We investigate the product form uncertainty relations of variances for $n\,(n\geq 3)$ quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented.
Uncertainty relations are fundamental in quantum mechanics. Here I propose state-independent variance-based uncertainty relations for two or more arbitrary observables in finite dimensional spaces. The uncertainty relations provide near-optimal lower bounds in some typical examples, and are useful for entanglement detection.
We study the sum uncertainty relations based on variance and skew information for arbitrary finite N quantum mechanical observables. We derive new uncertainty inequalities which improve the exiting results about the related uncertainty relations. Detailed examples are provided to illustrate the advantages of our uncertainty inequalities.
Based on mixedness definition as M=1-tr(\r{ho}^2), we obtain a new variance-based uncertainty equality along with an inequality for Hermitian operators of a single-qubit system. The obtained uncertainty equality can be used as a measure of the system mixedness. A qubit system with feedback control is also exploited to demonstrate the new uncertainty.
We formulate uncertainty relations for arbitrary finite number of incompatible observables. Based on the sum of variances of the observables, both Heisenberg-type and Schrödinger-type uncertainty relations are provided. These new lower bounds are stronger in most of the cases than the ones derived from some existing inequalities. Detailed examples are presented.
Uncertainty relations play a significant role in drawing a line between classical physics and quantum physics. Since the introduction by Heisenberg, these relations have been considerably explored. However, the effect of quantum entanglement on uncertainty relations was not probed. Berta et al. [Nature Physics 6, 659-662 (2010)] removed this gap by deriving a conditional-entropic uncertainty relation in the presence of quantum entanglement. In the same spirit, using inferred-variance, we formulate uncertainty relations in the presence of entanglement for general two-qubit systems and arbitrary…
New sum and product uncertainty relations, containing variances of three or four observables, but not containing explicitly their covariances, are derived. One of consequences is the new inequality, giving a nonzero lower bound for the product of two variances in the case of zero mean value of the commutator between the related operators. Moreover, explicit examples show that in some cases this new bound can be better than the known Robertson--Schrödinger one.
Heisenberg-Robertson's uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two non-compatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg's uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as `uncertainty relation', plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and intro…
In quantum mechanics, the variance-based Heisenberg-type uncertainty relations are a series of mathematical inequalities posing the fundamental limits on the achievable accuracy of the state preparations. In contrast, we construct and formulate two quantum uncertainty equalities, which hold for all pairs of incompatible observables and indicate the new uncertainty relations recently introduced by L. Maccone and A. K. Pati [Phys. Rev. Lett. 113, 260401 (2014)]. Furthermore, we present an explicit interpretation lying behind the derivations and relate these relations to the so-called intelligent…
Uncertainty relations describe the lower bound of product of standard deviations of observables. By revealing a connection between standard deviations of quantum observables and numerical radius of operators, we establish a universal uncertainty relation for any $k$ observables, of which the formulation depends on the even or odd quality of $k$. This universal uncertainty relation is tight at least for the cases $k=2n$ and $k=3$. For two observables, the uncertainty relation is exactly a simpler reformulation of Schrödinger's uncertainty principle.
Uncertainty relation is a fundamental issue in quantum mechanics and quantum information theory. By using modified generalized variance (MGV), and modified generalized Wigner-Yanase-Dyson skew information (MGWYD), we identify the total and quantum uncertainty of quantum channels. The elegant properties of the total uncertainty of quantum channels are explored in detail. In addition, we present a trade-off relation between the total uncertainty of quantum channels and the entanglement fidelity and establish the relationships between the total uncertainty and entropy exchange/coherent informatio…
Variance is a ubiquitous quantity in quantum information theory. Given a basis, we consider the averaged variances of a fixed diagonal observable in a pure state under all possible permutations on the components of the pure state and call it the symmetrized variance. Moreover we work out the analytical expression of the symmetrized variance and find that such expression is in the factorized form where two factors separately depends on the diagonal observable and quantum state. By shifting the factor corresponding to the diagonal observable, we introduce the notion named the standard symmetrize…
The effect of the quantum feedback on the tightness of the variance-based uncertainty, the possibility of using quantum feedback to prepare the state with a better tightness, and the relationship between the tightness of the uncertainty and the mixedness of the system are studied. It is found that the tightness of Schrodinger-Robertson uncertainty (SUR) relation has a strict liner relationship with the mixedness of the system. As for the Robertson uncertainty relation (RUR), we find that the tightness can be enhanced by tuning the feedback at the beginning of the evolution. In addition, we ded…
We present an equivalence theorem to unify the two classes of uncertainty relations, i.e., the variance-based ones and the entropic forms, which shows that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators. That means an uncertainty relation in the language of entropy may be mapped onto a variance-based one, and vice versa. Employing the equivalence theorem, alternative formulations of entropic uncertainty relations stronger than existing ones in the literature are obtained for qubit system, and variance based uncertainty relations…
For angular observables pairs (angular momentum-angle and number-phase) the adequate reference element of normality is not the Robertson-Schrödinger uncertainty relation but a Schwarz formula regarding the quantum fluctuations. Beyond such a fact the traditional interpretation of the uncertainty relations appears as an unjustified doctrine.
We formulate uncertainty relations for arbitrary $N$ observables. Two uncertainty inequalities are presented in terms of the sum of variances and standard deviations, respectively. The lower bounds of the corresponding sum uncertainty relations are explicitly derived. These bounds are shown to be tighter than the ones such as derived from the uncertainty inequality for two observables [Phys. Rev. Lett. 113, 260401 (2014)]. Detailed examples are presented to compare among our results with some existing ones.
Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for $N$ incompatible observables stronger than the simple generalization of the uncertainty relation for two observables derived by Maccone and Pati [Phys. Rev. Lett. {\bf113}, 260401 (2014)]. Further comparisons of our uncertainty relation with other related ones for spin-$\frac{1}{2}$ and spin-$…
The local uncertainty relation (LUR) criteria for quantum entanglement, which is dependent on chosen observables, is developed recent. In the paper, applying the uncertainty principle, an entanglement criteria for multipartite Gaussian states is given, which is implemented by a minimum optimization computer program and independent on observalbes.
The uncertainty relation is a distinguishing feature of quantum theory, characterizing the incompatibility of noncommuting observables in the preparation of quantum states. Recently, many uncertainty relations were proposed with improved lower bounds and were deemed capable of incorporating multiple observables. Here we report an experimental verification of seven uncertainty relations of this type with single-photon measurements. The results, while confirming these uncertainty relations, show as well the relative stringency of various uncertainty lower bounds.
We define an uncertainty observable, acting on several replicas of a continuous-variable bosonic state, whose trivial uncertainty lower bound induces nontrivial phase-space uncertainty relations for a single copy of the state. By exploiting the Schwinger representation of angular momenta in terms of bosonic operators, we construct such an observable that is invariant under symplectic transformations (rotation and squeezing in phase space). We first design a two-copy uncertainty observable, which is a discrete-spectrum operator vanishing with certainty if and only if it is applied on (two copie…
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.
Quantum information-theoretic approach has been identified as a way to understand the foundations of quantum mechanics as early as 1950 due to Shannon. However there hasn't been enough advancement or rigorous development of the subject. In the following paper we try to find relationship between a general quantum mechanical observable and it von Neumann entropy. We find that the expectation values and the uncertainties of the observables have bounds which depend on the entropy. The results also show that von Neumann entropy is not just the uncertainty of the state but also encompasses the infor…
The variance of an observable in a quantum state is usually used to describe Heisenberg uncertainty relation. For mixed states, the variance includes quantum uncertainty and classical uncertainty. By means of the skew information and the decomposition of the variance, a stronger uncertainty relation was presented by Luo in [Phys. Rev. A 72, 042110 (2005)]. In this paper, by using Wigner-Yanase-Dyson information which is a generalization of the skew information, we propose a general uncertainty relation of mixed states.
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…
Conventional quantum uncertainty relations (URs) contain dispersions of two observables. Generalized URs are known which contain three or more dispersions. They are derived here starting with suitable generalized Cauchy inequalities. It is shown what new information the generalized URs provide. Similar interpretation is given to generalized Cauchy inequalities.
Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg-type uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state …
Constructive techniques to establish state-independent uncertainty relations for the sum of variances of arbitrary two observables are presented. We investigate the range of simultaneously attainable pairs of variances, which can be applied to a wide variety of problems including finding exact bound for the sum of variances of two components of angular momentum operator for any total angular momentum quantum number $j$ and detection of quantum entanglement. Resulting uncertainty relations are state-independent, semianalytical, bounded-error and can be made arbitrarily tight. The advocated appr…
Uncertainty principle is the basis of quantum mechanics. It reflects the basic law of the movement of microscopic particles. Wigner-Yanase skew information, as a measure of quantum uncertainties, is used to characterize the intrinsic features of the state and the observable. In this paper, we mainly investigate the sum uncertainty relations for both quantum mechanical observables and quantum channels based on skew information. We establish a new uncertainty relation in terms of Wigner-Yanase skew information for $n$ observables, which is saturated (thus it holds as equality) for two incompatib…
We prove a few novel state-dependent uncertainty relations for product as well the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Roberson-Schrödinger uncertainty relation and other ones existing in the current literature. Also, we derive state dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only we cannot prepare quantum states for which two incompatible observables can have sharp values, but also we have both…
We construct a multi-observable uncertainty equality as well as an inequality based on the sum of standard deviations in the qubit system. The obtained equality indicates that the uncertainty relation can be expressed more accurately, and also can be used to detect the mixedness of the system. Meanwhile, the new uncertainty inequality can provide a tighter lower bound, and the tightness can be maintained at a high level even in an open system. Furthermore, the deficiency of the uncertainty relation, that the lower bound of the product form uncertainty relations can be null even for two incompa…
We ask which is the best strategy to reveal uncertainty relations between comple- mentary observables of a continuous variable system for coarse-grained measurements. This leads to the derivation of new uncertainty relations for coarse-grained measurements that are always valid, even for detectors with low precision. These relations should be particularly relevant in experimental demonstrations of squeezing in quantum optics, quantum state reconstruction, and the development of trustworthy entanglement criteria.
Quantum Fisher information places the fundamental limit to the accuracy of estimating an unknown parameter. Here we shall provide the quantum Fisher information an operational meaning: a mixed state can be so prepared that a given observable has the minimal averaged variance, which equals exactly to the quantum Fisher information for estimating an unknown parameter generated by the unitary dynamics with the given observable as Hamiltonian. In particular we shall prove that the quantum Fisher information is the convex roof of the variance, as conjectured by Toth and Petz based on numerical and …
Determining the measurement uncertainty region is a difficult problem for generic sets of observables. For this reason the literature on exact measurement uncertainty regions is focused on symmetric sets of observables, where the symmetries are used to simplify the calculation. We provide a framework to systematically exploit available symmetries, formulated in terms of covariance, to simplify problems of measurement uncertainty. Our key result is that for a wide range figures of merit the optimal compatible approximations of covariant target observables are themselves covariant. This substant…
We have made the first experimental demonstration of the simultaneous minimum uncertainty product between two complementary observables for a two-state system (a qubit). A partially entangled two-photon state was used to perform such measurements. Each of the photons carries (partial) information of the initial state thus leaving a room for measurements of two complementary observables on every member in an ensemble.
A new lower boundary for the product of variances of two observables is obtained in the case, when these observables are entangled with the third one. This boundary can be higher than the Robertson--Schrödinger one. The special case of the two-dimensional pure Gaussian state is considered as an example.
The uncertainty relation, as one of the fundamental principles of quantum physics, captures the incompatibility of noncommuting observables in the preparation of quantum states. In this work, we derive two strong and universal uncertainty relations for $N(N\ge2)$ observables with discrete and bounded spectra, one in multiplicative form and the other in additive form. To verify their validity, for illustration, we implement in the spin-1/2 system an experiment with single-photon measurement. The experimental results exhibit the validity and robustness of these uncertainty relations, and indicat…
Universally valid uncertainty relations are proven in a model independent formulation for inherent and unavoidable extra noises in arbitrary joint measurements on single systems, from which Heisenber's original uncertainty relation is proven valid for any joint measurements with statistically independent noises.
The Heisenberg-Robertson uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of incompatible observables. However it does not capture the concept of incompatible observables because it can be trivial even for two incompatible observables. We experimentally demonstrate the new stronger uncertainty relations proposed by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)] relating on that sum of variances are valid in a state-dependent manner and the lower bound is guaranteed to be nontrivial for two observables being incompatible on the state of the sy…
Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the characterization of observables that allow strong uncertainty relations, and the construction of such relations for the case of several observables. We demonstrate how the stabilizer formalism can be applied to these questions. We show that the Maassen-Uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases. We comp…
A tight information-theoretic measurement uncertainty relation is experimentally tested with neutron spin-1/2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum …
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.
Uncertainty principle is one of the cornerstones of quantum theory. In the literature, there are two types of uncertainty relations, the operator form concerning the variances of physical observables and the entropy form related to entropic quantities. Both these forms are inequalities involving pairwise observables, and are found to be nontrivial to incorporate multiple observables. In this work we introduce a new form of uncertainty relation which may give out complete trade-off relations for variances of observables in pure and mixed quantum systems. Unlike the prevailing uncertainty relati…
Reality of quantum observables, a feature of long-standing interest within foundations of quantum mechanics, has recently been quantified and deeply studied by means of entropic measures [Phys. Rev. A 97, 022107 (2018)]. However, there is no state-independent "reality trade-off" between noncommuting observables, as in certain systems all observables are real [Europhys. Lett. 112, 40005 (2015)]. We show that the entropic uncertainty relation in the presence of quantum memory [Nature Phys. 6, 659 (2010)] perfectly supplements the discussed notion of reality, rendering trade-offs between reality …
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.
The canonical Robertson-Schrödinger uncertainty relation provides a loose bound for the product of variances of two non-commuting observables. Recently, several tight forward and reverse uncertainty relations have been proved which go beyond the traditional uncertainty relations. Here we experimentally test multifold of state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables for photonic qutrits. These uncertainty relations are independent of any optimization and still tighter than the Robertson-Schrödinger uncertainty relation and …
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…
Quantum uncertainty relations have deep-rooted significance on the formalism of quantum mechanics. Heisenberg's uncertainty relations attracted a renewed interest for its applications in quantum information science. Robertson derived a general form of Heisenberg's uncertainty relations for a pair of arbitrary observables represented by Hermitian operators. In the present work, we discover a temporal version of the Heisenberg-Robertson uncertainty relations for the measurement of two observables at two different times, where the dynamical uncertainties crucially depend on the time evolution of …
In this work we establish rigorously a measurement uncertainty relation (MUR) for three unbiased qubit observables, which was previously shown to hold true under some presumptions. The triplet MUR states that the uncertainty, which is quantified by the total statistic distance between the target observables and the jointly implemented observables, is lower bounded by an incompatibility measure that reflects the joint measurement conditions. We derive a necessary and sufficient condition for the triplet MUR to be saturated and the corresponding optimal measurement. To facilitate experimental te…
Heisenberg's measurement uncertainty relations (MUR) of two quantum observables are essential for contemporary researches in quantum foundations and quantum information science. Going beyond, here we report the first experimental test of MURs for three quantum observables. Following the proposal of Bush, Lahti, and Werner [Phys. Rev. A 89, 012129 (2014)], we first establish rigorously MURs for triplets of unbiased qubit observables as combined approximation errors lower-bounded by an incompatibility measure. We then develop a convex programming protocol to numerically find the exact value of t…
