Abstract
One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the quantum marginal problem. The generality of the connection is harnessed by solving the marginal problem for Gaussian and Bell diagonal states, as well as for pure states under depolarizing noise. Furthermore, we derive entropic criteria for channel compatibility, and develop a converging hierarchy of semi-definite programs for quantifying the strength of quantum memories.
Highlights
Incompatibility of measurements is one of the most intriguing features of quantum theory [1]
We show that a recently introduced concept of channel incompatibility, which includes measurement incompatibility and no-broadcasting as special cases, is one-to-many connected to an instance of the socalled quantum marginal problem
We focus on an instance of the quantum marginal problem, that is the most relevant from the viewpoint of quantum correlations
Summary
Incompatibility of measurements is one of the most intriguing features of quantum theory [1]. Being intimately related to the quantum advantage in various tasks, it becomes desirable to search for the fundamental properties of quantum theory that allow for the existence of incompatible measurements. Motivated by this question, we attack the problem with a general approach towards the fact that not all quantum resources can be broadcasted. It is related to the monogamy of entanglement, which refers to the fact that entanglement cannot be shared arbitrarily [24] In this manuscript, we focus on an instance of the quantum marginal problem, that is the most relevant from the viewpoint of quantum correlations. We derive entropic criteria for channel incompatibility, characterize the antidegradability of qubit channels, and solve the marginal problem for the general Gaussian case, for pairs of Bell-diagonal states, and for pairs of general pure states under depolarizing noise
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