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CP maps

  1. Apr 8, 2009 #1
    I am trying to teach myself quantum mechanics and I have heard a lot about Completely Positive maps but I haven't been able to find anything on them could someone please tell me what they are and what they are good fore?


  2. jcsd
  3. Apr 8, 2009 #2


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    From linear algebra point of view, a bounded operator [tex]A[/tex] acting on a Hilbert space [tex]H[/tex] is said to be positive (P), if for all [tex]|x\rangle\in H[/tex], [tex]\langle x|A|x\rangle\geq0[/tex].
    An operator [tex]E[/tex] which maps density operators of a space [tex]H_1[/tex] to [tex]H_2[/tex] is called completely positive (CP). (Now you understand why they are impotent).
    Equivalently, [tex]E[/tex] is completely positive, if and only if [tex]I_n\otimes E[/tex] is a positive operator for all [tex]n\geq0[/tex]. [tex]I_n[/tex] is the identity operator. Testing an operator is CP or not is a difficult problem. The operators which are P but not CP can be used as entanglement witnesses.

    For more details read Nielsen and Chuang.
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