The entanglement criteria based on equiangular tight frames

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!

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.

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.

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.

A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension d, there is good evidence that there always exists an aligned SIC in dimension d(d − 2), having predictable symmetries and smaller equiangular tight frames embedded in them. We provide a recipe for how to calculate sets of vectors in dimension d(d − 2) that share these properties. They consist of maximally entangled vectors in certain subspaces defined by the numbers entering the d dimensional SIC. However, the construction contains free parameters and we have not proven that they can always be chosen so that one of these sets of vectors is a SIC. We give some worked examples that, we hope, may suggest to the reader how our construction can be improved. For simplicity we restrict ourselves to the case of odd dimensions.

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An efficient algorithm for designing projection matrix in compressive sensing based on alternating optimization

This survey paper introduces several topics in quantum information theory, especially positive operator valued measures (POVMs) and entangled states. POVMs are needed to mathematically express measurements in the quantum world, and there are two special POVMs, named symmetric informationally complete POVMs and mutually unbiased bases. These two POVMs have very beautiful symmetry. We give the definitions and examples of these POVMs and discuss the existence or non-existence problem of these POVMs. Entanglement is some kind of relationship between two or more quanta. When several quanta have entanglement, the state of the quanta is called an entangled state. In this paper, quantum teleportation is introduced as an application using entangled states. Moreover, we investigate the problem of determining whether a given quantum state is an entangled state.

Quantum properties are soon subject to decoherence once the quantum system interacts with the classical environment. In this paper we experimentally test how propagation losses, in a Gaussian channel, affect the bipartite Gaussian entangled state generated by a subthreshold type-II optical parametric oscillator. Experimental results are discussed in terms of different quantum markers, as teleportation fidelity, quantum discord, and mutual information, and continuous-variable (CV) entanglement criteria. To analyze state properties we have retrieved the composite system covariance matrix by a single homodyne detector. We experimentally found that, even in the presence of a strong decoherence, the generated state never disentangles and keeps breaking the quantum limit for the discord. This result proves that the class of CV entangled states discussed in this paper would allow, in principle, to realize quantum teleportation over an infinitely long Gaussian channel.

We analyze the quantum Fisher information (QFI) and entanglement of the ground state for the XY model. The QFI determines the precision in parameter estimations, moreover, it is a criterion of entanglement. The scaling behaviors of the QFI are studied both at the critical point and in different phases. In the symmetry-broken phase, the QFI scales proportional to the square of the system size, while in the symmetric phase, it is nearly independent of the system size. The study of QFI not only reveals the distinct entanglement properties of the XY model in different phases, but also shows that the ground state of the XY model can be used to perform Heisenberg-limit parameter estimation.

Quantum properties, such as entanglement and coherence, are indispensable resources in various quantum information processing tasks. However, given a quantum state, it is not a simple question to identify whether it is entangled or not. We study the entanglement of quantum systems in arbitrary dimensions using the Bloch representation of density matrix. This approach enables us to consider quantum entanglement geometrically. We first analyze the effect of local Bloch vector on determining whether a quantum state is entangled or separable, and obtain an entanglement criterion related to local Bloch vector, which is stronger than the criterion that only considers the correlation matrix. This criterion is equivalent to the optimized computable cross norm criterion in [14]. Our proof is considered from the perspective of Bloch representation, and there is no need to know the matrix relignment in advance. This criterion also be extended to the problem of the separability of tripartite quantum states. We then combine the local filtering transformation with this entanglement criterion. After applying a suitable local filtering transformation, the quantum state has at most one nonzero Bloch vector, and examples are given to illustrate that the local filtering transformation can help detect some bound states where the original entanglement criterion fails. Moreover, for a finite-dimensional quantum system in which one party is a qubit, we give an analytical form of the combined entanglement criterion.

Quantum mechanics, information theory, and relativity theory are the basic foundations of theoretical physics. The acquisition of information from a quantum system is the interface of classical and quantum physics. Essential tools for its description are Kraus matrices and positive operator valued measures (POVMs). Special relativity imposes severe restrictions on the transfer of information between distant systems. Quantum entropy is not a Lorentz covariant concept. Lorentz transformations of reduced density matrices for entangled systems may not be completely positive maps. Quantum field theory, which is necessary for a consistent description of interactions, implies a fundamental trade-off between detector reliability and localizability. General relativity produces new, counterintuitive effects, in particular when black holes (or more generally, event horizons) are involved. Most of the current concepts in quantum information theory may then require a reassessment.

Entanglement detection is essential in quantum information science and quantum many-body physics. It has been proved that entanglement exists almost surely for a random quantum state, while the realizations of effective entanglement criteria usually consume exponentially many resources with regard to system size or qubit number, and efficient criteria often perform poorly without prior knowledge. This fact implies a fundamental limitation might exist in the detectability of entanglement. In this work, we formalize this limitation as a fundamental trade-off between the efficiency and effectiveness of entanglement criteria via a systematic method to evaluate the detection capability of entanglement criteria theoretically. For a system coupled to an environment, we prove that any entanglement criterion needs exponentially many observables to detect the entanglement effectively when restricted to single-copy operations. Otherwise, the detection capability of the criterion will decay double exponentially. Furthermore, if multicopy joint measurements are allowed, the effectiveness of entanglement detection can be exponentially improved, which implies a quantum advantage in entanglement detection problems. Our results may shed light on why quantum phenomena are difficult to observe in large noisy systems.