Abstract We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure Q c is faithful since it vanishes only for commuting observables. We prove an upper bound for Q c for any set of observables in a finite-dimensional system and show that the upper bound is attained if and only if the observables are mutually unbiased. Finally, we use our formalism to obtain the optimal quantum cloner for a pair of qubit observables. Our work marks an important step in formalising the connection between two fundamental concepts in quantum information theory, namely, the no-cloning principle and the existence of incompatible observables in quantum theory.
Abstract We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure Q c is faithful since it vanishes only for commuting observables. We prove an upper bound for Q c for any set of observables in a finite-dimensional system and show that the upper bound is attained if and only if the observables are mutually unbiased. Finally, we use our formalism to obtain the optimal quantum cloner for a pair of qubit observables. Our work marks an important step in formalising the connection between two fundamental concepts in quantum information theory, namely, the no-cloning principle and the existence of incompatible observables in quantum theory.
After the appearance of the no-cloning theorem, approximate quantum cloning machines (QCMs) has become a well-studied subject in quantum information theory. Among several measures to quantify the performance of a QCM, single-qudit fidelity and global fidelity have been most widely used. In this paper we compute the optimal global fidelity for phase-covariant cloning machines via semidefinite programming optimization, thereby completing a remaining gap in the previous results on QCMs. We also consider optimal simulations of the cloning and transpose cloning map, both by a direct optimization and by a composition of component-wise optimal QCMs. For the cloning map the composition method is suboptimal whereas for the transpose cloning map the method is asymptotically optimal.
We pose the question whether the asymptotic equivalence between quantum cloning and quantum state estimation, valid at the single-clone level, still holds when all clones are examined globally. We conjecture that the answer is affirmative and present a large amount of evidence supporting our conjecture, developing techniques to derive optimal asymptotic cloners and proving their equivalence with estimation in virtually all scenarios considered in the literature. Our analysis covers the case of arbitrary finite sets of states, arbitrary families of coherent states, arbitrary phase- and multiphase-covariant sets of states, and two-qubit maximally entangled states. In all these examples we observe that the optimal asymptotic cloners enjoy a universality property, consisting in the fact that scaling of their fidelity does not depend on the specific details of the input states, but only on the number of free parameters needed to specify them.
We propose a class of incompatibility measures for quantum observables based on quantifying the effect of a measurement of one observable on the statistics of the outcomes of another. Specifically, for a pair of observables A and B with purely discrete spectra, we compare the following two probability distributions: one resulting from a measurement of A followed by a measurement of B on a given state and the other obtained from a measurement of B alone on the same state. We show that maximizing the distance between these two distributions over all states yields a valid measure of the incompatibility of observables A and B, which is zero if and only if they commute and is strictly greater than zero (and less than or equal to one) otherwise. For finite-dimensional systems, we obtain a tight upper bound on the incompatibility of any pair of observables and show that the bound is attained when the observables are totally nondegenerate and associated with mutually unbiased bases. In the process, we also establish an important relation between the incompatibility of a pair of observables and the maximal disturbances due to their measurements. Finally, we indicate how these measures of incompatibility and disturbance can be extended to the more general class of nonprojective measurements. In particular, we obtain a nontrivial upper bound on the incompatibility of one L\"uders instrument with another.
We discuss a recently proposed class of incompatibility measures for quantum measurements, which is based on quantifying the effect of measurements of one observable on the statistics of the outcome of another. We summarize the properties of this class of measures, and present a tight upper bound for the incompatibility of any set of projective measurements in finite dimensions. We also discuss non-projective measurements, and give a non-trivial upper bound on the mutual incompatibility of a pair of Luders instruments. Using the example of incompatible observables that commute on a subspace, we elucidate how this class of measures goes beyond uncertainty relations in quantifying the mutual incompatibility of quantum measurements.
Pivotal within quantum physics, the concept of quantum incompatibility is generally related to algebraic aspects of the formalism, such as commutation relations and unbiasedness of bases. Recently, the concept was identified as a resource in tasks involving quantum state discrimination and quantum programmability. Here, we link quantum incompatibility with the amount of information that can be extracted from a system upon successive measurements of noncommuting observables, a scenario related to communication tasks. This approach leads us to characterize incompatibility as a resource encoded in a physical context, which involves both the quantum state and observables. Moreover, starting with a measure of context incompatibility we derive a measurement-incompatibility quantifier that is easily computable, admits a geometrical interpretation, and is maximum only if the eigenbases of the involved observables are mutually unbiased.
This thesis is concerned with several topics related to concept of incompatibility of quan- tum observables. The operational description of quantum theory is given, in which incom- patibility is expressed in terms of joint measurability. A connection between symmetric informationally complete positive operator-valued measures and mutually unbiased bases is given, and examples of this connection holding based on investigations in Mathematica are presented. An extension of the Arthurs-Kelly measurement model is then given, where the measured observable is calculated, thereby generalising the results given previously in the literature. It is shown that in the case of prior correlations between measurement probes there exists the possibility that a measurement of both probes leads to marginal observables with smaller statistical spread than if measurements are performed on the individual probes. This concept is then highlighted by considering two probe states that allow for this reduction in spread, and the required conditions for success are given. Fi- nally, error-error relations for incompatible dichotomic qubit observables are considered in the case of state-dependent and independent error measures. Quantities that arise in the state-independent measures case, which were previously presented geometrically, have been given operational meaning, and optimal approximating schemes in both cases are compared. Limitations regarding the state-dependent optimal approximations, and experimental work built upon this construction are also discussed.