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arXiv:quant-ph/0601156 · Physical Review A

Information–disturbance tradeoff in continuous-variable Gaussian systems

Marco G. Genoni, Matteo G. A. Paris

Published Jul 6, 2006• 23 citations
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Abstract

We address the information--disturbance tradeoff for state measurements on continuous variable Gaussian systems and suggest minimal schemes for implementations. In our schemes, the symbols from a given alphabet are encoded in a set of Gaussian signals which are coupled to a probe excited in a known state. After the interaction the probe is measured, in order to infer the transmitted state, while the conditional state of the signal is left for the subsequent user. The schemes are minimal, i.e., involve a single additional probe, and allow for the nondemolitive transmission of a continuous real alphabet over a quantum channel. The tradeoff between information gain and state disturbance is quantified by fidelities and, after optimization with respect to the measurement, analyzed in terms of the energy carried by the signal and the probe. We found that transmission fidelity only depends on the energy of the signal and the probe, whereas estimation fidelity also depends on the alphabet size and the measurement gain. Increasing the probe energy does not necessarily lead to a better tradeoff, the most relevant parameter being the ratio between the alphabet size and the signal width, which in turn determine the allocation of the signal energy.

DOI
10.1103/physreva.74.012301

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We propose and experimentally demonstrate an optimal nonunity gain Gaussian scheme for partial measurement of an unknown coherent state that causes minimal disturbance of the state. The information gain and the state disturbance are quantified by the noise added to the measurement outcomes and to the output state, respectively. We derive the optimal trade-off relation between the two noises and we show that the tradeoff is saturated by nonunity gain teleportation. Optimal partial measurement is demonstrated experimentally using a linear optics scheme with feedforward.
We address the class of positive operator-valued measures (POVMs) for qubit systems that are obtained by coupling the signal qubit with a probe qubit and then performing a projective measurement on the sole probe system. These POVMs, which represent the simplest class of qubit POVMs, depends on 3 + 3 + 2 = 8 free parameters describing the initial preparation of the probe qubit, the Cartan representative of the unitary coupling, and the projective measurement at the output, respectively. We analyze in some detail the properties of the POVM matrix elements, and investigate their values for given ranges of the free parameters. We also analyze in detail the tradeoff between information and disturbance for different ranges of the free parameters, showing, among other things, that (i) typical values of the tradeoff are close to optimality and (ii) even using a maximally mixed probe one may achieve optimal tradeoff.
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Quantum channels are indispensable instruments for transmitting, extracting, and processing information. According to the spirit of the Bohr complementarity principle and the Heisenberg uncertainty principle, one expects that there are intrinsic tradeoff relations between state disturbance and information gain for any channel, which indeed have been widely studied and characterized from various angles. In this work, we investigate this issue from the perspective of information conservation. More specifically, we divide the information associated with a channel into three categories: transmitted information (quantified by operation fidelity), disturbance (quantified by the Hilbert-Schmidt norm), and extracted information (quantified by the increase of Tsallis-2 entropy), which are all derived naturally from a channel with direct physical motivations. We reveal basic properties of these three information-theoretic quantities and establish a triality relation between them. As applications, we apply these quantities to shed insights into the Mach-Zehnder interferometry and several prototypical channels.