Stability of Global Entanglement in Thermal States of Spin Chains
Stephen Bullock, G K. Brennen
The n-qubit concurrence canonical decomposition (CCD) is a generalization of the two-qubit canonical decomposition SU(4)=[SU(2) (x) SU(2)] ? [SU(2) (x) SU(2)], where ? is the commutative group which phases the maximally entangled Bell basis. A prequel manuscript creates the CCD as a particular example of the G=KAK metadecomposition theorem of Lie theory. We hence denote it by SU(2n)=KAK. If Cn(|?)= &< ?*| (-isy) (x)n| ?>| is the concurrence entanglement monotone, then computations in the K group are symmetries of a related bilinear form and so do not change the concurrence. Hence for a quantum computation v=k1 a k2, analysis of a in e A allows one to study one aspect of the entanglement dynamics of the evolution v, i.e. the concurrence dynamics. Note that analysis of such an a in e A is simpler than the generic case, since A is a commutative group whose dimension is exponentially less than that of SU(N). In this manuscript, we accomplish three main goals. First, we expand upon the treatment of the odd-qubit case of the sequel, in that we (i) present an algorithm to compute the CCD in case n=2p-1 and (ii) characterize the maximal odd-qubit concurrence capacity in terms of convex hulls. Second, we interpret the CCD in terms of a time-reversal symmetry operator, namely the quantum bit flip |?> ? (-i sy) (x)n | ?*>. In this context, the CCD allows one to write any unitary evolution as a two-term product of a time-reversal symmetric and anti-symmetric evolution; no Trotterization is required. Finally, we use these constructions to study time-reversal symmetric Hamiltonians. In particular, we show that any | ?> in the ground state of such an H must either develop a Kramer's degeneracy or be maximally entangled in the sense that Cn(| ?>)=1. Many time-reversal symmetric Hamiltonians are known to be nondegenerate and so produce maximally concurrent ground states.
Physical Review A (Atomic, Molecular and Optical Physics)