Jordan products of quantum channels and their compatibility

Abstract

Given two quantum channels, we examine the task of determining whether they are compatible-meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). We show several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-andprepare channels (i.e., entanglement-breaking channels) do not necessarily have a measureand-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolaring channels are compatible.

Quantum Channel Marginal Problem

“…We therefore introduce the natural dynamical generalization of the SMP; namely, the channel marginal problems (CMPs) and provide necessary and sufficient conditions to solve it that can be addressed using semi-definite programming. There exist a few previous works that have also considered the CMP [10][11][12][13], but under a different approach. We compare our work with previous studies below.…”

Relating compatibility and divisibility of quantum channels

“…Example 1: The most paradigmatic example of incompatibility, that will be used later on, is related to the no-broadcasting theorem [12,14,30]: consider ψ A|A = id A = φ A|A , where (id) A : D(H A ) → D(H A ) is the identity channel. If there existed a compatibilizer for this case, there would exist a linear map θ AA|A such that:…”

“…Compatibility constitutes an active research topic, and this short sub-section does not represent an exhaustive review of it. For recent developments on compatibility we suggest [12] and [14].…”

“…In this work we focus on the generalization of this notion to quantum channels. Intuitively, compatibility between two quantum channels can be seen as on-demand simulability of those channels via a third, larger CPTP map [11][12][13][14][15][16]. In other words, this third channel contains, at all times, the information about the two original maps.…”

“…Although we do not rule out the natural parallel with joint measurability [10,17,18], we do not define the compatibility of two compatible channels as the fact that they can be realized simultaneously. Rather than saying that two compatible channels can be jointly realizable, we define compatible quantum channels as those channels that can be recovered from a third quantum map via marginalization (figure 1) [14]. It is within this perspective that we associate compatibility and simulability.…”