scholarly journals Necessary and sufficient criterion for k-separability of N-qubit noisy GHZ states

2018 ◽  
Vol 16 (04) ◽  
pp. 1850037 ◽  
Author(s):  
Xiao-Yu Chen ◽  
Li-Zhen Jiang ◽  
Zhu-An Xu
Keyword(s):  
White Noise ◽  
Sufficient Conditions ◽  
Ghz State ◽  
Entangled State ◽  
Multipartite Entanglement ◽  
Quantum Code ◽  
Sufficient Criterion ◽  
Output State ◽  
Ghz States ◽  
Necessary And Sufficient

A Multipartite entangled state has many different kinds of entanglements specified by the number of partitions. The most essential example of multipartite entanglement is the entanglement of multi-qubit Greenberger–Horne–Zeilinger (GHZ) state in white noise. We explicitly construct the entanglement witnesses for these states with stabilizer generators of the GHZ states. For an [Formula: see text] qubit GHZ state in white noise, we demonstrate the necessary and sufficient criterion of separability when it is divided into [Formula: see text] parties with [Formula: see text] for arbitrary [Formula: see text] and [Formula: see text]. The criterion covers more than a half of all kinds of partial entanglements for [Formula: see text]-qubit GHZ states in white noise. For the rest of multipartite entanglement problems, we present a method to obtain the sufficient conditions of separability. As an application, we consider [Formula: see text] qubit GHZ state as a codeword of the degenerate quantum code passing through depolarizing channel. We find that the output state is neither genuinely entangled nor fully separable when the quantum channel capacity reduces from positive to zero.

scholarly journals ON THE UNITARITY OF STOCHASTIC EVOLUTIONS DRIVEN BY THE SQUARE OF WHITE NOISE

2001 ◽  
Vol 04 (04) ◽  
pp. 579-588 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ANDREAS BOUKAS ◽  
HUI-HSUNG KUO
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Using the closed Itô's table for the renormalized square of white noise, recently obtained by Accardi, Hida, and Kuo in Ref. 4, we consider the problem of providing necessary and sufficient conditions for the unitarity of the solutions of a certain type of quantum stochastic differential equations.

scholarly journals Multipartite entanglement and few-body Hamiltonians

2014 ◽  
Vol 12 (02) ◽  
pp. 1461003
Author(s):  
Francesco V. Pepe
Keyword(s):  
Short Range ◽  
Ghz State ◽  
Entangled States ◽  
Entangled State ◽  
Multipartite Entanglement ◽  
Multipartite Entangled State ◽  
Coupled Qubits

We investigate the possibility to obtain higly multipartite-entangled states as non-degenerate eigenstates of Hamiltonians that involve only short-range and few-body interactions. We study small-size systems (with a number of qubits ranging from three to five) and search for Hamiltonians with a maximally multipartite entangled state (MMES) as a non-degenerate eigenstate. We then find conditions, including bounds on the number of coupled qubits, to build a Hamiltonian with a Greenberger–Horne–Zeilinger (GHZ) state as a non-degenerate eigenstate. We finally comment on possible applications.

Separability and entanglement of n-qubit and a qubit and a qudit using Hilbert–Schmidt decompositions

2016 ◽  
Vol 14 (05) ◽  
pp. 1650030 ◽  
Author(s):  
Y. Ben-Aryeh ◽  
A. Mann
Keyword(s):  
Singular Value ◽  
Ghz State ◽  
Entangled State ◽  
Sufficient Criterion ◽  
Density Matrices ◽  
Negative Eigenvalues ◽  
Qubit System ◽  
Partial Transpose ◽  
Unit Density ◽  
Value Decomposition

Hilbert–Schmidt (HS) decompositions are employed for analyzing systems of [Formula: see text]-qubit, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary (PTU) transformations for one qubit from the whole system, are used for indicating entanglement/separability. A sufficient criterion for full separability of the [Formula: see text]-qubit and qubit–qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and [Formula: see text]-qubit systems, with emphasis on maximally disordered subsystems (MDS) (i.e. density matrices for which tracing over any subsystem gives the unit density matrix). A sufficient condition that [Formula: see text] (MDS) is not separable is that it has an eigenvalue larger than [Formula: see text] for a qubit and a qudit, and larger than [Formula: see text] for [Formula: see text]-qubit system. The PTU transformation does not change the eigenvalues of the [Formula: see text]-qubit MDS density matrices for odd [Formula: see text]. Thus, the Peres–Horodecki (PH) criterion does not give any information about entanglement of these density matrices. The PH criterion may be useful for indicating inseparability for even [Formula: see text]. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the [Formula: see text] state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8[Formula: see text]GHZ density matrices with probabilities [Formula: see text], is analyzed as function of these probabilities. In some cases, we show that the PH criterion is both sufficient and necessary.

On the Stability of Linear Stochastic Differential Equations

1973 ◽  
Vol 40 (1) ◽  
pp. 87-92 ◽  
Author(s):  
F. Kozin ◽  
C.-M. Wu
Keyword(s):  
White Noise ◽  
Stability Region ◽  
Sufficient Conditions ◽  
Stability Conditions ◽  
Density Functions ◽  
Close Approximation ◽  
Sample Stability ◽  
Gaussian Coefficient ◽  
The Stability ◽  
Necessary And Sufficient

In this paper we present a study of the almost-sure sample stability properties of second-order linear systems with stochastic coefficients. Using knowledge of the first density functions of the coefficient processes, stability conditions are obtained. Based upon recent necessary and sufficient conditions for white-noise coefficient systems, the conditions obtained may yield a close approximation of the exact stability region for the Gaussian coefficient case.

PERFECT TELEPORTATION OF UNKNOWN QUDIT BY A d-LEVEL GHZ CHANNEL

2009 ◽  
Vol 07 (04) ◽  
pp. 755-770 ◽  
Author(s):  
YINXIANG LONG ◽  
DAOWEN QIU ◽  
DONGYANG LONG
Keyword(s):  
General Scheme ◽  
Bell State ◽  
Ghz State ◽  
Quantum States ◽  
Entangled State ◽  
Input State ◽  
Quantum Channels ◽  
Maximally Entangled State ◽  
Ghz States ◽  
Measurement Basis

In the past decades, various schemes of teleportation of quantum states through different types of quantum channels (a prior shared entangled state between the sender and the receiver), e.g. EPR pairs, generalized Bell states, qubit GHZ states, standard W states and its variations, genuine multiqubit entanglement states, etc., have been developed. Recently, three-qutrit quantum states and two-qudit quantum states have also been considered as quantum channels for teleportation. In this paper, we investigate the teleportation of an unknown qudit using a d level GHZ state, i.e. a three-qudit maximally entangled state, as quantum channel. We design a general scheme of faithful teleportation of an unknown qudit using a d-level GHZ state shared between the sender and the receiver, or among the sender, the receiver and the controller; an unknown two-qudit of Schmidt form using a d level GHZ state shared between the sender and the receiver; as well as an unknown arbitrary two-qudit using two shared d level GHZ states between the sender, the receiver and the controller, or using one shared d level GHZ state and one shared generalized Bell state. We obtain the general formulas of Alice's measurement basis, Charlie's measurement basis and Bob's unitary operations to recover the input state of Alice. It is intuitionistic to generalize the protocols of teleporting an arbitrary two-qudit state to teleporting an arbitrary n-qudit state.

scholarly journals Bell-CHSH violation under global unitary operations: Necessary and sufficient conditions

2018 ◽  
Vol 16 (04) ◽  
pp. 1850040 ◽  
Author(s):  
Nirman Ganguly ◽  
Amit Mukherjee ◽  
Arup Roy ◽  
Some Sankar Bhattacharya ◽  
Biswajit Paul ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Comparative Study ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Criterion ◽  
Density Matrices ◽  
Cartan Decomposition ◽  
Chsh Inequality ◽  
Necessary And Sufficient ◽  
Local Filtering

The relation between Bell-CHSH violation and factorization of Hilbert space is considered here. That is, a state which is local in the sense of the Bell-CHSH inequality under a certain factorization of the underlying Hilbert space can be Bell-CHSH nonlocal under a different factorization. While this question has been addressed with respect to separability, the relation of the factorization with Bell-CHSH violation has remained hitherto unexplored. We find here that there is a set containing density matrices, which do not exhibit Bell-CHSH violation under any factorization of the Hilbert space brought about by global unitary operations. Using the Cartan decomposition of [Formula: see text], we characterize the set in terms of a necessary and sufficient criterion based on the spectrum of density matrices. Sufficient conditions are obtained to characterize such density matrices based on their bloch representations. For some classes of density matrices, necessary and sufficient conditions are derived in terms of bloch parameters. Furthermore, an estimation of the volume of such density matrices is achieved in terms of purity. The criterion is applied to some well-known class of states in two qubits. Since both local filtering and global unitary operations influence the Bell-CHSH violation of a state, a comparative study is made between the two operations. The inequivalence of the two operations (in terms of increasing Bell-CHSH violation) is exemplified through their action on some classes of states.

Stability of Dynamical Systems with a Multiplicative White Noise

2003 ◽  
Vol 03 (02) ◽  
pp. 187-212 ◽  
Author(s):  
Boris P. Belinskiy ◽  
Peter Caithamer
Keyword(s):  
White Noise ◽  
Mechanical System ◽  
Initial Conditions ◽  
Random Noise ◽  
Sufficient Conditions ◽  
Auxiliary Equation ◽  
Stability Of Dynamical Systems ◽  
Linear Mechanical System ◽  
Approach Zero ◽  
Necessary And Sufficient

We discuss the behavior, for large values of time, of a class of linear mechanical systems with a white noise in their parameters. The initial conditions may be random as well but are independent of white noise. It is well known that a deterministic linear mechanical system with viscous damping is stable, i.e. its energy approaches zero as time increases. We calculate the expected energy and check that this behavior takes place in the case when the initial conditions are random but the parameters are not. When the parameters contain a random noise the expected energy may be infinite, approach zero, remain bounded, or increase with no bound. This regime is similar to but more interesting than the known regime for the solutions of differential equations with time-dependent periodic coefficients that describe the behavior of a mechanical system whose characteristics are periodic functions of time. We give necessary and sufficient conditions for stability of the systems considered in terms of the roots of an auxiliary equation. We explain why our approach may not be applied to some other models.

scholarly journals Ent: A Multipartite entanglement measure, and parameterization of entangled states

2018 ◽  
Vol 18 (5&6) ◽  
pp. 389-442
Author(s):  
Samuel R. Hedemann
Keyword(s):  
Sufficient Conditions ◽  
Special Property ◽  
Entangled States ◽  
Multipartite Entanglement ◽  
Entanglement Measure ◽  
Mixed States ◽  
Schmidt Decomposition ◽  
Multipartite System ◽  
Necessary And Sufficient ◽  
Maximally Entangled Basis

A multipartite entanglement measure called the ent is presented and shown to be an entanglement monotone, with the special property of automatic normalization. Necessary and sufficient conditions are developed for constructing maximally entangled states in every multipartite system such that they are true-generalized X states (TGX) states, a generalization of the Bell states, and are extended to general nonTGX states as well. These results are then used to prove the existence of maximally entangled basis (MEB) sets in all systems. A parameterization of general pure states of all ent values is given, and proposed as a multipartite Schmidt decomposition. Finally, we develop an ent vector and ent array to handle more general definitions of multipartite entanglement, and the ent is extended to general mixed states, providing a general multipartite entanglement measure.

scholarly journals No semiconjugacy to a map of constant slope

2014 ◽  
Vol 36 (3) ◽  
pp. 875-889 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
SAMUEL ROTH
Keyword(s):  
Sufficient Conditions ◽  
Sufficient Criterion ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Piecewise Continuous ◽  
Necessary And Sufficient ◽  
Constant Slope

We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a non-decreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give examples of maps not semiconjugate to a map of constant slope via a non-decreasing map. Our examples are continuous and transitive.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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