A general framework for complete positivity

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• 作者：Jason M. Dominy ; Alireza Shabani ; Daniel A. Lidar
• 关键词：Open quantum systems ; Reduced dynamics ; Complete positivity
• 刊名：Quantum Information Processing
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：15
• 期：1
• 页码：465-494
• 全文大小：1,404 KB
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• 作者单位：Jason M. Dominy (1) (4)
  Alireza Shabani (5) (6)
  Daniel A. Lidar (1) (2) (3) (4)
  
  1. Department of Chemistry, University of Southern California, Los Angeles, CA, 90089, USA
  4. Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, CA, 90089, USA
  5. Department of Chemistry, University of California, Berkeley, CA, 94720, USA
  6. Google Research, Venice, CA, 90291, USA
  2. Department of Electrical Engineering, University of Southern California, Los Angeles, CA, 90089, USA
  3. Department of Physics, University of Southern California, Los Angeles, CA, 90089, USA
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Physics
  Mathematics
  Engineering, general
  Computer Science, general
  Characterization and Evaluation Materials
  
• 出版者：Springer Netherlands
• ISSN：1573-1332

文摘

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 (Phys Rev Lett 73:1060–1062, 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 such as 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. In particular, we correct the claim made by two of us (Shabani and Lidar in Phys Rev Lett 102:100402–100404, 2009) that vanishing discord is necessary for complete positivity, and explain that it is valid only for a particular class of initial states. The problem remains open, and may require fresh perspectives and new mathematical tools. The formalism presented herein may be one step in that direction.