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We propose and experimentally demonstrate an optimal non-unity 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 trade-off is saturated by non-unity gain teleportation. Optimal partial measurement is demonstrated experimentally using a linear optics scheme with feed-forward.
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We investigate continuous variable quantum teleportation. We discuss the methods presently used to characterize teleportation in this regime, and propose an extension of the measures proposed by Grangier and Grosshans cite{Grangier00}, and Ralph and Lam cite{Ralph98}. This new measure, the gain normalized conditional variance product $mathcal{M}$, turns out to be highly significant for continuous variable entanglement swapping procedures, which we examine using a necessary and sufficient criterion for entanglement. We elaborate on our recent experimental continuous variable quantum teleportation results cite{Bowen03}, demonstrating success over a wide range of teleportation gains. We analyze our results using fidelity; signal transfer, and the conditional variance product; and a measure derived in this paper, the gain normalized conditional variance product.
The trade-off between the information gain and the state disturbance is derived for quantum operations on a single qubit prepared in a uniformly distributed pure state. The derivation is valid for a class of measures quantifying the state disturbance and the information gain which satisfy certain invariance conditions. This class includes in particular the Shannon entropy versus the operation fidelity. The central role in the derivation is played by efficient quantum operations, which leave the system in a pure output state for any measurement outcome. It is pointed out that the optimality of efficient quantum operations among those inducing a given operator-valued measure is related to Davies characterization of convex invariant functions on hermitian operators.
We establish a quantitative relation between Hardys paradox and the breaking of uncertainty principle in the sense of measurement-disturbance relations in the conditional measurement of non-commuting operators. The analysis of the inconsistency of local realism with entanglement by Hardy is simplified if this breaking of measurement-disturbance relations is taken into account, and a much simplified experimental test of local realism is illustrated in the framework of Hardys thought experiment. The essence of Hardys model is identified as a combination of two conditional measurements, which give rise to definite eigenvalues to two non-commuting operators simultaneously in hidden-variables models. Better understanding of the intimate interplay of entanglement and measurement-disturbance is crucial in the current discussions of Hardys paradox using the idea of weak measurement, which is based on a general analysis of measurement-disturbance relations.
We investigate the optimal tradeoff between information gained about an unknown coherent state and the state disturbance caused by the measurement process. We propose several optical schemes that can enable this task, and we implement one of them, a scheme which relies on only linear optics and homodyne detection. Experimentally we reach near optimal performance, limited only by detection inefficiencies. In addition we show that such a scheme can be used to enhance the transmission fidelity of a class of noisy channels.
It is often said that measuring a systems position must disturb the complementary property, momentum, by some minimum amount due to the Heisenberg uncertainty principle. Using a weak-measurement, this disturbance can be reduced. One might expect this comes at the cost of also reducing the measurements precision. However, it was recently demonstrated that a sequence consisting of a weak position measurement followed by a regular momentum measurement can probe a quantum system at a single point, with zero width, in position-momentum space. Here, we study this joint weak-measurement and reconcile its compatibility with the uncertainty principle. While a single trial probes the system with a resolution that can saturate Heisenbergs limit, we show that averaging over many trials can be used to surpass this limit. The weak-measurement does not trade-away precision, but rather another type of uncertainty called predictability which quantifies the certainty of retrodicting the measurements outcome.
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