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Classical channel vs quantum channel

  1. Jan 23, 2016 #1
    I have a hard time understanding what a channel that transmits classical information is in the framework of quantum information theory. My textbook says the following:

    Let P(ylx) be a classical channel defined by a conditional probability distribution. We define the corresponding quantum channel by:

    x,y (Tr lx><xlX)P(ylx)ly><yl

    How should I interpret this formula and what is the role of the trace appearing? And in general what is a classical channel as opposed to a quantum channel? I guess the answer is that a classical channel can send definite states given by some probability distribution, while quantum channels are maps between states in general.

    Another place in the book it states that a classical channel of two bits is of the form:
    C(X) = ∑i,j lij><ijl Tr lij><ijlX
    Is this equivalent with the first formula?
     
  2. jcsd
  3. Jan 27, 2016 #2

    naima

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    Could you explain what is X in your question?
    Have you links to this subject?
     
  4. Jan 27, 2016 #3

    naima

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    The first thing which comes to my mind is that the trace of ##\rho X## is the mean value of X in the state ##\rho##
    Now if X is an operator it sends X to another operator. How can we describe it? What is the image of a projector |z><z|?
     
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