Understanding the relationship between various different forms of nonclassicality and their resource character is of great importance in quantum foundation and quantum information. Here we discuss a quantitative link between quantum entanglement and the anomalous or nonclassical nonreal values of Kirkwood-Dirac (KD) quasiprobability in a bipartite setting. We first construct an entanglement monotone for a pure bipartite state based on the nonreality of the KD quasiprobability defined over a pair of orthonormal bases, in which one of them is a product, and optimizations over these bases. It admits a closed expression as a Schur concave function of the state of the subsystem having a form of nonadditive quantum entropy. We then construct a bipartite entanglement monotone for generic quantum states using the convex roof extension. Its normalized value is upper bounded by the concurrence of formation, and for two-qubit systems, they are equal. We also derive lower and upper bounds in terms of different forms of uncertainty in the subsystem quantified by an extremal trace-norm asymmetry and a nonadditive quantum entropy, respectively. The entanglement monotone can be expressed as the minimum total state disturbance due to a nonselective local binary measurement. Finally, we discuss its estimation using weak value measurement and classical optimization, and its connection with strange weak value and quantum contextuality.
Understanding the relationship between various different forms of nonclassicality and their resource character is of great importance in quantum foundation and quantum information. Here we discuss a quantitative link between quantum entanglement and the anomalous or nonclassical nonreal values of Kirkwood-Dirac (KD) quasiprobability in a bipartite setting. We first construct an entanglement monotone for a pure bipartite state based on the nonreality of the KD quasiprobability defined over a pair of orthonormal bases, in which one of them is a product, and optimizations over these bases. It admits a closed expression as a Schur concave function of the state of the subsystem having a form of nonadditive quantum entropy. We then construct a bipartite entanglement monotone for generic quantum states using the convex roof extension. Its normalized value is upper bounded by the concurrence of formation, and for two-qubit systems, they are equal. We also derive lower and upper bounds in terms of different forms of uncertainty in the subsystem quantified by an extremal trace-norm asymmetry and a nonadditive quantum entropy, respectively. The entanglement monotone can be expressed as the minimum total state disturbance due to a nonselective local binary measurement. Finally, we discuss its estimation using weak value measurement and classical optimization, and its connection with strange weak value and quantum contextuality.
Abstract We propose a characterization and a quantification of the general quantum correlation which is exhibited even by a separable (unentangled) mixed bipartite state in terms of the nonclassical values of the associated Kirkwood–Dirac (KD) quasiprobability. Such a general quantum correlation, wherein entanglement is a subset, is not only intriguing from a fundamental point of view, but it has also been recognized as a resource in a variety of schemes of quantum information processing and quantum technology. Given a bipartite state, we construct a quantity based on the imaginary part the associated KD quasiprobability defined over a pair of orthonormal product bases and an optimization procedure over all pairs of such bases. We show that it satisfies certain requirements expected for a quantifier of general quantum correlations. It gives a lower bound to the total sum of the quantum standard deviation of all the elements of the product (local) basis, minimized over all such bases. It suggests an interpretation as the minimum genuine quantum share of uncertainty in all local von-Neumann projective measurements. Moreover, it is a faithful witness for entanglement and measurement-induced nonlocality of pure bipartite states. We then discuss a variational scheme for its estimation, and based on this, we offer information theoretical meanings of the general quantum correlation. Our results suggest a deep connection between the nonclassical concept of general quantum correlation and the nonclassical values of the KD quasiprobability and the associated strange weak values.
Abstract Just a few years after the inception of quantum mechanics, there has been a research program using the nonclassical values of some quasiprobability distributions to delineate the nonclassical aspects of quantum phenomena. In particular, in Kirkwood–Dirac (KD) quasiprobability distribution, the distinctive quantum mechanical feature of noncommutativity which underlies many nonclassical phenomena, manifests in the nonreal values and/or the negative values of the real part. Here, we develop a faithful quantifier of quantum coherence based on the KD nonclassicality which captures simultaneously the nonreality and the negativity of the KD quasiprobability. The KD-nonclassicality coherence thus defined, is upper bounded by the uncertainty of the outcomes of measurement described by a rank-1 orthogonal projection-valued measure (PVM) corresponding to the incoherent orthonormal basis which is quantified by the Tsallis 1 2 -entropy. Moreover, they are identical for pure states so that the KD-nonclassicality coherence for pure state admits a simple closed expression in terms of measurement probabilities. We then use the Maassen–Uffink uncertainty relation for min-entropy and max-entropy to obtain a lower bound for the KD-nonclassicality coherence of a pure state in terms of optimal guessing probability in measurement described by a PVM noncommuting with the incoherent orthonormal basis. We also derive a trade-off relation for the KD-nonclassicality coherences of a pure state relative to a pair of noncommuting orthonormal bases with a state-independent lower bound. Finally, we sketch a variational scheme for a direct estimation of the KD-nonclassicality coherence based on weak value measurement and thereby discuss its relation with quantum contextuality.
Kirkwood-Dirac (KD) quasiprobability is a quantum analog of classical phase space probability. It offers an informationally complete representation of quantum state wherein the quantumness associated with quantum noncommutativity manifests in its nonclassical values, i.e., the nonreal and/or negative values of the real part. This naturally raises a question: how does such form of quantumness comply with the uncertainty principle which also arises from quantum noncommutativity? Here, first, we obtain sufficient conditions for the KD quasiprobability defined relative to a pair of projection-valued measure (PVM) bases to have nonclassical values. Using these nonclassical values, we then introduce two quantities which capture the amount of KD quantumness in a quantum state relative to a single PVM basis. They are defined, respectively, as the nonreality and the classicality---which captures both the nonreality and negativity---of the associated KD quasiprobability over the PVM basis of interest, and another PVM basis, and maximized over all possible choices of the latter. We obtain their lower bounds, and derive trade-off relations respectively reminiscent of the Robertson and Robertson-Schr\"odinger uncertainty relations but with lower bounds maximized over the convex sets of Hermitian operators whose complete sets of eigenprojectors are given by the PVM bases. We discuss their measurement using weak value measurement and classical optimization. We then suggest an information theoretical interpretation of the KD nonreality relative to a PVM basis as a lower bound to the maximum total root-mean-squared error in an optimal estimation of the PVM basis, and thereby obtain a lower bound and a trade-off relation for the root-mean-squared error. Finally, we suggest an interpretation of the KD nonclassicality relative to a PVM basis as a lower bound to the total state disturbance caused by a nonselective projective binary measurement associated with the PVM basis, and derive a lower bound and a trade-off relation for the disturbance.
Abstract Kirkwood-Dirac (KD) quasiprobability distributions serve as quantum analog representations of the classical phase-space probability, capable of exhibiting nonclassical behavior through negativity or nonreal values. Recent studies have demonstrated that the nonclassicality of KD quasiprobability distributions plays a crucial role in quantum information tasks and provides deep insights into foundational aspects of quantum mechanics. In this work, we introduce an interpolated KD quasiprobability distribution parametrized by a continuous variable α , which smoothly connects different KD quasiprobability representations. Based on the interpolated KD quasiprobability distribution, we propose a quantitative measure of nonclassicality and rigorously analyze its key properties, including unitary covariance, convexity, and monotonicity. Furthermore, we derive an upper bound for this measure and conduct a comparison with other nonclassicality measures. Finally, we evaluate the nonclassicality measures for several prototypical interpolated KD quasiprobability distributions, illustrating their applicability in quantum information theory.
Kirkwood-Dirac (KD) quasiprobability is a quantum analog of phase space probability of classical statistical mechanics, allowing negative or/and nonreal values. It gives an informationally complete representation of a quantum state. Recent works have revealed the important roles played by the KD quasiprobability in the broad fields of quantum science and quantum technology. In the present work, we use the KD quasiprobability to access the quantum coherence in a quantum state. We show that the l1 norm of the imaginary part of the KD quasiprobability over an incoherent reference basis and a second basis, maximized over all possible choices of the latter, can be used to quantify quantum coherence, satisfying certain desirable properties. It is upper bounded by the quantum uncertainty, i.e., the quantum standard deviation, of the incoherent basis in the state. It gives a lower bound to the l1 norm quantum coherence, and for a single qubit, they are identical. We discuss the measurement of the KD coherence based on the measurement of the KD quasiprobability and an optimization procedure in hybrid quantum-classical schemes, and suggest statistical interpretations. We also discuss its relevance in the physics of linear response regime.
Quantum theory is fundamentally different from its classical counterpart. This difference has led to striking new insights in the foundations of physics and the theories of computation and information. Nevertheless, characterising the difference between classical and quantum theory has proven to be a difficult problem. In this thesis, I present my progress towards a general solution for this open problem. My main tool is the Kirkwood-Dirac (KD) quasiprobability distribution. The KD distribution provides an alternative description of quantum states, operations and measurements in terms of quasiprobability distributions. The KD quasiprobability distribution obeys all properties of probability distributions except that its values may, in general, lie in the complex unit disc instead of the interval [0,1]. For some experiments, however, all associated KD distributions are genuine probability distributions. For such `KD-positive’ states, operations and measurements, the KD distribution enables the construction of a classical model. The absence of such a construction can provide a natural criterion for the definition of the quantum-classical boundary. In the first part of my thesis, I present my results on the mathematical structure of the KD distribution. I summarise previous literature, and prove several new results, including a proof that the set of KD-positive states is almost always minimal, which was published in Physical Review A; novel bounds on the entries of the KD distribution and KD nonpositivity, a measure of how far a KD distribution is from being positive; and the construction of exotic states, mixed KD-positive states that cannot be written as the convex combination of pure KD-positive states, for Hilbert spaces of any odd dimension. These results lay the foundation for the objective of the remainder of this thesis: the application of the KD distribution in quantum foundations, computation and information. In the foundational setting, I study contextuality, a notion of nonclassicality often used in quantum foundations. An experiment is contextual if it is impossible to describe its indistinguishable procedures identically classically. I design an experiment that is contextual if and only if the input state of the experiment is not KD-positive. By analysing this connection with respect to the aforementioned exotic states, I obtain a quantum protocol that allows an experimenter to verify contextuality using a noncontextual experiment, thereby uncovering a previously unknown feature of contextuality. This section of the thesis is based on my publication in Physical Review A `Verification of contextuality by noncontextual experiments’ KDContextualityPaper. In the computational setting, I study the Delfosse-Guerin-Bian-Raussendorf (DGBR) model of quantum computation with a restricted set of input states. For certain sets of input states, this model is computationally as powerful as a universal quantum computer. However, for other sets of input states, this model is efficiently simulable on a classical computer and thus cannot be as powerful as a universal quantum computer, unless quantum computers offer no advantage over their classical counterparts. I prove that if all states in the set of input states are KD-positive, then the DGBR model is classically efficiently simulable. In this setting, some exotic states are bound-magic states, magic states that are nevertheless simulable input states in the DGBR model. I use my previous work on exotic states to construct the first bound-magic states on qubits. Furthermore, I show that KD nonpositivity is a resource monotone for quantum computation, establishing KD nonpositivity as a necessary resource for quantum computational advantage. This section of the thesis is based on my article `Kirkwood-Dirac nonpositivity is a necessary resource for quantum computing’, which has been accepted for publication in Physical Review Letters. In the information-theoretical setting, I study the uncertainty principle. In modern quantum information theory, the uncertainty principle is formulated as an `entropic lower bound’ on the sum of the entropies of the outcome distributions of two observables. I extend the definition of the standard entropic functions to the KD distribution and the Margenau-Hill (MH) distribution, which is the real part of the KD distribution. I introduce the MH Shannon entropy and the KD Rényi entropy, and establish their basic properties. For the MH Shannon entropy, I prove a generalised subadditivity relation, which yields a state-dependent entropic uncertainty relation that strengthens the previous best bound in most regimes and implies the finite-dimensional analogue of the Van Herstraeten--Cerf conjecture. For the KD Rényi entropy, I prove an entropic uncertainty relation strengthening the Maassen--Uffink bound, in which the KD nonpositivity appears as an additive correction. This yields a new interpretation of KD nonpositivity as a measure of quantum uncertainty, as well as a new entropic witness of KD positivity. Applying these results to cryptographic games, I show that KD nonpositivity is the key resource governing the optimal strategy in each case. This section is based on my article `Kirkwood-Dirac nonpositivity is sufficient for quantum uncertainty'. These results demonstrate that the KD distribution provides an effective formulation of quantum theory for studying the quantum-classical boundary in various settings. My work establishes explicit connections between the KD distribution and quantum foundations, computation, and information, opening up these fields to be studied using the extensive KD toolkit.