Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and Gradient-Flow-Based Numerical Checks

Details Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and Gradient-Flow-Based Numerical Checks Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and Gradient-Flow-Based Numerical Checks

2006-10-09

arxiv.org/abs/quant-ph/0610061v1

Physics

Quantum Physics

19 pages, 7 figures; comments welcome

Scientific paper

Generalising in the sense of Hahn's spin echo, we completely characterise those unitary propagators of effective multi-qubit interactions that can be inverted solely by {\em local} unitary operations on $n$ qubits (spins-$\tfrac{1}{2}$). The subset of $U\in \mathbf{SU}(2^n)$ satisfying $U^{-1}=K_1 U K_2$ with pairs of local unitaries $K_1, K_2\in\mathbf{SU}(2)^{\otimes n}$ comprises two classes: in type-I, $K_1$ and $K_2$ are inverse to one another, while in type-II they are not. {Type-I} consists of one-parameter groups that can jointly be inverted for all times $t\in\R{}$ because their Hamiltonian generators satisfy $K H K^{-1} = \Ad K (H) = -H$. As all the Hamiltonians generating locally invertible unitaries of type-I are spanned by the eigenspace associated to the eigenvalue -1 of the {\em local} conjugation map $\Ad K$, this eigenspace can be given in closed algebraic form. The relation to the root space decomposition of $\mathfrak{sl}(N,\C{})$ is pointed out. Special cases of type-I invertible Hamiltonians are of $p$-quantum order and are analysed by the transformation properties of spherical tensors of order $p$. Effective multi-qubit interaction Hamiltonians are characterised via the graphs of their coupling topology. {Type-II} consists of pointwise locally invertible propagators, part of which can be classified according to the symmetries of their matrix representations. Moreover, we show gradient flows for numerically solving the decision problem whether a propagator is type-I or type-II invertible or not by driving the least-squares distance $\norm{K_1 e^{-itH} K_2 - e^{+itH}}^2_2$ to zero.

Affiliated with

Also associated with

No associations

Rating

Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and Gradient-Flow-Based Numerical Checks does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and Gradient-Flow-Based Numerical Checks, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Which Quantum Evolutions Can Be Reversed by Local Unitary Operations? Algebraic Classification and Gradient-Flow-Based Numerical Checks will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-343594