We address the "major open problem" of evaluating how much increased efficiency in estimation is possible using non-separable measurements, rather than separable ones, of N copies of two-level quantum systems. To proceed, we compute the the 3 x 3 Fisher information matrices for the optimal measurements recently devised by Vidal et al (quant-ph/9812068) for N = 2,...,7. We obtain simple polynomial expressions for the traces of the products of the inverse of the quantum Helstrom information matrix and the computed Fisher information matrices. These traces all have minima of 2 N - 1 in the pure state limit, while for separable measurements, as established by Gill and Massar (quant-ph/9902063), they are bound above by N. This indicates both that there is a large discontinuity at the pure states (where the computed Fisher information matrices are indeterminate) and that the gain in using joint measurements over separate measurements is substantial for all impure states. In seeking to extend our results to N > 7, we investigate possible relationships between the Fisher information matrices generated here and the class of monotone Riemannian metrics, studied by Petz and Sudar as well as others.
We address the "major open problem" of evaluating how much increased efficiency in estimation is possible using non-separable measurements, rather than separable ones, of N copies of two-level quantum systems. To proceed, we compute the the 3 x 3 Fisher information matrices for the optimal measurements recently devised by Vidal et al (quant-ph/9812068) for N = 2,...,7. We obtain simple polynomial expressions for the traces of the products of the inverse of the quantum Helstrom information matrix and the computed Fisher information matrices. These traces all have minima of 2 N - 1 in the pure state limit, while for separable measurements, as established by Gill and Massar (quant-ph/9902063), they are bound above by N. This indicates both that there is a large discontinuity at the pure states (where the computed Fisher information matrices are indeterminate) and that the gain in using joint measurements over separate measurements is substantial for all impure states. In seeking to extend our results to N > 7, we investigate possible relationships between the Fisher information matrices generated here and the class of monotone Riemannian metrics, studied by Petz and Sudar as well as others.
We address the `major open problem' of evaluating how much increased efficiency in estimation is possible using non-separable —as opposed to separable—measurements of N copies of m -level quantum systems. First, we study the six cases m = 2, N = 2, ..., 7 by computing the 3 × 3 Fisher information matrices for the corresponding optimal measurements recently devised by Vidal et al . We obtain simple polynomial expressions for the (`Gill–Massar') traces of the products of the inverse of the quantum Helstrom information matrix and these Fisher information matrices. The six traces all have minima of 2 N − 1 in the pure state limit—while for separable measurements, the traces can equal N , but not exceed it. Then, the result of an analysis for m = 3, N = 2 leads us to conjecture that for optimal measurements for all m and N , the Gill–Massar trace achieves a minimum of (2 N − 1)( m − 1) in the pure state limit.
We generalize the approach by Braunstein and Caves, [Phys. Rev. Lett. 72, 3439 (1994)]. to quantum multiparameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cram\'er-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cram\'er-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.
We consider the problem of estimating the state of a large but finite number N of identical quantum systems. As N becomes large the problem simplifies dramatically. The only relevant measure of the quality of estimation becomes the mean quadratic error matrix. Here we present a bound on this quantity: a quantum Cram\'er-Rao inequality. This bound succinctly expresses how in the quantum case one can trade information about one parameter for information about another. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states---a striking example of nonlocality without entanglement for mixed but not for pure states. Cram\'er-Rao bounds are generally only derived for unbiased estimators. Here we give a version of our bound for biased estimators, and a simple asymptotic version for large N. Finally we prove that when the unknown state belongs to a two-dimensional Hilbert space our quantum Cram\'er-Rao bound can always be attained, and we provide an explicit measurement strategy that attains it. Thus we have a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. The same is true for qubits in the same mixed state if one restricts oneself to separable measurements, but nonseparable measurements allow dramatic increase of efficiency. Exactly how much increase is possible is a major open problem.
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it.
A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties.
Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement.
Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output.
In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
Given a large number N of copies of a qubit state of which we wish to estimate its purity, we prove that separable-measurement protocols can be as efficient as the optimal joint-measurement one if classical communication is used. This shows that the optimal estimation of the entanglement of a two-qubit state can also be achieved asymptotically with fully separable measurements. Thus, quantum memories provide no advantage in this situation. The relationship between our global Bayesian approach and the quantum Cramér-Rao bound is discussed.
Quantum state estimation is important for various quantum information processes, including quantum communications, computation, and metrology, which require the characterization of quantum states for evaluation and optimization. We present a three-stage adaptive method for estimating arbitrary d-dimensional pure quantum states using locally informationally complete Fisher symmetric measurements (FSM) and a single-shot measurement basis. We derive finite-sample high-probability error bounds for the protocol and demonstrate that our approach scales as O(d/N) for large sample sizes, thereby guaranteeing the advantage of adaptation. Moreover, numerical simulations indicate that the protocol achieves an average infidelity close to the optimal given by the Gill-Massar lower bound (GMB). The total number of measurement outcomes scales linearly with 7d−3, avoiding the need for collective measurements on multiple copies of the unknown state. This work highlights the potential of adaptive estimation techniques in quantum state characterization while maintaining efficiency in the number of measurement outcomes.