Conditional quantum entropy

<h2 id="definition">Definition</h2>
<p>Given a bipartite quantum state 
  
    
      
        
          ρ
          
            A
            B
          
        
      
    
    {\displaystyle \rho ^{AB}}
  
, the entropy of the joint system AB is 
  
    
      
        S
        (
        A
        B
        
          )
          
            ρ
          
        
         
        
          
            
              
                =
              
              
                
                  d
                  e
                  f
                
              
            
          
        
         
        S
        (
        
          ρ
          
            A
            B
          
        
        )
      
    
    {\displaystyle S(AB)_{\rho }\ {\stackrel {\mathrm {def} }{=}}\ S(\rho ^{AB})}
  
, and the entropies of the subsystems are 
  
    
      
        S
        (
        A
        
          )
          
            ρ
          
        
         
        
          
            
              
                =
              
              
                
                  d
                  e
                  f
                
              
            
          
        
         
        S
        (
        
          ρ
          
            A
          
        
        )
        =
        S
        (
        
          
            t
            r
          
          
            B
          
        
        
          ρ
          
            A
            B
          
        
        )
      
    
    {\displaystyle S(A)_{\rho }\ {\stackrel {\mathrm {def} }{=}}\ S(\rho ^{A})=S(\mathrm {tr} _{B}\rho ^{AB})}
  
 and 
  
    
      
        S
        (
        B
        
          )
          
            ρ
          
        
      
    
    {\displaystyle S(B)_{\rho }}
  
.  The von Neumann entropy measures an observer's uncertainty about the value of the state, that is, how much the state is a <a href="/facts/Mixed_state_(physics)/7jbxd7Dx">mixed state</a>.
</p><p>By analogy with the classical conditional entropy, one defines the conditional quantum entropy as 
  
    
      
        S
        (
        A
        
          |
        
        B
        
          )
          
            ρ
          
        
         
        
          
            
              
                =
              
              
                
                  d
                  e
                  f
                
              
            
          
        
         
        S
        (
        A
        B
        
          )
          
            ρ
          
        
        −
        S
        (
        B
        
          )
          
            ρ
          
        
      
    
    {\displaystyle S(A|B)_{\rho }\ {\stackrel {\mathrm {def} }{=}}\ S(AB)_{\rho }-S(B)_{\rho }}
  
.
</p><p>An equivalent operational definition of the quantum conditional entropy (as a measure of the <a href="/facts/Quantum_communication/JDuxpHp8">quantum communication</a> cost or surplus when performing <a href="/facts/Quantum_state/7jbxd7Dx">quantum state</a> merging) was given by <a href="/facts/Micha%C5%82_Horodecki/kekMHskT">Michał Horodecki</a>, <a href="/facts/Jonathan_Oppenheim/DH87mzBG">Jonathan Oppenheim</a>, and <a href="/facts/Andreas_Winter/qbhdM8dd">Andreas Winter</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="properties">Properties</h2>
<p>Unlike the classical <a href="/facts/Conditional_entropy/gHzPezGB">conditional entropy</a>, the conditional quantum entropy can be negative.  This is true even though the (quantum) von Neumann entropy of single variable is never negative. The negative conditional entropy is also known as the <a href="/facts/Coherent_information/nDd1Awj5">coherent information</a>, and gives the additional number of bits above the classical limit that can be transmitted in a quantum dense coding protocol. Positive conditional entropy of a state thus means the state cannot reach even the classical limit, while the negative conditional entropy provides for additional information.
</p>

<ul><li><a href="/facts/Michael_Nielsen/hedbpYkS">Nielsen, Michael A.</a>; <a href="/facts/Isaac_Chuang/qEgqj9tt">Chuang, Isaac L.</a> (2010). <a href="/facts/Quantum_Computation_and_Quantum_Information_(book)/l2pYpSSj"><i>Quantum Computation and Quantum Information</i></a> (2nd ed.). Cambridge: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-107-00217-3. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/844974180">844974180</a>.</li>
<li>Wilde, Mark M. (2017), "Preface to the Second Edition", <i>Quantum Information Theory</i>, Cambridge University Press, pp. xi–xii, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1106.1445">1106.1445</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2011arXiv1106.1445W">2011arXiv1106.1445W</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2F9781316809976.001">10.1017/9781316809976.001</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781316809976, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:2515538">2515538</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Cerf, N. J.; Adami, C. (1997). "Negative Entropy and Information in Quantum Mechanics". Physical Review Letters. 79 (26): 5194–5197. arXiv:quant-ph/9512022. Bibcode:1997PhRvL..79.5194C. doi:10.1103/physrevlett.79.5194. S2CID 14834430. <a href="/wiki/Physical_Review_Letters" target="_blank">/wiki/Physical_Review_Letters</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Cerf, N. J.; Adami, C. (1999-08-01). "Quantum extension of conditional probability". Physical Review A. 60 (2): 893–897. arXiv:quant-ph/9710001. Bibcode:1999PhRvA..60..893C. doi:10.1103/PhysRevA.60.893. S2CID 119451904. <a href="/wiki/Physical_Review_A" target="_blank">/wiki/Physical_Review_A</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Horodecki, Michał; Oppenheim, Jonathan; Winter, Andreas (2005). "Partial quantum information". Nature. 436 (7051): 673–676. arXiv:quant-ph/0505062. Bibcode:2005Natur.436..673H. doi:10.1038/nature03909. PMID 16079840. S2CID 4413693. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Conditional quantum entropy open-in-new

Conditional quantum entropy