Information-disturbance tradeoff for spin coherent state estimation

The tradeoff between information retrieved from a quantum measurement and the disturbance caused on the state of a quantum system is a fundamental concept of quantum mechanics and has received a lot of attention in the literature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Such an issue is studied for both foundations and its enormous relevance in practice, in the realm of quantum key distribution and quantum cryptography [21, 22]. Quantitative derivations of such a tradeoff have been obtained in the scenario of quantum state estimation [23, 24]. The optimal tradeoff has been derived in the following cases: in estimating a single copy of an unknown pure state [7], many copies of identically prepared pure qubits [9] and qudits [14], a single copy of a pure state generated by independent phase-shifts [13], an unknown maximally entangled state [18], an unknown coherent state [17] and Gaussian state [19]. Experimental realization of minimal-disturbing measurements has been also reported [15, 17]. Recently, the optimal tradeoff has been also derived for quantum state discrimination [20]. The problem is typically the following. One performs a measurement on a quantum state picked (randomly, or according to an assigned a priori distribution) from a known set, and evaluates the retrieved information along with the disturbance caused on the state. To quantify the tradeoff between information and disturbance, one can adopt two mean fidelities [7]: the estimation fidelity G, which evaluates on average the best guess we can do of the original state on the basis of the measurement outcome, and the operation fidelity F , which measures the average resemblance of the state of the system after the measurement to the original one. In this paper, we study the optimal tradeoff between estimation and operation fidelities when the state is a completely unknown spin coherent state. Our results will be obtained by exploiting the group simmetry of the problem, which allows us to restrict our analysis on covariant measurement instruments. In fact, the property of covariance generally leads to a striking