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Ref. 17 it is stated without proof that any Gaussian LOCC operation can be decomposed into a series of steps which involve additional ancillas in Gaussian states, local unitary Gaussian operations, projections into pure Gaussian states, partial traces, and “mixings such that the resulting state is Gaussian.” There and in Refs. 10 11 it is described how the CM changes under the first four of these operations
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In Ref. 17 it is stated without proof that any Gaussian LOCC operation can be decomposed into a series of steps which involve additional ancillas in Gaussian states, local unitary Gaussian operations, projections into pure Gaussian states, partial traces, and “mixings such that the resulting state is Gaussian.” There and in Refs. 1011 it is described how the CM changes under the first four of these operations.
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Although this maximally entangled state does not belong to the Hilbert space, it can always be considered as a limit of a proper pure state
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Although this maximally entangled state does not belong to the Hilbert space, it can always be considered as a limit of a proper pure state.
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44
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85037179249
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A mathematically correct way of dealing with the “projections into improper eigenstates of (Formula presented)” would be via a limit procedure: define (Formula presented) and a “Bell-POVM” (Formula presented) Then we define (Formula presented) and finally (Formula presented) The limit can be performed directly on the level of correlation matrices yielding the expressions Eqs. (10)
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A mathematically correct way of dealing with the “projections into improper eigenstates of (Formula presented)” would be via a limit procedure: define (Formula presented) and a “Bell-POVM” (Formula presented) Then we define (Formula presented) and finally (Formula presented) The limit can be performed directly on the level of correlation matrices yielding the expressions Eqs. (10).
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The correspondence of Gaussian operations and Gaussian states has been used in Ref. 35 to optimize continuous variable teleportation with Gaussian means
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The correspondence of Gaussian operations and Gaussian states has been used in Ref. 35 to optimize continuous variable teleportation with Gaussian means.
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47
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calculating the inverses of block matrices the formula (Formula presented)is very useful. See, e.g., R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1987)
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In calculating the inverses of block matrices the formula (Formula presented)is very useful. See, e.g., R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1987).
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