Point estimation of states of finite quantum systems

Quantum Probability and Infinite Dimensional Analysis, 2007

In the paper the Bayesian and the least squares methods of quantum state tomography are compared for a single qubit. The quality of the estimates are compared by computer simulation when the true state is either mixed or pure. The fidelity and the Hilbert-Schmidt distance are used to quantify the error. It was found that in the regime of low measurement number the Bayesian method outperforms the least squares estimation. Both methods are quite sensitive to the degree of mixedness of the state to be estimated, that is, their performance can be quite bad near pure states.

Physical Review A, 2005

We analyze the estimation of a qubit pure state by means of local measurements on N identical copies and compare its average fidelity for an isotropic prior probability distribution to the absolute upper bound given by collective measurements. We discuss two situations: the first one, where the state is restricted to lie on the equator of the Bloch sphere, is formally equivalent to phase estimation; the second one, where there is no constrain on the state, can also be regarded as the estimation of a direction in space using a quantum arrow made out of N parallel spins. We discuss various schemes with and without classical communication and compare their efficiency. We show that the fidelity of the most general collective measurement can always be achieved asymptotically with local measurements and no classical communication.

Journal of Applied Mathematics and Physics

An optimal estimator of quantum states based on a modified Kalman's filter is proposed in this work. Such estimator acts after a state measurement, allowing us to obtain an optimal estimate of the quantum state resulting in the output of any quantum algorithm. This method is much more accurate than other types of quantum measurements, such as, weak measurement, strong measurement, and quantum state tomography, among others.

An optimal estimator of quantum states based on a modified Kalman's Filter is proposed in this work. Such estimator acts after state measurement, allowing obtain an optimal estimation of quantum state resulting in the output of any quantum algorithm. This method is much more accurate than other types of quantum measurements, such as, weak measurement, strong measurement, quantum state tomography, among others.

2008

The properties of the indirect measurement scheme is investigated in this report in the discrete time case. The simplest possible case is considered, where both the unknown and the measurement quantum systems are quantum bits. The measurements applied on the measurement qubit are the classical von Neumann measurements using the Pauli matrices as observables. The statistical properties of the estimate in terms of the variance of the ML estimator and the non-demolition probability are analytically calculated in a simple case when the measurements are applied in the x direction while the qubits interact in the y direction and the initial state of the measurement qubit is [0, 0, c]. A way of finding an optimal compromising measurement strategy between the asymptotic variance and the non-demolition probability is also proposed. The efficiency of the results have been compared with a classical 'standard' state estimation procedure available in the literature. Although the classical one performs better by means of the variance, the indirect one gives a degree of freedom in the above mentioned trade-off problem. The estimation method has also been modified in a few ways to improve its precision.

arXiv (Cornell University), 2023

Quantum state estimation is an important task of many quantum information protocols. We consider two families of unitary evolution operators, one with a one-parameter and the other with a two-parameter, which enable the estimation of a single spin component and all spin components, respectively, of a two-level quantum system. To evaluate the tomographic performance, we use the quantum tomographic transfer function (qTTF), which is calculated as the average over all pure states of the trace of the inverse of the Fisher information matrix. Our goal is to optimize the qTTF for both estimation models. We find that the minimum qTTF for the one-parameter model is achieved when the entangling power of the corresponding unitary operator is at its maximum. The models were implemented on an IBM quantum processing unit, and while the estimation of a single-spin component was successful, the whole spin estimation displayed relatively large errors due to the depth of the associated circuit. To address this issue, we propose a new scalable circuit design that improves qubit state tomography when run on an IBM quantum processing unit.

We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N , we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N . We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.

Physical Review A, 2004

We analyze the estimation of a qubit pure state by means of local measurements on $N$ identical copies and compare its averaged fidelity for an isotropic prior probability distribution to the absolute upper bound given by collective measurements. We discuss two situations: the first one, where the state is restricted to lie on the equator of the Bloch sphere, is

Current Science, 2015

We explore the possibility of using "weak measurements" without "weak value" for quantum state estimation. Since for weak measurements the disturbance caused during each measurement is small, we can rescue and recycle the state, unlike for the case of projective measurements. We use this property of weak measurements and design schemes for quantum state estimation for qubits and for Gaussian states. We show, via numerical simulations, that under certain circumstances, our method can outperform the estimation by projective measurements. It turns out that ensemble size plays an important role and the scheme based on recycling works better for small ensembles. A. Von Neumann's measurement model for discrete basis Consider the measurement of an observable A of a quantum system with eigenvectors {|a j } and eigenvalues {a j }, j = 1 • • • n. Imagine an apparatus with continuous pointer positions described by a variable q and its

Physical Review Letters, 1998

We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable positive operator valued measurement (POVM) on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of N independent and identically prepared two-level systems (qubits).