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Quantum Entanglement in Electron Optics pp 13–46Cite as

Quantum Information: Basic Relevant Concepts and Applications

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Part of the book series: Springer Series on Atomic, Optical, and Plasma Physics ((SSAOPP,volume 67))

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Notes

  1. 1.

    The remaining discussion in the present Sect. 2.2.3 has been adapted from  [99, 100].

  2. 2.

    It (see, e.g., [90, 96, 97, 101, 102], etc) is related to the spatial separation of the detectors used by the parties A and B in their experiment to test Bell’s inequality (2.21).

  3. 3.

    This loophole (see, [90, 94, 101] and references therein) refers to the problem arising due to low total detection efficiency in an experiment performed to test the inequality (2.21). One, therefore, assumes in such experiments that the detected pairs of particles is a fair representation of the emitted pairs.

  4. 4.

    A criterion applicable to test nonseparability of a mixed states can always be used for pure states as well, but not vice-versa.

  5. 5.

    The following demonstration is based upon that given in [134].

  6. 6.

    An operator, say, Ω is said to be positive if each of its eigenvalues is real and non-negative. It is denoted as Ω  ≥ 0. The expectation value of a positive operator for any state is also positive. If each of the eigenvalues of this operator is strictly greater than zero, then Ω is said to be positive definite. See Appendix A for more details.

  7. 7.

    Although, the form of a partially transposed matrix (\(e.g.,{\rho }^{\mbox{ T}_{1}},or{\rho }^{\mbox{ T}_{2}},\) etc) depends on the choice of the bases, its eigenvalues are, however, always independent of the chosen bases.

  8. 8.

    Sect. 4.2.2 on pages 108–114 in Chap. 4 contains a brief description of bound entanglement.

  9. 9.

    Rank of a matrix is equal to the number of its nonzero eigenvalues [132].

  10. 10.

    For ρ to represent a product state, it should be possible to write it in the form of (2.39) [or, equivalently, (A.4)].

  11. 11.

    A non-separable state ρ cannot be written in the form of (2.39) [or, equivalently, (A.4)].

  12. 12.

    It consists of unitary transformations or (generalized) measurements, say, Ω A and Ω B, performed separately by A and B on their respective qubits of a bipartite state. Such a local transformation/operation is written as Ω A ⊗ Ω B.

  13. 13.

    It is the local operations performed in (ii) which have been correlated by classical communications. That is, the parties A and B communicate to each other the results of their measurements on their respective qubits of a shared bipartite state using presently available telecom technologies based on classical information. For more on LOCC see, for example, [126] and references therein.

  14. 14.

    A maximally entangled mixed state, on the other hand, is the one [142] which, for a given mixedness, achieves the greatest possible entanglement. See, for example, discussion given on pages  128 and  129.

  15. 15.

    The trace norm of a matrix M is defined by (see, e.g., Sect. VI 6 in [158]) \(\parallel M \parallel _{1}\ \equiv \ \mbox{ Tr}\sqrt{M\,{M}^{\dag }}\). Square roots of the eigenvalues of M M  †  are called singular values of the matrix M. Remembering that the trace of a matrix is always equal to the sum of its eigenvalues, the trace norm of a matrix is, therefore, the sum of its singular values. If M is a Hermitian matrix then, obviously, \(\parallel M \parallel _{1}\ =\sum \, \vert \text{eigenvalues of}\ M\vert \).

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  1. Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur, 721302, West Bengal, India

    N. Chandra

  2. c/o B.N. Panda, 1/A–46(H.I.G.), Lingraj Vihar, Pokhariput, 751020, Orissa, India

    R. Ghosh

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Chandra, N., Ghosh, R. (2013). Quantum Information: Basic Relevant Concepts and Applications. In: Quantum Entanglement in Electron Optics. Springer Series on Atomic, Optical, and Plasma Physics, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24070-6_2

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