...

Entropic measurement uncertainty relations for all the infinite components of a spin vector

“…In writing the decompositions (32) we have decided to have the maximum possible value for the visibility h l q , , without imposing conditions on the class of allowed noises. As we remarked above, the last class of noises discussed in [32] is indeed the one of general POVMs. If the class of noises is restricted, the value of the visibility could diminish, as we can see in the example of = s 1 2, section 2.3.1.…”

“…A different approach [9,10,30,29,31,32] to the construction of compatible observables is to consider noisy versions of the target observables.…”

“…where p j is a classical probability, independent of the system state [9,30,31]. A review of some choices for the noise classes introduced in the literature is given in [32]; the typical choices are: (a) classical noises, (b) noises represented by compatible POVMs, (c) general POVMs.…”

“…Our aim in introducing the device information loss (46) and the minimum information loss (54), (55) was to have uncertainty measures, based on information theory, by which MURs could be expressed in a simple way, section 3.3; this construction produced also an incompatibility measure, the minimum information loss. The result above gives a link with the robustness measures [9,10,[29][30][31][32] which quantify the incompatibility by maximizing the visibility; in other terms, these measures are based on the ability of the target observables to maintain incompatibility against noise.…”

“…The visibilities above have been obtained by allowing for general noises, not only classical ones. Inside the noise robustness approach to incompatibility, the two visibilities h 1 2 r for spin 1/2 have already been obtained in [32]; they are in the class called incompatibility generalized robustness, which means that general POVMs are allowed for noises. By comparing with section 3.4, we can say that we have shown how to generalize this approach to the case of infinitely many observables, such as the spin vector.…”