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 Unified Monogamy Relations of Multipartite Entanglement

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Awais Khan ◽  
Junaid ur Rehman ◽  
Kehao Wang ◽  
Hyundong Shin
Keyword(s):  
Analytic Formula ◽  
Entangled States ◽  
Multipartite Entanglement ◽  
Entanglement Measure ◽  
Mixed States ◽  
Bipartite Entanglement ◽  
Entanglement Of Formation ◽  
Special Cases ◽  
Qubit States ◽  
Monogamy Relations

Abstract Unified-(q, s) entanglement $$({{\mathscr{U}}}_{q,s})$$ ( U q , s ) is a generalized bipartite entanglement measure, which encompasses Tsallis-q entanglement, Rényi-q entanglement, and entanglement of formation as its special cases. We first provide the extended (q; s) region of the generalized analytic formula of  $${{\mathscr{U}}}_{q,s}$$ U q , s . Then, the monogamy relation based on the squared  $${{\mathscr{U}}}_{q,s}$$ U q , s for arbitrary multiqubit mixed states is proved. The monogamy relation proved in this paper enables us to construct an entanglement indicator that can be utilized to identify all genuine multiqubit entangled states even the cases where three tangle of concurrence loses its efficiency. It is shown that this monogamy relation also holds true for the generalized W-class state. The αth power $${{\mathscr{U}}}_{q,s}$$ U q , s based general monogamy and polygamy inequalities are established for tripartite qubit states.

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 Separability of mixed states: necessary and sufficient conditions

1996 ◽  
Vol 223 (1-2) ◽  
pp. 1-8 ◽  
Author(s):  
Michał Horodecki ◽  
Paweł Horodecki ◽  
Ryszard Horodecki
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Mixed States ◽  
Necessary And Sufficient

Generalized concurrences do not provide necessary and sufficient conditions for entanglement detection

2009 ◽  
Vol 9 (1&2) ◽  
pp. 166-180
Author(s):  
L. Cattaneo ◽  
D. D'Alessandro
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary And Sufficient Conditions ◽  
Entangled States ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bipartite Quantum Systems ◽  
Rank 2 ◽  
Entanglement Detection ◽  
Necessary And Sufficient

We study generalized concurrences as a tool to detect the entanglement of bipartite quantum systems. By considering the case of 2x4 states of rank 2, we prove that generalized concurrences do not, in general, give a necessary and sufficient condition of separability. We identify a set of entangled states which are undetected by this method.

scholarly journals MIXING AND DECOHERENCE TO NEAREST SEPARABLE STATES

2009 ◽  
Vol 07 (04) ◽  
pp. 829-846
Author(s):  
AVIJIT LAHIRI ◽  
GAUTAM GHOSH ◽  
SANKHASUBHRA NAG
Keyword(s):  
Quantum Correlation ◽  
Entangled States ◽  
Entanglement Measure ◽  
Measuring Apparatus ◽  
Separable State ◽  
Mixed States ◽  
Classical Correlation ◽  
Separable States ◽  
System A ◽  
Pure States

We consider a class of entangled states of a quantum system (S) and a second system (A) where pure states of the former are correlated with mixed states of the latter, and work out the entanglement measure with reference to the nearest separable state. Such "pure-mixed" entanglement is expected when the system S interacts with a macroscopic measuring apparatus in a quantum measurement, where the quantum correlation is destroyed in the process of environment-induced decoherence whereafter only the classical correlation between S and A remains, the latter being large compared to the former. We present numerical evidence that the entangled S–A state drifts towards the nearest separable state through decoherence, with an additional tendency of equimixing among relevant groups of apparatus states.

scholarly journals Locally undetermined states, generalized Schmidt decomposition, and application in distributed computing

2009 ◽  
Vol 9 (11&12) ◽  
pp. 997-1012
Author(s):  
Y. Feng ◽  
R. Duan ◽  
M. Ying
Keyword(s):  
Distributed Computing ◽  
Quantum State ◽  
Fault Tolerant ◽  
A Priori ◽  
Sufficient Conditions ◽  
Quantum States ◽  
Consensus Problem ◽  
Schmidt Decomposition ◽  
Pure States ◽  
Necessary And Sufficient

Multipartite quantum states that cannot be uniquely determined by their reduced states of all proper subsets of the parties exhibit some inherit `high-order' correlation. This paper elaborates this issue by giving necessary and sufficient conditions for a pure multipartite state to be locally undetermined, and moreover, characterizing precisely all the pure states sharing the same set of reduced states with it. Interestingly, local determinability of pure states is closely related to a generalized notion of Schmidt decomposition. Furthermore, we find that locally undetermined states have some applications to the well-known consensus problem in distributed computing. To be specific, given some physically separated agents, when communication between them, either classical or quantum, is unreliable, then there exists a totally correct and completely fault-tolerant protocol for them to reach a consensus if and only if they share a priori a locally undetermined quantum state.

scholarly journals SEPARABILITY OF PURE N-QUBIT STATES: TWO CHARACTERIZATIONS

2003 ◽  
Vol 14 (05) ◽  
pp. 797-814 ◽  
Author(s):  
PHILIPPE JORRAND ◽  
MEHDI MHALLA
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Quantum System ◽  
Pure State ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Entangled States ◽  
Necessary And Sufficient ◽  
Qubit States

Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.

Multipartite entanglement and hyperdeterminants

2002 ◽  
Vol 2 (Special) ◽  
pp. 540-555
Author(s):  
A. Miyake ◽  
M. Wadati
Keyword(s):  
Entangled States ◽  
Multipartite Entanglement ◽  
Maximal Dimension ◽  
Ordered Structure ◽  
Mixed States ◽  
Separable States ◽  
Entanglement Measures ◽  
Pure States ◽  
Maximally Entangled States ◽  
Bipartite Case

We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the \(n \!\geq\! 4\) qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical \(n\)-partite entangled states. Our classification is also useful for that of mixed states.

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