Alberto Barchielli, Giancarlo Lupieri
Some bounds on the entropic informational quantities related to a quantum continual measurement are obtained and the time dependencies of these quantities are studied.
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This note shows how to apply the James-Stein estimator to the case of entropic uncertainty relations of more than two observables. A better result is found compared to applying the ordinary estimator and we find a more optimal model compared with experimental data. The theoretical bounds for entropic relations for entropic relations for more than two observables are shown to be improved.
For a system randomly prepared in a number of quantum states, we present a lower bound for the distinguishability of the quantum states, that is, the success probability of determining the states in the form of entropy. When the states are all pure, acquiring the entropic lower bound requires only the density operator and the number of the possible states. This entropic bound shows a relation between the von Neumann entropy and the distinguishability.
The uncertainty principle is one of the key concepts in quantum theory. This principle states that it is not possible to measure two incompatible observables simultaneously and accurately. In quantum information theory, the uncertainty principle is formulated using the concept of entropy. In this work we consider the entropic uncertainty relation in the presence of quantum memory. We study the dynamics of entropic uncertainty bound for a two-qubit quantum system coupled to a spin chain with Dzyaloshinskii-Moriya interaction and we investigate the effect of environmental parameters on the entro…
Entropic uncertainty relations place nontrivial lower bounds to the sum of Shannon information entropies for noncommuting observables. Here we obtain a novel lower bound on the entropy sum for general pairs of observables in finite-dimensional Hilbert space, which improves on the best bound known to date [Maassen and Uffink, Phys. Rev. Lett. 60, 1103 (1988)] for a wide class of observables. This result follows from another formulation of the uncertainty principle, the Landau-Pollak inequality, whose relationship to the Maassen-Uffink entropic uncertainty relation is discussed.
We define the entropic bounds, i.e minimal uncertainty for pairs of unitary testers in distinguishing between unitary transformations not unlike the well known entropic bounds for observables. We show that in the case of specific sets of testers which pairwise saturate the trivial zero bound, the testers are all equivalent in the sense their statistics are the same. On the other hand, when maximal bounds are saturated by such sets of testers, the unitary operators would form unitary bases which are mutually unbiased. This resembles very much the role of mutually unbiased bases in maximizing th…
The quantum marginal problem asks, given a set of reduced quantum states of a multipartite system, whether there exists a joint quantum state consistent with these reduced states. The quantum marginal problem is known to be hard to solve in general as it is a variant of the N-representability problem. We provide entropic bounds on the number of orthogonal solutions to the quantum marginal problem.
Entropic uncertainty relations are underpinning to compute the quantitative security bound in quantum cryptographic applications, such as quantum random number generation (QRNG) and quantum key distribution (QKD). All security proofs derive a relation between the information accessible to the legitimate group and the maximum knowledge that an adversary may have gained, Eve, which exploits entropic uncertainty relations to lower bound Eve's uncertainty about the raw key generated by one party, Alice. The standard entropic uncertainty relations is to utilize the smooth min- and max-entropies to …
We provide an analytical proof of the entropic uncertainty relations presented by de Vicente and Sanchez-Ruiz in [Phys. Rev. A 77, 042110 (2008)] and also show that the replacement of Eq. (27) by Eq. (29) in that reference introduces solutions that do not take fully into account the contraints of the problem, which in turn leads to some mistakes in their treatment.
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 discuss the information entropy for a general open pointer-based simultaneous measurement and show how it is bound from below. This entropic uncertainty bound is a direct consequence of the structure of the entropy and can be obtained from the formal solution of the measurement dynamics. Furthermore, the structural properties of the entropy allow us to give an intuitive interpretation of the noisy influence of the pointers and the environmental heat bath on the measurement results.
Quantum discord and quantum uncertainty are two important features of the quantum world. In this work, the relation between entropic uncertainty relation and the shareability of quantum discord is studied. First, by using tripartite quantum-memory-assisted entropic uncertainty relation, an upper bound for the shareability of quantum discord among different partites of a composite system is obtained. It is also shown that, for a specific class of tripartite states, the obtained relation could be expressed as monogamy of quantum discord. Moreover, it is illustrated that the relation could be gen…
Entropic Uncertainty for Biased Measurements
针对两个测量基彼此有偏时标准熵不确定性关系无法提供最优下界的问题，推导出一种新的熵不确定性关系，并在包括有偏测量设备的BB84等量子密码协议中证明其能产生更高的密钥率。
Entropic Bounds as Uncertainty Measure of Unitary Operators
本文在猜测博弈框架下重新定义酉算子对的不确定性，推导其熵不确定性关系，并揭示算子可区分性、相容性及最大不相容性与相互无偏基之间的联系。
One of the defining traits of quantum mechanics is the uncertainty principle which was originally expressed in terms of the standard deviation of two observables. Alternatively, it can be formulated using entropic measures, and can also be generalized by including a memory particle that is entangled with the particle to be measured. Here we consider a realistic scenario where the memory particle is an open system interacting with an external environment. Through the relation of conditional entropy to mutual information, we provide a link between memory effects and the rate of change of conditi…
We have obtained the optimal upper bound of entropic uncertainty relation for $N$ Mutually Unbiased Bases (MUBs). We have used the methods of variational calculus for the states that can be written in terms of $N$ MUBs. Our result is valid for any state when $N$ is $d+1$, where $d$ is the dimension of the related system. We provide a quantitative criterion for the extendibilty of MUBs. In addition, we have applied our result to the mutual information of $d+1$ observables conditioned with a classical memory.
In this paper, we study entropic uncertainty relations on a finite-dimensional Hilbert space and provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an {\it admixture bound} for entropic uncertainty relations for multi-measurements. Comparisons among all bounds for multi-measurements are shown in figures in our favor.
In this paper we present a new proof technique for semi-quantum key distribution protocols which makes use of a quantum entropic uncertainty relation to bound an adversary's information. Our new technique provides a more optimistic key-rate bound than previous work relying only on noise statistics (as opposed to using additional mismatched measurements which increase the noise tolerance of this protocol, but at the cost of requiring four times the amount of measurement data). Our new technique may hold application in the proof of security of other semi-quantum protocols or protocols relying on…
We derive an optimal bound on the sum of entropic uncertainties of two or more observables when they are sequentially measured on the same ensemble of systems. This optimal bound is shown to be greater than or equal to the bounds derived in the literature on the entropic uncertainties of two observables which are measured on distinct but identically prepared ensembles of systems. In the case of a two-dimensional Hilbert Space, the optimal bound for successive measurements of two spin components, is seen to be strictly greater than the optimal bound for the case when they are measured on distin…
The uncertainty principle is an important principle in quantum theory. Based on this principle, it is impossible to predict the measurement outcomes of two incompatible observables, simultaneously. Uncertainty principle basically is expressed in terms of the standard deviation of the measured observables. In quantum information theory, it is shown that the uncertainty principle can be expressed by Shannon's entropy. The entopic uncertainty lower bound can be altered by considering a particle as the quantum memory which is correlated with the measured particle. We assume that the quantum memory…
We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an entropic bound for the which-way information, which we derive in this letter. We show that this latter crucially depends on the choice of the measurement strategy to obtain the path information. In particular, we present results for two strategies: zero-error identification of the path-detector states, which never produces an error but sometimes fails to return a conclusive answer, and a mixed stra…
The uncertainty principle determines the distinction between the classical and quantum worlds. This principle states that it is not possible to measure two incompatible observables with the desired accuracy simultaneously. In quantum information theory, Shannon entropy has been used as an appropriate measure to express the uncertainty relation. According to the applications of entropic uncertainty relation, studying and trying to improve the bound of this relation is of great importance. Uncertainty bound can be altered by considering an extra quantum system as the quantum memory $B$ which is …
The uncertainty principle sets a bound on our ability to predict the measurement outcomes of two incompatible observables which are measured on a quantum particle simultaneously. In quantum information theory, the uncertainty principle can be formulated in terms of the Shannon entropy. Entropic uncertainty bound can be improved by adding a particle which correlates with the measured particle. The added particle acts as a quantum memory. In this work, a method is provided for obtaining the entropic uncertainty relations in the presence of a quantum memory by using quantum coherence. In the meth…
The quantum entropic uncertainty relation and entanglement witness in the two-atom system coupling with the non-Markovian environments are studied by the time-convolutionless master-equation approach. The influence of non-Markovian effect and detuning on the lower bound of the quantum entropic uncertainty relation and entanglement witness is discussed in detail. The results show that, only if the two non-Markovian reservoirs are identical, increasing detuning and non-Markovian effect can reduce the lower bound of the entropic uncertainty relation, lengthen the time region during which the enta…
We study entropic uncertainty relations by using stepwise linear functions and quadratic functions. Two kinds of improved uncertainty lower bounds are constructed: the state-independent one based on the lower bound of Shannon entropy and the tighter state-dependent one based on the majorization techniques. The analytical results for qubit and qutrit systems with two or three measurement settings are explicitly derived, with detailed examples showing that they outperform the existing bounds. The case with the presence of quantum memory is also investigated.
Tight bounds from multiple-observable entropic uncertainty relations
该研究利用基于局部测量联合香农熵的熵不确定关系，分析二部与多粒子系统的可加性，提出两类熵不等式，证明可加性仅对两可观测量EUR成立，并将其应用于二部及三量子比特系统。
The uncertainty principle is the most important feature of quantum mechanics, which can be called the heart of quantum mechanics. This principle sets a lower bound on the uncertainties of two incompatible measurement. In quantum information theory, this principle is expressed in terms of entropic measures. Entropic uncertainty bound can be altered by considering a particle as a quantum memory. In this work we investigate the entropic uncertainty relation under the relativistic motion. In relativistic uncertainty game Alice and Bob agree on two observables, $\hat{Q}$ and $\hat{R}$, Bob prepares…
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…
We present a family of entropic uncertainty relations for pointer-based simultaneous measurements of conjugate observables. The lower bounds of these relations explicitly incorporate the influence of the measurement apparatus. We achieve this by using a mathematical theorem which states that the information entropy of convoluted probability distributions is bound from below. As a consequence of these results we can straightforwardly show that appropriately squeezed states are minimal entropy states for simultaneous measurements.
We investigate the uncertainty associated with a joint quantum measurement of two components of spin of a spin-1/2 particle and quantify this in terms of entropy. We consider two entropic quantities: the joint entropy and the sum of the marginal entropies, and obtain lower bounds for each of these quantities. For the case of joint measurements where we measure each spin observable equally well, these lower bounds are tight.
The uncertainty principle is a fundamental principle in quantum physics. It implies that the measurement outcomes of two incompatible observables can not be predicted simultaneously. In quantum information theory, this principle can be expressed in terms of entropic measures. Berta \emph{et al}. [\href{http://www.nature.com/doifinder/10.1038/nphys1734}{ Nature Phys. 6, 659 (2010) }] have indicated that uncertainty bound can be altered by considering a particle as a quantum memory correlating with the primary particle. In this article, we obtain a lower bound for entropic uncertainty in the pre…
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 …
Tighter entropic uncertainty relations in the presence of quantum memories for complete sets of mutually unbiased bases
本文提出更严格的量子记忆辅助熵不确定性关系，通过观测互补性、态纯度、条件von-Neumann熵、霍夫曼量和互信息等，推导出量子不确定性的上下界，并通过实例展示其优于现有结果。
We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observable $G$ and the displacement $r$ of the $G$-generated evolution $e^{-ir G}$. We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coles et al.. Our derivation uses as a subroutine a recently proposed novel algebraic method, which can in general be used to d…
Strong and general entropic and geometric Heisenberg limits are obtained, for estimates of multiparameter unitary displacements in quantum metrology, such as the estimation of a magnetic field from the induced rotation of a probe state in three dimensions. A key ingredient is the Holevo bound on the Shannon mutual information of a quantum communication channel. This leads to a Bayesian bound on performance, in terms of the prior distribution of the displacement and the asymmetry of the input probe state with respect to the displacement group. A geometric measure of performance related to entro…
We studied the dynamics of entropic uncertainty in Markovian and non-Markovian systems during the charging of open quantum batteries (QBs) mediated by a common dissipation environment. In the non-Markovian regime, the battery is almost fully charged efficiently, and the strong non-Markovian property is beneficial for improving the charging power. In addition, the results show that the energy storage is closely related to the couplings of the charger-reservoir and battery-reservoir; that is, the stronger coupling of a charger-reservoir improves energy storage. In particular, entanglement is req…
The effects of detuning on entropic uncertainty bound and quantum correlations in dissipative environment
本文在耗散环境下研究量子记忆与腔频失谐对熵不确定度界及量子关联的影响，发现增大失谐可保持量子关联并降低不确定度界。
Recently, Xu et al. [Phys. Rev. A 86, 012113(2012)] explored the behavior of the entropic uncertainty relation under the influence of local unital and nonunital noisy channels for a class of Bell-diagonal states. We here reform their results and investigate the entropic uncertainty relation under the influence of unital and nonunital noisy channels with memory. Different types of noisy channels with memory, such as amplitude damping channel(nonunitary), phase-damping and depolarizing channels(unitary) have been taken into account. Some analytical or numerical results are presented. The effect …
We derive new inequalities for the probabilities of projective measurements in mutually unbiased bases of a qudit system. These inequalities lead to wider ranges of validity and tighter bounds on entropic uncertainty inequalities previously derived in the literature.
We prove tight entropic uncertainty relations for a large number of mutually unbiased measurements. In particular, we show that a bound derived from the result by Maassen and Uffink for 2 such measurements can in fact be tight for up to sqrt{d} measurements in mutually unbiased bases. We then show that using more mutually unbiased bases does not always lead to a better locking effect. We prove that the optimal bound for the accessible information using up to sqrt{d} specific mutually unbiased bases is log d/2, which is the same as can be achieved by using only two bases. Our result indicates t…
The Heisenberg uncertainty principle shows that no one can specify the values of the non-commuting canonically conjugated variables simultaneously. However, the uncertainty relation is usually applied to two incompatible measurements. We present tighter bounds on both entropic uncertainty relation and information exclusion principle for multiple measurements in the presence of quantum memory. As applications, three incompatible measurements on Werner state and Horodecki's bound entangled state are investigated in details.
How much of the uncertainty in predicting measurement outcomes for non-commuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two non-commuting observables into a classical component, and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits "purity-based" lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a non-trivial inter…
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…
We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. Rényi entropy is used as uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of Rényi entropies for any couple of (positive) entropic indices (α,β). Thus, we have overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz-Thorin theorem. In addition, we present an analytical expression for …
While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measurement are obtained in a unified manner. In particular, it is shown that such bounds can all be stated as inequalities between mutual entropies. This approach based on channels gives rise to a unified picture of known and new bounds on the classical infor…
In presence of quantum memory [M. Berta, M. Christandl, R. Colbeck, J.M. Renes, and R. Renner, Nature Phys. 6, 659 (2010)] the lower bound of entropic uncertainty relation depends on amount of entanglement between the particle (on which two incompatible observable are measured) and the quantum memory. On other hand, fine grained uncertainty relation [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)] gives the upper bound of winning non-local (CHSH) game. We construct similar kind of quantum game and relate two uncertainty relation for measurement of any two observable on the quantum system…
Recently, an entropic uncertainty relation for multiple measurements has been proposed by Liu et al. in [Phys. Rev. A 91, 042133 (2015)]. However, the lower bound of the relation is not always tight with respect to different measurements. Herein, we improve the lower bound of the entropic uncertainty relation for multiple measurements, termed as simply constructed bound (SCB). We verify that the SCB is tighter than Liu et al.'s result for arbitrary mutually unbiased basis measurements, which might play a fundamental and crucial role in practical quantum information processing. Moreover, we opt…
Entropic uncertainty and entanglement are two distinct aspects of quantum mechanical procedures. To estimate entropic uncertainty relations, entropies are used: the greater the entropy bound, the less effective the quantum operations and entanglement are. In this regard, we analyze the entropic uncertainty, entropic uncertainty lower bound, and concurrence dynamics in two non-interacting qubits. The exposure of two qubits is studied in two different qubit-noise configurations, namely, common qubit-noise and independent qubit-noise interactions. To include the noisy effects of the local externa…
We analyze entropic uncertainty relations in a finite dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained in projective measurements with respect to any two orthogonal bases. We improve the recent bounds by Coles and Piani, which are known to be stronger than the well known result of Maassen and Uffink. Furthermore, we find a novel bound based on majorization techniques, which also happens to be stronger than the recent results involving largest singular values of submatrices of the unitary matrix connecting both bases. The first set of new bounds g…
We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches [S. Friendland, V. Gheorghiu and G. Gour, Phys. Rev. Lett. 111, 230401 (2013); A. E. Rastegin and K. Życzkowski, J. Phys. A, 49, 355301 (2016)], particularly by extending the direct-sum majorization relation first introduced in [Ł. Rudnick…
For the power-law quantum wave packet in configuration space, the variance of the position observable may be divergent. Accordingly, the information-entropic formulation of the uncertainty principle becomes more appropriate than the Heisenberg-type formulation, since it involves only the finite quantities. It is found that the total amount of entropic uncertainty converges to its lower bound in the limit of a large value of the exponent.
The positivity conditions of the relative entropy between two thermal equilibrium states $\hatρ_1$ and $\hatρ_2$ are used to obtain upper and lower bounds for the subtraction of their entropies, the Helmholtz potential and the Gibbs potential of the two systems. These limits are expressed in terms of the mean values of the Hamiltonians, number operator, and temperature of the different systems. In particular, we discuss these limits for molecules which can be represented in terms of the Franck--Condon coefficients. We emphasize the case where the Hamiltonians belong to the same system at two d…
Finite-Size Security Bounds in Semi-Quantum Key Distribution: Spectral, Operator-Theoretic, and Entropic Perspectives
本文从谱扰动、算子约化和熵权衡三个视角推导半量子密钥分发的有限尺寸安全性界，给出基于集中不等式的实用参数估计方法。
Quantum uncertainty relations are formulated in terms of relative entropy between distributions of measurement outcomes and suitable reference distributions with maximum entropy. This type of entropic uncertainty relation can be applied directly to observables with either discrete or continuous spectra. We find that a sum of relative entropies is bounded from above in a nontrivial way, which we illustrate with some examples.
Given two orthornormal bases A and B, the basic form of the entropic uncertainty principle is stated in terms of the sum of the Shannon entropies of the probabilities of measuring A and B onto a given quantum state. State independent lower bounds for this sum encapsulate the degree of incompatibility of the observables diagonal in the A and B bases, and are usually derived by extracting as much information as possible from the unitary operator U connecting the two bases. Here we show a strategy to derive sequences of lower bounds based on alternating sequences of measurements onto A and B. The…
Uncertainty relations lie at the very core of quantum mechanics, and form the cornerstone of essentially all quantum cryptographic applications. In particular, they play an important role in cryptographic protocols in the bounded-quantum-storage model, where proving the security of all existing protocols ultimately reduces to bounding such relations. Yet, very little is known about such uncertainty relations for more than two measurements. Here, we prove optimal entropic uncertainty relations for anti-commuting binary observables for the Shannon entropy, and nearly optimal relations for the co…
The measurement range problem, where one cannot determine the data outside the range of the detector, limits the characterization of entanglement in high-dimensional quantum systems when employing, among other tools from information theory, the entropic uncertainty relations. Practically, the measurement range problem weakens the security of entanglement-based large-alphabet quantum key distribution (QKD) employing degrees of freedom including time-frequency or electric field quadrature. We present a modified entropic uncertainty relation that circumvents the measurement range problem under ce…
The uncertainty principle sets limit on our ability to predict the values of two incompatible observables measured on a quantum particle simultaneously. This principle can be stated in various forms. In quantum information theory, it is expressed in terms of the entropic measures. Uncertainty bound can be altered by considering a particle as a quantum memory correlating with the primary particle. In this work, we provide a method for converting the entropic uncertainty relation in the absence of quantum memory to that in its presence. It is shown that the lower bounds obtained through the meth…
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…
In analogy with its classical counterpart, a noisy quantum channel is characterized by a loss, a quantity that depends on the channel input and the quantum operation performed by the channel. The loss reflects the transmission quality: if the loss is zero, quantum information can be perfectly transmitted at a rate measured by the quantum source entropy. By using block coding based on sequences of n entangled symbols, the average loss (defined as the overall loss of the joint n-symbol channel divided by n, when n tends to infinity) can be made lower than the loss for a single use of the channel…
In Coles-Piani's recent remarkable version of the entropic uncertainty principle, the entropic sum is controlled by the first and second maximum overlaps between the two projective measurements. We generalize the entropic uncertainty relation in the presence of quantum memory and find the exact dependence on all $d$ largest overlaps between two measurements on any $d$-dimensional Hilbert space. The corresponding entropic uncertainty principle in the absence of quantum memory is also derived. Our bounds are strictly tighter than previous entropic bounds in the presence of quantum memory, which …
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