arXiv subjects

Rajesh Pereira

Publications and source records attributed to Rajesh Pereira.

At least 37 records · Page 2Linked to original sources

arXiv2020

Coherence Concurrence for X States

We study the properties of coherence concurrence and present a physical explanation analogous to the coherence of assistance. We give an optimal pure state decomposition which attains the coherence concurrence for qubit states. We prove the additivity of coherence concurrence under direct sum operations in another way. Using these results, we calculate analytically the coherence concurrence for X states and show its optimal decompositions. Moreover, we show that the coherence concurrence is exactly twice the convex roof extended negativity of the Schmidt correlated states, thus establishing a direct relation between coherence concurrence and quantum entanglement.

quant-ph↗

arXiv2020

The l1 Norm of Coherence of Assistance

We introduce and study the l1 norm of coherence of assistance both theoretically and operationally. We first provide an upper bound for the l1 norm of coherence of assistance and show a necessary and sufficient condition for the saturation of the upper bound. For two and three dimensional quantum states, the analytical expression of the l1 norm of coherence of assistance is given. Operationally, the mixed quantum coherence can always be increased with the help of another party' s local measurement and one way classical communication since the l1 norm of coherence of assistance, as well as the relative entropy of coherence of assistance, is shown to be strictly larger than the original coherence. The relation between the l1 norm of coherence of assistance and entanglement is revealed. Finally, a comparison between the l1 norm of coherence of assistance and the relative entropy of coherence of assistance is made.

quant-ph↗

arXiv2019

Isoclinic Subspaces and Quantum Error Correction

We exhibit equivalent conditions for subspaces of an inner product space to be isoclinic, including a characterization based on the classical notion of canonical angles. We identify a connection with quantum error correction, showing that every quantum error correcting code is associated with a family of isoclinic subspaces, and we prove a converse for pairs of such subspaces. We also show how the canonical angles for isoclinic subspaces arise in the structure of the higher rank numerical ranges of the corresponding orthogonal projections.

math.QA↗

arXiv2019

Approximate Quasiorthogonality of Operator Algebras and Relative Quantum Privacy

We show that the approximate quasiorthogonality of two operator algebras is equivalent to the algebras being approximately private relative to their conditional expectation quantum channels. Our analysis is based on a characterization of the measure of orthogonality in terms of Choi matrices and Kraus operators for completely positive maps. We present examples drawn from different areas of quantum information.

quant-ph↗

arXiv2019

A simplified and unified generalization of some majorization results

We consider positive, integral-preserving linear operators acting on $L^1$ space, known as stochastic operators or Markov operators. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of stochastic integral operators (such operators arise naturally when considering matrix majorization in $L^1$). We collect a number of results for vector-valued functions on $L^1$, simplifying some proofs found in the literature. In particular, matrix majorization and multivariate majorization are related in $\mathbb{R}^n$. In $\mathbb{R}$, these are also equivalent to convex function inequalities.

math.FA↗

arXiv2019

Which quartic polynomials have a hyperbolic antiderivative?

Every linear, quadratic or cubic polynomial having all real zeros is the derivative of a polynomial having all real zeros. The statement is false for higher degree polynomials. In particular, not every fourth degree polynomial with real zeros is the derivative of a polynomial having all real zeros. We derive a necessary and sufficient condition for a quartic polynomial to be the derivative of a polynomial having all real zeros. This condition is a single quadratic form inequality involving the zeros of the quartic polynomial.

math.CA↗

arXiv2019

Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials

A simple method for transmitting quantum states within a quantum computer is via a quantum spin chain---that is, a path on $n$ vertices. Unweighted paths are of limited use, and so a natural generalization is to consider weighted paths; this has been further generalized to allow for loops (\emph{potentials} in the physics literature). We study the particularly important situation of perfect state transfer with respect to the corresponding adjacency matrix or Laplacian through the use of orthogonal polynomials. Low-dimensional examples are given in detail. Our main result is that PST with respect to the Laplacian matrix cannot occur for weighted paths on $n\geq 3$ vertices nor can it occur for certain symmetric weighted trees. The methods used lead us to a conjecture directly linking the rationality of the weights of weighted paths on $n>3$ vertices, with or without loops, with the capacity for PST between the end vertices with respect to the adjacency matrix.

quant-ph↗

arXiv2018

Quantum Error Correction and One-Way LOCC State Distinguishability

We explore the intersection of studies in quantum error correction and quantum local operations and classical communication (LOCC). We consider one-way LOCC measurement protocols as quantum channels and investigate their error correction properties, emphasizing an operator theory approach to the subject, and we obtain new applications to one-way LOCC state distinguishability as well as new derivations of some established results. We also derive conditions on when states that arise through the stabilizer formalism for quantum error correction are distinguishable under one-way LOCC.

quant-ph↗

arXiv2017

Quantum Privacy and Schur Product Channels

We investigate the quantum privacy properties of an important class of quantum chan-nels, by making use of a connection with Schur product matrix operations and associated correlationmatrix structures. For channels implemented by mutually commuting unitaries, which cannot priva-tise qubits encoded directly into subspaces, we nevertheless identify private algebras and subsystemsthat can be privatised by the channels. We also obtain further results by combining our analysiswith tools from the theory of quasiorthogonal operator algebras and graph theory.

math-ph↗

arXiv2017

The DJL Conjecture for CP Matrices over Special Inclines

Drew, Johnson and Loewy conjectured that for $n \geq 4$, the CP-rank of every $n \times n$ completely positive real matrix is at most $\left[n^{2}/4\right]$. While this conjecture is false for completely positive real matrices, we show that this conjecture is true for $n \times n$ completely positive matrices over certain special types of inclines. In addition, we prove an incline version of Markham's theorems which gives sufficient conditions for completely positive matrices over special inclines to have triangular factorizations.

math.RA↗

arXiv2017

Anticoherent Subspaces

We extend the notion of anticoherent spin states to anticoherent subspaces. An anticoherent subspace of order t, is a subspace whose unit vectors are all anticoherent states of order t. We use Klein's description of algebras of polynomials which are invariant under finite subgroups of SU(2) to provide constructions of anticoherent subspaces. Furthermore, we show a connection between the existence of these subspaces and the properties of the higher-rank numerical range for a set of spin observables. We also note that these constructions give us subspaces of spin states all of whose unit vectors have Majorana representations which are spherical designs of order at least t.

math-ph↗

arXiv2016

Operator Structures and Quantum One-Way LOCC Conditions

We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator systems, operator algebras, and Hilbert C*-modules all naturally arise in this setting, and we make use of these structures to derive new results and new derivations of some established results in the study of LOCC. We also show that perfect distinguishability under one-way LOCC and under arbitrary operations is equivalent for several families of operators that appear jointly in matrix and operator theory and quantum information theory.

quant-ph↗

arXiv2015

Private Quantum Subsystems and Quasiorthogonal Operator Algebras

We generalize a recently discovered example of a private quantum subsystem to find private subsystems for Abelian subgroups of the $n$-qubit Pauli group, which exist in the absence of private subspaces. In doing so, we also connect these quantum privacy investigations with the theory of quasiorthogonal operator algebras through the use of tools from group theory and operator theory.

quant-ph↗

arXiv2015

Some geometric interpretations of quantum fidelity

We consider quantum fidelity between two states $ρ$ and $σ$, where we fix $ρ$ and allow $σ$ to be sent through a quantum channel. We determine the minimal fidelity where one minimizes over (a) all unital channels, (b) all mixed unitary channels, and (c) arbitrary channels. We derive results involving the minimal eigenvalue of $ρ$, which we can interpret as a convex combination coefficient. As a consequence, we give a new geometric interpretation of the minimal fidelity with respect to the closed, convex set of density matrices and with respect to the closed, convex set of quantum channels. We further investigate the geometric nature of fidelity by considering density matrices arising as normalized projections onto subspaces; in this way, fidelity can be viewed as a geometric measure of distance between two spaces. We give a connection between fidelity and the canonical (principal) angles between the subspaces.

quant-ph↗

arXiv2015

Trumping and Power Majorization

Majorization is a basic concept in matrix theory that has found applications in numerous settings over the past century. Power majorization is a more specialized notion that has been studied in the theory of inequalities. On the other hand, the trumping relation has recently been considered in quantum information, specifically in entanglement theory. We explore the connections between trumping and power majorization. We prove an analogue of Rado's theorem for power majorization and consider a number of examples.

math.FA↗

arXiv2014

Extending a characterization of majorization to infinite dimensions

We consider recent work linking majorization and trumping, two partial orders that have proven useful with respect to the entanglement transformation problem in quantum information, with general Dirichlet polynomials, Mellin transforms, and completely monotone sequences. We extend a basic majorization result to the more physically realistic infinite-dimensional setting through the use of generalized Dirichlet series and Riemann-Stieltjes integrals.

quant-ph↗

arXiv2014

Taking all positive eigenvectors is suboptimal in classical multidimensional scaling

It is hard to overstate the importance of multidimensional scaling as an analysis technique in the broad sciences. Classical, or Torgerson multidimensional scaling is one of the main variants, with the advantage that it has a closed-form analytic solution. However, this solution is exact if and only if the distances are Euclidean. Conversely, there has been comparatively little discussion on what to do in the presence of negative eigenvalues: the intuitive solution, prima facie justifiable in least-squares terms, is to take every positive eigenvector as a dimension. We show that this, minimizing least-squares to the centred distances instead of the true distances, is suboptimal - throwing away positive eigenvectors can decrease the error even as we project to fewer dimensions. We provide provably better methods for handling this common case.

math.ST↗

arXiv2013

Dirichlet polynomials, Majorization, and Trumping

Majorization and trumping are two partial orders which have proved useful in quantum information theory. We show some relations between these two partial orders and generalized Dirichlet polynomials, Mellin transforms, and completely monotone functions. These relations are used to prove a succinct generalization of Turgut's characterization of trumping.

quant-ph↗