Entropic Uncertainty Relations Research Articles

Realization of a source-device-independent quantum random number generator secured by nonlocal dispersion cancellation

Quantum random number generators (QRNGs) can provide genuine randomness by exploiting the intrinsic probabilistic nature of quantum mechanics, which play important roles in many applications. However, the true randomness acquisition could be subjected to attacks from untrusted devices involved or their deviations from the theoretical modeling in real-life implementation. We propose and experimentally demonstrate a source-device-independent QRNG, which enables one to access true random bits with an untrusted source device. The random bits are generated by measuring the arrival time of either photon of the time–energy entangled photon pairs produced from spontaneous parametric downconversion, where the entanglement is testified through the observation of nonlocal dispersion cancellation. In experiment, we extract a generation rate of 4 Mbps by a modified entropic uncertainty relation, which can be improved to gigabits per second by using advanced single-photon detectors. Our approach provides a promising candidate for QRNGs with no characterization or error-prone source devices in practice.

Entanglement based six-state quantum cryptography protocol: exact proof of security

A novel quantum key distribution (QKD) protocol is proposed, which uses six states of three states in two bases. The protocol provides a greater range of key distribution compared to known QKD protocols. The protocol has an accurate proof of secrecy based on fundamental entropy uncertainty relations, the use of which is possible with the entanglement version of the protocol. The implementation of the protocol does not require adjustment of the polarization state at the output of the communication line, since the receiving side does not use active polarization-sensitive elements—phase modulators, which virtually excludes active probing of the active elements of the system.

Quantum Brascamp–Lieb Dualities

Brascamp–Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory—including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of ‘geometric’ Brascamp–Lieb inequalities.

Controllable entropic uncertainty relation in an effective two-level non-Hermitian system

Under the large detuning conditions, a three-level atom in Λ-type configuration reduced to an effective two-level system has been investigated by Shahriar and Hemmer (1990 Phys. Rev. Lett. 65 1865). By introducing unbalanced gain and loss, we in detail investigate the behaviors of entropic uncertainty relation (EUR) in an effective two-level non-Hermitian system. Compared with the dynamics governed by a Hermitian Hamiltonian, we find the EUR in non-Hermitian system is well defined based on the exceptional points of the effective non-Hermitian system, and displays two distinct behaviors. More specifically, when systems possess an exceptional point above which the EUR undergoes an oscillatory behavior, while below which the EUR increases asymptotically to a finite constant value.

Quantum-memory-assisted entropic uncertainty and quantum correlation in structured reservoir

In this work, we investigate the dynamics of quantum-memory-assisted entropic uncertainty relation (EUR) and quantum correlation quantified by entropic discord and geometric discord within the Markovian reservoir and non-Markovian reservoir. Particularly, when the initial state is a pure sate or mixed state the EUR gradually increases during the evolution at first, but subsequently tends to stable in the long-time limit. On other hand, for the separable state the EUR almost keeps constant. In contrast, quantum discord decay asymptotically to zero for all types of initial states. As an application, we prove that both two quantum discords witnessed by the EUR are determined by the version of discord chosen, which regardless of the reservoir.

Quantitative description of uncertainty andentropic uncertainty relation

Uncertainty is a fundamental characteristic of quantum system. The degree of uncertainty of an observable has long been investigated by the standard deviation of the observable. In recent years, however, by analyzing some special examples, researchers have found that the Shannon entropy of the measurement outcomes of an observable is more suitable to quantify its uncertainty. Formally, Shannon entropy is a special limit of a more general Rényi entropy. In this paper, we discuss the problem of how to predict the measurement outcome of an observable by the existing measurement results of the observable, and how to quantitatively describe the uncertainty of the observable from the perspective of the repeatable probability of the measurement results of this observable in an unknown state. We will argue that if the same observable of different systems in the same state is repeatedly and independently measured many times, then the probability of obtaining an identical measurement result is a decaying function of the number of measurements of obtaining the same result, and the decay rate of the repeatable probability for obtaining the same measurement results and the repeatable number of measurements can represent the degree of uncertainty of the observable in this state. It means that the greater the uncertainty of an observable, the faster the repeatable probability decays with the number of repeatable measurements; conversely, the smaller the uncertainty, the slower the repeatable probability decays with the number of repeatable measurements. This observation enables us to give the Shannon entropy and the Rényi entropy of an observable uniformly by the functional relation between the repeatable probability and the number of repeatable measurements. We show that the Shannon entropy and the Rényi entropy can be formally regarded as the “decay index” of the repeatable probability with the number of repeatable measurements. In this way we also define a generalized Rényi entropy by the repeatable probability for consecutively observing identical results of an observable, and therefore we give a proof of the Maassen-Uffink type entropic uncertainty relation by using this generalized Rényi entropy. This method of defining entropy shows that entropic uncertainty relation is a quantitative limitation for the decay rate of the total probability for obtaining identical measurement results when we simultaneously measure two observables many times.

Tighter generalized entropic uncertainty relations in multipartite systems

The uncertainty principle, which demonstrates the intrinsic uncertainty of nature from an information-theory perspective, is at the heart of quantum information theory. In the realm of quantum information theory, Shannon entropy is used to depict the uncertainty relation in general. A tighter lower bound for uncertainty relations facilitates more accurate predictions of measurement outcomes and more robust quantum information processing. Interestingly, the tripartite entropic uncertainty relation (EUR) can be further optimized. Renes et al. proposed a tripartite EUR [J. M. Renes and J.-C. Boileau, Phys. Rev. Lett. 103, 020402 (2009)], and subsequently, Ming et al. strengthened its lower bound in [F. Ming, D. Wang, X.-G. Fan, W.-N. Shi, L. Ye, and J.-L. Chen, Phys. Rev. A 102, 012206 (2020)]. Specifically, we derive a tighter lower bound of the tripartite EUR using the Holevo quantity. Furthermore, we generalize the tripartite EUR, that is, the generalized entropic uncertainty relation for multiple measurements in multipartite systems. As illustrations, we provide several typical examples to show that our bound is tight and outperforms the previous bound. Furthermore, our findings pave the way for using the tighter bound for the quantum secret key rate in quantum key distribution protocols and are essential for quantum precision measurements in the framework of genuine multipartite systems. By providing a close peek at the nature of uncertainty, our results may find broad applications in the security analysis of quantum cryptography.

Non-locality and entropic uncertainty relations in neutrino oscillations

Using the wave-packet approach to neutrino oscillations, we analyze quantum-memory-assisted entropic uncertainty relations and show that uncertainty and the non-local advantage of quantum coherence are anti-correlated. Furthermore, we explore the hierarchy among three different definitions of NAQC, those based on l_{1}-norm, relative entropy and skew information coherence measures, and we find that the coherence content detected by the l_{1}-norm-based NAQC overcomes the other two. The connection between QMA-EUR and NAQC could provide a better understanding of the physical meaning of the results so far obtained and suggest their extension to quantum field theory.

Better Heisenberg Limits, Coherence Bounds, and Energy-Time Tradeoffs via Quantum Rényi Information.

An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form RMSE≥f(α)/(⟨N⟩+12), bounding the root mean square error of any estimate of a random optical phase shift in terms of average photon number, where f(α) is maximised for non-Shannon entropies. Related simple yet strong uncertainty relations linking phase uncertainty to the photon number distribution, such as ΔΦ≥maxnpn, are also obtained. These results are significantly strengthened via upper and lower bounds on the Rényi mutual information of quantum communication channels, related to asymmetry and convolution, and applied to the estimation (with prior information) of unitary shift parameters such as rotation angle and time, and to obtain strong bounds on measures of coherence. Sharper Rényi entropic uncertainty relations are also obtained, including time-energy uncertainty relations for Hamiltonians with discrete spectra. In the latter case almost-periodic Rényi entropies are introduced for nonperiodic systems.

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States.

The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.

Entropic uncertainty relation of two-qubit detectors coupled to scalar fields in an expanding spacetime

The principles of quantum information provide new avenues to investigate the cosmos. The uncertainty principle is an important trait of the nonclassical world, and it characterizes a significant lower bound (LB) that can be used to estimate measurement results for two noncommuting observables. Subsequently, the uncertainty principle is generalized to a new version by using the entropy and the quantum memory, that is, the quantum-memory-assisted entropic uncertainty relations (Q-M-A-E-U-Rs). Here, considering two qubit detectors coupled to scalar fields, we explore the effects of different cosmic parameters on Q-M-A-E-U-Rs and reveal the influence of the Holevo quantity on the LB of a Q-M-A-E-U-R. It is revealed that an increase in the expansion rapidity of spacetime can enhance the entropic uncertainty and decrease the ability to accurately predict the measurement outcome. The volume expansion leads to the invariance of the entropic uncertainty. An increase in the particle mass of the scalar field causes degradation in entropic uncertainty. In addition, the influence of the Holevo quantity on the Q-M-A-E-U-R’s LB can be ignored if one considers ({sigma _x},{sigma _y}) as two complementary observables. Therefore, one can use the Adabi bound and Berta bound to equivalently predict the left-hand side (LHS) of the Q-M-A-E-U-R in this situation. However, when ({sigma _x},{sigma _z}) are chosen as two noncommuting observables, the effect of the Holevo quantity on the LB of the uncertainty relation cannot be ignored, and the Adabi bound can always precisely achieve the LHS and predict entropic uncertainty.

Tripartite quantum-memory-assisted entropic uncertainty relations for multiple measurements

Quantum uncertainty relations are typically analyzed for a pair of incompatible observables, however, the concept per se naturally extends to situations of more than two observables. In this work, we obtain tripartite quantum-memory-assisted entropic uncertainty relations for multiple measurements and show that the lower bounds of these relations have three terms that depend on the complementarity of the observables, the conditional von-Neumann entropies, the Holevo quantities, and the mutual information. The saturation of these inequalities is analyzed.

Crossed-product entangled states

This paper demonstrates a new formalism of producing some entangled states attached to a two-particle system. We explain how these entangled states come directly from a new algebraic method through the cross-product of two spin coherent states. They lead to various quantum states with considerable nonclassical properties, and are capable candidates to minimize the entropic uncertainty relation, too. We will also examine and optimize the quantum properties of these states, for example by selecting the appropriate parameters one can control quantum (classical) correlations.

Witnessing criticality in non-Hermitian systems via entopic uncertainty relation

Non-Hermitian systems with exceptional points lead to many intriguing phenomena due to the coalescence of both eigenvalues and corresponding eigenvectors, in comparison to Hermitian systems where only eigenvalues degenerate. In this paper, we propose an alternative and accurate proposal based on the entropy uncertainty relation (EUR) to detect the exceptional points and identify different phases of the non-Hermitian systems. In particular, we reveal a general connection between the EUR and the exceptional points of non-Hermitian system. Compared to the unitary Hermitian dynamics, the behaviors of EUR in the non-Hermitian system are well defined into two different ways depending on whether the system is located in unbroken or broken phase regimes. In the unbroken phase regime where EUR undergoes an oscillatory behavior, while in the broken phase regime where the oscillation of EUR breaks down. Moreover, we identify the critical phenomena of non-Hermitian systems in terms of the EUR in the dynamical limit. It is found that the EUR can detect exactly the critical points of non-Hermitian systems beyond (anti-)PT symmetric systems. Finally, we comment on the prospective experimental situation.

Quantum memory and coherence dynamics of two dipole-coupled qubits interacting with two cavity fields under decoherence effect

This paper investigates the intrinsic decoherence effect on quantum memory and coherence dynamics of two dipole-coupled qubits in a non-degenerate bimodel cavity. Entropic uncertainty, entropy, and concurrence are utilized to examine the dynamics of quantum-memory-assisted entropic uncertainty relations, mixedness, and entanglement. The results indicate that the nonlinear interactions between the two qubits and the cavity generate quantum-memory-assisted entropic uncertainty, mixedness, and entanglement, which are reliant not only on the qubit–qubit interaction but also on the intrinsic decoherence and coherence intensity of the initial two-mode cavity states. For specific values of the dipole–dipole interaction, non-classical correlations can be enhanced By reducing the initial intensity coherence, the stability of the quantum memory-assisted entropic uncertainty relations, as well as the two-qubit mixedness of the intrinsic decoherence, are accelerated.

Generalized uncertainty relations for multiple measurements

The uncertainty relation is regarded as a remarkable feature of quantum mechanics differing from the classical counterpart, and it plays a backbone role in the region of quantum information theory. In principle, the uncertainty relation offers a nontrivial limit to predict the outcome of arbitrarily incompatible observed variables. Therefore, to pursue a more general uncertainty relations ought to be considerably important for obtaining accurate predictions of multi-observable measurement results in genuine multipartite systems. In this article, we derive a generalized entropic uncertainty relation (EUR) for multi-measurement in a multipartite framework. It is proved that the bound we proposed is stronger than the one derived from Renes et al. in [Phys. Rev. Lett. 103,020402(2009) ] for the arbitrary multipartite case. As an illustration, we take several typical scenarios that confirm that our proposed bound outperforms that presented by Renes et al. Hence, we believe our findings provide generalized uncertainty relations with regard to multi-measurement setting, and facilitate the EUR’s applications on quantum precision measurement regarding genuine multipartite systems.

Security of a High Dimensional Two‐Way Quantum Key Distribution Protocol

AbstractTwo‐way quantum key distribution (QKD) protocols utilize bi‐directional quantum communication to establish a shared secret key. Due to the increased attack surface, security analyses remain challenging. Here a high‐dimensional variant of the Ping Pong protocol is investigated and an information theoretic security analysis in the finite‐key setting is performed. The main contribution in this work is to show a new proof methodology for two‐way quantum key distribution protocols based on the quantum sampling framework of Bouman and Fehr introduced in 2010 and also sampling‐based entropic uncertainty relations introduced by the authors in 2019. The Ping Pong protocol is only investigated here, but these methods may be broadly applicable to other QKD protocols, especially those relying on two‐way channels. Along the way, some fascinating benefits to high‐dimensional quantum states applied to two‐way quantum communication are also showed.

Witnessing entanglement between two two-level atoms moving inside a leaky cavity under classical control

Based on the quantum memory-assisted entropic uncertainty relation (QMA-EUR), we investigate the Markovian and non-Markovian dynamics of entanglement witness in a two-atom system that asymmetrically are coupled to a leaky cavity with two-photon relaxation. The atoms are two-level systems, they move inside the cavity and interact with a classical driving laser field. We discuss in detail the effects of the translational movement of the atoms, classical driving field and atom-cavity coupling on the protection of entanglement witness. Our results show that with a regular increase in the intensity of the classical driving field, the entanglement region witnessed by lower bound of QMA-EUR increases and the time of entanglement witness gets longer. In the Markovian regime, we find that quantum entanglement between the moving atoms can be witnessed for a longer time by increasing the velocity of the atoms. However, in the non-Markovian regime, higher speeds are required to witness entanglement over long time intervals. In addition, we conclude that the entanglement region witnessed by the lower bound of QMA-EUR can be protected in both the symmetrical and asymmetrical atom-cavity coupling regimes. In the asymmetrical regime, a better protection can be achieved when the memory atom is coupled to the cavity field stronger than the other atom.

Dense Coding and Quantum Memory Assisted Entropic Uncertainty Relations in a Two‐Qubit State Influenced by Dipole and Symmetric Cross Interactions

AbstractOptimal dense coding and entropic uncertainty relation of quantum memory (EUR‐QM) of the two‐qubit Heisenberg chain model influenced by atomic dipole and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions are studied. The temporal evolution of the two functions at the phase decoherence environment with the Werner state as the initial state is discussed. The results imply that the dense coding and EUR‐QM have a contrary relationship, and an increase in the phase decoherence rate reduces the behavior of the two functions. The growth in coupling dipole interaction plays a role in oscillating and increasing the minimum bounds of the dense coding capacity. On the other hand, the effect of a thermal bath on the two functions at finite temperatures is investigated. The coding capacity suffers from sudden death as the temperature increases. The possibility of restraining the sudden death induced by the temperature may be enhanced by increasing the strength of the symmetric cross interaction. As the coupling of dipole increases, the EUR‐QM decreases while the coding capacity rises to its maximum value.

Quantifying incompatibility of quantum measurements through non-commutativity

The existence of incompatible measurements, i.e. measurements which cannot be performed simultaneously on a single copy of a quantum state, constitutes an important distinction between quantum mechanics and classical theories. While incompatibility might at first glance seem like an obstacle, it turns to be a necessary ingredient to achieve the so-called quantum advantage in various operational tasks like random access codes or key distribution. To improve our understanding of how to quantify incompatibility of quantum measurements, we define and explore a family of incompatibility measures based on non-commutativity. We investigate some basic properties of these measures, we show that they satisfy some natural information-processing requirements and we fully characterize the pairs which achieve the highest incompatibility (in a fixed dimension). We also consider the behavior of our measures under different types of compositions. Finally, to link our new measures to existing results, we relate them to a robustness-based incompatibility measure and two operational scenarios: random access codes and entropic uncertainty relations.