Review for "Entropic uncertainty relations from equiangular tight frames and their applications"

Decision letter for "Entropic uncertainty relations from equiangular tight frames and their applications"

Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite states. Here we derive some entanglement criteria based on positive operator-valued measurements built from equiangular tight frames. We also present a class of entanglement witnesses based on the equiangular tight frames. At last, we generalize the entanglement criterion for bipartite systems to multipartite systems.

We present the theory of equiangular tight frames (ETFs) from association schemes as a generalization of harmonic ETFs. This theory has the pleasing consequence of including not only harmonic ETFs from abelian groups but also real ETFs with centroidal symmetry and the ETFs discovered by Renes and Strohmer coming from skew-symmetric conference matrices. Finally we present a theory of nonabelian harmonic ETFs from the perspective of association schemes. An infinite class of ETFs arising as group frames from nonabelian groups is presented.

An equiangular tight frame (ETF) is an M N matrix which has orthogonal equal norm rows, equal normcolumns, and the inner products of all pairs of columns have the same modulus. In this paper we studyETFs in which all of the entries are unimodular, and in particular pth roots of unity. A new constructionof unimodular ETFs based on resolvable Steiner systems is presented. This construction gives many newexamples of unimodular ETFs. In particular, an new in nite class of ETFs with entries in f1; 1g ispresented.Keywords: tight frames, equiangular, Hadamard matrices, Steiner systems, resolvable 1. INTRODUCTION Let F be an M N matrix with real or complex entries. If there exists A > 0 such that FF = AI , where Iis the identity matrix, then F is called a tight frame . If F F has constant diagonal and all of the o -diagonalentries have the same modulus, then F is called equiangular. If all of the entries of F have modulus 1 thenwe call F unimodular . If all of the entries of F are pth roots of unity, then we call F unimodular of degree p.This article is concerned with equiangular tight frames (ETFs) which are unimodular of degree p. Inparticular, we are concerned with nding unimodular ETFs of degree 2, which we shall also refer to asHadamard ETFs. Note that in the usual de nition of ETFs the columns are taken to have norm 1. However,since we are concerned with unimodular ETFs we will avoid normalizing, thus the columns of an M Nunimodular ETF have norm M .The paper is organized as follows. In Section 2 we present some necessary conditions on the existence ofunimodular ETFs. In Section 3 we review the construction of harmonic ETFs, which is an important classof ETFs that also happen to be unimodular. In Section 4 we review the construction of the Steiner ETFs ofFickus, Mixon, and Tremain,

Detecting or classifying already known sparse signals contaminated by Gaussian noise from compressive measurements is different from reconstructing sparse signals, as its objective is to minimize the error probability which describes performance of the detectors or classifiers. This paper is concerned about the performance improvement of a commonly used Compressive Classifier. We prove that when the arbitrary sensing matrices used to get the Compressive Measurements are transformed into Equi-Norm Tight Frames, i.e. the matrices that are row-orthogonal, The Compressive Classifier achieves better performance. Although there are other proofs that among all Equi-Norm Tight Frames the Equiangular tight Frames (ETFs) bring best worst-case performance, the existence and construction of ETFs on some dimensions is still an open problem. As the construction of Equi-Norm Tight Frames from any arbitrary matrices is very easy and practical compared with ETF matrices, the result of this paper can also provide a practical method to design an improved sensing matrix for Compressive Classification. We can conclude that: Tight is Better!

Non-Hermitian systems with exceptional points lead to many intriguing phenomena due to the coalescence of both eigenvalues and corresponding eigenvectors, in comparison to Hermitian systems where only eigenvalues degenerate. In this paper, we have investigated entropic uncertainty relation (EUR) in a non-Hermitian system and revealed a general connection between the EUR and the exceptional points of non-Hermitian system. Compared to the unitarity dynamics determined by a Hermitian Hamiltonian, the behaviors of EUR can be well defined in two different ways depending on whether the system is located in unbroken phase or broken phase regimes. In unbroken phase regime, EUR undergoes an oscillatory behavior while in broken phase regime where the oscillation of EUR breaks down. The exceptional points mark the oscillatory and non-oscillatory behaviors of the EUR. In the dynamical limit, we have identified the witness of critical behavior of non-Hermitian systems in terms of the EUR. Our results reveal that the witness can detect exactly the critical points of non-Hermitian systems beyond (anti-) PT-symmetric systems. Our results may have potential applications to witness and detect phase transition in non-Hermitian systems.

In this paper, we investigate the tightness of entropic uncertainty relation in the absence (presence) of the quantum memory which the memory particle being weakly coupled to a decohering Davies-type Markovian environment. The results show that the tightness of the quantum uncertainty relation can be controlled by the energy relaxation time F, the dephasing time G and the rescaled temperature p, the perfect tightness can be arrived by dephasing and energy relaxation satisfying F = 2G and p = 1/2. In addition, the tightness of the memory-assisted entropic uncertainty relation and the entropic uncertainty relation can be influenced mainly by the purity. While in memory-assisted model, the purity and quantum correlation can also influence the tightness actively while the quantum entanglement can influence the tightness slightly.

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. By using tripartite quantum-memory-assisted entropic uncertainty relation, an upper bound for the shareability of quantum discord among different parties 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 generalized and an upper bound for the shareability of quantum discord for multipartite states is derived.

The uncertainty principle provides a nontrivial bound to expose the precision for the outcome of the measurement on a pair of incompatible observables in a quantum system. Therefore, it is of essential importance for quantum precision measurement in the area of quantum information processing. Herein, we investigate quantum-memory-assisted entropic uncertainty relation (QMA-EUR) in a two-qubit Heisenberg spin chain. Specifically, we observe the dynamics of QMA-EUR in a realistic model there are two correlated sites linked by a thermal entanglement in the spin chain with an inhomogeneous magnetic field. It turns out that the temperature, the external inhomogeneous magnetic field and the field inhomogeneity can lift the uncertainty of the measurement due to the reduction of the thermal entanglement, and explicitly higher temperature, stronger magnetic field or larger inhomogeneity of the field can result in inflation of the uncertainty. Besides, it is found that there exists distinct dynamical behaviors of the uncertainty for ferromagnetism and antiferromagnetism chains. Moreover, we also verify that the measuring uncertainty is dramatically anti-correlated with the purity of the bipartite spin system, the greater purity can result in the reduction of the measuring uncertainty, vice versa. Therefore, our observations might provide a better understanding of the dynamics of the entropic uncertainty in the Heisenberg spin chain, and thus shed light on quantum precision measurement in the framework of versatile systems, particularly solid states.

Quantum entanglement is regarded as one of the core concepts, which is used to describe the non-classical correlation between subsystems, and entropic uncertainty relation plays a vital role in quantum precision measurement. It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states. An interesting question is naturally raised: is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement? Or if the relation can be applied to estimate the entanglement. In this work, we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation. The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system, and specifically the larger uncertainty will induce the weaker entanglement of the probed system, and vice versa. Besides, we use randomly generated states as illustrations to verify our results. Therefore, we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement.

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension d=22l+1 over the field F32. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.


 
 
 
 We apply the generalised uncertainty principle (GUP) to the entropic uncertainty relation conditions on quantum entanglement. In particular, we study the GUP corrections to the Shannon entropic uncertainty condition for entanglement. We combine previous work on the Shannon entropy entanglement criterion for bipartite systems and the GUP corrections to the Shannon entropy for a single system to calculate the GUP correction for an entangled bipartite system. As in an earlier paper of the second author, which dealt with variance relations, it is shown that there is an increase in the upper bound for the entanglement condition upon the application of the generalised uncertainty principle. Necessary fundamental concepts of the generalised uncertainty principle, entanglement and the entropic uncertainty relations are also discussed. This paper puts together the concepts of entanglement, entropic uncertainty relations and the generalised uncertainty principle all of which have been separately discussed in pedagogical papers by Schroeder, Majernik et al., Blado et al. and Sprenger.
 
 
 
 

Uncertainty relations and quantum entanglement are pivotal concepts in quantum theory. Beyond their fundamental significance in shaping our understanding of the quantum world, they also underpin crucial applications in quantum information theory. In this article, we investigate entropic uncertainty relations and entanglement detection with an emphasis on quantum measurements with design structures. On the one hand, we derive improved Rényi entropic uncertainty relations for design-structured measurements, exploiting the property that the sum of powered (e.g., squared) probabilities of obtaining different measurement outcomes is now invariant under unitary transformations of the measured system and can be easily computed. On the other hand, the above property essentially imposes a state-independent upper bound, which is achieved at all pure states, on one’s ability to predict local outcomes when performing a set of design-structured measurements on quantum systems. Realizing this, we also obtain criteria for detecting multi-partite entanglement with design-structured measurements.

Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, little is known about entropic uncertainty relations with more than two measurement settings. In the present survey, we review known results and open questions.

The relationship between equiangular tight frames and strongly regular graphs has been known for several years. This relationship has been exploited to construct many of the latest examples of new strongly regular graphs. Recently it was shown that there is a similar relationship between two-distance tight frames and strongly regular graphs. In this paper we present a new tensor like construction of two-distance tight frames, and hence a family of strongly regular graphs. While graphs with these parameters were known to exist, this new construction is very simple, requiring only the existence of an affine plane, whereas the original constructions often require more complicated objects such as generalized quadrangles.