A possible mechanism of time is formulated by developing an idea of time replaced by quantum correlations, with the aid of modern quantum information theory. We invent a microscopic model, where correlations of a closed system are steadily read out as internal, quantum clocks that define time via their relative phases. The model could realize emergent time evolutions which exhibit unitarity of quantum theory, while its underlying process is driven entropically. The key quantity turns out to be the amount of accessible information about the clocks recording past events. By postulating the so-called data-processing inequality (or strong subadditivity of entropy) as a fundamental, physical limitation about how information decays, we propose that conditional entropy about this past information should be constrained to be a positive constant. The proposal implies a holographic property of this conditional entropy in an analogous manner with the area law of entanglement entropy.
We introduce the concept of entropic nonsignaling correlations, i.e., entropies arising from probabilistic theories that are compatible with the fact that we cannot transmit information instantaneously. We characterize and show the relevance of these entropic correlations in a variety of different scenarios, ranging from typical Bell experiments to more refined descriptions such as bilocality and information causality. In particular, we apply the framework to derive the first entropic inequality testing genuine tripartite nonlocality in quantum systems of arbitrary dimension and also prove the…
A framework for categorizing entropic measures of nonclassical correlations in bipartite quantum states is presented. The measures are based on the difference between a quantum entropic quantity and the corresponding classical quantity obtained from measurements on the two systems. Three types of entropic quantities are used, and three different measurement strategies are applied to these quantities. Many of the resulting measures of nonclassical correlations have been proposed previously. Properties of the various measures are explored, and results of evaluating the measures for two-qubit qua…
Quantum correlation often refers to correlations exhibited by two or more local subsystems under a suitable measurement. These correlations are beyond the framework of classical statistics and the associated classical probability distribution. Quantum entanglement is the most well known of such correlations and plays an important role in quantum information theory. However, there exist non-entangled states that still possess quantum correlations which cannot be described by classical statistics. One such measure that captures these nonclassical correlations is discord. Here we introduce a new …
We investigate whether the entropic regularisation of the strictly-correlated-electrons problem can be used to build approximations for the kinetic correlation energy functional at large coupling strengths and, more generally, to gain new insight in the problem of describing and understanding strong correlation within Density Functional Theory.
Recently, it has been shown that there exist quantum states for which quantum correlations dominate over classical correlations. Inspired by this observation, we investigate the problem of quantum correlations dominance for two-qubit Bell diagonal states in the Ollivier--Zurek paradigm, using both entropic and geometric approach to quantification of classical and quantum correlations. In particular, we estimate numerically the amount of two-qubit Bell diagonal states for which quantum correlations dominate over classical correlations, and vice versa. Moreover, we show that in general these two…
The existence of a global causal order between events places constraints on the correlations that parties may share. Such "causal correlations" have been the focus of recent attention, driven by the realization that some extensions of quantum mechanics may violate so-called causal inequalities. In this paper we study causal correlations from an entropic perspective, and we show how to use this framework to derive entropic causal inequalities. We consider two different ways to derive such inequalities. Firstly, we consider a method based on the causal Bayesian networks describing the causal rel…
We extend the Hilbert-Schmidt (square norm) distance, previously used to define the geometric quantum discord, to define also geometric quantifiers of total and classical correlations. We then compare the dynamics of geometric and entropic quantifiers of the different kinds of correlations in a non-Markovian open two-qubit system under local dephasing. We find that qualitative differences occur only for quantum discords. This is taken to imply that geometric and entropic discords are not, in general, equivalent in describing the dynamics of quantum correlations. We then show that also geometri…
We introduce quantum correlations measures based on the minimal change in unified entropies induced by local rank-one projective measurements, divided by a factor that depends on the generalized purity of the system in the case of non-additive entropies. In this way, we overcome the issue of the artificial increasing of the value of quantum correlations measures based on non-additive entropies when an uncorrelated ancilla is appended to the system without changing the computability of our entropic correlations measures with respect to the previous ones. Moreover, we recover as limiting cases t…
The uncertainty principle is one of the comprehensive and fundamental concept in quantum theory. This principle states that it is not possible to simultaneously measure two incompatible observatories with high accuracy. Uncertainty principle has been formulated in various form. The most famous type of uncertainty relation is expressed based on the standard deviation of observables. In quantum information theory the uncertainty principle can be formulated using Shannon and von Neumann entropy. Entropic uncertainty relation in the presence of quantum memory is one of the most useful entropic unc…
Heisenberg's uncertainty principle is a fundamental element in quantum mechanics. It sets a bound on our ability to predict the measurement outcomes of two incompatible observables simultaneously. In quantum information theory, the uncertainty principle can be expressed using entropic measures. The entropic uncertainty relation can be improved by considering an additional particle as a memory particle. The presence of quantum correlation between the memory particle and the measured particle reduces the uncertainty. In a curved space-time, the presence of the Hawking radiation can reduce quantu…
We formulate entropic Leggett-Garg inequalities, which place constraints on the statistical outcomes of temporal correlations of observables. The information theoretic inequalities are satisfied if macrorealism holds. We show that the quantum statistics underlying correlations between time-separated spin component of a quantum rotor mimics that of spin correlations in two spatially separated spin-$s$ particles sharing a state of zero total spin. This brings forth the violation of the entropic Leggett-Garg inequality by a rotating quantum spin-$s$ system in similar manner as does the entropic B…
We propose a general measure of non-classical correlations for bipartite systems based on generalized entropic functions and majorization properties. Defined as the minimum information loss due to a local measurement, in the case of pure states it reduces to the generalized entanglement entropy, i.e., the generalized entropy of the reduced state. However, in the case of mixed states it can be non-zero in separable states, vanishing just for states diagonal in a general product basis, like the Quantum Discord. Simple quadratic measures of quantum correlations arise as a particular case of the p…
Quantum correlations have fundamental and technological interest, and hence many measures have been introduced to quantify them. Some hierarchical orderings of these measures have been established, e.g., discord is bigger than entanglement, and we present a class of bipartite states, called premeasurement states, for which several of these hierarchies collapse to a single value. Because premeasurement states are the kind of states produced when a system interacts with a measurement device, the hierarchy collapse implies that the uncertainty of an observable is quantitatively connected to the q…
We discuss a generalization of the conditional entropy and one-way information deficit in quantum systems, based on general entropic forms. The formalism allows to consider simple entropic forms for which a closed evaluation of the associated optimization problem in qudit-qubit systems is shown to become feasible, allowing to approximate that of the quantum discord. As application, we examine quantum correlations of spin pairs in the exact ground state of finite $XY$ spin chains in a magnetic field through the quantum discord and information deficit. While these quantities show a similar behav…
We show how to quantify tri-partite entanglement using entropies derived from experimental correlations. We use a multi-partite generalization of the entanglement of formation that is greater than zero if and only if the state is genuinely multi-partite entangled. We develop an entropic witness for tripartite entanglement, and show that the degree of violation of this witness places a lower limit on the tripartite entanglement of formation. We test our results in the three-qubit regime using the GHZ-Werner state and the W-Werner state, and in the high-dimensional pure-state regime using the tr…
With the aid of a quantum memory, the uncertainty about the measurement outcomes of two incompatible observables of a quantum system can be reduced. We investigate this measurement uncertainty bound by considering an additional quantum system connected with both the quantum memory and the measured quantum system. We find that the reduction of the uncertainty bound induced by a quantum memory, on the other hand, implies its increasing for a third participant. We also show that the properties of the uncertainty bound can be viewed from perspectives of both quantum and classical correlations, in …
For noncomposite systems in classical and quantum domains, we obtain new inequalities such as the subadditivity and strong subadditivity conditions for Shannon entropies and information determined by the probability distributions and for von Neumann entropies of quantum states determined by the density operators. We extend the relations of Shannon and Tsallis entropies to the entropies of conditional probability distributions known for composite systems to the case of noncomposite systems. We give a review of the approach to construct the tomographic-probability distributions for qudit systems…
We use entropic uncertainty relations to formulate inequalities that witness Einstein-Podolsky-Rosen (EPR) steering correlations in diverse quantum systems. We then use these inequalities to formulate symmetric EPR-steering inequalities using the mutual information. We explore the differing natures of the correlations captured by one-way and symmetric steering inequalities, and examine the possibility of exclusive one-way steerability in two-qubit states. Furthermore, we show that steering inequalities can be extended to generalized positive operator valued measures (POVMs), and we also derive…
Recently, an inequality satisfied by non-contextual hidden-variable models and violated by quantum mechanics for all states of a four-level system has been derived based on information-theoretic distance approach to non-classical correlations. In this work, we experimentally demonstrate violation of this inequality with single photons. Our experiment offers a method to study a distinction between quantum and classical correlations from an information-theoretic perspective.
We discuss the notion of hidden correlations in classical and quantum indivisible systems along with such characteristics of the correlations as the mutual information and conditional information corresponding to the entropic subadditivity condition and the entropic strong subadditivity condition. We present an analog of the Bayes formula for systems without subsystems, study entropic inequality for von Neumann entropy and Tsallis entropy of the single-qudit state, and discuss the inequalities for qubit and qutrit states as an example.
We introduce a multipartite extension of an information-theoretic distance first introduced in [Nature 341, 119 (1989)]. We use this new distance to derive entropic tests of multipartite nonlocality for three and for an arbitrary even number of qubits as well as a test of state-independent contextuality. In addition, we re-derive the tripartite Mermin inequality and a state-independent non-contextuality inequality by Cabello [Phys. Rev. Lett. 101, 210401 (2008)]. This suggests that the information-theoretic distance approach to multipartite nonlocality and state-independent contextuality can p…
We derive a new memory-assisted entropic uncertainty relation for non-degenerate Hermitian observables where both quantum correlations, in the form of conditional von Neumann entropy, and quantum discord between system and memory play an explicit role. Our relation is `universal', in the sense that it does not depend on the specific observable, but only on properties of the quantum state. We contrast such an uncertainty relation with previously known memory-assisted relations based on entanglement and correlations. Further, we present a detailed comparative study of entanglement- and discord-a…
Evidence for cavity-induced metallic phase from entropic electron correlation effects
Max Planck Institute for the Structure and Dynamics of Matter（马克斯·普朗克物质与结构动力学研究所）；Center for Free-Electron Laser Science（自由电子激光科学中心）；The Hamburg Center for Ultrafast Imaging（汉堡超快成像中心）；Zurich University of Applied Sciences, School of Engineering（苏黎世应用科技大学工程学院）
The aim of this work is to verify the new entropic and information inequalities for non-composite systems using experimental $5 \times 5$ density matrix of the qudit state, measured by the tomographic method in a multi-level superconducting circuit. These inequalities are well-known for bipartite and tripartite systems, but have never been tested for superconducting qudits. Entropic inequalities can also be used to evaluate the accuracy of experimental data and the value of mutual information, deduced from them, may charachterize correlations between different degrees of freedom in a noncompos…
We will discuss the generalization of entropic uncertainty principles in terms of a game. The game involves k-players, each measuring one of k possible observables. The question is, what is the maximum number of players that can play such that their joint entropic uncertainties are tightened by the presence of non-classical correlations? We answer this question and relate it to the ratio of quantum mutual information to classical correlations. This game hence serves to construct an operational interpretation of the aforementioned ratio. It provides another example of quantum correlations provi…
The celebrated Bell-Kochen-Specker no-go theorem asserts that quantum mechanics does not present the property of realism, the essence of the theorem is the lack of a joint probability distributions for some experiment settings. In this work, we exploit the information theoretic form of the theorem using information measure instead of probabilistic measure and indicate that quantum mechanics does not present such entropic realism neither. The entropic form of Gleason's no-disturbance principle is developed and it turns out to be characterized by the intersection of several entropic cones. Entro…
New entropic inequalities are obtained for Franck-Condon factors (FCF-s) of diatomic molecules in adiabatic harmonic oscillator approximation. The probability distributions associated with FCF-s of diatomic molecules are presented in the form of joint-probability distributions of either two or three random variables. In view of this representation information inequalities corresponding to subadditivity condition is found and in case of harmonic oscillator approximation is obtained in explicit form of new inequalities for Hermite polynomials. Physical meaning of obtained entropic-information in…
We present a derivation of the Von Neumann entropy and mutual information of arbitrary two--mode Gaussian states, based on the explicit determination of the symplectic eigenvalues of a generic covariance matrix. The key role of the symplectic invariants in such a determination is pointed out. We show that the Von Neumann entropy depends on two symplectic invariants, while the purity (or the linear entropy) is determined by only one invariant, so that the two quantities provide two different hierarchies of mixed Gaussian states. A comparison between mutual information and entanglement of format…
The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find entanglement-dependent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem~\ref{th:bound}. Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on t…
Spin-outcome correlations and memory-assisted entropic uncertainty in Bhabha scattering with an entangled spectator
State Key Laboratory of Opto-Electronic Information Acquisition and Protection Technology, School of Physics, Anhui University（安徽大学光电信息获取与保护技术国家重点实验室物理学院）
Classical and quantum physics impose different constraints on the joint probability distributions of observed variables in a causal structure. These differences mean that certain correlations can be certified as non-classical, which has both foundational and practical importance. Rather than working with the probability distribution itself, it can instead be convenient to work with the entropies of the observed variables. In the Bell causal structure with two inputs and outputs per party, a technique that uses entropic inequalities is known that can always identify non-classical correlations. …
We consider information characteristics of single qudit state (spin j=9/2), such as von Neumann entropy, von Neumann mutual information. We review different mathematical properties of these information characteristics: subadditivity and strong subadditivity conditions, Araki-Lieb inequality. The inequalities are entropic inequalities for composite systems (bipartite, tripartite), but they can be written for noncomposite systems. Using the density matrix, describing the noncomposite qudit system state in explicit matrix form we proved new entropic inequalities for single qudit state (spin j=9/2…
For a pair of incompatible quantum measurements, the total uncertainty can be bounded by a state-independent constant. However, such a bound can be violated if the quantum system is entangled with another quantum system (called memory); the quantum correlation between the systems can reduce the measurement uncertainty. On the other hand, in a curved spacetime, the presence of the Hawking radiation can increase the uncertainty in quantum measurement. The interplay of quantum correlation in the curved spacetime has become an interesting arena for studying quantum uncertainty relations. Here we d…
By using relative entropy of coherence, we characterize the coherence gain induced by some quantum evolutions, including the cohering power of unitary operations and the decohering power of quantum operations. We find that the cohering power of the controlled unitary operation can be reduced to the cohering power of the corresponding unitary operation. We observe that the global coherence generated via incoherent operation applied to the system and an incoherent ancilla do not exceed the amount of coherence contained in the initial system. Our result provides a much tighter lower bound of cohe…
The uncertainty of measurement on a quantum system can be reduced in presence of quantum memory [M. Berta et. al. Nature Phys. {\bf 6}, 659 (2010)]. By measurement on quantum memory, some information (non-classical information) is inaccessible, hence, it limits the lower bound of entropic uncertainty relation. In the present work, we improve the lower bound of entropic uncertainty relation in presence of quantum memory by considering inaccessibility of non classical correlation at Bob side and also compare the lower bound given by M. Berta et. al. and our prescription for two different kind of…
We address the experimental determination of entropic quantum discord for systems made of a pair of polarization qubits. We compare results from full and partial tomography and found that the two determinations are statistically compatible, with partial tomography leading to a smaller value of discord for depolarized states. Despite the fact that our states are well described, in terms of fidelity, by families of depolarized or phase-damped states, their entropic discord may be largely different from that predicted for these classes of states, such that no reliable estimation procedure beyond …
Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position $x$ and momentum $p$. This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take $x$-$p$ correlations into account and admit all Gaussian pure states as minimum uncertaint…
Entropic Efficiency of Bayesian Inference Protocols
MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Laboratory（MajuLab，CNRS-南加州大学-新加坡国立大学-南洋理工大学国际联合研究实验室）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
The uncertainty principle can be expressed in entropic terms, also taking into account the role of entanglement in reducing uncertainty. The information exclusion principle bounds instead the correlations that can exist between the outcomes of incompatible measurements on one physical system, and a second reference system. We provide a more stringent formulation of both the uncertainty principle and the information exclusion principle, with direct applications for, e.g., the security analysis of quantum key distribution, entanglement estimation, and quantum communication. We also highlight a f…
The information causality principle is a generalisation of the no-signalling principle which implies some of the known restrictions on quantum correlations. But despite its clear physical motivation, information causality is formulated in terms of a rather specialised game and figure of merit. We explore different perspectives on information causality, discussing the probability of success as the figure of merit, a relation between information causality and the non-local `inner-product game', and the derivation of a quadratic bound for these games. We then examine an entropic formulation of in…
The uncertainty principle brings out intrinsic quantum bounds on the precision of measuring non-commuting observables. Statistical outcomes in the measurement of incompatible observables reveal a trade-off on the sum of corresponding entropies. Massen-Uffink entropic uncertainty relation (Phys. Rev. Lett. 60, 1103 (1988)) constrains the sum of entropies associated with incompatible measurements. The entropic uncertainty principle in the presence of quantum memory (Nature Phys. 6, 659 (2010)) brought about a fascinating twist by showing that quantum side information, enabled due to entanglement…
The uncertainty principle is one of the most important issues that clarify the distinction between classical and quantum theory. This principle sets a bound on our ability to predict the measurement outcome of two incompatible observables precisely. Uncertainty principle can be formulated via Shannon entropies of the probability distributions of measurement outcome of the two observables. It has shown that the entopic uncertainty bound can be improved by considering an additional particle as the quantum memory $B$ which has correlation with the measured particle $A$. In this work we consider t…
Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of nonlocal correlations. The uncertainty principle states that there exist pairs of incompatible measurements on quantum systems such that their outcomes cannot both be predicted. On the other hand, nonlocal correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in the presence of entanglement. Here, we show that these two fundamental quantum effects are quantitatively related. Namely, we provide an entropic uncertainty relation for …
In the course of the last decades entropic uncertainty relations have attracted much attention not only due to their fundamental role as manifestation of non-classicality of quantum mechanics, but also as major tools for applications of quantum information theory. Amongst the latter are protocols for the detection of quantum correlations or for the secure distribution of secret keys. In this work we show how to derive entropic uncertainty relations for sets of measurements whose effects form quantum designs. The key property of quantum designs is their indistinguishability from truly random qu…
The Leggett-Garg inequalities probe the classical-quantum boundary by putting limits on the sum of pairwise correlation functions between classical measurement devices that consecutively measured the same quantum system. The apparent violation of these inequalities by standard quantum measurements has cast doubt on quantum mechanics' ability to consistently describe classical objects. Recent work has concluded that these inequalities cannot be violated by either strong or weak projective measurements [1]. Here I consider an entropic version of the Leggett-Garg inequalities that are different f…
In the dynamics of open quantum systems, the backflow of information to the reduced system under study has been suggested as the actual physical mechanism inducing memory and thus leading to non-Markovian quantum dynamics. To this aim, the trace-distance or Bures-distance revivals between distinct evolved system states have been shown to be subordinated to the establishment of system-environment correlations or changes in the environmental state. We show that this interpretation can be substantiated also for a class of entropic quantifiers. We exploit a suitably regularized version of Umegaki'…
Heisenberg's uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heis…
The use of the so-called entropic inequalities is revisited in the light of new quantum correlation measures, specially nonlocality. We introduce the concept of {\it classicality} as the non-violation of these classical inequalities by quantum states of several multiqubit systems and compare it with the non-violation of Bell inequalities, that is, {\it locality}. We explore --numerically and analytically-- the relationship between several other quantum measures and discover the deep connection existing between them. The results are surprising due to the fact that these measures are very differ…
The measurement outcomes of two incompatible observables on a particle can be precisely predicted when it is maximally entangled with a quantum memory, as quantified recently [Nature Phys. 6, 659 (2010)]. We explore the behavior of the uncertainty relation under the influence of local unital and nonunital noisy channels. While the unital noises only increase the amount of uncertainty, the amplitude-damping nonunital noises may amazingly reduce the amount of uncertainty in the long-time limit. This counterintuitive phenomenon could be justified by different competitive mechanisms between quantu…
By using the quantum-memory-assisted entropic uncertainty relation (EUR), we derive a computable tight upper bound for quantum discord, which applies to an arbitrary bipartite state. Detailed examples show that this upper bound is tighter than other known bounds in a wide regime. Furthermore, we show that for any tripartite pure state, the quantum-memory-assisted EUR imposes a constraint on the shareability of quantum correlations among the constituent parties. This conclusion amends the well accepted result that quantum discord is not monogamous.
We show that the density-matrix states of noncomposite qudit systems satisfy entropic and information relations like the subadditivity condition, strong subadditivity condition, and Araki--Lieb inequality, which characterize hidden quantum correlations of observables associated with these indivisible systems. We derive these relations employing a specific map of the entropic inequalities known for density matrices of multiqudit systems to the inequalities for density matrices of single-qudit systems. We present the obtained relations in the form of mathematical inequalities for arbitrary Hermi…
We demonstrate that by using a pair of photodetectors endowed with internal gain we are able to quantify the correlation coefficient between the two components of a pulsed bipartite state in the mesoscopic intensity regime (less than 100 mean photons).
The uncertainty principle is an inherent characteristic of quantum mechanics. This principle can be formulated in various form. Fundamentally, this principle can be expressed in terms of the standard deviation of the measured observables. In quantum information theory the preferred mathematical quantity to express the entropic uncertainty relation is the Shannon's entropy. In this work, we consider the generalized entropic uncertainty relation in which there is an additional particle as a quantum memory. Alice measures on her particle $A$ and Bob, with memory particle $B$, predicts the Alice's…
The uncertainty principle sets lower bound on the uncertainties of two incompatible observables measured on a particle. The uncertainty lower bound can be reduced by considering a particle as a quantum memory entangled with the measured particle. In this paper, we consider a tripartite scenario in which a quantum state has been shared between Alice, Bob, and Charlie. The aim of Bob and Charlie is to minimize Charlie's lower bound about Alice's measurement outcomes. To this aim, they concentrate their correlation with Alice in Charlie's side via a cooperative strategy based on local operations …
We examine the quantum correlations of spin pairs in the ground state of finite XY chains in a transverse field, by evaluating the quantum discord as well as other related entropic measures of quantum correlations. A brief review of the latter, based on generalized entropic forms, is also included. It is shown that parity effects are of crucial importance for describing the behavior of these measures below the critical field. It is also shown that these measures reach full range in the immediate vicinity of the factorizing field, where they become independent of separation and coupling range. …
