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arXiv:quant-ph/0409101 · Journal of the Optical Society of America B

Quantum and classical fidelities for Gaussian states

Hyunseok Jeong, Timothy C. Ralph, Warwick P. Bowen

Published Jan 26, 2007• 14 citations
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Abstract

We examine the physical significance of fidelity as a measure of similarity for Gaussian states by drawing a comparison with its classical counterpart. We find that the relationship between these classical and quantum fidelities is not straightforward, and in general does not seem to provide insight into the physical significance of quantum fidelity. To avoid this ambiguity we propose that the efficacy of quantum information protocols be characterized by determining their transfer function and then calculating the fidelity achievable for a hypothetical pure reference input state.

DOI
10.1364/josab.24.000355

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We examine the physical significance of fidelity as a measure of similarity for Gaussian states by drawing a comparison with its classical counterpart. We find that the relationship between these classical and quantum fidelities is not straightforward, and in general does not seem to provide insight into the physical significance of quantum fidelity. To avoid this ambiguity we propose that the efficacy of quantum information protocols be characterized by determining their transfer function and then calculating the fidelity achievable for a hypothetical pure reference input state.
In this paper, from the original definition of fidelity in a pure state, we first give a well-defined expansion fidelity between two Gaussian mixed states. It is related to the variances of output and input states in quantum information processing. It is convenient to quantify the quantum teleportation (quantum clone) experiment since the variances of the input (output) state are measurable. Furthermore, we also give a conclusion that the fidelity of a pure input state is smaller than the fidelity of a mixed input state in the same quantum information processing.
From the original definition of fidelity in a pure state,we first give the well defined expansion fidelity between two Gaussian mixed states.It is related to the variances of output and input states in quantum information processing.It could be convenient to quantify the quantum teleportation(quantum clone) experiment since the variances of input(output) state are measurable.Furthermore,we also show that the fidelity of a pure input state is smaller than that of a mixed input state in the same quantum information processing.
Applications of quantum technology often require fidelities to quantify performance. These provide a fundamental yardstick for the comparison of two quantum states. While this is straightforward in the case of pure states, it is much more subtle for the more general case of mixed quantum states often found in practice. A large number of different proposals exist. In this review, we summarize the required properties of a quantum fidelity measure, and compare them, to determine which properties each of the different measures has. We show that there are large classes of measures that satisfy all the required properties of a fidelity measure, just as there are many norms of Hilbert space operators, and many measures of entropy. We compare these fidelities, with detailed proofs of their properties. We also summarize briefly the applications of these measures in teleportation, quantum memories and quantum computers, quantum communications, and quantum phase-space simulations.
We analyze the Uhlmann fidelity of a pair of nnn-mode Gaussian states of the quantum radiation field. This quantity is shown to be the product of an exponential function depending on the relative average displacement and a factor fully determined by the symplectic spectrum of the covariance matrix of a specific Gaussian state. However, it is difficult to handle our general formula unless the Gaussian states commute or at least one of them is pure. On the contrary, in the simplest cases n=1n=1n=1 and n=2n=2n=2, it leads to explicit analytic formulas. Our main result is a calculable expression of the fidelity of two arbitrary two-mode Gaussian states. This can be applied to build reliable measures of quantum correlations between modes in various branches of quantum physics.