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arXiv:2007.15893 · Journal of Physics A Mathematical and Theoretical

Nullspaces of entanglement breaking channels and applications

David W. Kribs, Jeremy Levick, Katrina Olfert, Rajesh Pereira, Mizanur Rahaman

Published Jan 20, 2021• 6 citations
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Abstract

Abstract Quantum entanglement breaking channels are a fundamental class of quantum operations; originally investigated for quantum information theoretic reasons, their study has since grown to touch on many aspects of quantum information science. Here we investigate the nullspace structures of entanglement breaking channels and we derive a pair of related applications. We show that every operator space of trace zero matrices is the nullspace of an entanglement breaking channel. We derive a test for mixed unitarity of quantum channels based on complementary channel behaviour and entanglement breaking channel nullspaces. We identify conditions that guarantee the existence of private algebras for certain classes of entanglement breaking channels.

DOI
10.1088/1751-8121/abddea

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Abstract Quantum entanglement breaking channels are a fundamental class of quantum operations; originally investigated for quantum information theoretic reasons, their study has since grown to touch on many aspects of quantum information science. Here we investigate the nullspace structures of entanglement breaking channels and we derive a pair of related applications. We show that every operator space of trace zero matrices is the nullspace of an entanglement breaking channel. We derive a test for mixed unitarity of quantum channels based on complementary channel behaviour and entanglement breaking channel nullspaces. We identify conditions that guarantee the existence of private algebras for certain classes of entanglement breaking channels.
Quantum channels are central objects of study in quantum information, of which entanglement breaking channels are an interesting subset. Entanglement is considered a fundamental resource and property of quantum mechanics that we can use as a driving force for new theories and to continue research in this area. We exhibit the connection between stochastic matrix theory and the iterative behaviour of entanglement breaking channels and relate the Jordan forms of the entanglement breaking channels and stochastic matrices. We build on this perspective to study the fixed point theory for such channels. We further consider the nullspace structure of entanglement breaking channels, in particular proving that every operator space of trace zero matrices is the nullspace of such a channel. We connect the nullspace of entanglement breaking channels with private subspaces and present examples and discuss connections between quantum privacy and trace zero matrices. Finally, we draw a connection between random unitary channels and entanglement breaking channels.
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This note introduces a family of circulant quantum channels -- a subclass of the mixed-permutation channels -- and investigates its key structural and operational properties. We show that the image of the circulant quantum channel is precisely the set of circulant matrices. This characterization facilitates the analysis of arbitrary nnn-th order Bargmann invariants. Furthermore, we prove that the channel is entanglement-breaking, implying a substantially reduced resource cost for erasing quantum correlations compared to a general mixed-permutation channel. Applications of this channel are also discussed, including the derivation of tighter lower bounds for ℓp\ell_pℓp​-norm coherence and a characterization of its action in bipartite systems.