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 that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect, and we need to look for additional properties.
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 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…
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.
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.
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…
We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of en…
We derive uncertainty relation inequalities according to the mutually unbiased measurements. Based on the calculation of the index of coincidence of probability distribution given by $d+1$ MUMs on any density operator $ρ$ in $\mathbb{C}^{d}$, both state-dependent and state-independent forms of lower entropic bounds are given. Furthermore, we formulate uncertainty relations for MUMs in terms of Rényi and Tsallis entropies.
We consider entropic uncertainty relations for outcomes of the measurements of a quantum state in 3 or more mutually unbiased bases (MUBs), chosen from the standard construction of MUBs in prime dimension. We show that, for any choice of 3 MUBs and at least one choice of a larger number of MUBs, the best possible entropic uncertainty relation can be only marginally better than the one that trivially follows from the relation by Maassen and Uffink (PRL, 1987) for 2 bases.
We construct inequalities between Rényi entropy and the indexes of coincidence of probability distributions, based on which we obtain improved state-dependent entropic uncertainty relations for general symmetric informationally complete positive operator-valued measures (SIC-POVM) and mutually unbiased measurements (MUM) on finite dimensional systems. We show that our uncertainty relations for general SIC-POVMs and MUMs can be tight for sufficiently mixed states, and moreover, comparisons to the numerically optimal results are made via information diagrams.
Squashed entanglement and entanglement of purification are quantum mechanical correlation measures and defined as certain minimisations of entropic quantities. We present the first non-trivial calculations of both quantities. Our results lead to the conclusion that both measures can drop by an arbitrary amount when only a single qubit of a local system is lost. This property is known as "locking" and has previously been observed for other correlation measures, such as the accessible information, entanglement cost and the logarithmic negativity. In the case of squashed entanglement, the results…
We consider two (natural) families of observables $O_k$ for systems with dimension $d=3,4,5$: the spin observables $S_x$, $S_y$ and $S_z$, and the observables that have mutually unbiased bases as eigenstates. We derive tight entropic uncertainty relations for these families, in the form $\sum_kH(O_k)\geqslantα_d$, where $H(O_k)$ is the Shannon entropy of the measurement outcomes of $O_k$ and $α_d$ is a constant. We show that most of our bounds are stronger than previously known ones. We also give the form of the states that attain these inequalities.
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…
We establish a general connection between entropic uncertainty relations, Einstein-Podolsky-Rosen steering, and joint measurability. Specifically, we construct steering inequalities from any entropic uncertainty relation, given that the latter satisfies two natural properties. We obtain steering inequalities based on Rényi entropies. These turn out to be tight in many scenarios, using max- and min-entropy. Considering steering tests with two noisy measurements, our inequalities exactly recover the noise threshold for steerability. This is the case for any pair of qubit 2-outcome measurements, …
We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long standing problem of the existence of MUB. We derive their general form, and show that in a finite, $d$-dimensional Hilbert space, one can construct a complete set of $d+1$ mutually unbiased measurements. Beside of their intrinsic link to MUB, we show, that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unb…
We study two operational approaches to quantifying incompatibility that depart significantly from the well known entropic uncertainty relation (EUR) formalism. Both approaches result in incompatibility measures that yield non-zero values even when the pair of incompatible observables commute over a subspace, unlike EURs which give a zero lower bound in such cases. Here, we explicitly show how these measures go beyond EURs in quantifying incompatibility: For any set of quantum observables, we show that both incompatibility measures are bounded from below by the corresponding EURs for the Tsalli…
Uncertainty relations are among the unique fingerprints of quantum physics, being direct expression of non-commutativity and complementarity. Entropic uncertainty relations arise in quantum information theory as the most natural expression of the uncertainty principle and have proven themselves important tools for cryptographic security proofs. An open question has been how strong entropic uncertainty relations can be. Here we answer this question by showing that there exist observables that obey maximally strong entropic uncertainty relations in high dimensional Hilbert spaces. Actually, we s…
In this work we present some new properties of fiducial states in prime dimensions. We parameterize fiducial operators on eigenvectors bases of displacement operators, which allows us to find a manifold $Ω$ of hermitian operators satisfying $\mathrm{Tr}(ρ)=\mathrm{Tr}(ρ^2)=1$ for any $ρ$ in $Ω$. This manifold contains the complete set of fiducial pure states in every prime dimension. Indeed, any quantum state $ρ\geq0$ belonging to $Ω$ is a fiducial pure state. Also, we present an upper bound for every probability associated to mutually unbiased decomposition of fiducial states. This bound allo…
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…
A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions ($d=p^{2}$) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of $p+1…
Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood. Here, we construct special sets of up to 2n+1 mutually unbiased bases (MUBs) in dimension d=2^n which have particularly beautiful symmetry properties derived from the Clifford algebra. More precisely, we show that there exists a unitary transformation that cyclically permutes such bases. This unitary can be understood as a generalization of the Fourier transform, which exchanges two MUBs, to multiple complementary aspects. We proceed to pr…
A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete POVMs and mutually unbiased bases. Additionally, it provides a natural way to define two new families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.
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 provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interestin…
Uncertainty relation usually is one of the most important features in quantum mechanics, and is the backbone of quantum theory, which distinguishes from the rule in classical counterpart. Specifically, entropy-based uncertainty relations are of fundamental importance in the region of quantum information theory, offering one nontrivial bound of key rate towards quantum key distribution. In this work, we experimentally demonstrate the entropic uncertainty relations and coherence-based uncertainty relations in an all-optics platform. By means of preparing two kinds of bipartite initial states wit…
We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size $N$ in $L$ orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For $L=2$ the maximal average entropy saturates at $\log N$ as there exists a mutually coherent state, but certainty relations are shown to be nontrivial for $L \ge 3$ measurements. In the case of a prime power dimension, $N=p^k$, and the number of measurements $L=N+1$, the upper bound for the average entropy becomes minimal for…
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…
Separability conditions for a bipartite quantum system of finite-dimensional subsystems are formulated in terms of Rényi and Tsallis entropies. Entropic uncertainty relations often lead to entanglement criteria. We propose new approach based on the convolution of discrete probability distributions. Measurements on a total system are constructed of local ones according to the convolution scheme. Separability conditions are derived on the base of uncertainty relations of the Maassen-Uffink type as well as majorization relations. On each of subsystems, we use a pair of sets of subnormalized vecto…
We prove that magic states from the Clifford hierarchy give optimal solutions for tasks involving nonlocality and entropic uncertainty with respect to Pauli measurements. For both the nonlocality and uncertainty tasks, stabilizer states are the worst possible pure states so our solutions have an operational interpretation as being highly non-stabilizer. The optimal strategy for a qudit version of the Clauser-Horne-Shimony-Holt (CHSH) game in prime dimensions is achieved by measuring maximally entangled states that are isomorphic to single-qudit magic states. These magic states have an appealin…
Heisenberg's uncertainty principle is formulated for a set of generalized measurements within the framework of majorization theory, resulting in a partial uncertainty order on probability vectors that is stronger than those based on quasi-entropic measures. The theorem that results from this formulation guarantees that the uncertainty of the results of a set of generalized measurements without a common eigenstate has an inviolable lower bound which depends on the measurement set but not the state. A corollary to this theorem yields a parallel formulation of the uncertainty principle for genera…
The existence of quantum uncertainty relations is the essential reason that some classically impossible cryptographic primitives become possible when quantum communication is allowed. One direct operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of inf…
We study experimentally accessible lower bounds on entanglement measures based on entropic uncertainty relations. Experimentally quantifying entanglement is highly desired for applications of quantum simulation experiments to fundamental questions, e.g., in quantum statistical mechanics and condensed-matter physics. At the same time it poses a significant challenge because the evaluation of entanglement measures typically requires the full reconstruction of the quantum state, which is extremely costly in terms of measurement statistics. We derive an improved entanglement bound for bipartite sy…
We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation sugge…
We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finite-dimensional system and for any approximate joint measurement of two target discrete observables, we define the entropic divergence as the maximal total loss of information occurring in the approximation at hand. For fixed target observables, we study the joint measurements minimizing the entropic divergence, and we prove the general properties of its minimum value. Such a minimum is our uncertainty …
To find the essential nature of quantum theory has been an important problem for not only theoretical interest but also applications to quantum technologies. In those studies on quantum foundations, the notion of uncertainty plays a primary role among several stunning features of quantum theory. The purpose of this thesis is to investigate fundamental aspects of uncertainty. In particular, we address this problem focusing on convexity, which has an operational origin. We first try to reveal why in quantum theory similar bounds are often obtained for two types of uncertainty relations, namely, …
