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arXiv:2608.13088

Quantifying nonclassicality in qubit systems via positive operator-valued measures

Abdul Sattar Khan, Mehdi Abdi

quant-phSubmitted Aug 13, 2026• Announced Aug 14, 2026• 0 citations
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Abstract

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 $a_z$. 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.

Comments
10 pages, 4 figures

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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 aza_zaz​. 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 1/41/41/4, 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.
Determining the conditions under which positive operator-valued measures (POVMs), the most general class of quantum measurements, outperform projective measurements remains a challenging and largely unresolved problem. Of particular interest are projectively simulable POVMs, which can be realized through probabilistic mixtures of projective measurements, and therefore offer no advantage over projective schemes. Characterizing the boundary between simulable and non-simulable POVMs is, however, a difficult task, and existing tools either fail to scale efficiently, provide limited experimental feasibility or work only for specific POVMs. Here, we introduce and demonstrate a general method to certify non-simulability of a POVM by introducing a complete hierarchy of semidefinite programs. It provides upper bounds on the non-simulability measure of critical visibility of arbitrary POVMs which are tight in many cases and outperform previously known criteria. We experimentally certify the non-simulability of two- and three-dimensional POVMs using a trapped-ion qudit quantum processor by constructing non-simulability witnesses and introduce a modification of our framework that makes them robust against state preparation errors. Finally, we extend our results to the setting where an additional ancilla system is available.
In this paper we address the issue of the interplay between non-Markovianity and nonclassicality of a system. We consider the action of non-Markovian environments on a qubit in terms of two models, a classical noise model and a quantum microscopic collision model. For both models we use the violation of the Leggett-Garg inequality (LGI) as a criterion of quantumness, and use the conditional past-future (CPF) correlation to measure non-Markovianity. Besides for the collision model, we also use violation of the nondiscord generating and detecting (NDGD) dynamics to indicate the nonclassicality. And LGI, NDGD, and CPF are probed sequentially by three projective measurements of some observable. In this paper we do not use the assumption of the quantum regression theorem and especially for the collision model the measurement backaction on the environment is explicitly considered. We find that compared with the Markovian dynamics the LGI can be violated for a longer time interval, i.e., non-Markovianity can preserve the quantumness of the system. For the collision model we find that for a specific measurement process, although the non-Markovianity cannot be detected, the effect of non-Markovianity of the system dynamics on nonclassicality still exists, and for some specific measurement operator the violation of both LGI and NDGD will never occur for the Markovian dynamics while with the enhancement of non-Markovianity the quantumness can appear. We also find that the NDGD can be violated for a much wider parameter regime than the LGI.
Quantifying non-Markovianity is a fundamental problem in the theory of open quantum systems, as it is essential for exploiting environmental memory as a resource in quantum technologies. Existing resource-theoretic and operational quantifiers, however, generally require complete knowledge of the intermediate dynamical map and are therefore experimentally demanding. Here, we develop a witness-based framework for the quantification of non-Markovianity that bypasses complete reconstruction of the Choi state associated with the intermediate dynamical map. We first show that every non-Markovianity witness provides a certifiable lower bound on the geometric resource measure of non-Markovianity and identify the necessary and sufficient conditions under which this bound is tight. Motivated by the fact that non-positive-semidefinite witnesses are generally difficult to measure in real experimental scenario, we then focus on positive-semidefinite witnesses, which admit a natural implementation as positive operator valued measure (POVM) elements. We prove that every positive witness yields a certified lower bound on the Rivas--Huelga--Plenio (RHP) measure, thereby establishing a complete hierarchy connecting witness-based quantification, the RHP measure, and the geometric distance measure. Applying our framework to amplitude-damping and general Pauli dynamics, we find that the optimal positive witnesses reduce to simple Bell-parity projectors. Furthermore, we show that the resulting witness values can be obtained directly from a small number of experimentally accessible correlation measurements, thereby eliminating the need for complete Choi-state reconstruction and providing an efficient and experimentally feasible approach for the quantification of non-Markovianity in open quantum systems.
Abstract In the standard Bell scenario, when making a local projective measurement on each system component, the amount of randomness generated is restricted. However, this limitation can be surpassed through the implementation of sequential measurements. Nonetheless, a rigorous definition of random numbers in the context of sequential measurements is yet to be established, except for the lower quantification in device-independent scenarios. In this paper, we define quantum intrinsic randomness in sequential measurements and quantify the randomness in the Collins–Gisin–Linden–Massar–Popescu inequality sequential scenario. Initially, we investigate the quantum intrinsic randomness of the mixed states under sequential projective measurements and the intrinsic randomness of the sequential positive-operator-valued measure (POVM) under pure states. Naturally, we rigorously define quantum intrinsic randomness under sequential POVM for arbitrary quantum states. Furthermore, we apply our method to one-Alice and two-Bobs sequential measurement scenarios, and quantify the quantum intrinsic randomness of the maximally entangled state and maximally violated state by giving an extremal decomposition. Finally, using the sequential Navascues–Pironio–Acin hierarchy in the device-independent scenario, we derive lower bounds on the quantum intrinsic randomness of the maximally entangled state and maximally violated state.
In the heart of every physical theory lies measurement theory. While classical measurements are passive probes, QMs can be viewed as active physical processes that shape the available information. Depending on the measurement type, different aspects of the information about a system are retrieved. All possible measurements in quantum physics are described by mathematical objects called Positive Operator-Valued Measures (POVMs). The most familiar standard measurements are called Projective-Valued Measures (PVMs) and make up a subset of POVMs. The complementary subset of POVMs is non-projective. However, some non-projective POVMs can be expressed as a probabilistic mixture of PVMs. Such POVMs are called projectively simulable, while non-simulable POVMs are genuinely non-projective. These can be made projectively simulable by adding noise to the measurements. If a genuinely non-projective POVM requires a large amount of noise before it becomes projectively simulable, it is more non-projective than a POVM that requires less noise. Hence, the degree of non-projectiveness is determined by the required noise strength. The act of measuring a quantum system changes the quantum state. While quantum measurements only take into account the readable classical output value, quantum instruments account for both the classical outcome and the post-measurement state. Hence, instruments are associated with measurements, and in an analogous way, projective and projectively simulable instruments can be defined. For many purposes, non-projective POVMs are more useful than PVMs, and the more non-projective a POVM is, the greater the advantage of using them over PVMs. Therefore, the aim of the thesis is to investigate the genuinely non-projective POVMs. More precisely, the degree of non-projectiveness in qubit POVMs is studied. This is done by convex optimization, specifically by solving semidefinite programs. Furthermore, the thesis addresses the open problem of finding the most non-projective qubit quantum instrument by a witness-based see-saw algorithm.
We characterize a class of superclassical non-Markovian open quantum system dynamics that are defined by their lack of measurement invasiveness when the corresponding observable commutates with the premeasurement state. This diagonal noninvasiveness guarantees that joint probabilities for measurement outcomes fulfill classical Kolmogorov consistency conditions. These features are fulfilled regardless of the previous (measurement) system history and are valid at arbitrary later times after an arbitrary system initialization. It is shown that a subclass of depolarizing dynamics, which are based on a (time-irreversible) nonunitary system-environment coupling, satisfy the required properties. The relationship with other operational [S. Milz et al., Phys. Rev. X 10, 041049 (2020)] and nonoperational [M. Banacki et al., Phys. Rev. A 107, 032202 (2023)] notions of classicality in non-Markovian open quantum systems is studied in detail and exemplified through different examples.