Abstract
Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between complementary pieces of information about the system. In quantum multiparameter estimation, such trade-offs occur for the precision achievable for different parameters characterizing a density matrix: an uncertainty relation emerges between the achievable variances of the different estimators. This is in contrast to classical multiparameter estimation, where simultaneous optimal precision is attainable in the asymptotic limit. We study trade-off relations that follow from known tight bounds in quantum multiparameter estimation. We compute trade-off curves and surfaces from Cramér–Rao type bounds which provide a compelling graphical representation of the information encoded in such bounds, and argue that bounds on simultaneously achievable precision in quantum multiparameter estimation should be regarded as measurement uncertainty relations. From the state-dependent bounds on the expected cost in parameter estimation, we derive a state-independent uncertainty relation between the parameters of a qubit system.
Highlights
We study trade-off relations that follow from known tight bounds in quantum multiparameter estimation
This paper illustrated the fact that the unsaturability of the quantum Fisher information Cramér–Rao bound for multi parameter estimation gives rise to a rich variety of quantum uncertainty relations in the form of trade-off curves
This can be seen as a parameter estimation analogue of the quantum Chernoff and Hoeffding bounds [64] in quantum hypothesis testing, where trade-off curves are obtained for the error exponents αi for the error of the rst versus the second kind—scaling as exp (−αiN)
Summary
In classical estimation theory [40] we are given a family of probability distributions with probability density p (θ) parametrized by a vector of parameters θ = The task is to estimate the unknown values θ0 by sampling from p(θ0). In order to do so we shall pick an estimator, a function that produces an estimated value θ(x1, x2, . XN) given the N samples drawn {xi}. The estimation statistics are described by the random variable θ(X1, X2, . XN), where the random variables Xi are distributed according to p(xi|θ0) := p(X = xi|θ0). An estimator is called locally unbiased if Eθ = θ0, where E is the expectation value is with respect to p(θ0)
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