Question

Fix V $V$ a complex inner product vector space of finite dimension n. $n$. Consider A $A$ the *-algebra End(V)End($V$). Take Gamma $\Gamma = Z^d \mathbb{Z}^d$ a lattice. Define the *-algebra A_Gamma $A_\Gamma$ (quantum cellular automaton observables) in the following manner. Assign to each element x $x$ of Gamma $\Gamma$ a copy A_x $A_x$ of A. $A$. To each finite subset S $S$ of Gamma $\Gamma$ we can correspond the *-algebra A_S $A_S$ defined by

A_S

$A_S$ := Tensor product of A_x $A_x$ over $x \in SS$

For two finite subsets S, R $S$, $R$ of Gamma $\Gamma$ with S $S$ contained in R $R$ we have the morphism

i_S,R:

$i_{S,R}$: $A_S -> A_R\rightarrow A_R$

obtained by tensoring with $1 \in A_y A_y$ for all $y \in R\SR\backslash S$. We define A_Gamma $A_\Gamma$ to be the direct limit of A_S $A_S$ w.r.t. S. $S$.

Denote T_Gamma $T_\Gamma$ the group of translations of Gamma $\Gamma (Z^d). T_Gamma Z^d)$. $T_\Gamma$ acts on A_Gamma $A_\Gamma$ in the obvious manner.

A quantum cellular automaton is defined to be a *-endomorphism of A_Gamma $A_\Gamma$ commuting with the action of T_Gamma. $T_\Gamma$. An invertible quantum cellular automaton is defined to be a *-automorphism of A_Gamma $A_\Gamma$ commuting with the action of T_Gamma. $T_\Gamma$. The 1st question is:

Any unit vector $v \in V V$ defines a state

phi_v: A_Gamma -> C

$\phi_v$: $A_\Gamma \rightarrow C$

in the following manner. Suppose S $S$ is a finite subset of Gamma $\Gamma$ and for any $x \in SS$, a_x $a_x$ is an element of A_x. $A_x$. Then we have a_S $a_S$ an element of A_S $A_S$ (and hence of A_Gamma) $A_\Gamma$) defined by

a_S

$a_S$ := tensor product of a_x $a_x$ over $x \in SS$

phi_v(a_S)

$\phi_v(a_S)$ := product of (v, $(v, a_x v) v)$ over $x \in SS$

It is easy to see this extends uniquely to a linear map phi_v: A_Gamma -> C $\phi_v$: $A_\Gamma \rightarrow C$ and that the map is a state.

Fix v $v$ in V. $V$. We construct the Hilbert space H_v $H_v$ in the following manner. Choose v_1 $v_1$ ... v_n $v_n$ an orthonormal basis of V $V$ s.t. $v_1 = vv$. Consider maps

alpha:

$\alpha: \Gamma -> \rightarrow {1 ... n}n}$

s.t. alpha(x) $\alpha(x) = 1 1$ for all x $x$ except a finite set. Define J $J$ to be the set of such alpha. $\alpha$. To each alpha $\alpha \in J J$ we assign the basis vector Psi_alpha $\Psi_\alpha$ of H_v, $H_v$, thought of as

Psi_alpha

$\Psi_\alpha$ := tensor product of v_alpha(x) $v_\alpha(x)$ over $x \in Gamma\Gamma$

Thus H_v $H_v$ is defined to be l^2(J). $l^2(J)$. It is easy to see that H_v $H_v$ thus defined depends only on v $v$ and not on v_2 $v_2$ ... v_n $v_n$ i.e. that for any two choices of v_2 $v_2$ ... v_n, $v_n$, there is a canonical isomorphism between the corresponding Hilbert spaces.

rho: A_Gamma -> B(H_v)

$\rho$: $A_\Gamma \rightarrow B(H_v)$

where B(H_v) $B(H_v)$ is the *-algebra of bounded operators on H_v. $H_v$. Thus, any unit vector Psi $\Psi \in H_v H_v$ defines a state

phi_Psi: A_Gamma -> C

$\phi_\Psi$: $A_\Gamma \rightarrow C$

phi_Psi(a)

$\phi_\Psi(a) = (Psi, rho(a) Psi)\Psi, \rho(a) \Psi)$

Now, fix an invertible quantum cellular automaton f: A_Gamma -> A_Gamma$f$: $A_\Gamma \rightarrow A_\Gamma$. Suppose v $v$ in V $V$ is s.t. phi_v $\phi_v$ is f-invariant. $f$-invariant. Then f $f$ is called v-representable$v$-representable if there exists

U: H_v ->

$U$: $H_v \rightarrow H_v$

a unitary operator s.t. for any Psi $\Psi \in H_v H_v$ we have

phi_U

$\phi_U \Psi = f*(phi_Psi)f^*(\phi_\Psi)$

It is clear that if such U $U$ exists it is unique.

Is any invertible quantum cellular automaton v-representable $v$-representable for any v $v$ with phi_v f-invariant?$\phi_v$ $f$-invariant?