A general framework for complete positivity

Abstract

Complete positivity of quantum dynamics is often viewed as a litmus test for physicality, yet it is well known that correlated initial states need not give rise to completely positive evolutions. This observation spurred numerous investigations over the past two decades attempting to identify necessary and sufficient conditions for complete positivity. Here we describe a complete and consistent mathematical framework for the discussion and analysis of complete positivity for correlated initial states of open quantum systems. This formalism is built upon a few simple axioms and is sufficiently general to contain all prior methodologies going back to Pechakas, PRL (1994). The key observation is that initial system-bath states with the same reduced state on the system must evolve under all admissible unitary operators to system-bath states with the same reduced state on the system, in order to ensure that the induced dynamical maps on the system are well-defined. Once this consistency condition is imposed, related concepts like the assignment map and the dynamical maps are uniquely defined. In general, the dynamical maps may not be applied to arbitrary system states, but only to those in an appropriately defined physical domain. We show that the constrained nature of the problem gives rise to not one but three inequivalent types of complete positivity. Using this framework we elucidate the limitations of recent attempts to provide conditions for complete positivity using quantum discord and the quantum data-processing inequality. The problem remains open, and may require fresh perspectives and new mathematical tools. The formalism presented herein may be one step in that direction.Comment: 15 pages, 8 figure

Complete Positivity, Markovianity, and the Quantum Data-Processing Inequality, in the Presence of Initial System-Environment Correlations

“…As it turns out, quantum operations can always be modeled as interactions of the input system with an environment, initially factorized from (and independent of) the input system, and discarded after the interaction took place [1][2][3][4][5]. Such a model, however, is not universally valid, but relies on an initial factorization condition.The question then naturally arises [6][7][8][9][10][11][12][13][14][15][16]: what happens when the initial factorization condition does not hold, namely, when system and environment are, already before the interaction is turned on, correlated? While this question arguably originated from practical motivations (e.g., the difficulty to experimentally enforce the initial factorization assumption), it soon moved to a more fundamental level, in an attempt to challenge the very physical arguments often put forth to promote CP dynamics as the only "physically reasonable" reduced dynamics.…”

“…As one would expect, by allowing the input system and its environment to start in a correlated state, it is possible that the reduced dynamics of the system are not CP anymore. The possibility of exploring phenomena outside the CP framework attracted considerable interest [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], in particular in connection with the possibility of circumventing thermodynamic or information-theoretic tenet like, e.g., the second law of thermodynamics (by anomalous heat flow [17,18]) or the data-processing inequality (by anomalous increase of distinguishability [20], by entanglement revivals via local operations [9,[21][22][23], or by violating the no-cloning theorem [19]). In the language of the theory of open quantum systems, all such violations are interpreted as signatures of the fact that the underlying global evolution is non-divisible [20][21][22][23][24], i.e., it cannot be decomposed into a chain of CP maps across successive time intervals.…”

“…A rigorous framework for this problem has recently been provided in Ref. [16], but a complete characterization of which iniarXiv:1307.0363v3 [quant-ph] 18 Sep 2014 2 tial system-environment correlations lead to CP reduced dynamics is not available yet.Here, we propose a different way to tackle this problem starting from a simple initial idea: if, on one hand, non-CP reduced dynamics lead to the violation of data/energy-processing principles (as seen above), can complete positivity, on the other hand, be recovered by assuming that a suitably chosen data-processing inequality always holds? In other words, is the absence of anomalous backward flows of information/heat only necessary, or is it also sufficient for the complete positivity of the reduced dynamics?…”

Biexponential decay and ultralong coherence of a qubit

“…Indeed, there exists a one-by-one link between this inequality and CP [53]. Moreover, the standard notion of CP only makes sense for linear dynamics [49,52,90] and, as mentioned by Pechukas [50] as well as by Shaji and Sudarshan [54], it does not need to be a physical requirement in spite of its mathematical attractiveness.…”

Witness for initial correlations among environments

“…Shabani and Lidar showed that the above-mentioned condition is not only sufficient but also necessary for complete positivity of the corresponding map [9,10]. Recently, some examples were provided to show that the relation between complete positivity and quantum discord is not generalized to all cases [11][12][13].…”