In a recent paper (Buscemi et al 2005 J. Math. Phys. 46 082109) a notion of clean positive operator-valued measures (POVMs) was defined. We characterize those POVMs which are clean in a class that we call quasi-qubit POVMs, namely POVMs whose elements are all rank-1 or full-rank elements. We give an algorithm to check whether a given quasi-qubit POVM satisfies this condition. We describe explicitly all the POVMs that are clean for the qubit. Meanwhile, we give a sufficient condition for a general POVM to be clean.
In a recent paper (Buscemi et al 2005 J. Math. Phys. 46 082109) a notion of clean positive operator-valued measures (POVMs) was defined. We characterize those POVMs which are clean in a class that we call quasi-qubit POVMs, namely POVMs whose elements are all rank-1 or full-rank elements. We give an algorithm to check whether a given quasi-qubit POVM satisfies this condition. We describe explicitly all the POVMs that are clean for the qubit. Meanwhile, we give a sufficient condition for a general POVM to be clean.
In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.
A quantum probability measure–or quantum measurement–is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al. [“Clean positive operator valued measures,” J. Math. Phys. 46(8), 082109 (2005)10.1063/1.2008996] for finite-dimensional Hilbert space, and was studied subsequently by Kahn [“Clean positive operator-valued measures for qubits and similar cases,” J. Phys. A 40(18), 4817–4832 (2007)10.1088/1751-8113/40/18/009] and Pellonpää [“Complete characterization of extreme quantum observables in finite dimensions,” J. Phys. A 44(8), 085304 (2011)10.1088/1751-8113/44/8/085304]. The present paper provides new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of “approximately clean quantum probability measures” and characterise this property for measures whose range determines a finite-dimensional operator system.
Various forms of optimality for quantum observables described as normalized positive-operator-valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables that are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient conditions are imposed on informationally complete POVMs within the set of pure states. We also discuss joint and sequential measurements of optimal quantum observables.
There has been much interest in so-called SIC-POVMs, i.e., rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs that are symmetric and informationally complete, but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM that is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of constructing rank 1, symmetric, informationally complete POVMs. In this connection we derive an extremal condition alternative to the one derived by Renes et al.
Sufficient and necessary conditions are presented for the existence of ( N, M)-positive operator valued measures (( N, M)-POVMs) valid for arbitrary-dimensional quantum systems. Firstly, a sufficient condition for the existence of ( N, M)-POVMs is presented. It yields a simple relation determining an upper bound on the continuous parameter of an arbitrary ( N, M)-POVM, below which all its POVM elements are guaranteed to be positive semidefinite. Secondly, for arbitrary optimal ( N, M)-POVMs conditions on their existence are derived, which exhibit a close connection to the existence of sets of ( M − 1) N isospectral, traceless, orthonormal, and hermitian operators with particular common spectra. The specific form of the two possible types of spectra depends on whether M is smaller than the dimension of the quantum system under consideration or not. For the special case of M = 2 and dimensions, which are powers of two, these sets of operators always exist and can be expressed in terms of the Clifford algebra of tensor products of Pauli operators.
We introduce an operational measure of nonclassicality for qubit systems based on the violation of Kolmogorov consistency conditions in sequential measurements. In contrast to previous work by Milz et al.~\cite{Milz-2020}, who characterized classicality via NCGD maps, and Sakuldee et al.~\cite{Sakuldee-2022a, Sakuldee-2022b}, who studied quantum correlations under measurement disturbance, our witness explicitly quantifies nonclassicality in terms of the POVM unsharpness parameter az. For projective measurements on an initially diagonal state, the witness vanishes identically, showing that such measurements cannot reveal nonclassicality. However, by generalizing to positive operator-valued measures (POVMs), we find that non-projective measurements can reveal nonclassicality even for diagonal initial states. We derive explicit expressions for the witness for general initial states, including coherences, and show that its maximum value is 1/4, which is a new result achieved for unbiased POVMs, maximal dephasing, and equal initial populations. Our results connect Kolmogorov consistency, Leggett-Garg inequalities, and POVMs, providing an experimentally accessible tool for detecting nonclassicality in qubit systems with potential applications in quantum technology certification.