Abstract
The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.
Highlights
“Quantum” or “noncommutative” graphs appear in the setting of quantum error correction [3,10,11]
The usual construction starts with a quantum channel, which in the Schrodinger picture is modelled by a completely positive trace preserving (CPTP) map : Mm → Mn
Quantum channels transform mixed states to mixed states, and in error correction problems one is interested in determining which input states can be distinguished with certainty after passing through the channel
Summary
Let us see why the definitions of restrictions, pushforwards, and pullbacks given above are natural. The states connected by −→V are just the states whose images under ∗ are connected by V, and the states connected by ← W− are just the states whose images under are connected by W This characterization shows that the definitions of pushforward and pullback only depend on the map , not the choice of Kraus matrices. We can use the idea of connecting mixed states to give an intrinsic characterization of the “noncommutative (directed) bipartite graphs” of Duan, Severini, and Winter [3]. The noncommutative bipartite graph associated to a CPTP map : Mm → Mn connects mixed states A ∈ Mm ⊗ Mk and C ∈ Mn ⊗ Mk if and only if ( A)C = 0, i.e., there is a possibility of confusing the image of A with C. The reverse direction follows from Lemma 7.3 (cf. the proof of Theorem 7.4)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
