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Register a new userJoint measurement of complementary observables in moment tomography
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Wigner and Husimi quasi-distributions, owing to their functional regularity, give the two archetypal and equivalent representations of all observable-parameters in continuous-variable quantum information. Balanced homodyning and heterodyning that correspond to their associated sampling procedures, on the other hand, fare very differently concerning their state or parameter reconstruction accuracies. We present a general theory of a now-known fact that heterodyning can be tomographically more powerful than balanced homodyning to many interesting classes of single-mode quantum states, and discuss the treatment for two-mode sources.
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We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity - some of which new - in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position-momentum, position-energy, number-phase, as well as periodic observables relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Pauls work on operational formulation of quantum measurements and central to his philosophy of unsharp reality.
We present a single inequality as the necessary and sufficient condition for two unsharp observables of a two-level system to be jointly measurable in a single apparatus and construct explicitly the joint observables. A complementarity inequality arising from the condition of joint measurement, which generalizes Englerts duality inequality, is derived as the trade-off between the unsharpnesses of two jointly measurable observables.
Heisenbergs uncertainty relations for measurement quantify how well we can jointly measure two complementary observables and have attracted much experimental and theoretical attention recently. Here we provide an exact tradeoff between the worst-case errors in measuring jointly two observables of a qubit, i.e., all the allowed and forbidden pairs of errors, especially asymmetric ones, are exactly pinpointed. For each pair of optimal errors we provide an optimal joint measurement that is realizable without introducing any ancilla and entanglement. Possible experimental implementations are discussed and Toronto experiment [Rozema et al., Phys. Rev. Lett. 109, 100404 (2012)] can be readily adapted to an optimal joint measurement of two orthogonal observables.
Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the states expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.
The two observables are complementary if they cannot be measured simultaneously, however they become maximally complementary if their eigenstates are mutually unbiased. Only then the measurement of one observable gives no information about the other observable. The spin projection operators onto three mutually orthogonal directions are maximally complementary only for the spin 1/2. For the higher spin numbers they are no longer unbiased. In this work we examine the properties of spin 1 Mutually Unbiased Bases (MUBs) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator onto some direction z, the states of the other MUBs have to be squeezed. Then, we introduce the analogs of momentum and position operators and interpret what information about the spin vector the observer gains while measuring them. Finally, we study the generation and the measurement of MUBs states by introducing the Fourier like transform through spin squeezing. The higher spin numbers are also considered.
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