Search

We provide an analytic method for estimating the entanglement of the non-Gaussian energy eigenstates of disordered harmonic oscillator systems. We invoke the explicit formulas of the eigenstates of the oscillator systems to establish bounds for their ϵ -Rényi entanglement entropy ϵ ∈ (0 , 1) . Our methods result in a logarithmically corrected area law for the entanglement of eigenstates, corresponding to one excitation, of the disordered harmonic oscillator systems. [ABSTRACT FROM AUTHOR]

The rank of a tensor is analysed in the context of quantum entanglement. A pure quantum state $ \bf v $ v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for the characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state $ \mathbf{v} $ v is called entangled, if it cannot be written in the product form, $ \mathbf{v} \ne \mathbf{v}_1 \otimes \mathbf{v}_2 \otimes \cdots \otimes \mathbf{v}_d $ v ≠ v 1 ⊗ v 2 ⊗ ⋯ ⊗ v d , what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed. [ABSTRACT FROM AUTHOR]

We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer et al. (Phys Rev A 95:032308, 2017. ), and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces. [ABSTRACT FROM AUTHOR]

We discuss dense coding with n copies of a specific preshared state between the sender and the receiver when the encoding operation is limited to the application of group representation. Typically, to act on multiple local copies of these preshared states, the receiver needs quantum memory, because in general the multiple copies will be generated sequentially. Depending on available encoding unitary operations, we investigate what preshared state offers an advantage of using quantum memory on the receiver’s side. [ABSTRACT FROM AUTHOR]

We introduce a Steane-like enlargement procedure for entanglement-assisted quantum error-correcting codes (EAQECCs) obtained by considering Euclidean inner product. We give formulae for the parameters of these enlarged codes and apply our results to explicitly compute the parameters of enlarged EAQECCs coming from some BCH codes. [ABSTRACT FROM AUTHOR]

We analyze a quantum version of the Monge–Kantorovich optimal transport problem. The quantum transport cost related to a Hermitian cost matrix C is minimized over the set of all bipartite coupling states ρ AB with fixed reduced density matrices ρ A and ρ B of size m and n. The minimum quantum optimal transport cost T C Q (ρ A , ρ B) can be efficiently computed using semidefinite programming. In the case m = n the cost T C Q gives a semidistance if and only if C is positive semidefinite and vanishes exactly on the subspace of symmetric matrices. Furthermore, if C satisfies the above conditions, then T C Q induces a quantum analogue of the Wasserstein-2 distance. Taking the quantum cost matrix C Q to be the projector on the antisymmetric subspace, we provide a semi-analytic expression for T C Q Q for any pair of single-qubit states and show that its square root yields a transport distance on the Bloch ball. Numerical simulations suggest that this property holds also in higher dimensions. Assuming that the cost matrix suffers decoherence and that the density matrices become diagonal, we study the quantum-to-classical transition of the Monge–Kantorovich distance, propose a continuous family of interpolating distances, and demonstrate that the quantum transport is cheaper than the classical one. Furthermore, we introduce a related quantity—the SWAP-fidelity—and compare its properties with the standard Uhlmann–Jozsa fidelity. We also discuss the quantum optimal transport for general d-partite systems. [ABSTRACT FROM AUTHOR]

We introduce a notion of teleportation scheme between subalgebras of semi-finite von Neumann algebras in the commuting operator model of locality. Using techniques from subfactor theory, we present unbiased teleportation schemes for relative commutants N ′ ∩ M of a large class of finite-index inclusions N ⊆ M of tracial von Neumann algebras, where the unbiased condition means that no information about the teleported observables is contained in the classical communication sent between the parties. For a large class of subalgebras N of matrix algebras M n (C) , including those relevant to hybrid classical/quantum codes, we show that any tight teleportation scheme for N necessarily arises from an orthonormal unitary Pimsner–Popa basis of M n (C) over N ′ , generalising work of Werner (J Phys A 34(35):7081–7094, 2001). Combining our techniques with those of Brannan–Ganesan–Harris (J Math Phys 63(11): 112204, 2022) we compute quantum chromatic numbers for a variety of quantum graphs arising from finite-dimensional inclusions N ⊆ M . [ABSTRACT FROM AUTHOR]

In Christandl et al. (Ann Henri Poincaré 20(7):2295–2322, 2019), the authors introduced k-entanglement breaking linear maps to understand the entanglement breaking property of completely positive maps on taking composition. In this article, we do a systematic study of k-entanglement breaking maps. We prove many equivalent conditions for a k-positive linear map to be k-entanglement breaking, thereby study the mapping cone structure of k-entanglement breaking maps. We discuss examples of k-entanglement breaking maps and some of their significance. As an application of our study, we characterize the completely positive maps that reduce Schmidt number on taking composition with another completely positive map. Finally, we extend a spectral majorization result for separable states. [ABSTRACT FROM AUTHOR]

We explicitly construct an Archimedean order unit space whose state space is affinely isomorphic to the set of quantum commuting correlations. Our construction only requires fundamental techniques from the theory of order unit spaces and operator systems. Our main results are achieved by characterizing when a finite set of positive contractions in an Archimedean order unit space can be realized as a set of projections on a Hilbert space. [ABSTRACT FROM AUTHOR]

The Hermitian tensor is an extension of Hermitian matrices and plays an important role in quantum information research. It is known that every symmetric tensor has a symmetric CP-decomposition. However, symmetric Hermitian tensor is not the case. In this paper, we obtain a necessary and sufficient condition for symmetric Hermitian decomposability of symmetric Hermitian tensors. When a symmetric Hermitian decomposable tensor space is regarded as a linear space over the real number field, we also obtain its dimension formula and basis. Moreover, if the tensor is symmetric Hermitian decomposable, then the symmetric Hermitian decomposition can be obtained by using the symmetric Hermitian basis. In the application of quantum information, the symmetric Hermitian decomposability condition can be used to determine the symmetry separability of symmetric quantum mixed states. [ABSTRACT FROM AUTHOR]

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory Φ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra H FG cut (Φ) and lattices of intermediate structures. Feynman diagrams in H FG (Φ) are applied to describe states in Φ where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space S graphon Φ , M ⊆ [ 0 , ∞) ([ 0 , 1 ]) . We associate Hopf subalgebras of H FG (Φ) generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green's functions of Φ . These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson–Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales. [ABSTRACT FROM AUTHOR]

Quantum states are represented by positive semidefinite Hermitian operators with unit trace, known as density matrices. An important subset of quantum states is that of separable states, the complement of which is the subset of entangled states. We show that the problem of deciding whether a quantum state is entangled can be seen as a moment problem in real analysis. Only a small number of such moments are accessible experimentally, and so in practice the question of quantum entanglement of a many-body system (e.g, a system consisting of several atoms) can be reduced to a truncated moment problem. By considering quantum entanglement of n identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of n copies of unit balls in R d . We work with moments up to degree 2 only, since these are most readily available experimentally. We derive necessary and sufficient conditions for belonging to the moment cone, which can be expressed via a linear matrix inequality of size at most 2 d + 2 , which is independent of n. The linear matrix inequalities can be converted into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone, and show that the two conditions approach each other in the limit of large n. The inequalities are derived via considering the dual cone of nonnegative polynomials, and its sum-of-squares relaxation. We show that the sum-of-squares relaxation of the dual cone is asymptotically exact, and using symmetry reduction techniques (Blekherman and Riener: Symmetric nonnegative forms and sums of squares. arXiv:1205.3102, 2012; Gatermann and Parrilo: J Pure Appl Algebra 192(1–3):95–128. https://doi.org/10.1016/j.jpaa.2003.12.011, 2004), it can be written as a small linear matrix inequality of size at most 2 d + 2 , which is independent of n. For the cone of symmetric nonnegative polynomials with the relevant support we also prove an analogue of the half-degree principle for globally nonnegative symmetric polynomials (Riener: J Pure Appl Algebra 216(4): 850–856. https://doi.org/10.1016/j.jpaa.2011.08.012, 2012; Timofte: J Math Anal Appl 284(1):174–190. https://doi.org/10.1016/S0022-247X(03)00301-9, 2003). [ABSTRACT FROM AUTHOR]

We investigate the number of zero entries in a unitary matrix. We show that the sets of numbers of zero entries for n × n unitary and orthogonal matrices are the same. They are both the set { 0 , 1 , ... , n 2 − n − 4 , n 2 − n − 2 , n 2 − n } for n>4. We explicitly construct examples of orthogonal matrices with the numbers in the set. We apply our results to construct a necessary condition by which a multipartite unitary operation is a product operation. The latter is a fundamental problem in quantum information. We also construct an n × n orthogonal matrix of Schmidt rank n 2 − 1 with many zero entries, and it solves an open problem in Muller-Hermes and Nechita [Operator Schmidt ranks of bipartite unitary matrices. Linear Algebra Appl. 2018;557:174—187]. [ABSTRACT FROM AUTHOR]

A geometric measure for the entanglement of a unit length tensor T ∈ (C n ) ⊗ k is given by − 2 log 2 ⁡ ∥ T ∥ σ , where ∥ ⋅ ∥ σ denotes the spectral norm. A simple induction gives an upper bound of (k − 1) log 2 ⁡ (n) for the entanglement. We show the existence of tensors with entanglement larger than k log 2 ⁡ (n) − log 2 ⁡ (k) − o (log 2 ⁡ (k)). Friedland and Kemp (Proc. AMS, 2018) have similar results in the case of symmetric tensors. Our techniques give improvements in this case. [ABSTRACT FROM AUTHOR]

Can vectors with low Schmidt rank form mutually unbiased bases? Can vectors with high Schmidt rank form positive under partial transpose states? In this work, we address these questions by presenting several new results related to Schmidt rank constraints and their compatibility with other properties. We provide an upper bound on the number of mutually unbiased bases of C m ⊗ C n (m ≤ n) formed by vectors with low Schmidt rank. In particular, the number of mutually unbiased product bases of C m ⊗ C n cannot exceed m + 1 , which solves a conjecture proposed by McNulty et al. Then, we show how to create a positive under partial transpose entangled state from any state supported on the antisymmetric space and how their Schmidt numbers are exactly related. Finally, we show that the Schmidt number of operator Schmidt rank 3 states of M m ⊗ M n (m ≤ n) that are invariant under left partial transpose cannot exceed m - 2 . [ABSTRACT FROM AUTHOR]

This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. Such functionals are related to more conservative behaviour and robustness of systems with respect to statistical uncertainty, which makes the challenging problems of their computation and minimization practically important. To this end, we obtain an integro-differential equation for the time evolution of the quadratic–exponential functional, which is different from the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaussian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite-horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. Further approximations of the cost functional, based on higher-order cumulants and their growth rates, are applied to large deviations estimates in the form of upper bounds for tail distributions. We discuss an auxiliary classical Gaussian–Markov diffusion process in a complex Euclidean space which reproduces the quantum system variables at the level of covariances but has different fourth-order cumulants, thus showing that the risk-sensitive criteria are not reducible to quadratic–exponential moments of classical Gaussian processes. The results of the paper are illustrated by a numerical example and may find applications to coherent quantum risk-sensitive control problems, where the plant and controller form a fully quantum closed-loop system, and other settings with nonquadratic cost functionals. [ABSTRACT FROM AUTHOR]

We show that the mathematical structures in a recent work of Bultinck–Mariëna–Williamson–Şahinoğlu–Haegemana–Verstraete are the same as those of flat symmetric bi-unitary connections and the tube algebra in subfactor theory. More specifically, a system of flat symmetric bi-unitary connections arising from a subfactor with finite index and finite depth satisfies all their requirements for tensors and the tube algebra for such a subfactor, and the anyon algebra for such tensors are isomorphic up to the normalization constants. Furthermore, the matrix product operator algebras arising from tensors corresponding to possibly non-flat symmetric bi-unitary connections are isomorphic to those arising from flat symmetric bi-unitary connections for subfactors. [ABSTRACT FROM AUTHOR]

A symmetric n × n matrix X is completely positive semidefinite (cpsd) if there exist d × d positive semidefinite matrices { P i } i = 1 n (for some d ∈ N ) such that X i j = Tr (P i P j) , for all i , j ∈ { 1 , ... , n }. The cpsd − rank of a cpsd matrix is the smallest d ∈ N for which such a representation is possible. It was shown independently in Prakash A, Sikora J, Varvitsiotis A, et al. [Completely positive semidefinite rank. 2016 Apr. arXiv:1604.07199] and Gribling S, de Laat D, Laurent M. [Matrices with high completely positive semidefinite rank. Linear Algebra Appl. 2017 May;513:122–148] that there exist completely positive semidefinite matrices with sub-exponential cpsd-rank. Both proofs were obtained using fundamental results from the quantum information literature as a black-box. In this work we give a self-contained and succinct proof of the existence of completely positive semidefinite matrices with sub-exponential cpsd-rank. For this, we introduce matrix valued Gram decompositions for correlation matrices and show that for extremal correlations, the matrices in such a factorization generate a Clifford algebra. Lastly, we show that this fact underlies and generalizes Tsirelson's results concerning the structure of quantum representations for extremal quantum correlation matrices. [ABSTRACT FROM AUTHOR]

In this short note, we show two completely opposite methods of constructing bipartite entangled states. Given a bipartite state γ ∈ M k ⊗ M k , define γ S = (I d + F) γ (I d + F) , γ A = (I d - F) γ (I d - F) , where F ∈ M k ⊗ M k is the flip operator. In the first method, entanglement is a consequence of the inequality rank (γ S) < rank (γ A) . In the second method, there is no correlation between γ S and γ A . These two methods show how diverse is quantum entanglement. We show that any bipartite state γ ∈ M k ⊗ M k satisfies SN (γ) ≥ max rank (γ L) rank (γ) , rank (γ R) rank (γ) , SN (γ S) 2 , SN (γ A) 2 , where SN (γ) stands for the Schmidt number of γ and γ L and γ R are the marginal states of γ . These inequalities are useful to compute the Schmidt number of many bipartite states. We prove that SN (γ) = min { rank (γ L) , rank (γ R) } , if rank (γ) = max { rank (γ L) , rank (γ R) } min { rank (γ L) , rank (γ R) } . We also present a family of PPT states in M k ⊗ M k , whose members have Schmidt number equal to n, for any given 1 ≤ n ≤ k 2 . This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the problem of computing unitary eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of symmetric embedding of complex tensors, the relationship between U-eigenpairs of a non-symmetric complex tensor and unitary symmetric eigenpairs (US-eigenpairs) of its symmetric embedding tensor is established. An Algorithm 3.1 is given to compute the U-eigenvalues of non-symmetric complex tensors by means of symmetric embedding. Another Algorithm 3.2, is proposed to directly compute the U-eigenvalues of non-symmetric complex tensors, without the aid of symmetric embedding. Finally, a tensor version of the well-known Gauss–Seidel method is developed. Efficiency of these three algorithms are compared by means of various numerical examples. These algorithms are applied to compute the geometric measure of entanglement of quantum multipartite non-symmetric pure states. [ABSTRACT FROM AUTHOR]

We present a signaling scheme for information embedding into the illumination of radar using frequency-hopping pulses. A frequency-hopping-based joint radar-communication system enables implementing a primary radar operation and a secondary communication function simultaneously. Then, we consider the problems of radar codes optimization under a peak-to-average-power ratio and an energy constraint. These radar codes design problems can be converted into non-convex quadratic programs with a finite or an infinite number of quadratic constraints. All problems are proved to be NP-hard optimization problems. Therefore, we develop optimization approaches, resorting to semi-definite programming relaxation technique along with to the idea of trigonometric polynomials, offering expected approximate solutions with a polynomial time calculation burden. We assess the capability of the proposed schemes, considering both the detection probability and the robustness in correspondence of Doppler shifts offered by the Neyman–Pearson detector. Simulation results show an improvement in detection performance as the average signal-to-noise ratio value increases, while still maintaining low symbol error rates between the proposed system nodes. [ABSTRACT FROM AUTHOR]

In this work, we adapt Sinkhorn–Knopp theorem for rectangular positive maps. We extend their concepts of support and total support to these maps. We show that a positive map is equivalent to a doubly stochastic map if and only if is equivalent to a positive map with total support. Moreover, if k and m are coprime then support is sufficient for the equivalence with a doubly stochastic map. This result provides a necessary and sufficient condition for the filter normal form, which is commonly used in Quantum Information Theory. Let be a state and be the positive map. We show that A can be put in the filter normal form if and only if is equivalent to a positive map with total support. We prove that any state such that , if k=m, and , if , can be put in the filter normal form. [ABSTRACT FROM AUTHOR]

Given a PPT state A = ∑ i = 1 n A i ⊗ B i ∈ M k ⊗ M k and a rank k tensor v within the image of A, we provide an algorithm that checks whether the positive map G A : M k → M k , G A (X) = ∑ i = 1 n t r (A i X) B i , is equivalent to a doubly stochastic map. This procedure is based on the search for Perron eigenvectors of completely positive maps and unique solutions of, at most, k unconstrained quadratic minimization problems. As a corollary, we can check whether this state can be put in the filter normal form. This normal form is an important tool for studying quantum entanglement. An extension of this procedure to PPT states in M k ⊗ M m is also presented. [ABSTRACT FROM AUTHOR]

We review quantum mappings used in problems of characterization of entanglement of two-part and multi-particle systems. Together with positive and n-tensorial constant positive mappings, we consider physical dynamical processes that lead to quantum channels that break entanglement, annihilate entanglement, dissociate entanglement of multi-particle states, and prohibit distillation of output states. We introduce a new class of absolutely disentangling channels that provide absolutely separable states at the output, and also characterize a new class of entanglement-imposing channels whose output states are entangled. We present states that are most resistant to loss of entanglement and prove that they may differ from maximally entangled states. [ABSTRACT FROM AUTHOR]