Quantum Markov chain

<h2 id="introduction">Introduction</h2>
<p>Very roughly, the theory of a quantum Markov chain resembles that of a <a href="/facts/Quantum_finite_automaton/SjRwos3V">measure-many automaton</a>, with some important substitutions: the initial state is to be replaced by a <a href="/facts/Density_matrix/A1XdylJa">density matrix</a>, and the projection operators are to be replaced by <a href="/facts/POVM/ouBuwcck">positive operator valued measures</a>.
</p>
<h2 id="formal-statement">Formal statement</h2>
<p>More precisely, a quantum Markov chain is a pair 
  
    
      
        (
        E
        ,
        ρ
        )
      
    
    {\displaystyle (E,\rho )}
  
 with 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
 a <a href="/facts/Density_matrix/A1XdylJa">density matrix</a> and 
  
    
      
        E
      
    
    {\displaystyle E}
  
 a <a href="/facts/Quantum_channel/o7DKPKJc">quantum channel</a> such that
</p>

E
        :
        
          
            B
          
        
        ⊗
        
          
            B
          
        
        →
        
          
            B
          
        
      
    
    {\displaystyle E:{\mathcal {B}}\otimes {\mathcal {B}}\to {\mathcal {B}}}

<p>is a <a href="/facts/Completely_positive_trace-preserving/o7DKPKJc">completely positive trace-preserving</a> map, and 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 a <a href="/facts/C-star_algebra/B3tnGHIo">C*-algebra</a> of bounded operators. The pair must obey the quantum Markov condition, that
</p>

Tr
        ⁡
        ρ
        (
        
          b
          
            1
          
        
        ⊗
        
          b
          
            2
          
        
        )
        =
        Tr
        ⁡
        ρ
        E
        (
        
          b
          
            1
          
        
        ,
        
          b
          
            2
          
        
        )
      
    
    {\displaystyle \operatorname {Tr} \rho (b_{1}\otimes b_{2})=\operatorname {Tr} \rho E(b_{1},b_{2})}

<p>for all 
  
    
      
        
          b
          
            1
          
        
        ,
        
          b
          
            2
          
        
        ∈
        
          
            B
          
        
      
    
    {\displaystyle b_{1},b_{2}\in {\mathcal {B}}}
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Quantum_walk/KhIasMUL">Quantum walk</a></li></ul>

<ul><li>Gudder, Stanley. "<a href="https://www.researchgate.net/profile/Stan_Gudder/publication/228697748_Quantum_Markov_chains/links/004635213c09cea606000000/Quantum-Markov-chains.pdf">Quantum Markov chains</a>." Journal of Mathematical Physics 49.7 (2008): 072105.</li></ul>

Quantum Markov chain open-in-new

Quantum Markov chain