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.

15 pages, 8 figures

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• Initial Correlations in Open Quantum Systems: Constructing Linear Dynamical Maps and Master Equations
• Structure of correlated initial states that guarantee completely positive reduced dynamics
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• Completely Positive Maps for Reduced States of Indistinguishable Particles
• Effects of symmetry breaking of the structurally-disordered Hamiltonian ensembles on the anisotropic decoherence of qubits
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• Protocol for nonlinear state discrimination in rotating condensate
• Reduced quantum dynamics with initial system-environment correlations characterized by pure Markov states
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• Requiring linearity leads to complete positivity
• Complete positivity on the subsystems level
• Initial correlations in open quantum systems are always detectable
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• Adapted projection operator technique for the treatment of initial correlations
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