Minimum-cost quantum measurements for quantum information

Physical Review Letters, 1998

Optimal and finite positive operator valued measurements on a finite number N of identically prepared systems have been presented recently. With physical realization in mind we propose here optimal and minimal generalized quantum measurements for two-level systems. We explicitly construct them up to N = 7 and verify that they are minimal up to N = 5. We finally propose an expression which gives the size of the minimal optimal measurements for arbitrary N .

Physical Review A, 2004

We provide optimal measurement schemes for estimating relative parameters of the quantum state of a pair of spin systems. We prove that the optimal measurements are joint measurements on the pair of systems, meaning that they cannot be achieved by local operations and classical communication. We also demonstrate that in the limit where one of the spins becomes macroscopic, our results reproduce those that are obtained by treating that spin as a classical reference direction.

Physical Review A, 2012

Knowledge of optimal quantum measurements is important for a wide range of situations, including quantum communication and quantum metrology. Quantum measurements are usually optimised with an ideal experimental realisation in mind. Real devices and detectors are, however, imperfect. This has to be taken into account when optimising quantum measurements. In this paper, we derive the optimal minimum-cost and minimum-error measurements for a general model of imperfect detection.

We consider the problem of a state determination for a two-level quantum system which can be in one of two nonorthogonal mixed states. It is proved that for the two independent identical systems the optimal combined measurement (which considers the pair as one system) cannot be less optimal than the corresponding sequential one (local measurements, accompanying by transfer of classical information). The case of equality is achieved only when the mixed states have the same eigenvalues or the same eigenvectors. Further, we consider a case then the two systems are entangled: measurement of one system induces a reduction of the another one's state. The conclusion about optimal character of combined measurement takes place again, and conditions where the abovementioned methods coincide are derived.

Physical review letters, 2007

We consider the general measurement scenario in which the ensemble average of an operator is determined via suitable data processing of the outcomes of a quantum measurement described by a positive operator-valued measure. We determine the optimal processing that minimizes the statistical error of the estimation.

The purpose of this short note is to utilize work on isotropic lines in [1], on Wigner distributions for finite-state systems in [2], estimation of the state of a finite level quantum system based on Weyl operators in the L 2 -space over a finite field in [3] to display maximal abelian subsets of certain unitary bases for the matrix algebra M d of complex square matrices of order d > 3; and then, combine these special forms with constrained elementary measurements to obtain optimal ways to determine a d-level quantum state. This enables us to generalise illustrations and strengthen results related to quantum tomography in [4].

Physical Review Letters, 2006

We consider the problem of discriminating between states of a specified set with maximum confidence. For a set of linearly independent states unambiguous discrimination is possible if we allow for the possibility of an inconclusive result. For linearly dependent sets an analogous measurement is one which allows us to be as confident as possible that when a given state is identified on the basis of the measurement result, it is indeed the correct state.

Journal of Physics A-mathematical and General, 2005

Similarly to quantum states, also quantum measurements can be 'mixed', corresponding to a random choice within an ensemble of measuring apparatuses. Such mixing is equivalent to a sort of hidden variable, which produces a noise of purely classical nature. It is then natural to ask which apparatuses are indecomposable, i.e. do not correspond to any random choice of apparatuses. This problem is interesting not only for foundations, but also for applications, since most optimization strategies give optimal apparatuses that are indecomposable. Mathematically the problem is posed describing each measuring apparatus by a positive operator-valued measure (POVM), which gives the statistics of the outcomes for any input state. The POVMs form a convex set, and in this language the indecomposable apparatuses are represented by extremal points-the analogous of 'pure states' in the convex set of states. Differently from the case of states, however, indecomposable POVMs are not necessarily rank-one, e.g. von Neumann measurements. In this paper we give a complete classification of indecomposable apparatuses (for discrete spectrum), by providing different necessary and sufficient conditions for extremality of POVMs, along with a simple general algorithm for the decomposition of a POVM into extremals. As an interesting application, 'informationally complete' measurements are analysed in this respect. The convex set of POVMs is fully characterized by determining its border in terms of simple algebraic properties of the corresponding POVMs.

Physical Review A, 1999

Frontiers in Optics 2013 Postdeadline, 2013