Initial system–environment correlations
For a specified correlated preparation family and class of joint unitaries, characterize the induced reduced maps on their compatibility domain and when they admit a CP extension, distinguishing domain positivity, extension positivity and intervention-based de
Problem statement
For a specified correlated preparation family and class of joint unitaries, characterize the induced reduced maps on their compatibility domain and when they admit a CP extension, distinguishing domain positivity, extension positivity and intervention-based descriptions.
Lean 4
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Background
A fixed product-state environment suffices for a CPTP reduced map on all system inputs; it is not necessary for every correlated preparation family. With initial correlations, different joint states may have the same system marginal, so even a single-valued reduced map requires an assignment or consistency condition. Positivity on the physical compatibility domain is different from a CP extension to the full operator space.
What is known: Pechukas's full-domain linear, positive and consistent assignment assumptions force a fixed-product assignment, not a trivial interaction Hamiltonian. Buscemi gives an information-theoretic CP characterization under specified preparation assumptions. Dominy–Shabani–Lidar distinguish inequivalent notions of CP on restricted domains and explain why discord or information-backflow criteria do not answer every assignment problem. The 2009 Shabani–Lidar claim must be read with its 2016 erratum, not cited as an unrestricted necessary-and-sufficient zero-discord criterion. Process tensors already treat correlations operationally via interventions.
Remaining tasks include checkable CP-extension criteria for physically structured preparation families and robust inference of the admissible domain from limited preparation data. Fix the assignment assumptions and allowed joint dynamics before formulating an iff statement; do not relabel the characterized special cases as open. Related: O5, O3.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Pechukas, 'Reduced dynamics need not be completely positive', PRL 73, 1060 (1994) link
- [2]Shabani & Lidar, 'Vanishing quantum discord is necessary and sufficient for completely positive maps', PRL 102, 100402 (2009; see 2016 erratum) link
- [3]Shabani & Lidar, Erratum, PRL 116, 049901 (2016) link
- [4]Buscemi, 'Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations', PRL 113, 140502 (2014) link
- [5]Dominy, Shabani & Lidar, 'A general framework for complete positivity', Quantum Information Processing 15 (2016) link
Recorded progress
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