title: Quantum Markov chain
text: # Quantum Markov chain
  
  In mathematics, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain, replacing the classical definitions of probability with quantum probability.
  
  ## Introduction
  
  Very roughly, the theory of a quantum Markov chain resembles that of a measure-many automaton, with some important substitutions: the initial state is to be replaced by a density matrix, and the projection operators are to be replaced by positive operator valued measures.
  
  ## Formal statement
  
  More precisely, a quantum Markov chain is a pair $(E,\rho )$ with $\rho$ a density matrix and $E$ a quantum channel such that
  
      $E:{\mathcal {B}}\otimes {\mathcal {B}}\to {\mathcal {B}}$
  
  is a completely positive trace-preserving map, and ${\mathcal {B}}$ a $C^*-algebra$ of bounded operators. The pair must obey the quantum Markov condition, that
  
      $\operatorname {Tr} \rho (b_{1}\otimes b_{2})=\operatorname {Tr} \rho E(b_{1},b_{2})$
  
  for all $b_{1},b_{2}\in {\mathcal {B}}$.
  
  