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arXiv:1011.0987 · Physical Review A

Two local observables are sufficient to characterize maximally entangled states of N qubits

Fengli Yan, Ting Gao, Eric Chitambar

Published Feb 22, 2011• 47 citations
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Abstract

Maximally entangled states (MES) represent a valuable resource in quantum information processing. In $N$-qubit systems the MES are $N$-GHZ states [i.e., the collection of $|{\mathrm{GHZ}}_{N}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|00\dots{}0\ensuremath{\rangle}+|11\dots{}1\ensuremath{\rangle})]$ and its local unitary (LU) equivalences. While it is well known that such states are uniquely stabilized by $N$ commuting observables, in this article we consider the minimum number of noncommuting observables needed to characterize an $N$-qubit MES as the unique common eigenstate. Here, we prove, rather surprisingly, that in this general case any $N$-GHZ state can be uniquely stabilized by only two observables. Thus, for the task of MES certification, only two correlated measurements are required with each party observing the spin of his or her system along one of two directions.

DOI
10.1103/physreva.83.022319

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Maximally entangled states (MES) represent a valuable resource in quantum information processing. In NNN-qubit systems the MES are NNN-GHZ states [i.e., the collection of ∣GHZN\ensuremath⟩=12(∣00…0\ensuremath⟩+∣11…1\ensuremath⟩)]|{\mathrm{GHZ}}_{N}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|00\dots{}0\ensuremath{\rangle}+|11\dots{}1\ensuremath{\rangle})]∣GHZN​\ensuremath⟩=2​1​(∣00…0\ensuremath⟩+∣11…1\ensuremath⟩)] and its local unitary (LU) equivalences. While it is well known that such states are uniquely stabilized by NNN commuting observables, in this article we consider the minimum number of noncommuting observables needed to characterize an NNN-qubit MES as the unique common eigenstate. Here, we prove, rather surprisingly, that in this general case any NNN-GHZ state can be uniquely stabilized by only two observables. Thus, for the task of MES certification, only two correlated measurements are required with each party observing the spin of his or her system along one of two directions.
We present a comprehensive study of maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. A general criterion is provided to easily identify whether given symmetric states are maximally entangled in that respect or not. We show that these maximally entangled symmetric (MES) states are the only symmetric states for which the expectation value of the associated collective spin of the system vanishes, as well as in corollary the dipole moment of the Husimi function. We establish the link between this kind of maximal entanglement, the anticoherence properties of spin states, and the degree of polarization of light fields. We analyze the relationship between the MES states and the classes of states equivalent through stochastic local operations with classical communication (SLOCC). We provide a nonexistence criterion of MES states within SLOCC classes of qubit states and show in particular that the symmetric Dicke state SLOCC classes never contain such MES states, with the only exception of the balanced Dicke state class for even numbers of qubits. The 4-qubit system is analyzed exhaustively and all MES states of this system are identified and characterized. Finally the entanglement content of MES states is analyzed with respect to the geometric and barycentric measures of entanglement, as well as to the generalized NNN-tangle. We show that the geometric entanglement of MES states is ensured to be larger than or equal to 1/21/21/2, but also that MES states are not in general the symmetric states that maximize the investigated entanglement measures.
Entanglement is a resource in quantum information theory when state manipulation is restricted to local operations assisted by classical communication (LOCC). It is therefore of paramount importance to decide which LOCC transformations are possible and, particularly, which states are maximally useful under this restriction. While the bipartite maximally entangled state is well known (it is the only state that cannot be obtained from any other and, at the same time, it can be transformed to any other by LOCC), no such state exists in the multipartite case. In order to cope with this fact, we introduce here the notion of the maximally entangled set (MES) of n-partite states. This is the set of states which are maximally useful under LOCC manipulation; i.e., any state outside of this set can be obtained via LOCC from one of the states within the set and no state in the set can be obtained from any other state via LOCC. We determine the MES for states of three and four qubits and provide a simple characterization for them. In both cases, infinitely many states are required. However, while the MES is of measure zero for 3-qubit states, almost all 4-qubit states are in the MES. This is because, in contrast to the 3-qubit case, deterministic LOCC transformations are almost never possible among fully entangled four-partite states. We determine the measure-zero subset of the MES of LOCC convertible states. This is the only relevant class of states for entanglement manipulation.
We introduce a physically motivated classification of pure quantum states describing nnn qubits. We characterize all multipartite states which can be maximally entangled to local auxiliary systems using controlled operations. A state has this property if and only if one can construct out of it an orthonormal basis by applying independent local unitary operations. This implies that those states can be used to encode locally the maximum amount of nnn independent bits. Examples of these states are the so-called stabilizer states, which are used for quantum error correction and one-way quantum computing. We give a simple characterization of these states and construct a complete set of commuting unitary observables which characterize the state uniquely. Furthermore we show how these states can be prepared and discuss their applications.
In this paper, we characterize the maximal violation of Ardehali's inequality of nnn qubits by showing that GHZ's states and the states obtained from them by local unitary transformations are the unique states that maximally violate the Ardehali's inequalities. This concludes that Ardehali's inequalities can be used to characterize maximally entangled states of nnn qubits, as the same as Mermin's and Bell-Klyshko's inequalities.
Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the set and on the other hand, no state in the set can be obtained from any other state via LOCC. Recently, we studied LOCC transformations among pure multipartite states and derived the MES for three and generic four qubit states. Here, we consider the non-generic four qubit states and analyze their properties regarding local transformations. As already the most coarse grained classification, due to Stochastic LOCC (SLOCC), of four qubit states is much richer than in case of three qubits, the investigation of possible LOCC transformations is correspondingly more difficult. We prove that most SLOCC classes show a similar behavior as the generic states, however we also identify here three classes with very distinct properties. The first consists of the GHZ and W class, where any state can be transformed into some other state non-trivially. In particular, there exists no isolation. On the other hand, there also exist classes where all states are isolated. Last but not least we identify an additional class of states, whose transformation properties differ drastically from all the other classes. Although the possibility of transforming states into local-unitary inequivalent states by LOCC turns out to be very rare, we identify those states (with exception of the latter class) which are in the MES and those, which can be obtained (transformed) non-trivially from (into) other states respectively. These investigations do not only identify the most relevant classes of states for LOCC entanglement manipulation, but also reveal new insight into the similarities and differences between separable and LOCC transformations and enable the investigation of LOCC transformations among arbitrary four qubit states.
We consider mixed states of two qubits and show under which global unitary operations their entanglement is maximized. This leads to a class of states that is a generalization of the Bell states. Three measures of entanglement are considered: entanglement of formation, negativity, and relative entropy of entanglement. Surprisingly all states that maximize one measure also maximize the others. We give a complete characterization of these generalized Bell states and prove that these states for fixed eigenvalues are all equivalent under local unitary transformations. Furthermore we characterize all nearly entangled states closest to the maximally mixed state and derive a lower bound on the volume of separable mixed states.