Quantum Communication Through an Unmodulated Spin Chain

Definition .

Let $A$ be any complex associative algebra. A $\star$ structure on $A$ is an antilinear, involutive, algebra antiautomorphism and coalgebra automorphism:

(cx)^{\star}=\bar{c}(x^{\star}),\hskip 8.535827pt(x^{\star})^{\star}=x,\hskip 8% .535827pt(xy)^{\star}=y^{\star}x^{\star}

for any $c\in{\mathbb{C}}$ , $x,y\in A$ , where $\bar{x}$ is the complex conjugate of $x$ .

Definition 6 .

The mean value of potential $\mathbf{u}$ weighted by its own distribution is called entropy of the potential,

ent(\mathbf{u})=\mathbf{u}\,dist(\mathbf{u})

Definition 6.3 .

Two partial representations $\pi:G\to{\mathcal{B}}$ and $\pi^{\prime}:G\to{\mathcal{B}}^{\prime}$ are equivalent if there is an isomorphism $\varphi:{\mathcal{B}}^{\prime}\to{\mathcal{B}}$ such that

\pi(g)=\varphi(\pi^{\prime}(g))

for all $g\in G.$

Definition 1.3 .

( [ 11 ] , [ 10 ] , see e.g. [ 13 ] )

A Lie group $G$ of Lie algebra $\mathfrak{g}$ is Omori-Milnor regular (or regular for short) if and only if

1) for any $v\in C^{\infty}([0,1],\mathfrak{g})$ , the differential equation

g^{-1}{dg(t)\over dt}=v(t)

has an unique smooth solution $\gamma_{v}$ .

2) the map $v\mapsto\gamma_{v}(1)$ is smooth for the topology of uniform comvergence in $C^{\infty}([0,1],\mathfrak{g})$ .

3) $\gamma_{v}$ is obtained by product integral in $G$ (see [ 11 ] for the definition of the product integral).

Definition 1.1.1

[ 2 ] The system

\begin{array}[]{lllllllllllllllllllllllllllllllll}\dot{\xi}&=&f(\xi,u)\\ y&=&h(\xi,u)\end{array}

(1.1.6)

is uniformly observable for any input if there exist coordinates

\left\{x_{ij}:i=1,\ldots,p,\,j=1,\ldots,l_{i}\right\}

where $1\leq l_{1}\leq\ldots\leq l_{p}$ and $\sum l_{i}=n$ such that in these coordinates the system takes the form

\begin{array}[]{lllllllllllllllllllllllllllllllll}y_{i}&=&x_{i1}+h_{i}(u)\\ \dot{x}_{i1}&=&{x}_{i2}+f_{i1}(\underline{x}_{1},u)\\ &\vdots&\\ \dot{x}_{ij}&=&{x}_{ij+1}+f_{ij}(\underline{x}_{j},u)\\ &\vdots&\\ \dot{x}_{il_{i}-1}&=&{x}_{il_{i}}+f_{il_{i}-1}(\underline{x}_{l_{i}-1},u)\\ \dot{x}_{il_{i}}&=&f_{il_{i}}(\underline{x}_{l_{i}},u)\end{array}

(1.1.7)

for $i=1,\ldots,p$ where $\underline{x}_{j}$ is defined by

\displaystyle\underline{x}_{j}

\displaystyle=

\displaystyle(x_{11},\ldots,x_{1,j\wedge l_{1}},x_{21},\ldots,x_{pj}).

(1.1.8)

Notice that in $\underline{x}_{j}$ the indices range over $i=1,\ldots,p;\ k=1,\ldots,\min\{j,l_{i}\}$ and the coordinates are ordered so that second index moves faster than the first.

We also require that each $f_{ij}$ be Lipschitz continuous, there exists an $L$ such that for all $x,\xi\in{I\!\!R}^{n},u\in U$ ,

\begin{array}[]{lllllllllllllllllllllllllllllllll}|f_{i}(\underline{x}_{j},u)-% f_{i}(\underline{\xi}_{j},u)|&\leq&L|\underline{x}_{j}-\underline{\xi}_{j}|.% \end{array}

(1.1.9)

The symbol $|\cdot|$ denotes the Euclidean norm.

Definition 1.4 .

Let $G$ be a group. Two representations $\rho,\rho^{\prime}:B_{n}\to{\rm Aut}(G)$ are called equivalent if there exist automorphisms $\phi:G\to G$ and $\mu:B_{n}\to B_{n}$ such that

\rho^{\prime}(\mu(\beta))=\phi^{-1}\circ\rho(\beta)\circ\phi

for all $\beta\in B_{n}$ .

Definition 8.1 .

We say that a co-invariant subspace $\mathcal{L}$ is pure if the following implication holds:

x\in\mathcal{L},\;S^{\infty}x\subset\mathcal{L}\Longrightarrow x=0.

(8.1)

Definition 2.1 .

Given a functor $(\Phi,\Phi_{1})$ from a category ${\mathcal{X}}$ to the category of ${\mathbb{Z}}$ -graded projective $R$ -modules and degree $0$ maps, a twisting cochain $\psi$ on ${\mathcal{X}}$ with coefficients in $(\Phi,\Phi_{1})$ is a sum of cochains $\psi=\sum_{p\geq 0}\psi_{p}$ where $\psi_{p}$ is a $p$ -cochain on ${\mathcal{X}}$ with coefficients in the degree $p-1$ part $F_{p-1}$ of the $\operatorname{Hom}(\Phi,\Phi)$ bifunctor $F$ of ( 4 ) so that the following condition is satisfied.

\delta\psi=\psi\cup^{\prime}\psi.

Here $\cup^{\prime}$ is the cup product using the Koszul sign rule:

\psi_{p}\cup^{\prime}\psi_{q}(X_{0},\cdots,X_{p+q})=(-1)^{p}\psi_{p}(X_{0},% \cdots,X_{p})\psi_{q}(X_{p},\cdots,X_{p+q}).

since $\psi_{q}$ has total odd degree.

Definition 7.3 .

Given $(s,t)\in U$ , $(p,q)\in\mathbb{R}^{2}$ are defined by the rule

pu+qv=2\pi i.