Conditional quantum entropy

<p>The conditional quantum entropy is an entropy measure used in <a href="/facts/Quantum_information_theory/Bl9QIvHj">quantum information theory</a>.  It is a generalization of the <a href="/facts/Conditional_entropy/gHzPezGB">conditional entropy</a> of <a href="/facts/Classical_information_theory/qNLH3U5P">classical information theory</a>.   For a bipartite state 
  
    
      
        
          ρ
          
            A
            B
          
        
      
    
    {\displaystyle \rho ^{AB}}
  
, the conditional entropy is written 
  
    
      
        S
        (
        A
        
          |
        
        B
        
          )
          
            ρ
          
        
      
    
    {\displaystyle S(A|B)_{\rho }}
  
, or 
  
    
      
        H
        (
        A
        
          |
        
        B
        
          )
          
            ρ
          
        
      
    
    {\displaystyle H(A|B)_{\rho }}
  
, depending on the notation being used for the <a href="/facts/Von_Neumann_entropy/nzeJKowg">von Neumann entropy</a>. The quantum conditional entropy was defined in terms of a conditional density operator 
  
    
      
        
          ρ
          
            A
            
              |
            
            B
          
        
      
    
    {\displaystyle \rho _{A|B}}
  
 by <a href="/facts/Nicolas_Cerf/e7swsA8w">Nicolas Cerf</a> and <a href="/facts/Chris_Adami/Lg12q3rK">Chris Adami</a>, who showed that quantum conditional entropies can be negative, something that is forbidden in classical physics. The negativity of quantum conditional entropy is a sufficient criterion for quantum <a href="/facts/Separable_state/j0k8VWmF">non-separability</a>.
</p><p>In what follows, we use the notation 
  
    
      
        S
        (
        ⋅
        )
      
    
    {\displaystyle S(\cdot )}
  
 for the <a href="/facts/Von_Neumann_entropy/nzeJKowg">von Neumann entropy</a>, which will simply be called "entropy".
</p>

Conditional quantum entropy open-in-new

Conditional quantum entropy