Weakened claim concerning semi device-independence in the application to QKD. A full security proof for this setup without any restrictions on the measurement devices can be found in arXiv:1210.4359
Uncertainty relations give upper bounds on the accuracy by which the outcomes of two incompatible measurements can be predicted. While established uncertainty relations apply to cases where the predictions are based on purely classical data (e.g., a description of the system's state before measurement), an extended relation which remains valid in the presence of quantum information has been proposed recently [Berta et al., Nat. Phys. 6, 659 (2010)]. Here, we generalize this uncertainty relation to one formulated in terms of smooth entropies. Since these entropies measure operational quantities such as extractable secret key length, our uncertainty relation is of immediate practical use. To illustrate this, we show that it directly implies security of a family of quantum key distribution protocols including BB84. Our proof remains valid even if the measurement devices used in the experiment deviate arbitrarily from the theoretical model.
We formulate uncertainty relations for mutually unbiased bases and symmetric informationally complete measurements in terms of the Rényi and Tsallis entropies. For arbitrary number of mutually unbiased bases in a finite-dimensional Hilbert space, we give a family of Tsallis $α$-entropic bounds for $α\in(0;2]$. Relations in a model of detection inefficiences are obtained. In terms of Rényi's entropies, lower bounds are given for $α\in[2;\infty)$. State-dependent and state-independent forms of such bounds are both given. Uncertainty relations in terms of the min-entropy are separately considered…
The chain rule for the Shannon and von Neumann entropy, which relates the total entropy of a system to the entropies of its parts, is of central importance to information theory. Here we consider the chain rule for the more general smooth min- and max-entropy, used in one-shot information theory. For these entropy measures, the chain rule no longer holds as an equality, but manifests itself as a set of inequalities that reduce to the chain rule for the von Neumann entropy in the i.i.d. case.
Quantum mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the Renyi entropies. The proof of these inequalities requires the use of the exact expression for the (p,q)-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in N-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized Renyi entropies.
A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For given input state, different effects of each unraveling result in some probability distribution at the output. As it is shown, all Tsallis' entropies of positive order as well as some of Renyi's entropies of this distribution are minimized by the same unraveling of a super-operator. Entropic relations between a state ensemble and the generated density matrix are revisited in terms of both the adopted measures. Using Riesz'…
Uncertainty relations for a pair of arbitrary measurements and for a single measurement are posed in the form of inequalities using the Renyi entropies. The formulation deals with discrete observables. Both the relations with state-dependent and state-independent bounds are presented. The entropic bounds are illustrated within the distinction between non-orthogonal states.
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…
The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydroge…
This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography. We investigate the conditional min- and max-entropy for quantum states, generalizations of classical Rényi entropies. We introduce the purified distance, a novel metric for unnormalized quantum states and use it to define smooth entropies as optimizations of the min- and max-entropies over a ball of close states. We explore various properties of these entropies, including data-processing inequalities, chain rules and their classical limits. The most important prop…
Indeterminacy associated with probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes of repeated experiments. Here we express it as an effective amount (measure) of distinct outcomes instead. The resulting $μ$-uncertainties are described by the effective number theory [1] whose central result, the existence of a minimal amount, leads to a well-defined notion of intrinsic irremovable uncertainty. We derive $μ$-uncertainty formulas for arbitrary set of commuting operators, including the cases with continuous spectra. The associated entropy-l…
Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequa…
The Heisenberg uncertainty principle is known to be connected to the entropic uncertainty principle. This correspondence is obtained employing a Gaussian probability distribution for wave functions associated to the Shannon entropy. Independently, due to quantum gravity effects the Heisenberg uncertainty principle has been extended to a Generalized Uncertainty Principle (GUP). In this work, we show that GUP has been derived from considering non-extensive entropies, proposed by one of us. We found that the deformation parameters associated with $S_{+}$ and $S_-$ entropies are negative and posit…
We obtain uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Rényi entropies of order $α\in(0;1]$. It is shown that these entropic bounds are tight in the sense that they are always reached with certain pure states. A new result is the conditions for equality in Rényi-entropy uncertainty relations for the Pauli observables. Upper entropic bounds in the pure-state case are also novel. Combining the presented bounds leads to a band, in which the rescaled average Rényi $α$-entropy ranges for a pure measured state. A width of this band is …
Two new relative entropy quantities, called the min- and max-relative entropies, are introduced and their properties are investigated. The well-known min- and max- entropies, introduced by Renner, are obtained from these. We define a new entanglement monotone, which we refer to as the max-relative entropy of entanglement, and which is an upper bound to the relative entropy of entanglement. We also generalize the min- and max-relative entropies to obtain smooth min- and max- relative entropies. These act as parent quantities for the smooth Renyi entropies, and allow us to define the analogues o…
We study uncertainty and certainty relations for two successive measurements of two-dimensional observables. Uncertainties in successive measurement are considered within the following two scenarios. In the first scenario, the second measurement is performed on the quantum state generated after the first measurement with completely erased information. In the second scenario, the second measurement is performed on the post-first-measurement state conditioned on the actual measurement outcome. Induced quantum uncertainties are characterized by means of the Tsallis entropies. For two successive p…
For certain families of finite quantum graphs, we study the question of how eigenfunctions are distributed over the graph. To characterize properties of the distribution, generalized entropies of the Rényi and Tsallis types are considered. The presented approach is similar to entropic uncertainty relations of the Maassen-Uffink type. Using the Riesz theorem, we derive lower bounds on symmetrized generalized entropies of eigenfunctions. A quality of such estimates will depend on boundary conditions used at vertices of the given graph. Rényi and Tsallis entropies of eigenfunctions of star graphs…
The aim of the work is to give the explicit proofs of the Renyi-entropy uncertainty relations presented in the previous work [A. Rastegin, arXiv:0805.1777]. The relations with both the state-dependent and state-independent entropic bounds are proved. For a pair of POVM measurements the two relations are obtained. The first of them is generalization of the known results, whereas the second is quite alternative. It is shown that both these relations are meaty. The important case of POVM's with one-rank elements is extra discussed. The measurements designed for distinction between two non-orthogo…
The Rényi and von Neumann entropies of the thermal state in the generalized uncertainty principle (GUP)-corrected single harmonic oscillator system are explicitly computed within the first order of the GUP parameter $α$. While the von Neumann entropy with $α= 0$ exhibits a monotonically increasing behavior in external temperature, the nonzero GUP parameter makes the decreasing behavior of the von Neumann entropy at the large temperature region. As a result, the von Neumann entropy is maximized at the finite temperature if $α\neq 0$. The Rényi entropy $S_γ$ with nonzero $α$ also exhibits simila…
Entropic uncertainty relations for the position and momentum within the generalized uncertainty principle are examined. Studies of this principle are motivated by the existence of a minimal observable length. Then the position and momentum operators satisfy the modified commutation relation, for which more than one algebraic representation is known. One of them is described by auxiliary momentum so that the momentum and coordinate wave functions are connected by the Fourier transform. However, the probability density functions of the physically true and auxiliary momenta are different. As the …
Uncertainty relations for more than two observables have found use in quantum information, though commonly known relations pertain to a pair of observables. We present novel uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Tsallis $α$-entropies. For all real $α\in(0;1]$ and integer $α\geq2$, lower bounds on the sum of three $α$-entropies are obtained. These bounds are tight in the sense that they are always reached with certain pure states. The necessary and sufficient condition for equality is that the qubit state is an eigenstate o…
In classical and quantum information theory, operational quantities such as the amount of randomness that can be extracted from a given source or the amount of space needed to store given data are normally characterized by one of two entropy measures, called smooth min-entropy and smooth max-entropy, respectively. While both entropies are equal to the von Neumann entropy in certain special cases (e.g., asymptotically, for many independent repetitions of the given data), their values can differ arbitrarily in the general case. In this work, a recently discovered duality relation between (non-sm…
Many of the traditional results in information theory, such as the channel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are independent and identically distributed (i.i.d.) over a large number of uses. To overcome this limitation, two different techniques, the information spectrum method and the smooth entropy framework, have been developed independently. They are based on new entropy measures, called spectral entropy rates and smooth entropies, respectively, that generalize Shannon entropy (in the classical case) and von Neumann entro…
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 this study, we obtained the position-momentum uncertainties and some uncertainty relations for the Pöschl-Teller-type potential for any $\ell$. The radial expectation values of $r^{-2}$, $r^{2}$ and $p^{2}$ are obtained from which the Heisenberg Uncertainty principle holds for the potential model under consideration. The Fisher information is then obtained and it is observed that the Fisher-information-based uncertainty relation and the Cramer-Rao inequality hold for this even power potential. Some numerical and graphical results are displayed.
We address the generalized uncertainty principle in scenarios of successive measurements. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. Here, specific features of measurements of observables with continuous spectra should be taken into account. First, we formulated uncertainty relations in terms of Shannon entropies. Since such relations involve a state-dependent correction term, they generally differ from preparation uncertainty relations. This difference is revealed when position is measured by the first. In contrast, state-independent…
The Rényi entropies of Coulomb systems $R_{p}[ρ], 0 < p < \infty$ are logarithms of power functionals of the electron density $ρ(\vec{r})$ which quantify most appropriately the electron uncertainty and describe numerous physical observables. However, its analytical determination is a hard issue not yet solved except for the first lowest-lying energetic states of some specific systems. This is so even for the $D$-dimensional hydrogenic system, which is the main prototype of the multidimensional Coulomb many-body systems. Recently, the Rényi entropies of this system have been found in the two ex…
Tight relations and equivalences between smooth relative entropies
RIKEN Hakubi Research Team, RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC)（理化学研究所白兔研究团队、理学前驱性研究机构和理中心量子计算）；Scuola Normale Superiore（比萨高等师范学院）；QuSoft；Korteweg–de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特维格-德弗里斯数学研究所）；Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）；Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心应用数学与理论物理系）
Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In th…
The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We p…
We improve the entropic uncertainty relations for position and momentum coarse-grained measurements. We derive the continuous, coarse-grained counterparts of the discrete uncertainty relations based on the concept of majorization. The obtained entropic inequalities involve two Rényi entropies of the same order, and thus go beyond the standard scenario with conjugated parameters. In a special case describing the sum of two Shannon entropies the majorization-based bounds significantly outperform the currently known results in the regime of larger coarse graining, and might thus be useful for ent…
One of the most important and useful entropic uncertainty relations concerns a $d$ dimensional system and two mutually unbiased measurements. In such a setting, the sum of two information entropies is lower bounded by $\ln d$. It has recently been shown that projective measurements subject to operational mutual unbiasedness can also be constructed in a continuous domain, with the help of periodic coarse graining. Here we consider the whole family of Rényi entropies applied to these discretized observables and prove that such a scheme does also admit the uncertainty relation mentioned above.
In this work we determine and discuss the entropic uncertainty measures of Shannon type for all the discrete stationary states of the multidimensional harmonic systems directly in terms of the states' hyperquantum numbers, the dimensionality and the oscillator strength. We have found that these entropies have a monotonically increasing behavior when both the dimensionality and the population of the states are increasing
Entropic uncertainty relations, based on sums of entropies of probability distributions arising from different measurements on a given pure state, can be seen as a generalization of the Heisenberg uncertainty relation that is in many cases a more useful way to quantify incompatibility between observables. Of particular interest are relationships between `mutually unbiased' observables, which are maximally incompatible. Lower bounds on the sum of entropies for sets of two such observables, and for complete sets of observables within a space of given dimension, have been found. This paper explor…
Majorization uncertainty relations are generalized for an arbitrary mixed quantum state $ρ$ of a finite size $N$. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of $ρ$ and the entries of a unitary matrix $U$ relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the s…
There is a renewed interest in the uncertainty principle, reformulated from the information theoretic point of view, called the entropic uncertainty relations. They have been studied for various integrable systems as a function of their quantum numbers. In this work, focussing on the ground state of a nonlinear, coupled Hamiltonian system, we show that approximate eigenstates can be can be constructed within the framework of adiabatic theory. Using the adiabatic eigenstates, we estimate the information entropies and their sum as a function of the the nonlinearity parameter. We also briefly loo…
RIKEN Hakubi Research Team, RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC)（RIKEN 博慧研究团队、RIKEN 先锋研究所（PRI）和 RIKEN 量子计算中心）；Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
We revisit entropic formulations of the uncertainty principle for an arbitrary pair of positive operator-valued measures (POVM) $A$ and $B$, acting on finite dimensional Hilbert space. Salicrú generalized $(h,ϕ)$-entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet $…
This paper is prepared as a contribution to the proceedings after the 12th ICSSUR/Feynfest Conference held in Foz do Iguacu (Brazil) from 2 to 6 May 2011. In the first part I briefy report the topic of entropic uncertainty relations for position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of fnite number of detectors set to measure probability distributions in position and momentum spaces. I derive an uncertainty relation for the sum of the Shannon entropies which generalizes previous approaches [Phys. Lett. 103 A, 253 (1984)] based on an infni…
Uncertainty relations provide constraints on how well the outcomes of incompatible measurements can be predicted, and, as well as being fundamental to our understanding of quantum theory, they have practical applications such as for cryptography and witnessing entanglement. Here we shed new light on the entropic form of these relations, showing that they follow from a few simple entropic properties, including the data processing inequality. We prove these relations without relying on the exact expression for the entropy, and hence show that a single technique applies to several entropic quanti…
An uncertainty relation for the Rényi entropies of conjugate quantum observables is used to obtain a strong Heisenberg limit of the form ${\rm RMSE} \geq f(α)/(\langle N\rangle+\frac12)$, 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 $ΔΦ\geq \max_n p_n$, are also obtained. These results are significantly strengthened via upper and lower bounds on the Rény…
The uncertainty relation for continuous variables due to Byalinicki-Birula and Mycielski expresses the complementarity between two $n$-uples of canonically conjugate variables $(x_1,x_2,\cdots x_n)$ and $(p_1,p_2,\cdots p_n)$ in terms of Shannon differential entropy. Here, we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two $n$-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured varia…
An entropic approach to formulating uncertainty relations for the number-annihilation pair is considered. We construct some normal operator that traces the annihilation operator as well as commuting quadratures with a complete system of common eigenfunctions. Expanding the measured wave function with respect to them, one obtains a relevant probability distribution. Another distribution is naturally generated by measuring the number operator. Due to the Riesz-Thorin theorem, there exists a nontrivial inequality between corresponding functionals of the above distributions. We find the bound in t…
Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations which are thought (but not proven) to imply the existence of a minimal observable length. Here we prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, we make u…
Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies. There is a long history of such entropy inequalities between position and momentum. Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized. All the recently derived uncertainty relations utilize the strong subadditivity (SSA) theorem; our contribution relies on directly utilizing the proof technique…
The occupancies and entropic entanglement measures for the ground state of two particles in a two-dimensional harmonic anisotropic trap are studied. We implement a method to study the large interaction strength limit for different short- and long-range interaction potentials that allows to obtain the exact entanglement spectrum and several entropies. We show that for long-range interactions, the von Neumann, min-entropy and the family of Rényi entropies remain finite for the anisotropic traps and diverge logarithmically for the isotropic traps. In the short-range interaction case the entanglem…
Wave packet revivals and fractional revivals are studied by means of newly derived uncertainty relations that involve Rényi entropies and position and momentum dispersions.
Thermodynamic entropy, as defined by Clausius, characterizes macroscopic observations of a system based on phenomenological quantities such as temperature and heat. In contrast, information-theoretic entropy, introduced by Shannon, is a measure of uncertainty. In this Letter, we connect these two notions of entropy, using an axiomatic framework for thermodynamics [Lieb, Yngvason, Proc. Roy. Soc.(2013)]. In particular, we obtain a direct relation between the Clausius entropy and the Shannon entropy, or its generalisation to quantum systems, the von Neumann entropy. More generally, we find that …
We show that different entropic measures of fluctuations lead to contradictory uncertainty relations for two complementary observables. We apply Tsallis and Rényi entropies to the joint distribution emerging from a noisy simultaneous measurement of both observables as well as to the product of their individual statistics, either intrinsic or of operational origin.
The Rényi and Shannon entropies are information-theoretic measures which have enabled to formulate the position-momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density-functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment $\langle r^2\rangle$ for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the Rényi maximization procedure with a c…
We prove that the inequality used recently by Wilk and Włodarczyk [Phys. Rev. A {\bf 79}, 062108 (2009)] to find a better lower bound in the uncertainty relations for the Rényi entropies is invalid. Thus, the problem of improving the bound given in our paper [Phys. Rev. A {\bf 74}, 052101 (2006)] remains unsolved.
We explain several separability criteria which rely on uncertainty relations. For the derivation of these criteria uncertainty relations in terms of variances or entropies can be used. We investigate the strength of the separability conditions for the case of two qubits and show how they can improve entanglement witnesses.
Uncertainty relations emerging from the Tsallis entropy are derived and discussed. In particular we found a positively defined function that saturates the so called entropic inequalities for entropies characterizing the physical states under consideration.
One-parameter functionals of the Rényi $R_{ρ,γ}(α)$ and Tsallis $T_{ρ,γ}(α)$ types are calculated both in the position (subscript $ρ$) and momentum ($γ$) spaces for the azimuthally symmetric 2D nanoring that is placed into the combination of the transverse uniform magnetic field $\bf B$ and the Aharonov-Bohm (AB) flux $ϕ_{AB}$ and whose potential profile is modelled by the superposition of the quadratic and inverse quadratic dependencies on the radius $r$. Position (momentum) Rényi entropy depends on the field $B$ as a negative (positive) logarithm of $ω_{eff}\equiv\left(ω_0^2+ω_c^2/4\right)^{…
