Synthesis and reconstruction of quantum states (2)

In the second lesson, we set out to answer some fundamental questions: what is the optimum information that can be obtained from measurements made on a single copy of a quantum system? How can this information be quantified? More generally, what can we say about a system if we have a finite number N of copies? How does information about the state of the system increase with N? This problem is reminiscent of that of estimating a random variable in classical probability theory. The aim is to infer, from the results of the measurements, the value of the parameters that define the state. We began the lesson with a reminder of classical information theory, introducing Fisher's concept of information, the Cramer Rao limit and the maximum likelihood estimation method. In its simplest form, classical estimation theory associates an estimator θ(x) of any measurement result x of a random variable X obeying a probability distribution p(x|θ) dependent on an a priori unknown parameter θ. The variance of θ(x) averaged over a large number of measurements represents the precision of the estimate. If the mean of θ(x) over an infinite number of measurements corresponds to the true value of the parameter θ, the estimator is said to be "unbiased". The accuracy of an unbiased estimator is bounded below (Cramer Rao bound) by the quantity 1/I(θ) where I(θ), called Fisher information, is equal to the mean value of the square of the derivative with respect to θ of the log likelihood function p(x|θ). The greater the Fisher information, the smaller the lower bound on the variance of the estimate, in other words, the more potential information the statistical law contains for estimating θ. An estimate is optimal if its variance reaches the Cramer Rao bound. Based on these properties, we showed the additivity of Fisher information associated with independent measurements made on a set of N identical systems, which immediately leads to the well-known 1/√N variation in the standard deviation of the optimal estimate of the parameter θ when N measurements of the random variable are made. We then introduced the estimator θ(x) based on the maximum likelihood principle , which corresponds to the value of θ canceling the derivative of the likelihood function with respect to θ, and showed that this estimator is optimal in the limit of an infinite number of measurements. We conclude these reminders by illustrating them with a simple example, that of a binomial statistic (coin toss), showing that the well-known properties of this statistic can simply be found from an analysis based on Fisher information.

In the remainder of this lesson, we have applied these general considerations to the estimation of a qubit's state, the estimation problem being presented as a "quantum game" in which one player (Alice) presents another player (Bob) with N copies of the same qubit and asks him to make measurements on this system and deduce an estimate of the qubit's state. The score obtained by Bob after each deal is equal to the square of the dot product of the state proposed by Alice and that estimated by Bob. The final score of the game is the average of the scores over a very large number of deals. The questions we need to answer are, firstly, what is the maximum possible score as a function of N? And secondly, what measurement strategy can be used to reach this limit? We have shown that the optimal score is equal to (N + 1)/(N + 2) and that this score cannot be reached by individual measurements on the qubits, but requires a collective measurement strategy. Before demonstrating this general result, we considered successive qubit measurement strategies involving one, two or three components associated with the Pauli matrices _σx, _σy, _σz in the cases N = 1, 2 and 3, and then, in the limit where N tends to infinity, the measurement of N/3 qubits along each of the three directions Ox, Oy or Oz (so-called tomographic measurement). We showed that the score was then an increasing function of N, but was always less than (N + 1)/(N + 2). The last part of the lesson was devoted to demonstrating that (N + 1)/(N + 2) is indeed the optimum score. We began by describing a POVM collective measurement procedure for N qubits that enables this limit to be reached, then showed that this value cannot be exceeded by considering the most general possible collective measurements and establishing an upper bound on the score based on a variational method.