Quantum non-Markovianity: Overview and recent developments

Abstract

In the current era of noisy intermediate-scale quantum (NISQ) devices, research on the theory of open system dynamics has a crucial role to play. In particular, understanding and quantifying memory effects in quantum systems is critical to gain a better handle on the effects of noise in quantum devices. The main focus of this review is to address the fundamental question of defining and characterizing such memory effects—broadly referred to as quantum non-Markovianity—utilizing various approaches. We first discuss the two-time-parameter maps approach to open system dynamics and review the various notions of quantum non-Markovianity that arise in this paradigm. We then discuss an alternate approach to quantum stochastic processes based on the quantum combs framework, which accounts for multi-time correlations. We discuss the interconnections and differences between these two paradigms and conclude with a discussion on the necessary and sufficient conditions for quantum non-Markovianity.

Effects of non-Markovianity in quantum compositions of channels with indefinite causal order

“…For example, Ref. [95] showed that the absence of backflow of information in terms of mutual information is equivalent to the "CP-ness" of Φ, but we note that this contradicts some later works [99],…”

“…In our work, we do not consider these operational definitions of non-Markovianity, and will stick to the conventional CP-divisibility definition, also known as the RHP definition, named after Rivas, Huelga, and Plenio [84] (we refer the reader to Ref. [99] for a review of these operational definitions of non-Markovianity).…”

“…For example, an operational approach can allow the formulation of a resource theory of channel compositions [71], which enables association of quantifiable resources to how a channel is composed. Furthermore, an operational approach allows a stricter definition and quantification of non-Markovianity using process tensors or pseudo-density matrix [99]. This would adds to works studying the properties of ICO that relies on the process matrix framework which is closely related to the process tensor or quantum comb frameworks, e.g., works that studies ICO processes that violate causal inequalities [41,42], or ICO processes that are classical in nature and does not utilize the controlled superposition [46][47][48].…”

Quantum non-Markovianity of a qubit in presence of state dependent bath

“…Markovian evolutions have divisible dynamical maps, allowing for the representation of evolution in a series of continuous steps. Non-Markovianity arises when the dynamical map is not divisible, indicating interruptions or non-continuous evolution [34]. An important measure is given by the Fisher information flow [35][36][37].…”