Three exercises from Nielsen and Chuang

From: Kurio (thekurio_at_NOSPAMlibero.it)
Date: 03/26/05

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    Date: Sat, 26 Mar 2005 17:11:57 GMT

    I'm working my way through [NC00]:

    [NC00] M. Nielsen and I. Chuang, Quantum Computation and Quantum
    Information, Cambridge University Press, 2000.

    Currently, I'm in the first chapters and I couldn't solve three of the
    proposed exercises. Is there anyone in this newsgroup willing to give
    some hints or solutions? If there are more appropriate newsgroups,
    please redirect me there.

    Here are the exercises I'm having troubles with.

    * EX. 2.73 *
    Let r be a density operator.
    A minimal ensemble for r is an ensemble {p_i, |s_i>} containing a number
    of elements equal to the rank of r.
    Let |s> be any state in the support of r (i.e. in the vector space
    spanned by the eigenvectors of r with non-zero eigenvalues).
    Show that there is a minimal ensemble for r that contains |s>, and
    moreover that in any such ensemble |s> must appear with probability:

              p_i = 1 / (<s_i| r^(-1) |s_i>)

    where r^(-1) is defined to be the inverse of r, when r is considered as
    an operator acting only on the support of r.

    * EX. 2.77 *
    Suppose ABC is a three component quantum system.
    Show by example that there are quantum states |s> of such systems which
    cannot be written in the form:

              |s> = \Sigma_i c_i |i_A> |i_B> |i_C>

    where c_i are real numbers, and |i_A>, |i_B>, |i_C> are orthonormal
    bases of the respective sytems.

    * EX. 2.79 *
    Consider a composite system consisting of two qubits.
    Find the Schmidt decomposition of the state:

              (|00> + |01> + |10>) / 3^(1/2)

    Now, here are my ideas for possible solutions that I wasn't able to
    complete.

    In EX. 2.73, I can explain how to build the minimal ensemble for r
    containing |s> (basically, I think you should just consider a basis set
    that contains |s>), but I haven't much clues about how to find the p_i.
    Maybe, do I have to consider the unitary coefficients that link vectors
    from any two ensembles of the same density matrix?
    Actually, I find the notation in the exercise even a bit confusing (why
    |s_i> when you're considering |s>?), but maybe I'm just misunderstanding
    the problem.

    In EX. 2.77, I suppose I have to find an entangled state, otherwise it
    is a product state and the decomposition is then possible?
    Am I correct? Any hints on a suitable state and/or on how to prove it is
    not decomposable?

    In EX. 2.79, I can do it with the singular value decomposition procedure
    that is generally applied, but I'm asking if there are "smarter" ways to
    do that, especially by hand.

    Thanks in advance to anybody who's willing to help.

    Ajeje

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