Computable Cross Norm Research Articles

Universal Entanglement and Correlation Measure in Two-Dimensional Conformal Field Theories.

We calculate the amount of entanglement shared by two intervals in the ground state of a (1+1)-dimensional conformal field theory (CFT), quantified by an entanglement measure E based on the computable cross norm (CCNR) criterion. Unlike negativity or mutual information, we show that E has a universal expression even for two disjoint intervals, which depends only on the geometry, the central charge c, and the thermal partition function of the CFT. We prove this universal expression in the replica approach, where the Riemann surface for calculating E at each order n is always a torus topologically. By analytic continuation, the result of n=1/2 gives the value of E. Furthermore, the results of other values of n also yield meaningful conclusions: The n=1 result gives a general formula for the two-interval purity, which enables us to calculate the Rényi-2 N-partite information for N≤4 intervals; while the n=∞ result bounds the correlation function of the two intervals. We verify our findings numerically in the spin-1/2 XXZ chain, whose ground state is described by the Luttinger liquid.

Comparative Dynamical Study of a Bound Entangled State

The bound entangled state carries noisy entanglement and it is very hard to distill but the usefulness of bound entangled states has been depicted in different applications. This article represents a comparative dynamical study of an open quantum system for one of the bound entangled states proposed by Bennett et al. The study is conducted under the influence of Heisenberg, bi-linear bi-quadratic and Dzyaloshinskii–Moriya (DM) interaction. During the study, an auxiliary qutrit interacts with one of the qutrits of the selected two qutrit bound entangled state through different interactions. The computable cross-norm or realignment (CCNR) criterion has been used to detect the bound entanglement of the state and the negativity has been applied to measure the free entanglement. From this three-fold study it is observed that, although the auxiliary qutrit plays a significant role during the interaction, the probability amplitude of the qutrit does not affect the open quantum system. Further, it is found that the Dzyaloshinskii–Moriya (DM) interaction performs better to activate the chosen bound entangled state among all the interactions.

Correlation minor norms, entanglement detection and discord

In this paper we develop an approach for detecting entanglement, which is based on measuring quantum correlations and constructing a correlation matrix. The correlation matrix is then used for defining a family of parameters, named Correlation Minor Norms, which allow one to detect entanglement. This approach generalizes the computable cross-norm or realignment (CCNR) criterion, and moreover requires measuring a state-independent set of operators. Furthermore, we illustrate a scheme which yields for each Correlation Minor Norm a separable state that maximizes it. The proposed entanglement detection scheme is believed to be advantageous in comparison to other methods because correlations have a simple, intuitive meaning and in addition they can be directly measured in experiment. Moreover, it is demonstrated to be stronger than the CCNR criterion. We also illustrate the relation between the Correlation Minor Norm and entanglement entropy for pure states. Finally, we discuss the relation between the Correlation Minor Norm and quantum discord. We demonstrate that the CMN may be used to define a new measure for quantum discord.

Enhanced entanglement criterion via symmetric informationally complete measurements

We show that a special type of measurements, called symmetric informationally complete positive operator-valued measures (SIC POVMs), provide a stronger entanglement detection criterion than the computable cross-norm or realignment criterion based on local orthogonal observables. As an illustration, we demonstrate the enhanced entanglement detection power in simple systems of qubit and qutrit pairs. This observation highlights the significance of SIC POVMs for entanglement detection.

Realignment criteria for recognizing multipartite entanglement of quantum states

By multiple realignments of density matrices, we present a new separability criterion for the multipartite quantum state, which includes the computable cross-norm or realignment criterion and the multipartite partial realignment criterion as special cases. An example is used to show that the new criterion can be more efficient than the corresponding multipartite realignment criteria given in Horodecki et al. (Open Syst Inf Dyn 13:103---111, 2006) and Shen et al. (Phys Rev A 92:042332, 2015).

The Entanglement Level and the Detection of Quantum Data Transfer Correctness in Short Qutrit Spin Chains

The quantum entanglement is an important feature of many protocols in the field of quantum computing. In this paper we evaluate a level of entanglement in short qutrit chains. This evaluation is carried out with use of the computable cross norm and realignment criterion and the concurrence measure. We also present some explicit formulae describing the concurrence measure for exemplary short spin chains. Utilizing the obtained results, we indicate that analysing the level of entanglement allows to detect the noise or deviation in the transfer process, in comparison to the perfect transfer where only operation realizing transfer is present.

Separability criteria based on the realignment of density matrices and reduced density matrices

By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix, and the vectorization of its reduced density matrices, we present a family of separability criteria which are stronger than the computable cross-norm or realignment (CCNR) criterion. With linear contraction methods, the proposed criteria can be used to detect the multipartite entangled states that are biseparable under any bipartite partitions. Moreover, we show by examples that the presented multipartite separability criteria can be more efficient than the corresponding multipartite realignment criterion based on CCNR, multipartite correlation tensor criterion, and multipartite covariance matrix criterion.

Entanglement criterion of computable cross norm and realignment for continuous-variable bipartite symmetric states

We provide the entanglement criterion of computable cross norm and realignment for any two-mode continuous-variable state with permutational symmetry. The entanglement criterion for a non-Gaussian state is obtained from its Gaussian kernel by employing the technique of functional analysis. The entanglement conditions turn out to be necessary and sufficient for the non-Gaussian states prepared by one-photon subtracting from or adding to symmetric Gaussian states. The entanglement criterion of non-Gaussian states evolved in thermal noise and amplitude damping environment also is obtained.

ENTANGLEMENT CRITERION FOR COHERENT SUBTRACTION AND COHERENT ADDITION OF BIPARTITE CONTINUOUS VARIABLE STATES

We study the entanglement conditions of two quite general classes of two-mode non-Gaussian states. Using computable cross norm and realignment criterion, we obtain the sufficient conditions of entanglement for coherently added or subtracted two-mode squeezed thermal states, and the sufficient condition of entanglement for any photon number entangled state (PNES) evolving in a thermal noise and amplitude damping channel.

Realignment Operation and CCNR Criterion of Separability for States in Infinite-Dimensional Quantum Systems

The realignment operation and the computable cross norm or realignment (CCNR) criterion of separability for states in infinite-dimensional bipartite quantum systems are established. Let H A and H B be complex Hilbert spaces with dim H A ⊕ H B ≤ +∞. Let ρ be a quantum state acting on H A ⊕ H B and {δ k } be the Schmidt coefficients of ρ as a vector in the Hilbert space C 2 ( H A ) ⊕ C 2 ( H B ). We introduce the realignment operator ρ R and the computable cross norm ||ρ|| CCN of ρ and show that if ρ is separable, then ||ρ R || Tr = ||ρ|| CCN = Σ k δ ≤ 1. In particular, if ρ is a pure state, then ρ is separable if and only if ||ρ R || Tr = ||ρ|| CCN = Σ k δ k = 1. For the finite-dimensional case, this recovers the original computable cross norm criterion.

Entanglement detection beyond the CCNR criterion for infinite-dimensions

In this paper, in terms of the relation between the state and the reduced states of it, we obtain two inequalities which are valid for all separable states in infinite-dimensional bipartite quantum systems. One of them provides an entanglement criterion which is strictly stronger than the computable cross-norm or realignment (CCNR) criterion.

ENTANGLEMENT OF CONVEX LINEAR COMBINATION AND CONSTRUCTION OF PPT ENTANGLED STATES

Given two bipartite quantum states and the convex linear combination of them, we discuss the relation between the entanglement of the convex linear combination state and the entanglement of states being combined. This is achieved by characterizing quantum states quantitatively via the positive partial transpose (PPT) criterion and the computable cross-norm or realignment (CCNR) criterion. Inspired by the Horodecki's 3 ⊗ 3 quantum states, we also give explicit examples to illustrate all possible cases of convex linear combination. Finally, as an application of this method, we show how to construct new bipartite PPT entangled states from known PPT entangled states by convex linear combination.

Norm duality and the cross norm criteria for quantum entanglement

We use duality techniques to prove and generalize the cross norm and computable cross norm criteria for separability of quantum states. While the original proof of the cross norm criterion is long and involved, our new proof is short and elementary. Furthermore, our proof generalizes naturally to arbitrary Schmidt number. We also use these techniques to generalize the computable cross norm criterion to arbitrary Schmidt number and prove some results of independent interest along the way.

On the Relation Between Schmidt Coefficients and Entanglement

We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the 'realignment' or 'computable cross norm' criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance — with regard to the characterization of entanglement — of the 'symmetric polynomials' in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed.

A Class of Bound Entangled States of Two Qutrits

We provide a new class of states of two qutrits displaying the so-called bound entanglement. We investigate the states using various versions of computable cross-norm and realignment criteria recently introduced in [7].

A Lower Bound on Concurrence

We derive an analytical lower bound on the concurrence for bipartite quantum systems with an improved computable cross norm or realignment criterion and an improved positive partial transpose criterion respectively. Furthermore we demonstrate that our bound is better than that obtained from the local uncertainty relations criterion with optimal local orthogonal observables which is known as one of the best estimations of concurrence.

Unifying several separability conditions using the covariance matrix criterion

We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows us to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria that have been proposed to complement the positive partial transpose criterion---the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others---are in fact a corollary of the covariance matrix criterion.

Bipartite quantum systems: on the realignment criterion and beyond

Inspired by the ‘computable cross norm’ or ‘realignment’ criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states are derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.

A class of inequalities inducing new separability criteria for bipartite quantum systems

Inspired by the realignment or computable cross norm criterion, we present a new result about the characterization of quantum entanglement. Precisely, an interesting class of inequalities satisfied by all separable states of a bipartite quantum system is derived. These inequalities induce new separability criteria that generalize the realignment criterion.

Entanglement detection beyond the computable cross-norm or realignment criterion

Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the entanglement of many bound entangled states. The criterion is strictly stronger than the criterion based on Bloch representations, the computable cross-norm or realignment criterion and its optimal nonlinear entanglement witnesses. Furthermore, this criterion can be generalized to an analog of permutation separability criteria in the even-partite systems.