We investigate the intrinsic ambiguity in the definition of correlation functions arising from the inevitable invasiveness of quantum measurements. While algebraic correlations defined as expectation values of products of observables are widely used, their relationship to operational ones defined through actual measurement procedures remain unclear. We demonstrate that the differences among various definitions of correlation functions and those among their underlying (quasi-)joint probability distributions are bounded above by a quantitative measure of measurement invasiveness. We further obtain a lower bound on the discrepancy among operational and algebraic (quasi-)joint probability distributions, providing a new form of the uncertainty relation. In addition, we identify an equivalence condition under which operational and algebraic correlations coincide. As an application, we analyze the quantum violation of the Leggett-Garg inequality and clarify the structural origin of the equivalence among different approaches to observing the violation, including sequential projective measurements and weak-measurement. Our results provide an operational foundation for the commonly used algebraic concepts of quantum theory.
We investigate the intrinsic ambiguity in the definition of correlation functions arising from the inevitable invasiveness of quantum measurements. While algebraic correlations defined as expectation values of products of observables are widely used, their relationship to operational ones defined through actual measurement procedures remain unclear. We demonstrate that the differences among various definitions of correlation functions and those among their underlying (quasi-)joint probability distributions are bounded above by a quantitative measure of measurement invasiveness. We further obtain a lower bound on the discrepancy among operational and algebraic (quasi-)joint probability distributions, providing a new form of the uncertainty relation. In addition, we identify an equivalence condition under which operational and algebraic correlations coincide. As an application, we analyze the quantum violation of the Leggett-Garg inequality and clarify the structural origin of the equivalence among different approaches to observing the violation, including sequential projective measurements and weak-measurement. Our results provide an operational foundation for the commonly used algebraic concepts of quantum theory.
The origin of non-classical correlations is difficult to identify since the uncertainty principle requires that information obtained about one observable invariably results in the disturbance of any other non-commuting observable. Here, this problem is addressed by investigating the uncertainty trade-off between measurement errors and disturbance for measurement interactions controlled by the state of a single qubit, where the measurement is described by a quantum coherent superposition of a fully projective measurement and the identity operation. It is shown that the measurement statistics obtained from a quantum controlled measurement of followed by a projective measurement of can be explained in terms of a simple combination of resolution and back-action errors acting on an intrinsic joint probability of the non-commuting observables defined by the input state of the system. These intrinsic joint probabilities are consistent with the complex-valued joint probabilities recently observed in weak measurements of quantum systems and provide direct evidence of non-commutativity in the form of imaginary correlations between the non-commuting operators. In quantum controlled measurements, these imaginary correlations can be converted into well-defined contributions to the real measurement statistics, allowing a direct experimental observation of the less intuitive aspects of quantum theory.
The origin of non-classical correlations is difficult to identify since the uncertainty principle requires that information obtained about one observable invariably results in the disturbance of any other non-commuting observable. Here, this problem is addressed by investigating the uncertainty trade-off between measurement errors and disturbance for measurement interactions controlled by the state of a single qubit, where the measurement is described by a quantum coherent superposition of a fully projective measurement and the identity operation. It is shown that the measurement statistics obtained from a quantum controlled measurement of followed by a projective measurement of can be explained in terms of a simple combination of resolution and back-action errors acting on an intrinsic joint probability of the non-commuting observables defined by the input state of the system. These intrinsic joint probabilities are consistent with the complex-valued joint probabilities recently observed in weak measurements of quantum systems and provide direct evidence of non-commutativity in the form of imaginary correlations between the non-commuting operators. In quantum controlled measurements, these imaginary correlations can be converted into well-defined contributions to the real measurement statistics, allowing a direct experimental observation of the less intuitive aspects of quantum theory.
A general prescription for the construction of quasi-joint-probability distributions intended to describe the "joint behaviour" of an arbitrary pair of (generally non-commuting) quantum observables, along with its application to the geometric/statistical interpretation of Aharonov's weak value and a proposal for new uncertainty relations, are presented.The introduction of such distributions is motivated by two complementary approaches: one from a bottom-up and strictly operational construction by carefully examining the mathematical framework of the conditioned measurement scheme, and the other from a top-down viewpoint by applying the results of spectral theorem for normal operators and its Fourier transforms.It is revealed that, for a pair of simultaneously measurable observables, the distribution reduces to the unique standard joint-probability distribution of the pair as expected, whereas for a non-commuting pair, there exists an inherent indefiniteness for the choice of such distributions, admitting a multitude of candidates that qualify to describe the "joint behaviour" of the pair.The L 2 structures induced by the quasi-joint-probability distributions are then found to provide statistical interpretation of the geometric structures that can be naturally introduced on the space of operators on the underlying Hilbert space.As possible applications of our findings, we focus on orthogonal projections and inner products of observables, and observe that such geometric concepts can respectively be given statistical interpretations as "conditionings" and "correlations".Incidentally, weak values A w are given a geometric/statistical interpretation as either the orthogonal projection of an observable A on the subspace generated by another observable B (or equivalently, as the conditioning of A given B with respect to the quasi-joint-probability distribution under consideration).Moreover, the Cauchy-Schwartz inequality applied on inner products of two observables (or equivalently, correlations between the pair) leads to novel inequalities interpreted as uncertainty relations for approximation/estimation, which entails the time-energy uncertainty relation as its special case.
This dissertation investigates statistical constraints on measurement outcomes in quantum mechanics, addressing two fundamental features that distinguish quantum theory from classical physics: non-reality and uncertainty.We examine these features through two perspectives: observable-dependent statistical constraints on quantum correlations and estimationbased measurement uncertainty in infinite-dimensional quantum systems.In the first part, we study realizability problems in both local realism and quantum theory, focusing on observable-dependent constraints.While previous studies have mainly characterized quantum correlations independently of observables, we derive a necessary and sufficient condition for quantum realizability of correlations in the Clauser-Horne-Shimony-Holt (CHSH) scenario when observables are fixed in advance.Our main result provides an explicit inequality that imposes stronger constraints than previously known observable-independent bounds.We demonstrate that the maximum CHSH inequality violation depends on observable choice and apply our results to Bell's original scenario, revealing that whether Bell's original inequality is violated depends sensitively on observable selection.These findings demonstrate that non-reality does not manifest equally in all measurement configurations.In the second part, we extend the estimation-based formulation of measurement uncertainty by Watanabe, Sagawa, and Ueda to infinite-dimensional quantum systems.The original formulation provided clear physical interpretation but excluded fundamental observables such as position and momentum.Using pseudo-inverse forms of positive-semidefinite sesquilinear forms, we develop classical and quantum estimation theory for statistical models whose parameter space is the set of full-rank quantum states on infinite-dimensional Hilbert spaces.We derive classical and quantum Cramér-Rao inequalities and define estimationbased measurement errors for both bounded and unbounded operators.For each error type, we establish error-error uncertainty relation inequalities stronger than the original Watanabe-Sagawa-Ueda bound due to an improved derivation utilizing the full Cauchy-Schwarz inequality.Our work provides a universal definition of measurement errors with a clear physical interpretation while maintaining mathematical rigor.Our results contribute to quantum foundations by deepening the understanding of how measurement outcome constraints arise from quantum structure and observable properties.The methodological tools introduced-including characteristic functions, quasi-joint probability distributions, symmetry-based reduction techniques, and pseudo-inverse forms-may prove useful for future investigations in quantum foundations and quantum information theory.
We show that the joint behavior of an arbitrary pair of (generally noncommuting) quantum observables can be described by quasi-probabilities, which are an extended version of the standard probabilities used for describing the outcome of measurement for a single observable. The physical situations that require these quasi-probabilities arise when one considers quantum measurement of an observable conditioned by some other variable, with the notable example being the weak measurement employed to obtain Aharonov’s weak value. Specifically, we present a general prescription for the construction of quasi-joint probability (QJP) distributions associated with a given combination of observables. These QJP distributions are introduced in two complementary approaches: one from a bottom-up, strictly operational construction realized by examining the mathematical framework of the conditioned measurement scheme, and the other from a top-down viewpoint realized by applying the results of the spectral theorem for normal operators and their Fourier transforms. It is then revealed that, for a pair of simultaneously measurable observables, the QJP distribution reduces to the unique standard joint probability distribution of the pair, whereas for a noncommuting pair there exists an inherent indefiniteness in the choice of such QJP distributions, admitting a multitude of candidates that may equally be used for describing the joint behavior of the pair. In the course of our argument, we find that the QJP distributions furnish the space of operators in the underlying Hilbert space with their characteristic geometric structures such that the orthogonal projections and inner products of observables can be given statistical interpretations as, respectively, “conditionings” and “correlations”. The weak value Aw for an observable A is then given a geometric/statistical interpretation as either the orthogonal projection of A onto the subspace generated by another observable B, or equivalently, as the conditioning of A given B with respect to the QJP distribution under consideration.
The Leggett-Garg inequality is a widely used test of the ``quantumness'' of a system and involves correlations between measurements realized at different times. According to its widespread interpretation, a violation of the Leggett-Garg inequality disproves macroscopic realism and noninvasiveness. Nevertheless, recent results point out that macroscopic realism is a model-dependent notion and that one should always be able to attribute to invasiveness a violation of a Leggett-Garg inequality. This opens some natural questions: How do we provide such an attribution in a systematic way? How can apparent macroscopic realism violation be recast into a dimensional-independent invasiveness model? The present work answers these questions by introducing an operational model where the effects of invasiveness are controllable through a parameter associated with what is called the measurability of the physical system. Such a parameter leads to different generalized measurements that can be associated with the dimensionality of a system, to measurement errors, or to back action.