Transpose Map Research Articles

On Spherical Distributions

In this paper, we begin with the study of the hyperbolic spaces H G where ???? = ????(????. ????; ????) and ???? = ????(????; ????) × ????(???? − ????, ????; ????), ???? = ℝ, ℂ or ℍ denotes the set of real numbers, complex numbers and quaternions respectively. In the articles of J. Faraut [2] and M.T. Kosters and G. van Dijk [4], spherical distributions were derived following two different methods. The first method is to describe the behavior of spherical distributions making use of the Fourier transform of finite and infinite functions. The second is to express them asM'S where ????′:????′ → ????′(????) is a transpose map and S is a solution to the differential equation LS  a(t)S"b(t)S' and making use of the hypergeometric functions. Now we show that spherical distributions T can be obtained through a particular distribution S on ℝ by solving the equation . S LS   The technique of Methe̒ e’s [6] is instrumental for the context.

Detecting and quantifying entanglement on near-term quantum devices

Quantum entanglement is a key resource in quantum technology, and its quantification is a vital task in the current noisy intermediate-scale quantum (NISQ) era. This paper combines hybrid quantum-classical computation and quasi-probability decomposition to propose two variational quantum algorithms, called variational entanglement detection (VED) and variational logarithmic negativity estimation (VLNE), for detecting and quantifying entanglement on near-term quantum devices, respectively. VED makes use of the positive map criterion and works as follows. Firstly, it decomposes a positive map into a combination of quantum operations implementable on near-term quantum devices. It then variationally estimates the minimal eigenvalue of the final state, obtained by executing these implementable operations on the target state and averaging the output states. Deterministic and probabilistic methods are proposed to compute the average. At last, it asserts that the target state is entangled if the optimized minimal eigenvalue is negative. VLNE builds upon a linear decomposition of the transpose map into Pauli terms and the recently proposed trace distance estimation algorithm. It variationally estimates the well-known logarithmic negativity entanglement measure and could be applied to quantify entanglement on near-term quantum devices. Experimental and numerical results on the Bell state, isotropic states, and Breuer states show the validity of the proposed entanglement detection and quantification methods.

Relationship between Mullineux involution and the generalized regularization

The Mullineux involution is an important map on p-regular partitions that originates from the modular representation theory of Sn. In this paper we study the Mullineux transpose map and the generalized column regularization and prove a condition under which the two maps are exactly the same. Our results generalize the work of Bessenrodt, Olsson and Xu, and the combinatorial constructions is related to the Iwahori–Hecke algebra and the global crystal basis of the basic Uq(sl̂b)-module. In the conclusion, we provide several conjectures regarding the q-decomposition numbers and generalizations of results due to Fayers.

English

Let m and n be positive integers, and let R = (r1, . . . , rm) and S = (s1, . . . , sn) be nonnegative integral vectors. Let A(R,S) be the set of all m × n (0, 1)-matrices with row sum vector R and column vector S. Let R and S be nonincreasing, and let F(R) be the m × n (0, 1)-matrix, where for each i, the ith row of F(R,S) consists of ri 1’s followed by (n−ri) 0’s. Let A ∈ A(R,S). The discrepancy of A, disc(A), is the number of positions in which F(R) has a 1 and A has a 0. In this paper we investigate linear operators mapping m × n matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when m = n, the transpose mapping.

The Non-m-Positive Dimension of a Positive Linear Map

We introduce a property of a matrix-valued linear map Φ that we call its ``non-m-positive dimension'' (or ``non-mP dimension'' for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of Im⊗Φ. Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map Im⊗Φ∗ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

Scaling maps of s -ordered quasiprobabilities are either nonpositive or completely positive

Continuous-variable systems in quantum theory can be fully described through any one of the $s$-ordered family of quasiprobabilities ${\mathrm{\ensuremath{\Lambda}}}_{s}(\ensuremath{\alpha}), s\ensuremath{\in}[\ensuremath{-}1,1]$. We ask for what values of $(s,a)$ is the scaling map ${\mathrm{\ensuremath{\Lambda}}}_{s}(\ensuremath{\alpha})\ensuremath{\rightarrow}{a}^{\ensuremath{-}2}{\mathrm{\ensuremath{\Lambda}}}_{s}({a}^{\ensuremath{-}1}\ensuremath{\alpha})$ a positive map? Our analysis based on a duality we establish settles this issue: (i) the scaling map generically fails to be positive, showing that there is no useful entanglement witness of the scaling type beyond the transpose map, and (ii) in the two particular cases $(s=1,|a|\ensuremath{\le}1)$ and $(s=\ensuremath{-}1,|a|\ensuremath{\ge}1)$, and only in these two nontrivial cases, the map is not only positive but also completely positive as seen through the noiseless attenuator and amplifier channels. We also present a ``phase diagram'' for the behavior of the scaling maps in the $s\ensuremath{-}a$ parameter space with regard to its positivity, obtained from the viewpoint of symmetric-ordered characteristic functions. This also sheds light on similar diagrams for the practically relevant attenuation and amplification maps with respect to the noise parameter, especially in the range below the complete-positivity (or quantum-limited) threshold.

A generalization of the matrix transpose map and its relationship to the twist of the polynomial ring by an automorphism

A generalization of the notion of symmetric matrix was introduced by Cassidy and Vancliff in 2010 and used by them in a construction that produces quadratic regular algebras of finite global dimension that are generalizations of graded Clifford algebras. In this article, we further their ideas by introducing a generalization of the matrix transpose map and use it to generalize the notion of skew-symmetric matrix. With these definitions, an analogue of the result that every n × n matrix is a sum of a symmetric matrix and a skew-symmetric matrix holds. We also prove an analogue of the result that the transpose map is an antiautomorphism of the algebra of n × n matrices, and show that the antiautomorphism property of our generalized transpose map is related to the notion of twisting the polynomial ring on n variables by an automorphism.

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.

Optimal approximate transpose map via quantum designs and its applications to entanglement detection

We show that quantum designs characterize the general structure of the optimal approximation of the transpose map on quantum states. Based on this characterization, we propose an implementation of the approximate transpose map by a measurement-and-preparation scheme. The results show that state-manipulation in quantum two-designs suffices for transpose-based quantu3 information applications. In particular, we present how these results can be applied to the framework of detecting multipartite entangled states, for instance, when local measurements or interferometry-based experimental approaches are applied.

Generation of mapping cones from small sets

We answer in the affirmative a recently-posed question that asked if there exists an “untypical” convex mapping cone – i.e., one that does not arise from the transpose map and the cones of k-positive and k-superpositive maps. We explicitly construct such a cone based on atomic positive maps. Our general technique is to consider the smallest convex mapping cone generated by a single map, and we derive several results on such mapping cones. We use this technique to also present several other examples of untypical mapping cones, including a family of cones generated by spin factors. We also provide a full characterization of mapping cones generated by single elements in the qubit case in terms of their typicality.

Characterizing operations preserving separability measures via linear preserver problems

We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied in quantum information theory. We see, in particular, that for k ≥ 2 the operator norms induced by states with Schmidt rank k are invariant only under local unitaries, the swap operator and the transpose map. However, in the k = 1 case there is an additional isometry: the partial transpose map.

THETA SERIES OVER THE HURWITZ QUATERNIONS

There are six theta constants over the Hurwitz quaternions on the quaternion half-space of degree 2. The paper describes the behavior of these theta constants under the transpose mapping, which can be derived from the Fourier expansions. The results are applied to the theta series of the first and second kind.

Norm Estimates of the Partial Transpose Map on the Tensor Products of Matrices

We present a norm estimate for the partial transpose map Θ on the tensor product \(M_m \otimes M_n\) $$\Theta \left( \sum_k A_k \otimes B_k \right) := \sum_k A_k \otimes B_k^T$$ with respect to a unitarily invariant norm. This is related to the norm estimates of the following maps on Mm,n in terms of the spectral norm of \(\displaystyle \sum_k A_k \otimes B_k\) : $$X \longmapsto \sum_k A_k X B_k \quad \textrm{and} \quad X \longmapsto \sum_k A_k X B_k^T.$$ We show further that in the special case of \(A \otimes I_n + I_m \otimes B\) as well as AX + XB and AX + XBT those estimates are much improved and that $$\| A \otimes I_n + I_m \otimes B^T \|_p = \| A \otimes I_n + I_m \otimes B \|_p$$ for certain Schatten p-norms.

Constrained bounds on measures of entanglement

Entanglement measures constructed from two positive, but not completely positive, maps on density operators are used as constraints in placing bounds on the entanglement of formation, the tangle, and the concurrence of $4N$ mixed states. The maps are the partial transpose map and the phi map introduced by Breuer [H.-P. Breuer, Phys. Rev. Lett. 97, 080501 (2006)]. The norm-based entanglement measures constructed from these two maps, called negativity and phi negativity, respectively, lead to two sets of bounds on the entanglement of formation, the tangle, and the concurrence. We compare these bounds and identify the sets of $4N$ density operators for which the bounds from one constraint are better than the bounds from the other. In the process, we present a derivation of the already known bound on the concurrence based on the negativity. We compute bounds on the three measures of entanglement using both the constraints simultaneously. We demonstrate how such doubly constrained bounds can be constructed. We discuss extensions of our results to bipartite states of higher dimensions and with more than two constraints.

Realization of universal optimal quantum machines by projective operators and stochastic maps

Optimal quantum machines can be implemented by linear projective operations. In the present work a general qubit symmetrization theory is presented by investigating the close links to the qubit purification process and to the programmable teleportation of any generic optimal anti-unitary map. In addition, the contextual realization of the N ->M cloning map and of the teleportation of the N->(M-N) universal NOT gate is analyzed by a novel and very general angular momentum theory. An extended set of experimental realizations by state symmetrization linear optical procedures is reported. These include the 1->2 cloning process, the UNOT gate and the quantum tomographic characterization of the optimal partial transpose map of polarization encoded qubits.

There is no analog of the transpose map for infinite matrices

In this note we show that there are no ring anti-isomorphism between row finite matrix rings. As a consequence we show that row finite and column finite matrix rings cannot be either isomorphic or Morita equivalent rings. We also show that anti-isomorphisms between endomorphism rings of infinitely generated projective modules may exist.

Parallel prime-factor discrete Hartley transform on the orthogonal multiprocessor

This paper proposes parallel implementation of the prime-factor discrete Hartley transform algorithm on the orthogonal-multiprocessor (OMP) architecture when the transform size N is decomposable into two mutually prime numbers N = N 1 N 2. The implementation shows that the existing prime-factor DHT algorithm can be mapped easily on the architecture without modification. The proposed algorithms include input index mapping, summation, scaling, matrix transpose, adjust-add and output index mapping. The resulting algorithm can be accomplished in N 1 + N 2 multiplication steps on a J processors OMP, where J is the maximum dimension of scaling matrices ( C ̂ N 1 , C ̂ N 2 ).

On the geometry of positive maps in matrix algebras. II

Geometric relations among the identity map, the transpose map, and the map τ(x)− 1 n (x) Tr(x)1 n in the matrix algebra M n are illustrated, providing parametric examples of maps which are not k+1-positive but which satisfy the matrix Schwartz inequality of order k for 1 ⩽ k ⩽ n − 1.

On the Transpose Map of Matrix Algebras

It is shown that for the transpose map $\theta (n)$ of the $n \times n$ matrix algebra ${M_n}$, its $k$ th multiplicity map $\theta {(n)_k}$ has exactly the norm $k$ if $k \leqslant n$, hence the completely bounded norm of $\theta (n)$ written ${\left \| {\theta (n)} \right \|_{cb}}$ equals $n$. Some applications and related results are also proved.

On the transpose map of matrix algebras

It is shown that for the transpose map θ ( n ) \theta (n) of the n × n n \times n matrix algebra M n {M_n} , its k k th multiplicity map θ ( n ) k \theta {(n)_k} has exactly the norm k k if k ⩽ n k \leqslant n , hence the completely bounded norm of θ ( n ) \theta (n) written ‖ θ ( n ) ‖ c b {\left \| {\theta (n)} \right \|_{cb}} equals n n . Some applications and related results are also proved.