Random walk and quantum decoherence

10.6:Random walk and quantum decoherence

10.6.1: Diffusion process

Example 10.15 [Random walk]

Let the state space $\Omega$ be ${\mathbb Z}=\{0,\pm1,\pm2,\ldots\}$ with the counting measure $\nu$. Define the dual {{causal operator}} ${\Phi}^*: {\cal M}_{+1}({\mathbb Z}) \to {\cal M}_{+1}({\mathbb Z})$ such that

\begin{align} {\Phi^*}( \delta_i ) = \frac{\delta_{i-1} + \delta_{i+1}}{2} \qquad (i \in {\mathbb Z}) \end{align}

where $\delta_{(\cdot )} (\in {\cal M}_{+1}({\mathbb Z}) )$ is a point measure. Therefore, the causal operator $\Phi: L^\infty ({\mathbb Z}) \to L^\infty ({\mathbb Z})$ is defined by

\begin{align} [\Phi ( F ) ](i)= \frac{F({i-1}) + F({i+1})}{2} \qquad (\forall F \in L^\infty ({\mathbb Z}), \forall i \in {\mathbb Z}) \end{align}

and the pre-dual causal operator $\Phi_*: L^1 ({\mathbb Z}) \to L^1 ({\mathbb Z})$ is defined by

\begin{align} [\Phi_* ( f ) ](i)= \frac{f({i-1}) + F({i+1})}{2} \qquad (\forall f \in L^1 ({\mathbb Z}), \forall i \in {\mathbb Z}) \end{align}

Now, consider the discrete time $T=\{0,1,2,\ldots,N\}$, where the parent map $\pi: T \setminus \{0\} \to T$ is defined by $\pi ( t ) = t-1$ $(t =1,2,...)$. For each $t (\in T )$, a state space $\Omega_t$ is define by $\Omega_t= {\mathbb Z}$. Then, we have the {{sequential causal operator}} $\{ \Phi_{\pi(t), t }(=\Phi ){}: $ ${L^\infty (\Omega_t)} \to {L^\infty (\Omega_{\pi(t)})} \}_{ t \in T\setminus \{0\} }$

10.6.2: Quantum decoherence: non-deterministic causal operator

Consider the quantum basic structure:

\begin{align} [{\mathcal C}(H) \subseteq B(H) \subseteq {B(H)}] \end{align}

Let ${\mathbb P}=\{P_n \}_{n=1}^\infty$ be the spectrum decomposition in $B(H)$, that is,

\begin{align} P_n \mbox{ is a projection (i.e., $P_n=(P_n)^2$ ), and}, \sum_{n=1}^\infty P_n =I \end{align}

Define the operator $(\Psi_{\mathbb P})_*: {\mathcal Tr}(H) \to {\mathcal Tr}(H)$ such that

\begin{align} (\Psi_{\mathbb P})_* (|u \rangle \langle u |) = \sum_{n=1}^\infty |P_n u \rangle \langle P_n u | \quad (\forall u \in H) \end{align}

Clearly we see

\begin{align} \langle v, (\Psi_{\mathbb P})_* (|u \rangle \langle u |) v \rangle = \langle v, (\sum_{n=1}^\infty |P_n u \rangle \langle P_n u |) v \rangle =\sum_{n=1}^\infty |\langle v, |P_n u \rangle |^2 \ge 0 \qquad (\forall u,v \in H ) \end{align}

and,

\begin{align} & \mbox{Tr}((\Psi_{\mathbb P})_* (|u \rangle \langle u |)) \\ = & \mbox{Tr} (\sum_{n=1}^\infty |P_n u \rangle \langle P_n u |) = \sum_{n=1}^\infty \sum_{k=1}^\infty |\langle e_k , P_n u \rangle|^2 = \sum_{n=1}^\infty \| P_n u \|^2 = \|u\|^2 \qquad (\forall u \in H ) \end{align}

where $\{ e_k \}_{k=1}^\infty$ is CONS in $H$.

And so,

\begin{align} (\Psi_{\mathbb P})_* ({\mathcal Tr}_{+1}^p(H)) \subseteq {\mathcal Tr}_{+1}(H) \end{align}

Therefore, $\Psi_{\mathbb P} (=((\Psi_{\mathbb P})_*)^*): B(H) \to B(H)$ is a causal operator, but it is not deterministic. In this note, a non-deterministic (sequential) causal operator is called a quantum decoherence.

Remark 10.16 [Quantum decoherence]

For the relation between quantum decoherence and quantum Zeno effect, see $\S$11.3. Also, for the relation between quantum decoherence and Schrödinger's cat, see $\S$11.4.

In tis note, we assume that the don-deterministic causal operator belongs to the mixed measurement theory. Thus, we consider that quantum language (= measurement theory ) is classified as follows.

$(A):$

$ \underset{(=\mbox{ quantum language})}{\mbox{ measurement theory}} \left\{\begin{array}{ll} \underset{\mbox{(A$_1$)}}{ \mbox{pure type}} \left\{\begin{array}{ll} \!\! \mbox{classical system} : \mbox{ Fisher statistics} \\ \!\! \mbox{ quantum system} : \mbox{ usual quantum mechanics } \\ \end{array}\right. \\ \\ \underset{\mbox{(A$_2$)}} {\mbox{mixed type}} \left\{\begin{array}{ll} \!\! \mbox{ classical system} : \mbox{ including Bayesian statistics,} \\ \qquad \qquad \qquad \mbox{ Kalman filter} \\ \!\! \mbox{ quantum system} : \mbox{ quantum decoherence } \\ \end{array}\right. \end{array}\right. $