Time Reversal and n-qubit Canonical Decompositions
arXiv:quant-ph/0402051 · doi:10.1063/1.1900293
Abstract
For n an even number of qubits and v a unitary evolution, a matrix decomposition v=k1 a k2 of the unitary group is explicitly computable and allows for study of the dynamics of the concurrence entanglement monotone. The side factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the a factor. In this work, we provide an explicit numerical algorithm computing v=k1 a k2 for n odd. Further, in the odd case we lift the monotone to a two-argument function, allowing for a theory of concurrence dynamics in odd qubits. The generalization may also be studied using the CCD, leading again to maximal concurrence capacity for most unitaries. The key technique is to consider the spin-flip as a time reversal symmetry operator in Wigner's axiomatization; the original CCD derivation may be restated entirely in terms of this time reversal. En route, we observe a Kramers' nondegeneracy: the existence of a nondegenerate eigenstate of any time reversal symmetric n-qubit Hamiltonian demands (i) n even and (ii) maximal concurrence of said eigenstate. We provide examples of how to apply this work to study the kinematics and dynamics of entanglement in spin chain Hamiltonians.
20 pages, 3 figures; v2 (17pp.): major revision, new abstract, introduction, expanded bibliography
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- Fully quantum fluctuation theorems
- Quantum simulation of interacting high-dimensional systems: the influence of noise
- Time-optimal synthesis of unitary transformations in coupled fast and slow qubit system
- A General Framework for Recursive Decompositions of Unitary Quantum Evolutions
- Constraints on correlations in multiqubit systems
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